Derived Smooth Manifolds

DERIVED SMOOTH MANIF OLDS DA VID I. SPIV AK Abstract. W e deﬁne a simplicial category called the catego r y of derived man- ifolds. It conta i ns the category of sm ooth manifolds as a full discrete sub cat- egory , and it is closed under taking arbitrary intersections in a manifold. A derive d manifold is a space together with a sheaf of lo cal C ∞ -rings that i s obtained b y patc hing togethe r homotop y zero-sets of smo oth functions on Eu- clidean spaces. W e show t hat deriv ed manifolds come equipp ed with a stable normal b un- dle and can b e imbedded into Euclidean space. W e deﬁne a cohomology theory called derived cobordis m, and use a P ontrjagin-Thom argument to show that the derived cobor dism theory is isomorphic to the classical cobor dism theory . This allo ws us t o deﬁne fundamen tal classes in cob ordism for all derived man- ifolds. In particular, the in tersection A ∩ B of s ubmanifolds A, B ⊂ X e xis ts on the categorical lev el in our theory , and a cup pro duct formula [ A ] ⌣ [ B ] = [ A ∩ B ] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a c ate goriﬁc ation of int ersection theory . Contents 1. Int r o duction 1 2. The axio ms 8 3. Main r e s ults 14 4. Lay out for the construction of dMan 21 5. C ∞ -rings 23 6. Lo cal C ∞ -ringed spaces and derived manifolds 26 7. Cotangent Complexes 32 8. Pro ofs o f technical re sults 39 9. Derived manifolds a r e go o d for doing intersection theory 50 10. Relationship to similar work 60 References 63 1. Intr odu ction Let Ω denote the unoriented cob ordism ring (though what we will sa y applies to other cobordis m theories as w ell, e.g . oriented co bo rdism). The fundamen tal class of a compa c t manifold X is an element [ X ] ∈ Ω. By the Pon trja g in-Thom construction, such an elemen t is classiﬁed b y a ho mo topy class of maps from a large enough sphere S n to a lar ge e nough Thom space M O . One ca n alwa y s choose a map f : S n → M O whic h represents this homotopy class, is smo oth (aw ay fr om the bas epo int ), and meets the zer o-section B ⊂ M O transversely . The pullback 1 2 DA VID I. SPIV AK f − 1 ( B ) is a compact manifold which is cob orda nt to X , s o we hav e a n equality [ X ] = [ f − 1 ( B )] of elements in Ω. This construction pr ovides a co rresp ondence which is homotopical in na ture: one beg ins with a ho motopy class o f maps and r eceives a cob or dism cla ss. H owev er, it is close to exis ting on the nose, in that a dense subset of all represe nting maps f : S n → M O will be t r ansverse to B and yield an imbedded manifold rather than merely its image in Ω. If tr ansversality were no t a n issue, Pon tr ja gin-Thom would indeed provide a corresp ondence b etw een smo o th maps S n → M O a nd their zero-se ts. The pur p os e o f this pa per is to in tro duce the category of deriv ed manifolds wherein non-transverse intersections mak e sense. In this setting, f − 1 ( B ) is a derived manifold whic h is derived cobor dant to X , r egardles s of one’s c ho ic e of smooth map f , and in terms of fundamental cob or dism class es we have [ f − 1 ( B )] = [ X ]. Our hop e is that by using derived manifolds, r esearchers can a void having to make annoying tra nsversalit y arguments. This could b e o f use in string to po logy or Flo er homology , for example. As an exa mple, let us pr ovide a shor t story that can only tak e place in the derived setting. Consider the case of a smo oth degree d h yp ers ur face X ⊂ C P 3 in complex pro jective space. One can expres s the K -theory fundamen ta l class of X as (1.0.1) [ X ] =  d 1  [ C P 2 ] −  d 2  [ C P 1 ] +  d 3  [ C P 0 ] . It may b e diﬃcult to see exactly where this formula comes from; let us give the derived p er s pec tive, whic h should make the formula more clear. The union Y of d distinct hyperplanes in C P 3 is not smo o th, but it do es exist as an o b ject in the categor y of derived manifolds. Moreover, as the zero- set of a section of the line bundle O ( d ), o ne has that Y is a degree d derived s ubmanifold of C P 3 which is der ived cob ordant to X . As such, the fundamental class of Y is equal to that of X , i.e. [ Y ] = [ X ]. The p oint is that the ab ov e formula (1.0.1) ta kes o n added signiﬁca nce as the fundamen ta l c la ss of Y , beca use it has the fo rm of an inclusion-exclusio n formula. One could say that the K -theo r y fundamental cla ss of Y is obtained by adding the fundamen ta l classes of d h yp er planes, subtracting oﬀ the fundamen tal cla sses of the over-counted  d 2  lines of 2-fold intersection, a nd a dding back the fundamen tal classes of the missed  d 3  po int s of 3-fold intersection. Ho p efully , this convinces the reader tha t derived manifolds may be of use. T o constr uct the virtual fundamental class on an arbitra ry intersection of com- pact subma nifo lds , we follow an idea of Ko ntsevich [15, Section 1.4], explained to us by J acob Lurie. Basically , we take our g iven intersection X = A ∩ B , realize it as the zero set of a section of a v ec to r bundle, and then deform that section until it is tra nsverse to zero . The result is a derived cob ordism b etw een X and a smo oth manifold. While disp ensing with the transversality r equirement for intersecting manifolds is app ealing , it do es come with a cost, namely that deﬁning the category of derived manifolds is a bit technical. Ho wever, a ny one fa miliar with ho motopy sheav es will not b e to o surprised by o ur constr uc tio n. One star ts with Lawv er e’s algebraic theory of C ∞ -rings, which ar e rings whose element s ar e clo sed under comp osition DERIVED SMOOTH MANIFOLDS 3 with smo oth functions . Simplicial (lax) C ∞ -rings ar e the a ppropriate homotopy- theoretic analogue and as such ar e ob jects in a simplicial mo del c a tegory . W e then form the category of loca l C ∞ -ringed spaces, wher ein an ob ject is a top ologica l spa ce together with a ho motopy sheaf of simplicial C ∞ -rings who s e stalks are lo ca l rings. Euclidean space, with its ( dis c rete) C ∞ -ring of smooth real-v alued functions is suc h an ob ject, a nd the zer o-set of ﬁnitely many s mo o th functions on Euclidea n space is called an aﬃne der ived manifold. A derived manifold is a loca l C ∞ -ringed space which is obtained b y patching together aﬃne derived manifolds . See Deﬁnition 6.15. Notation 1.1. Let sSets denote the monoida l categ ory of simplicial sets. A sim- plicial category C is a categ ory enriched ov er sSets ; w e denote the mapping space functor Ma p C ( − , − ). If all of the mapping spaces in C ar e ﬁbra n t (i.e. Kan com- plexes), we call C ﬁbr ant a s a simplicial category; in the following discussion we will be considering only this case . By a map b et ween ob jects X and Y in C , we mean a 0 -simplex in Map C ( X, Y ). An ob ject X ∈ C is calle d homotopy initial if for every Y ∈ C , the mapping space Map( X, Y ) is contractible. Similarly , X is called homotopy terminal if for every Y ∈ C , the mapping space Map( Y , X ) is contractible. W e say that a vertex in a contractible spa c e is homotop y-u nique . W e sometimes a buse notation and refer to a contractible space a s though it were just a single po int , saying something like “ the natural map Y → X .” Let C be a simplicial categ ory . The homoto p y pullback of a dia gram A f − → B g ← − C is a diagra m A × B C f ′ / / g ′   ⋄ C g   A f / / B . (1.1.1) By this we mean an ob ject A × B C equipped with maps g ′ , h ′ , and f ′ to A, B , and C res p ectively , and further e q uipped with homotopies b etw een g f ′ and h ′ and betw e e n f g ′ and h ′ . Finally , we requir e that A × B C is homotopy terminal in the category of such ob jects. More succinctly , Diagr am 1.1.1 expres ses that fo r any ob ject X ∈ C the natural map Map( X , A × B C ) → Map( X , A ) × h Map( X,B ) Map( X, C ) is a weak equiv alence in the usual m o de l categ o ry of simplicial s ets ( s ee [11, 7.10.8]), where by × h we mean the ho mo topy pullbac k in sSets . The diamond in the upper left cor ne r o f the square in Diagra m 1 .1.1 ser ves to remind the r eader that ob ject in the upp er left corner is a homotopy pullback, a nd that the diagra m do es not commute on the nose but up to chosen homotopies. W e can deﬁne homotopy pushouts simila rly . Two ob jects X and Y in C are sa id to b e e qu ivalent if there exist maps f : X → Y and g : Y → X such that g ◦ f and f ◦ g are homoto pic to the identit y maps on X and Y . By [20, 1.2.4.1], this is equiv alent to the asser tion that the map Map( Z, X ) → Map( Z, Y ) is a weak equiv alence for a ll Z ∈ C . If C is a discrete simplicial ca teg ory (i.e. a ca tegory in the us ual sense) then the homotopy pullbac k of a diagram in C is the pullback of that diagram. The pullback 4 DA VID I. SPIV AK of a diagra m A f − → B g ← − C is a commutativ e diagra m A × B C f ′ / / g ′   p C g   A f / / B . The symbo l in the upper left corner ser ves to remind the reader that the ob ject in the upp er left co rner is a pullback in the usual sense. Two ob jects are equiv alent if and only if they a re is o morphic. R emark 1.2 . If C is a simplicial mo del categ ory (see, e.g. [11] for an in tr o duction to this sub ject), then the full sub categ ory of coﬁbra nt-ﬁbrant ob jects is a simplicial category in which a ll ma pping spaces are ﬁbra nt . Mor e ov er, we can re pla ce any diagram with a diagram of the same s hap e in which all ob jects are coﬁbra n t-ﬁbr ant. Our deﬁnitions o f homotopy pullback, homoto p y pushout, and equiv alence match the mo del ca tegory termino lo gy . In keeping with this, if we ar e in the setting o f mo del catego ries, the result of a cons truction (such as tak ing a homotopy limit) will always b e assumed coﬁbrant and ﬁbr ant. Whenev er w e sp eak of pullbac ks in a si mplicial category , w e are al- w a ys referring to homotop y pullbac ks unl ess otherwise sp eciﬁed. Simi- larly , whenever we sp eak of terminal (res p. initial) ob jects, we a re alwa ys referring to homotopy terminal (resp. homotop y initial) ob jects. Finally , we sometimes use the word “ca tegory” to mean “simplicial c ategory .” Given a functor F : C → D , w e say that an ob ject D ∈ D is in the essential image of F if there is an ob ject C ∈ C such that F ( C ) is equiv alent to D . W e de no te the ca tegory of smo o th ma nifolds by Man ; whenever we r e fer to a manifold, it is a lwa ys assumed s mo oth. It is discrete as a simplicial categor y . In other words, we do not include any kind of homo topy information in Man . W e now recall a few w e ll- known facts and deﬁnitions a bo ut M an . E very mani- fold A has a tangent bundle T A → A which is a v e c tor bundle whose r ank is equal to the dimension o f A . A morphism of smo oth manifolds f : A → B induces a mor - phism T f : T A → f ∗ T B of vector bundles o n A , fro m the tang ent bundle of A to the pullback of the tangent bundle on B . W e say that f is an immersion if T f is injec- tive and w e say that f is a submersion if T f is surjective. A pa ir of maps f : A → B and g : C → B ar e called tr ansverse if the induced map f ∐ g : A ∐ C → B is a sub- mersion. If f and g are trans verse, then their ﬁber pro duct (over B ) exists in Man . 1.1. Results. In this pape r we hop e to con v ince the reader that we have a reas o n- able catego ry in which to do intersection theory on smo o th manifolds. The following deﬁnition expr esses what we mean b y “re asonable.” Deﬁnition 1.7 expresse s what we mean by “doing intersection theory” on s uch a catego ry . The main result of the pap er, Theorem 1.8, is tha t there is a simplicial category which satisﬁes Deﬁnition 1.7 Deﬁnition 1 .3. A simplicial ca tegory C is ca lled ge ometric if it s atisﬁes the fol- lowing a xioms: DERIVED SMOOTH MANIFOLDS 5 (1) Fibran t. F o r a n y tw o ob jects X , Y ∈ C , the mapping space Map C ( X , Y ) is a ﬁbra n t simplicial set. (2) Smo oth m anifolds. There exists a fully faithful functor i : Man → C . W e say that M ∈ C is a manifold if it is in the e s sential imag e of i . (3) Manifold limits. The functor i commutes with transverse intersections. That is, if A → M and B → M a re transverse, then a homotopy limit i ( A ) × i ( M ) i ( B ) exists in C and the natur a l map i ( A × M B ) → i ( A ) × i ( M ) i ( B ) is an equiv alence in C . F urthermore, the functor i preserves the terminal o b ject (i.e. the o b j ec t i ( R 0 ) is homo topy termina l in C ). (4) Underlying spaces. Let CG deno te the discrete simplicial category of compactly generated Hausdorﬀ s paces. There exists an “underlying space” functor U : C → CG , such that the dia g ram Man i / /   C U | | z z z z z z z z z CG commutes, wher e the vertical arr ow is the usua l underlying space functor on smo oth manifolds. F ur thermore, the functor U commutes with ﬁnite limits when they exist. R emark 1.4 . When we sp eak of an ob ject (r esp ectively , a morphism or a set o f morphisms) in C having some topolog ical prop erty (e.g. Haus do rﬀ or compact ob ject, prop er mo rphism, op en cov er , etc.), we mea n that the under lying ob j ec t (resp. the underlying morphism o r se t o f morphisms) in CG has that prop erty . Since any discrete simplicial ca tegory has ﬁbra nt mapping spaces, it is clear that Man and CG are g e ometric. If C is a geometr ic catego ry a nd M ∈ Man is a manifold, we gener ally a buse notation a nd write M in place of i ( M ), as though M were itself an ob ject o f C . R emark 1.5 . Note that in Axiom 3 we ar e no t requiring that i commute with a ll limits which exist in M an , only those whic h we hav e deemed appropr ia te. F o r example, if one has a line L and a parab ola P which meet tangentially in R 2 , their ﬁber product L × R 2 P do es exist in the catego ry of manifolds (it is a p oint). Howev er, limits like thes e a re precisely the kind we wish to av oid! W e ar e sear chin g for a category in which intersections are in so me sense stable under p e r turbations (see Deﬁnition 1 .7, Condition (4)), and th us w e should not ask i to pr eserve intersections which a re no t stable in this sens e . R emark 1.6 . In all of the axio ms of Deﬁnition 1.3, we are working with simplicial categorie s, so when we sp eak of pullbacks and puhouts, we mean homotopy pull- backs and homotop y puhouts. Axiom 4 r equires special comment how ever. W e take CG , the categ ory of compactly g enerated hausdo rﬀ spaces, as a discr et e s implicia l category , so homotopy pullbacks and pusho uts are just pullbacks and pushouts in the usual se nse. The underlying space functor U takes ﬁnite homo to p y pullbacks in C to pullbacks in CG . 6 DA VID I. SPIV AK Again, our goa l is to ﬁnd a ca tegory in which in ter sections of arbitrary subman- ifolds exist at the categorica l level and descend corr ectly to the level of cob ordism rings. W e make this precise in the following deﬁnition. Deﬁnition 1.7. W e say that a simplicia l catego ry C has the gener al cu p pr o duct formula in c ob or dism if the following conditions hold. (1) Geometric. The simplicial ca teg ory C is geo metric in the sense of Deﬁ- nition 1.3. (2) In tersections. If M is a manifold and A and B are s ubmanifolds (po ssibly not trans verse), then there exis ts a homotopy pullback A × M B in C , which we denote by A ∩ B a nd call the derive d interse ct ion of A a nd B in M . (3) Deriv ed cob ordism . There exists an equiv alence r e la tion o n the com- pact ob jects of C called derived cob ordis m, which extends the cob ordism relation on manifolds. That is, for any manifold T , ther e is a ring Ω der ( T ) called the derive d c ob or dism ring over T , and the fun c to r i : Man → C induces a homo mo rphism o f cob ordism rings ov er T , i ∗ : Ω( T ) → Ω der ( T ) . W e further impose the condition that i ∗ be an injection. (4) Cup pro duct fo rmula. If A a nd B ar e compact submanifolds of a manifold M with derived intersection A ∩ B := A × M B , then the cup pro duct formula (1.7.1) [ A ] ⌣ [ B ] = [ A ∩ B ] holds, where [ − ] is the functor ta k ing a compact derived submanifold of M to its image in the derived cobo rdism ring Ω der ( M ), and where ⌣ denotes the multiplication op e r ation in that ring (i.e. the cup pr o duct). Without the requirement (Condition 3) that i ∗ : Ω( T ) → Ω der ( T ) be an injection, the general cup pro duct formula could be triv ially attained. F or example, one could extend Man by including non-tr a nsverse intersections which were given no mor e structure than their under lying space, and the derived cob ordism relation could b e chosen to b e maxima l (i.e. one equiv alence cla ss); then the cup pro duct formula would trivially hold. In fact, when we even tually prove that there is a category which has the general cup product form ula, we will ﬁnd that i ∗ is not just an injection but an isomorphism (see Theorem 2.6). W e did not include that as an axio m here, how ever, b ecause it do es no t seem to us to be an inhere n tly necessary asp ect o f a go o d intersection theory . The catego ry of smo oth manifolds do es not hav e the genera l cup pro duct for- m ula b eca use it do es not satisfy condition (2). Indeed, supp ose that A and B are submanifolds of M . If A and B meet tr a nsversely , then their in ter section will naturally hav e the structure of a manifold, and the cup pro duct formula 1.7 .1 will hold. If they do not, then one of t wo things can happ en: either their intersection cannot b e given the structure o f a ma nifold (so in the classica l setting, Equation 1.7.1 do es not hav e meaning ) or their intersection can b e given the str ucture o f a smo oth manifold, but it wil l not sa tisfy Equa tion 1.7 .1. Therefore, we said that a categor y which satisﬁes the conditio ns o f Deﬁnition 1.7 satisﬁes the gener al cup pro duct for m ula b ecause co ndition (4) holds even for non-transverse intersections. Of course, to accomplish this, one needs to ﬁnd a DERIVED SMOOTH MANIFOLDS 7 more reﬁned notio n of intersection, i.e. ﬁnd a setting in which homotopy limits will hav e the desired prop erties. Suppo se that a simplicial category C has the general cup pro duct fo r mula in cob ordism. It follows that an y cohomology theory E which has fu nda men ta l classes for compa ct manifolds (i.e. for whic h there exists a map M O → E ) also has fundamen ta l class e s for compa c t ob jects of C , and th a t these satisfy t he cup product formula as well. Returning to our pr evious e xample, the union o f d hyperplane s in co mplex pro jective space is a derived ma nifo ld which, we will see , is derived cob ordant to a smo oth degre e d hypers urface (see Example 2 .7). Thus, these tw o subspaces hav e the same K -theory fundamental classes , which justiﬁes Equation 1.0.1. Our main r esult is that the conditions of Deﬁnition 1 .7 ca n b e satisﬁed. Theorem 1.8. Ther e exists a simplicial c ate gory dMan , c al le d the catego r y of derived manifolds , which has the gener al cup pr o duct formula in c ob or dism, in t he sense of Deﬁnition 1.7. The ca tegory dMan is deﬁned in Deﬁnition 6.15, and the ab ov e theorem is prov ed a s Theo rem ?? . See also Deﬁnition 2 .1 for a list of axio ms sa tisﬁed b y dMan . R emark 1.9 . W e do no t oﬀer a uniqueness co un ter part to Theorem 1.8. W e pur- po sely left Deﬁnition 1.7 loo se, b ecause we could not convince the reader that any more structure on a category C was necessa ry in order to sa y th a t it “has the general cup pro duct formula.” F or example, we could hav e required that the morphism i ∗ be an isomor phism instead of just a n injection (this is indeed the case for dMan , see Corollary 3.1 3); how ever, doing so would b e hard to justify a s being nec es- sary . Beca use of the gener a lity of Deﬁnition 1 .7, we are pre c luded from o ﬀering a uniqueness r esult here . Finally , the following pr op osition justiﬁes the need for simplicial categ ories in this work. Prop ositio n 1.10. If C is a discr ete s implicial c ate gory (i.e. a c ate gory in the usual sense), then C c annot have t he gener al cup pr o duct formula in c ob or dism. Pr o of. W e assume that Conditions (1), (2), and (3) hold, a nd we show that Con- dition (4) cannot. Since C is g eometric, the o b ject R 0 (tec hnica lly i ( R 0 )) is homotopy termina l in C . Since C is discr ete, R 0 is terminal in C , a ll equiv alences in C are isomor phisms, and all homotopy pullbacks in C are c ategorica l pullbac ks . Let 0 : R 0 → R b e the origin, a nd let X b e deﬁned a s the pullback in the diag ram X / /   p R 0 0   R 0 0 / / R . A mor phism from an arbitra ry ob ject Y to X consists o f t wo morphisms Y → R 0 which agr ee under comp osition with 0 . F or any Y , there is exactly one such morphism Y → X beca use R 0 is terminal in C . That is, X represents the same functor a s R 0 do es, so X ∼ = R 0 , i.e. R 0 ∩ R 0 = R 0 . This e quation forces Condition (4) to fail. 8 DA VID I. SPIV AK Indeed, to s ee that the cup pro duct formula [ R 0 ] ⌣ [ R 0 ] = ? [ R 0 ∩ R 0 ] do es not hold in Ω( R ), note tha t the left-hand side is homogeneo us of degree 2, whereas the right-hand side is ho mo geneous of degr e e 1 in the cohomolo g y ring.  1.2. Struc ture of the pap e r. W e hav e decide d to pr esent this pap er in a hier- archical fa shion. In the introductio n, we pr e sented the g oal: to ﬁnd a g eometric category tha t has the general cup pro duct formula in cob ordism (se e Deﬁnition 1.7). In Section 2, we pre sent a se t of a xioms that s uﬃce to achiev e this g oal. In other w o rds, a n y catego ry that satisﬁes the a x ioms o f Deﬁnition 2.1 is said to b e “go o d fo r doing intersection theory on manifolds,” and we prove in Theo rem 2.6 that such a catego ry has the general cup pro duct for mula. O f course, we could hav e chosen our axio ms in a trivial wa y , but this would no t have given a useful lay er of abstr action. Instead, we chose axio ms that resemble ax ioms sa tis ﬁe d by smo oth ma nifo lds . Thes e axioms imply the gener al cup pro duct for mu la , but a re not implied by it. In Sections 5 - 9, we give an explicit formulation of a categor y that is go o d for doing intersection theory . This categor y can b e succinc tly desc rib ed as “the category o f homotopical C ∞ -schemes of ﬁnite type.” T o mak e this precis e and prov e that it satisﬁes the axio ms of Deﬁnition 2.1 takes ﬁve sections. W e lay out our metho dolog y for this undertaking in Section 4. Finally , in Section 10, w e discuss related co nstructions. Firs t we see the wa y that derived manifolds are related to Jacob Lurie’s “ structured space s” ([21]). Then w e discuss ma nifolds with singularities, Chen spaces, diﬀ e o logical s pa ces, and synthetic C ∞ -spaces, all of which ar e generalizatio ns of the catego ry Man of s mo oth mani- folds. In this section we hop e to show how the theory of derived manifolds ﬁts into the e xisting liter a ture. 1.3. Ac kno wledgments. This paper is a reformulation and a simpliﬁcation of my PhD dissertation. Essentially , my advisor for that pr o ject was Jacob Lurie, whom I thank for many enlightening and helpful conv er sations, as well as for s uggesting the pr o ject. I w ould like to thank Dan Dugger for suggestions whic h improv ed the rea dabil- it y o f this pap er tremendously , as well as for his a dv ice throug ho ut the r ewriting pro cess. The re fer ee rep orts were also quite useful in debugging and c la rifying the pap er. I thank Peter T eichner for sug g esting that I mov e to Boston to w ork directly with J acob, as well as for many useful conv ersatio ns. Finally , I would like to thank Mathieu Anel, T om Gra b er , Phil Hirschhorn, Mike Hopkins, Andr´ e Joy a l, Dan Ka n, Grace Ly o, Ha ynes Miller, and Dev Sinha for useful con versations and encourag e men t a t v arious stages o f this pro ject. 2. The a xioms Theorem 1.8 makes clear our ob jectives: to ﬁnd a simplicial category in which the g e neral cup pro duct formula ho lds. In this section, s peciﬁca lly in Deﬁnition 2.1, we provide a set of axioms which • naturally extend cor resp onding prop erties of smo o th manifolds and DERIVED SMOOTH MANIFOLDS 9 • together imply Theore m 1.8. This is a n attempt to give the reader the a bilit y to work with der ived manifolds (at least a t a surface level) without fully under standing their technical underpinnings. In the following section, Sectio n 3, we will prove Theore m 1.8 from the a xioms presented in Deﬁnition 2.1. Then in Section 4 we will give a n outline of the internal structure of a s implicial categor y which satisﬁes the axioms in Deﬁnition 2.1. Fi- nally , in the remaining sections we will fulﬁll the outline given in Section 4, proving our main result in Sec tio n 9. Deﬁnition 2.1. A simplicial category dM is called go o d for doing i n terse ction the ory on manifolds if it satisﬁes the following axioms: (1) Geometric. The simplicial c a tegory dM is geometric in the sense of Deﬁnition 1.3. That is, roug hly , dM has ﬁbra n t mapping spaces , contains the categ ory Man of smo oth manifolds, has reasonable limits, and has underlying Hausdorﬀ spa ces. (2) Ope n sub ob jects. Deﬁnition 2.2. Suppose that X ∈ dM is an ob ject with underlying space X = U ( X ) and tha t j : V ֒ → X is a o p en inclusion of top olog ical spaces. W e say tha t a map k : V → X in dM is an op en su b obje ct over j if it is Ca rtesian over j ; i.e. • U ( V ) = V , • U ( k ) = j , a nd • If k ′ : V ′ → X is a map with U ( V ′ ) = V ′ and U ( k ′ ) = j ′ , suc h that j ′ factors throug h j , then k ′ factors homotopy-uniquely throug h k ; that is, the s pa ce of dotted arrows ma king the diagr am V ′ _ U   k ′ ' ' / / V _ U   k / / X _ U   V ′ j ′ 7 7 / / V j / / X , commute is co nt r actible. F or any X ∈ dM and any op en inclusion j as ab ove, there exists an op en sub o b ject k o ver j . Moreover, if a map f : Z → X of top olo gical s paces underlies a map g : Z → X in dM , then for any op en neighborho o d U of f ( Z ), the map g factors thro ugh the op en sub ob ject U over U . (3) Unions. (a) Suppos e tha t X a nd Y are ob jects of dM with underly ing spaces X and Y and that i : U → X a nd j : V → Y are op en sub ob jects with underlying spaces U and V . If a : U → V is an equiv alence, and if the union of X and Y a long U ∼ = V is Ha usdorﬀ (so X ∪ Y exists in CG ) then the union X ∪ Y (i.e. the colimit of j ◦ a a nd i ) exists in dM , and one has as exp ected U ( X ∪ Y ) = X ∪ Y . 10 DA VID I. SPIV AK (b) If f : X → Z and g : Y → Z a re mor phisms whose r estrictions to U agree, then there is a morphis m X ∪ Y → Z whic h r estricts to f and g r esp ectively on X and Y . (4) Finite limits. Given ob jects X , Y ∈ dM , a smo oth manifold M , and maps f : X → M and g : Y → M , there exis ts an ob ject Z ∈ dM and a homotopy pullback diagra m Z / / i   ⋄ Y g   X f / / M . (2.2.1) W e denote Z b y X × M Y . If Y = R 0 , M = R k , and g : R 0 → R k is the origin, we deno te Z by X f =0 , and we ca ll i the canonical inclusion of the zeroset of f into X . (5) Local mo de l s. W e sa y that an o b ject U ∈ dM is a lo c al mo del if there exists a smo oth function f : R n → R k such tha t U ∼ = R n f =0 . The virtual dimension of U is n − k . F or a ny o b j ec t X ∈ dM , the underlying space X = U ( X ) can b e cov ered by op en subsets in such a wa y that the corres p onding op en sub ob jects of X a re all lo cal mo dels. Mor e gener ally , any open cover of U ( X ) by op en sets can b e reﬁned to an op en cover whose corres p onding ope n s ubo b j ec ts are lo cal models . (6) Local extensions for imbeddi ng s. Deﬁnition 2.3. F or an y map f : Y → R n in dM , the canonical inclusion of the zer oset Y f =0 → Y is called a mo del imb e dding . A map g : X → Y is called an imb e dding if there is a cover of Y by op en sub ob jects Y i such that, if we s et X i = g − 1 ( Y i ), the induced maps g | X i : X i → Y i are mo del im b eddings . Such op en sub ob jects Y i ⊂ Y a re called t rivializi n g neighb orho o ds of the imbedding. Let g : X → Y be a n imbedding and let h : X → R be a map in dM . Then there exists a dotted arrow such that the diagram X h / / g   R Y ? ?     commutes up to homoto p y . (7) Normal bundle. Let M b e a smooth manifold a nd X ∈ dM with underlying space X = U ( X ). If g : X → M is a n imbedding, then there exists a n o p en neighborho o d U ⊂ M of X , a r eal vector bundle E → U , and a sectio n s : U → E such that X g / / g   ⋄ U z   U s / / E , DERIVED SMOOTH MANIFOLDS 11 is a homoto p y pullback diagram, where z : U → E is the zero section of the vector bundle. Let g = U ( g ) a ls o denote the underlying map X → U ; then the pullback bundle g ∗ ( E ) on X is unique up to is o morphism. W e call g ∗ ( E ) the normal bund le of X in M and s a deﬁning se ction . Ob jects in dM will be called derive d manifolds (of typ e dM ) and morphisms in dM will b e called m orphisms of derive d manifolds (of typ e dM ) . R emark 2.4 . W e de ﬁned the v ir tual dimension of a local mo del U = R n f =0 in Axiom 5. W e often dro p the word “virtual” a nd refer to the vir tual dimension of U simply as t he dimension o f U . W e will even tually deﬁne the virtual dimension of an arbitrar y derived manifold as the Euler characteristic of its cotangent complex (Deﬁnition 7 .5). F o r now, the reader o nly needs to know that if Z , X , Y , a nd M are a s in Diagr a m (2.2.1) a nd these ob jects hav e consta n t dimension z , x, y , and m respectively , then z + m = x + y , as e x pected. Let us brieﬂy explain the deﬁnition of im b edding (Deﬁnition 2.3) given in Axiom 6. If we add the word “transverse” in the appropria te pla ces, we are left with the usual deﬁnition of imbedding for smo o th manifo lds . This is pr oven in Prop os ition 2.5. Recall that a map of manifolds is called a (smo oth) imbedding if the induced map on the total s pa ces of their resp ective tangent bundles is a n injection. Sa y that a map o f ma nifo lds X → U is the inclu s ion of a level manifold if there ex ists a smo oth function f : U → R n , transverse to 0 : R 0 → R n , suc h tha t X ∼ = U f =0 ov er U . Prop ositio n 2.5. L et X and Y b e smo oth manif olds, and g : X → Y a sm o oth map. Then g is an imb e dding if and only if ther e is a c over of Y by op en submanifolds U i such that, if we set X i = g − 1 ( U i ) , e ach of the induc e d maps g | X i : X i → U i is the inclusion of a level manifold. Sketch of Pr o of. W e may a ssume X is connected. If g is a smo oth imbedding o f co dimension d , let U b e a tubular neigh b orho o d, tak e the U i ⊂ U to b e op en subsets that trivializ e the normal bundle of X , a nd take the f i : U i → R k to be identit y on the ﬁb ers . The zero sets of the f i are op en subsets of X , namely U i f i =0 ∼ = X i . Since they are smo oth o f the cor rect co dimension, the f i are transverse to zer o. F or the conv er se, no te that the pr op erty of b eing an imbedding is lo cal o n the target, so we ma y assume that X is the preimage of the origin under a map f : U → R k that is transverse to zero, where U ⊂ Y is some op en subset. The induced map X → U is clear ly injective on tangent bundles.  W e now present a r eﬁnement of Theorem 1.8, w hich we will prove as Corolla ry 3.13 in the following section. Reca ll that i : Man → dMan denotes the inclusio n guaranteed by Axiom 1 of Deﬁnition 2.1. Theorem 2.6. If dM is go o d for doing int erse ction the ory on m anifolds, then dM has the gener al cup pr o duct formula in c ob or dism, in the sense of Deﬁn ition 1.7. Mor e over, for any manifold T , the functor i ∗ : Ω( T ) → Ω der ( T ) 12 DA VID I. SPIV AK is an isomorphism b etwe en the classic al c ob or dism ring and the derive d c ob or dism ring (over T ). Example 2.7 . Let dM denote a simplicial catego ry which is go o d for do ing inter- section theory . By the unproven theo rem (2.6), we hav e a cob ordism theo r y Ω der and a nice cup pro duct formula for do ing intersection theory . W e now give several examples which illustra te v arious types of in ters ections. The last few examples are cautionary . T ransv ers e planes: Consider a k -pla ne K a nd an ℓ -plane L inside of pro jec- tive spac e P n . If K and L meet transversely , then they do so in a ( k + ℓ − n )- plane, whic h we denote A . In dM there is an equiv a lence A ∼ = K × P n L. Of course, this descends to an eq ua lit y [ A ] = [ K ] ⌣ [ L ] b oth in the cob ordism ring Ω( P n ) a nd in the derived cob ordism ring Ω der ( P n ). Non-transv erse planes: Suppose no w that K and L a r e as ab ove but do no t meet transversely . Their topo logical in ter section A ′ will hav e the structure of a smo oth manifold but of “ the wrong dimensio n” ; i.e. dim( A ) > k + ℓ − n . Moreov er , the for mula [ A ′ ] = ? [ K ] ⌣ [ L ] will not hold in Ω( P n ). How ever, the intersection of K a nd L as a derived manifo ld is diﬀerent from A ′ ; le t us denote it A . Although the underlying spaces U ( A ′ ) = U ( A ) will be the s a me, the virtual dimensio n of A will be k + ℓ − n as e xpe c ted. Moreov er , the form ula [ A ] = [ K ] ⌣ [ L ] holds in the derived cob ordism ring Ω der ( P n ). (The formula do es not makes sense in Ω( P n ) b eca use A is no t a smo oth manifold.) Fib er pro ducts and zerosets: Let M → P ← N b e a dia g ram of smo oth manifolds with dimensions m, p, and n . The ﬁber pr o duct X of this diagram exists in dM , and the (vir tual) dimension of X is m + n − p . F or ex a mple, if f 1 , . . . f k is a ﬁnite set o f smo oth functions M → R on a manifold M , then their zer oset is a derived manifold X , even if the f 1 , . . . , f k are not tr ansverse. T o see this, let f = ( f 1 , . . . , f k ) : M → R k and realize X as the ﬁber pr o duct in the diagram X / /   ⋄ R 0 0   M f / / R k . The dimension of X is m − k , whe r e m is the dimensio n of M . F or example, le t T deno te the 2 -dimensional torus a nd let f : T → R denote a Morse function. If p ∈ R is a cr itical v alue of index 1, then the pullback f − 1 ( p ) is a “ﬁg ure 8 ” (as a top olog ical space). It co mes with the structure of a derived manifold of dimension 1. It is der ived c o bo rdant bo th to a pair o f disjoint circles and to a s ingle circle; howev er, it is not isomorphic a s a der ived manifold to either of these b ecause its underlying top ological s pace is diﬀerent. Euler classes: Let M denote a compa c t smo oth manifold and let p : E → M denote a smo o th vector bundle; consider M as a submanifold of E by way of the zer o section z : M → E . The Euler clas s e ( p ) is o ften tho ug ht of as the cohomo logy class represented by the intersection of M with itself inside E . How ever, classically one must always b e care ful to per turb M ⊂ E DERIVED SMOOTH MANIFOLDS 13 befo re taking this self intersection. In the theo ry of derived manifolds, it is not necess ary to cho ose a per turbation. The ﬁber pro duct M × E M exists as a compact derived submanifold of M , and one ha s e ( p ) = [ M × E M ] . V ector bundles: More generally , let M denote a smo oth manifold and let p : E → M deno te a smo oth vector bundle and z : M → E the zero section. Given an arbitrar y section s : M → E , the zer o set Z ( s ) := z ( M ) ∩ s ( M ) of s is a derived submanifold of M . If s is tr ansverse to z inside E , then Z ( s ) is a submanifold of M , and its ma nifold structure coincides with its derived manifold str uc tur e. O therwise, Z ( s ) is a derived manifold that is not equiv alen t to any smoo th manifold. Changing s by a linear automorphis m of E (ov er M ) does not c hange the homotopy type of the derived manifold Z ( s ). Arbitrary changes of section do change the homotop y type: if s and t a re any tw o sectio ns of E , then Z ( s ) is not gener ally equiv alent to Z ( t ) as a derived ma nifold. How ever these t wo derived ma nifolds will b e derive d c ob or dant . The derived cob ordism can be given by a straight-line homotopy in E . F ail ure of Null stellensatz: Sup p os e that X and Y are v arieties (resp. man- ifolds), and that X → Y is a closed im b edding. Let I ( X ) denote the ideal of functions on Y which v anish o n X ; given an idea l J , let Z ( J ) deno te the zeroset of J in Y . A classical v er s ion of the Nullstellensatz states tha t if k is an algebr aically clos e d ﬁeld, then I induces a bijection b etw een the Za riski closed subsets of aﬃne space A n = Sp ec k [ x 1 , . . . , x n ] and the radical ideals of the ring k [ x 1 , . . . , x n ]. A co r ollary is that for any closed subset X ⊂ Y , one has X = Z ( I ( X )); i.e. X is the zero -set of the ide a l of functions which v anish on X . This radically fails in the derived se tting. The simult a neous zero-set of n functions Y → R a lw ays has co dimensio n n in Y . F or example, the x -axis in R 2 is the zer o-set of a single function y : R 2 → R , as a derived manifold. How ever, y is not the only function that v anishes on the x -axis – for example, so do the functions 2 y, 3 y , y 2 , and 0 . If we ﬁnd the simultaneous zero set of a ll ﬁv e functions, the resulting deriv ed manifold has codimension 5 inside o f R 2 . Its underlying to po logical space is indeed the x -axis, but its structure sheaf is very diﬀerent from that of R . Thu s , if we take a clo sed subv ariety o f Y and quotient the co ordinate ring of Y b y the inﬁnitely many fu nctio ns which v anish on it, the re s ult will hav e inﬁnite co dimension. The fo r mula X = Z ( I ( X )) fails in the der ived setting (i.e. b oth for derived manifolds and for derived schemes). W e no te o ne upshot of this. Given a close d submanifold N ⊂ M , one cannot identify N w ith the zeros et o f the ideal shea f of functions o n M which v anis h on N . Given an a rbitrary clos ed s ubset X of a manifold, o ne may wish to ﬁnd a n appropr iate derived ma nifold str ucture on X – this cannot be done in a ca no nical wa y as it can in clas s ical a lg ebraic geo metry , unless X is a lo cal complete intersection (see “V ector Bundles” exa mple ab ov e). Unions not i ncluded: Note that the union of manifolds alo ng closed sub- ob jects do es not gener ally come e quipped with the structure of a derived 14 DA VID I. SPIV AK manifold. The rea son we include this c a utionary no te is that, in the intro- duction, we sp oke of the union Y of d hyperplane s in C P n . How ever, w e were secretly r egarding Y as the zeros et of a single section of the bundle O ( d ) o n C P n , and not as a union. W e referred to it a s a union only so as to aid the readers imagination of Y . See the “V ector bundles” exa mple ab ov e for more information o n Y . Twisted cubic: W e include o ne more cautiona ry example. Let C denote the t wis ted cubic in P 3 ; i.e. the ima g e of the map [ t 3 , t 2 u, tu 2 , u 3 ] : P 1 → P 3 . Scheme-theoretically , the cur ve C cannot be deﬁned a s the zero set of tw o homogeneous p olynomials on P 3 , but it can b e deﬁned as the zeroset of three homog e ne o us po ly nomials on P 3 (or more pre c isely 3 sections of the line bundle O (2) on P 3 ); namely C is the zero-se t of the p olynomia ls f 1 = xz − y 2 , f 2 = y w − z 2 , f 3 = xw − y z . This mig h t lea d one to conclude that C is 3 − 3 = 0 dimensiona l as a derived manifold (see Axiom 5). It is true that the ze r oset o f f 1 , f 2 , f 3 is a zero-dimensio nal derived man- ifold, how ever this zeros et is probably not what one means b y C . Instead, one sho uld think of C as loca lly the zeroset of tw o functions. That is, C is the zeroset o f a certa in section o f a r a nk 2 vector bundle o n P 3 . As such, C is a o ne-dimensional der ived manifold. The reas on for the discr e pancy is this. The scheme-theoretic in ter sec- tion do es not take into account the dependency relations among f 1 , f 2 , f 3 . While these three functions are globally indepe ndent, they a re lo ca lly de- pendent everywhere. That is, for ev ery p oint p ∈ C , there is an op en neighborho o d on which t wo o f thes e three p olynomials genera te the third (ideal-theoretica lly ). This is not a n issue scheme-theoretically , but it is an issue in the derived setting. 3. Main resul ts In this section, we pr ov e Theor em 2.6 , which says that any s implicial ca tegory that satisﬁes the axioms presented in the last section (Deﬁnition 2.1) has the general cup pro duct formula (Deﬁnition 1.7). Before w e do so , we pr ov e an im b edding theorem (Prop osition 3.3) for compact derived manifolds, which says that a ny compact derived manifold can b e imbedded int o a larg e enough Euclidea n space. The pro of very clo sely mimics the co rresp ond- ing pro o f for co mpa ct smo oth manifolds, except we do not hav e to worry a bo ut the rank of Jaco bians. Fix a categ ory dM which is go o d for doing intersection theory on manifolds, in the sense of Deﬁnition 2 .1. In this section we re fer to ob jects in dM as derived manifolds a nd to mor phisms in dM as morphisms of derived manifolds. When we sp eak o f an “Axiom,” we ar e re fer ring to the axioms of Deﬁnition 2.1 Before proving Prop osition 3 .3, let us sta te a few lemmas. Lemma 3.1. L et f : X → Y and g : Y ′ → Y b e morphisms of derive d manifolds, such that g is an op en sub obje ct. Then ther e exists an op en sub obje ct X ′ → X and DERIVED SMOOTH MANIFOLDS 15 a homotopy pul lb ack diagr am X ′ / /   ⋄ Y ′ g   X f / / Y . Sketch of pr o of. Apply U and le t X ′ denote the pr eimage of U ( Y ′ ) in U ( X ). The corres p onding o p en sub ob ject X ′ → X , g uaranteed b y Axiom 2, s atisﬁes the uni- versal prop erty of the homotopy ﬁbe r pro duct.  W e state one more lemma a bo ut imbeddings . Lemma 3.2. (1) The pul lb ack of an imb e dding is an imb e dding. (2) If M is a manifold, X is a derive d m anifold, and f : X → M is a morphism, then the gr aph Γ( f ) : X → X × M is an imb e dding. Pr o of. The ﬁrst result is o b vio us by Deﬁnition 2.3, Lemma 3.1, and bas ic prop er ties of pullba cks. F or the se c ond r esult, we may assume that X is an a ﬃne der ived manifold and M = R p . W e hav e a ho motopy pullback diag ram X / /   ⋄ R 0   R n / / R k and by Axiom 6, the map f : X → R p is homoto pic to so me c ompo site X → R n f ′ − → R p . By P rop osition 2.5, the g raph Γ( f ′ ) : R n → R n × R p is an imbedding, and since the dia gram X Γ( f ) / /   ⋄ X × R p   R n Γ( f ′ ) / / R n × R p is a homotopy pullback, it follows from the ﬁrst re sult that the top map is an im b edding to o, as des ired.  The next theor em sa ys that any compact derived ma nifo ld c a n be im b edded into Euclidean space. This r e sult is prov ed for smoo th ma nifolds in [6, I I.10.7], and w e simply ada pt tha t pro of to the derived setting. Prop ositio n 3.3. L et dM b e go o d for interse ction t he ory, and let X ∈ dM b e an obje ct whose underlying sp ac e U ( X ) is c omp act. Then X c an b e imb e dde d into some Euclide an sp ac e R N . 16 DA VID I. SPIV AK Pr o of. By Axiom 5 , we can cov er X by lo cal mo dels V i ; let V i = U ( V i ) deno te the underlying space. F or each V i , there is a homotopy pullback diag ram V i / / x i   ⋄ R 0 0   R m i f i / / R n i in dM . F or each i , let β i : R m i → R be a smo oth function that is 1 on some op en disk D m i (cent er ed at the origin), 0 outside of s o me bigg er op en disk, and nonnegative everywhere. Deﬁne z i = β i ◦ x i : V i → R . Then z i is identically 1 on an op en subset V ′ i ⊂ V i (the preimage of D m i ) and iden tically 0 outside some closed neighborho o d of V ′ i in V i . Let V ′ i denote the op en sub ob ject of V i ov er V ′ i ⊂ V i . Deﬁne functions y i : V i → R m i by m ultiplication: y i = z i x i . By construction, we have y i | V ′ i = x i | V ′ i , and ea ch y i is co nstantly equa l to 0 outside o f a closed neig hbo rho o d of V ′ i in V i . W e ca n th us extend the z i and the y i to a ll of X by making them zer o outside of V i (Axiom 3(b)). By Axiom 5 and the compactness of X = U ( X ), we can choose a ﬁnite num b er o f indices i so that the V ′ i cov er a ll of X , say for indices 1 ≤ i ≤ k . F or each i , we hav e an all-C a rtesian diagra m V ′ i ⋄ / / y i   V i ⋄ / / x i   R 0 0   D m i / / R m i f i / / R n i , (3.3.1) by Lemma 3.1. Let N = P k i =1 m i . Let b 1 : R N → R m 1 denote the ﬁrst m 1 co ordinate pro- jections, b 2 : R N → R m 2 the next m 2 co ordinate pro jections, and s o on for ea ch 1 ≤ i ≤ k . T he seq ue nce W = ( y 1 , . . . , y k ) gives a map W : X → R N , such that for each 1 ≤ i ≤ k one has b i ◦ W = y i . W e will show that W is an imbedding in the sense of Deﬁnition 2.3; i.e. that the restrictio n of W to each V ′ i comes as the inclusion of the zeroset of smo o th functions c i on a c ertain op en subset of R N . The work has alr eady b een done; we just need to tease o ut what we alrea dy hav e. The c i should a ct like f i on the relev ant co ordina tes for V ′ i and should a ct like co o r dinate pro jections everywhere else. With that in mind, we deﬁne for each 1 ≤ i ≤ k , the function c i = ( b 1 , . . . , b i − 1 , f i , b i +1 , . . . , b k ) : R N → R N − m i + n i . DERIVED SMOOTH MANIFOLDS 17 W e construct the following diag r am V ′ i ⋄ / / y i   V i x i   / / ⋄ R 0 0   D m i ⋄   / / R m i   f i / / ⋄ R n i   D m i × R N − m i / / R N c i / / R ( N − m i )+ n i The low er r ig ht - hand vertical map is a co ordinate im b edding. The low er righ t square and the lo wer left square are pullbac ks in the category of manifolds, so they are homotopy pullback in dM by Axio m 1. The upp e r square s are pullbacks (see equation 3.3 .1). Therefore the diag ram is (homotopy) all-Cartesian. The vertical comp osite V ′ i → D m i × R N − m i is the restriction of W to V ′ i , and it is also the zero set of the horizontal comp osite D m i × R N − m i → R ( N − m i )+ n i . Since D m i × R N − m i → R N is the inclusio n of a n op en subset, we hav e s hown that W is an imbedding in the sense of Deﬁnition 2.3.  The following r elative version of Prop osition 3.3 is pr ov en in almost exactly the same way . Recall that a ma p f of top olo gical spaces is sa id to b e pr op er if the inv erse image under f of any compact subspace is compa ct. A mor phis m of derived manifolds is said to b e prop er if the underlying mor phism of top olo gical spa ces is prop er. Corollary 3.4 . L et dM b e go o d for interse ction the ory, let X ∈ dM b e a derive d manifold, and let M ∈ M an b e a manifold. S u pp ose t hat f : X → M is pr op er, A ⊂ M is a c omp act su bset and A ⊂ f ( U ( X )) . L et A ′ denote the interior of A , and let X ′ = f − 1 ( A ′ ) . Ther e exists an imb e dding W : X ′ → R N × A ′ such that the diagr am X ′ / / W   X f   R N × A ′ π 2   A ′ / / M c ommutes, and the c omp osite π 2 ◦ W : X ′ → A ′ is pr op er. Sketch of pr o of. Copy the pro of of Prop os ition 3 .3 verbatim until it requir es the compactness o f X = U ( X ), which do es not hold in our ca se. Instea d, use the compactness of f − 1 ( A ) and choo se ﬁnitely many indices 1 ≤ i ≤ k such tha t the union ∪ i V ′ i contains f − 1 ( A ). Con tinue to c o py the pro of verbatim, except replace instances of X with ∪ i V ′ i . In this w ay , we prov e that one has an imbedding o f the derived ma nifold W ′ : ∪ i V ′ i → R N . Let f ′ : X ′ → A ′ be the pullback of f , and note that X ′ = f − 1 ( A ′ ) is co nt a ined in ∪ i V i . The restriction W ′ | X ′ : X ′ → R N and the map f ′ together induce a 18 DA VID I. SPIV AK map W : X ′ → R N × A ′ , which is a n imbedding by Lemma 3.2. By c o nstruction, f ′ = π 2 ◦ W is the pullback of f , so it is prop er.  3.1. Deriv ed cobordis m. Fix a simplicial catego ry dM that is g o o d for doing int er section theory on manifolds in the sense of Deﬁnition 2.1. One g e nerally deﬁnes cob ordism using the idea of manifolds with b oundary . Under the a bove deﬁnitions, manifolds with b oundary are not derived manifolds . One ca n deﬁne derived manifolds with b oundary in a wa y that emu la tes Deﬁni- tion 2.1, i.e. one c o uld give axioms for a simplicial category with lo cal mode ls that lo ok like genera lizations of ma nifolds with b oundar y , etc. One could then prove that the lo cal deﬁnition was equiv alent to a glo bal one , i.e. one co uld pr ov e that derived manifolds with b oundary can b e im b edded into Euclidea n half space. All of this was the a ppr oach o f the author’s diss ertation, [27]. F or the curr ent pr e- sentation, in the interest o f spa ce, we cho ose to disp ense with all that and deﬁne cob ordism using the idea of prop er maps to R . T o o rient the r eader to Deﬁnition 3.6, w e reformulate the usual co bo rdism r ela- tion for manifolds using this appro ach. Two compact smo oth manifolds Z 0 and Z 1 are cob orda nt if there exists a manifold (without b oundar y ) X and a prop er map f : X → R such that • for the p oints i = 0 , 1 ∈ R , the map f is i -collar e d in the s e nse of Deﬁnition 3.5, and • for i = 0 , 1, there is an isomorphism Z i ∼ = f − 1 ( i ), also written Z i ∼ = X f = i . This is precise ly the deﬁnition we give for der ived co bo rdism, except that we allow Z 0 , Z 1 , and X to b e a derived manifolds. Deﬁnition 3.5. Let X denote a derived manifold and f : X → R a mo rphism. Given a p o int i ∈ R , we say that f is i -c ol lar e d if there exists an ǫ > 0 and an equiv alence X | f − i | <ǫ ≃ X f = i × ( − ǫ, ǫ ) , (3.5.1) where X | f − i | <ǫ denotes the op en sub ob ject of X ov e r ( i − ǫ, i + ǫ ) ⊂ R (see Lemma 3.1). Deﬁnition 3. 6 . Le t dM b e go o d for in ters ection theor y . Compact de r ived mani- folds Z 0 and Z 1 are said to b e derive d c ob or dant if there exists a derived manifold X and a pr op er map f : X → R , such that for i = 0 and for i = 1 , the map f is i -collar ed and there is a n equiv alence Z i ≃ X f = i . The map f : X → R is called a derive d c ob or dism b etwe en Z 0 and Z 1 . W e r efer to Z 0 ∐ Z 1 as the b oundary of the cob ordism. If T is a CW co mplex, then a derive d c ob or dism over T is a pair ( a, f ), wher e f : X → R is a derived cob or dism and a : U X → T is a cont inuous map to T . If T is a ma nifold, w e denote the r ing whose elements a re derived cob ordis m classes o ver T (with sum given by disjoint union a nd product given b y ﬁber pro duct ov er T ) by Ω der ( T ). W e use Ω der to denote Ω der ( R 0 ). R emark 3.7 . In Deﬁnition 3 .6 we sp eak of der ived cob ordism classes, even though we ha ve not yet shown that der ived cobo rdism is an equiv alence relation on compact derived ma nifolds. W e will prov e this fact in Prop osition 3.1 0. DERIVED SMOOTH MANIFOLDS 19 R emark 3.8 . W e did not deﬁne or ient e d derived manifolds, so a ll of our cob ordis m rings Ω( T ) and derived cob or dism r ings Ω der ( T ) should b e taken to b e uno r iented. How ever, the oriented ca se is no harder, we m ust simply deﬁne o riented derived manifolds and or ient ed der ived cob or disms. T o do so, imbed a co mpact derived manifold X in to so me R n (whic h is p ossible by Prop ositio n 3.3), and consider the normal bundle g uaranteed by Axio m 7. W e deﬁne a n orie n tatio n on X to b e an orientation on some such normal bundle. One o r ients derived cobo rdisms similarly . All results which w e prove in the unoriented case als o w o r k in the or iented case. R emark 3 .9 . One can show that if U is an op en neighborho o d of the closed interv al [0 , 1] ⊂ R then any pro per map f : X → U ca n be extended to a prop er map g : X → R with an isomorphism f − 1 ([0 , 1]) ∼ = g − 1 ([0 , 1]) over [0 , 1]. So, one can consider such an f to b e a derived cob or dism b etw een f − 1 (0) and f − 1 (1). Prop ositio n 3.10. L et dM b e go o d for int erse ction t he ory. D erive d c ob or dism is an e quivalenc e r elation on c omp act obje ct s in dM . Pr o of. Deriv ed cob ordism is clear ly symmetric. T o see that it is reﬂexive, let X b e a compact derived ma nifo ld and co nsider the pro jection map X × R → R . It is a der ived co b or dism b etw een X and X . Finally , w e must show that der ived cob ordism is transitive. Supp ose that f : X → R is a derived cob ordism betw ee n Z 0 and Z 1 and that g : Y → R is a der ived cob ordism betw een Z 1 and Z 2 . By Axiom 3 , w e can glue open subo b jects X f < 1+ ǫ ⊂ X and Y g> − ǫ ⊂ Y together along the common op en subset Z 1 × ( − ǫ, ǫ ) to obtain a derived manifold W tog ether with a pr o pe r map h : W → R suc h that for i = 0 , 2, we ha ve W h = i = Z i . The result follows.  Lemma 3.11. Le t dM b e go o d for interse ction the ory. The functor i : Man → dM induc es a homomorph ism on c ob or dism rings i ∗ : Ω → Ω der . Pr o of. It suﬃces to show that manifolds which are co b or dant a r e derived cob or - dant. If Z 0 and Z 1 are manifolds w hich a re co bo rdant, then there exists a compact manifold X with b oundary Z 0 ∐ Z 1 . It is well-known that one can imbed X in to R n in such a way that for i = 0 , 1 we hav e Z i ∼ = X x n = i , where x n denotes the last co ordinate o f R n . Because each Z i has a collar neighbor ho o d, we can a ssume that X | x n − i | <ǫ ∼ = X x n = i × ( − ǫ , ǫ ) for some ǫ > 0. The preimage e X ⊂ X of ( − ǫ, 1 + ǫ ) is a manifold (without bo undary), hence a derived manifold (under i ). By Rema r k 3.9, x n : e X → R is a derived cob or dism b et ween Z 0 and Z 1 .  Theorem 3.12. L et dM b e go o d for interse ction the ory. The functor i ∗ : Ω → Ω der is an isomorphi sm . Pr o of. W e ﬁrs t show that i ∗ is surjective; i.e. that every compact derived manifold Z is derived co bo rdant to a smo oth manifold. Let W : Z → R N denote a clo sed im b edding, which exists b y Pr o po sition 3 .3. Let U ⊂ R N , E → U , and s, z : U → E 20 DA VID I. SPIV AK be the o p en neig hbo rho o d, vector bundle, zer o se c tio n, a nd de ﬁning section fro m Axiom 7, so that the diagr am Z g / / g   ⋄ U z   U s / / E , is a homo topy pullback. Note that U and E are smo o th manifolds, and that the image z ( U ) ⊂ E is a clos ed subset. Since Z is compa ct, we can c ho o se a co mpact subset U ′ ⊂ U whos e in ter ior contains Z . Let A ⊂ U b e the complimen t of t he interior of U ′ . Then s ( A ) ∩ z ( U ) = ∅ , so in particular they ar e transverse as closed submanifolds of E . By [28, App 2, p. 24], there exists a regular ho mo topy H : [0 , 1] × U → E such that • H 0 = s : U → E , • H 1 = t , wher e t : U → E is transverse to the clo sed subset z ( U ) ⊂ E , and • for all 0 ≤ i ≤ 1, w e have H i | A = s | A . F or any ǫ > 0 we can extend H to a homo topy H : ( − ǫ, 1 + ǫ ) × U → E , with the same thr ee bulleted prop erties, by making it constant on ( − ǫ, 0] and [1 , 1 + ǫ ). Consider the a ll-Cartesian diagr am Z i ⋄ / /   P ⋄ / / a   U z   R 0 × U ⋄ i × id U / / π   ( − ǫ, 1 + ǫ ) × U H / / π   E R 0 i / / R for i = 0 , 1. Notice that Z 0 = Z and that Z 1 is a smo oth manifold. It suﬃces to show that the map π ◦ a : P → R is prop er, i.e. that for each 0 ≤ i ≤ 1, the int er section of H i ( U ) and z ( U ) is co mpact. Since H i ( A ) ∩ z ( U ) = ∅ , one has H i ( U ) ∩ z ( U ) = H i ( U ′ ) ∩ z ( U ) , and the right hand side is compact. Th us H is a derived cob or dism b etw een Z and a smo o th manifold. T o prov e that i ∗ is injective, we must show that if s mo o th manifolds M 0 , M 1 are derived cobo rdant, then they are smo o thly cob ordant. Let f : X → R denote a derived cob or dis m b etw een M 0 and M 1 . Let A ′ denote the o p en interv al ( − 1 , 2) ⊂ R , a nd let X ′ = f − 1 ( A ′ ) . By Co rollary 3.4 we can ﬁnd an imbedding W : X ′ → R N × A ′ such that f ′ := f | X ′ = π 2 ◦ W is pr op er. Note that for i = 0 , 1 w e still hav e X ′ f ′ = i = M i . By Remark 3.9, f ′ induces a derived cobo rdism betw een M 0 and M 1 with the added b eneﬁt o f factoring thro ugh an imbedding X ′ → R N int o Euclidean space. Again we use Axiom 7 to ﬁnd a vector bundle and section s on a n op en subset of R N × R whos e zeroset is X ′ . W e a pply [28, App 2, p.24] to ﬁnd a reg ular homotopy betw e e n s and a section t which is tra nsverse to zero , all the while keeping the DERIVED SMOOTH MANIFOLDS 21 collar closed submanifold M 0 × ( − ǫ, ǫ ) ∐ M 1 × (1 − ǫ, 1 + ǫ ) ﬁxed. The zeroset o f t is a smo oth cob ordism b etw een M 0 and M 1 .  Corollary 3.13. If dM is go o d for doing int erse ction the ory on m anifolds, then dM has the gener al cup pr o duct formula in c ob or dism, in the sense of Deﬁnitions 1.3 and 1.7. Mor e over, for any manifold T , the fun ctor i ∗ : Ω( T ) → Ω der ( T ) is an isomorphism b etwe en the classic al c ob or dism ring and the derive d c ob or dism ring (over T ). Pr o of. Suppose dM is go o d for doing intersection theory o n manifolds. Since T is assumed to b e a CW complex, if Z ⊂ X is a close d subset of a metrizable top ological space, any map f : Z → T extends to a map f ′ : U → T , where U ⊂ X is a n open neighborho o d of Z . O ne ca n now mo dify the pro of of Theorem 3.12 so tha t all constructions ar e suitably “ov er T ,” which implies that i ∗ is an iso morphism. Let us quickly expla in how to make this mo diﬁcation. W e begin with a compact de r ived ma nifold Z = ( Z, O Z ) a nd a map of top ological spaces σ : Z → T . Since Z is Ha usdorﬀ and paracompact, it is metriz a ble. It follows that the map σ extends to a map σ ′ : U → T , where U ⊂ R N is op en neighborho o d of T and σ ′ | Z = σ . By in ters ecting if necessa ry , we ca n take this U to b e the op en neig hborho o d given in the pr o of o f Theorem 3.1 2. The vector bundle p is canonically deﬁned over T (via σ ′ ), as are the s ections s, z . O ne can contin ue in this wa y and see that the pro of extends without any additional work to the r elative setting (over T ). Now that we hav e proved the s econd assertio n, that i ∗ is an isomo rphism, we go back and show the ﬁrst, i.e. that dM ha s the gener a l cup pro duct formula in cob ordism. The ﬁrst three co nditio ns of Deﬁnition 1.7 follow from Axiom 1, Axio m 4, a nd Theore m 3.12. T o prov e the last conditio n, let M b e a manifold and supp ose that j : A → M and k : B → M a re co mpact subma nifo lds . There is a ma p H : A × R → M such that H 0 = j a nd such that H 1 : A → M is transverse to k . F o r i = 0 , 1, let X i denote the in ter section of (the images of ) H i and k . Note that X 0 = A × M B is a compact derived manifold which we also denote A ∩ B . F urther, X 1 is a compact smo oth manifold, often called the “transverse intersection of A a nd B ” (b ecause A is “ma de trans verse to B ”), and X 0 and X 1 are derived co b or dant over M . It is well known that [ X 1 ] = [ A ] ⌣ [ B ] as elements of Ω( M ). Since Ω( M ) ∼ = Ω der ( M ), and since [ X 0 ] = [ X 1 ] as elements of Ω der ( M ), the for m ula [ A ] ⌣ [ B ] = [ A ∩ B ] holds.  4. La yout for the co nstruct io n of dMan In the next few sections we will construct a simplicial categ ory dMan which is go o d for doing in ter section theo ry o n manifolds, in the sense of Deﬁnition 2.1. An ob ject of dM an is called a derive d manifold a nd a mor phism in dMan is ca lled a morphism of derive d manifolds . 22 DA VID I. SPIV AK A derived manifold could be called a “homotopical C ∞ -scheme of ﬁnite t yp e.” In the c oming sections w e will build up to a pr ecise deﬁnition. In the current section we will g ive a brief o utline o f the construction. Let us ﬁrst reca ll the pro cess by which o ne deﬁnes a scheme in algebr aic g eom- etry . One b egins with the categor y of commutativ e ring s , which can be deﬁned as the catego ry of a lgebras of a certain algebr aic theo ry (see [5, 3.3.5 .a]). One then deﬁnes a ringed space to b e a space X tog ether with a sheaf of rings O X on X . A lo cal ringed space is a ringed space in which all sta lk s ar e lo cal ring s. One must then functorially assig n to ea ch commutativ e ring A a lo cal ring ed space Spec A , called its prime s pectr um. Once that is done, a scheme is deﬁned as a lo cal r inged space which can b e cov er ed by open subob jects , each of which is is omorphic to the prime spe c trum of a ring. Note that the purp ose of deﬁning lo ca l ringed spaces is that morphisms of schemes hav e to b e morphisms of lo cal ringed spaces, not just morphisms of ringed spaces. If one is interested only in schemes of ﬁnite type (over Z ), one do es not need to de ﬁne prime sp ectra for all ring s. One co uld ge t awa y with just deﬁning the category of aﬃne spaces A n = Sp ec Z [ x 1 , . . . , x n ] as loc a l ring ed spaces . One then uses as lo cal mo dels the ﬁb er pro ducts o f aﬃne spaces, as taken in the c a tegory o f lo cal r inged spa ces. W e recall quickly the notion of an algebraic theor y [16]. An a lgebraic theory is a category T with ob jects { T i | i ∈ N } , such that T i is the i -fold pro duct of T 1 . A T - algebra is a pro duct preserving functor from T to Sets . F or example, the categor y of rings is the categor y of a lgebras on the alge br aic theory with o b jects A n Z and morphisms given by p olynomial maps b etw een aﬃne spac e s. A la x simplicial T - algebra is a functor from T to s Sets that descends to a pro duct-preser ving functor on the homotopy categor y . The ba s ic outline of our construction o f derived manifolds follows the a bove construction fair ly clos ely . The diﬀerences are tha t • Rings a re not suﬃcien t as our basic ob jects. W e need a smo o th version of the theo ry , whose alg ebras ar e called C ∞ -rings. • Ev er ything m ust be done homo topically . Our basic ob jects are in fact la x simplicial C ∞ -rings, which means tha t the deﬁning functor is not required to b e “pr o duct pre s erving on the nose” but instea d weakly pro duct pre- serving. • F urther more, our sheaves ar e homotopy sheaves, which mea ns that they satisfy a ho motopical version of descent. • Our aﬃne spaces a re quite familiar : they a re simply the Euclidean spaces R n (as smo oth manifolds). A morphism o f aﬃne spaces is a s mo o th function R n → R m . • W e use as our lo cal mo dels the homoto p y ﬁber pro ducts of a ﬃne spaces, as ta ken in the catego ry of loca l C ∞ -ringed spaces. A der ived manifold is hence a s mo o th, homotopic a l version of a scheme. In sec- tion 5 we deﬁne our basic ob jects, the lax C ∞ -rings, a nd prov e some lemmas ab out their r e lationship with C ∞ -rings (in the usual sense) and with commutativ e rings. In Section 6, we deﬁne lo ca l C ∞ -ringed spaces and derived manifo lds (Deﬁnition 6.15). W e then must dis c us s cota ngent complexes for der ived manifolds in Section 7. In Section 8, w e give the pro o fs of se veral technical res ults. Finally , in Sectio n DERIVED SMOOTH MANIFOLDS 23 9 we prov e that our catego r y of derived manifolds is go o d for doing int er section theory on manifolds, in the sense of Deﬁnition 2 .1. Convention 4 .1 . In this paper we will r ely hea vily on the theory of mo del ca tegories and their lo caliz a tions. See [11] o r [12] for a go o d in tro duction to this sub ject. If X is a categor y , ther e a r e tw o co mmon mo del structures on the simplicial presheaf category sS e ts X , c alled the injectiv e and the pro jective mo del structures. In this work, w e alw a ys use the injective mo del s tructure . If we s p ea k of a mo del structur e o n sSets X without s pecifying it, we mean the injective mo del structure. The injective mo del s tructure on s Sets X is given by ob ject-wise coﬁbra- tions, ob ject-wise weak equiv alences, and ﬁbrations determined by the rig h t- lifting prop erty with res pec t to acyclic coﬁbrations. With the injectiv e mo del structure, sSets X is a left prop er, combinatorial, sim- plicial mo del ca tegory . A s stated a bove, weak eq uiv alences a nd co ﬁbrations a re determined ob ject-wise; in particular every o b ject in sSets X is coﬁbra nt. See, for example, [4] or [20] for further details. 5. C ∞ -rings Let E denote the full sub c ategory of Man spanned by the Euclidean spa ces R i , for i ∈ N ; w e refer to E as the Euclide an c ate gory . Lawv ere, Dubuc, Mo erdijk and Reyes, and o thers hav e studied E as an a lgebraic theory (see [17], [9], [22]). The E -a lgebras, which are deﬁned as the pro duct prese r ving functor s fr o m E to sets, are calle d C ∞ -rings. W e use a homotopical version, in which we replace sets with simplicia l sets, and strictly pr o duct-preserving functors with functors which preserve pr o ducts up to weak equiv alence. Let sSets E denote the s implicial mo del catego ry of functors fro m E to the cat- egory of simplicial sets. As usual (see Conv ention 4.1), we use the injectiv e mo del structure on sSets E , in which co ﬁbrations and weak equiv alences a re determined ob ject-wise, and ﬁbratio ns are determined b y the r ight-lifting prop erty . F or i ∈ N , let H i ∈ sSets E denote the functor H i ( R n ) = Hom E ( R i , R n ) . F or each i, j ∈ N , let (5.0.1) p i,j : H i ∐ H j → H i + j denote the natural map induced by coo rdinate pro j ec tio ns R i + j → R i and R i + j → R j . W e also refer to H i as H R i (see E x ample 5.6). Note that if F : E → sSe ts is an y functor then the Y oneda lemma gives a natural isomorphism o f simplicia l sets Map( H i , F ) ∼ = F ( R i ) . Deﬁnition 5.1. With notatio n as ab ov e, deﬁne the categor y of lax simplici al C ∞ - rings , denoted s C ∞ , to b e the lo ca lization of s Se ts E at the set P = { p i,j | i, j ∈ N } (see [11, 3.1.1 ]). W e often r efer to the ob jects of s C ∞ simply as C ∞ -rings, dro pping the words “lax simplicial.” W e can identify a discrete ob ject in s C ∞ with a (strict) C ∞ -ring in the class ic al sense (see [22]), and thus we refer to these o b jects as d iscr ete C ∞ -rings. The mo del category s C ∞ is a left prop er, co ﬁbrantly-generated simplicial model category ([11, 3.4.4]), in which all ob jects are coﬁbrant. Let F ∈ sSets E be a 24 DA VID I. SPIV AK ﬁbrant ob ject, conside r ed as an o b ject of s C ∞ . Then F is ﬁbrant in s C ∞ if and only if it is lo ca l with resp ect to all the p i,j ; i.e. for ea ch i, j ∈ N the natural map Map( H i + j , F ) → Map( H i ∐ H j , F ) is a w eak equiv alence . Since eac h H i is the functor represent ed b y R i , this is equiv alent to the condition that the natura l map F ( R i + j ) → F ( R i ) × F ( R j ) be a weak equiv alence. In o ther words, F : E → sSets is ﬁbrant in s C ∞ if and only if it is w ea kly pr o duct preser ving (and ﬁbrant as a n ob ject of sSets E ). R emark 5.2 . There is a mo del category o f s trict simplicial C ∞ -rings, consisting of (strictly) pro duct preserving functors fro m E to sSets . It is Q uillen equiv alent to our s C ∞ (see [19, 15.3]). The reason w e use the lax version is that, a s a localiza tion of a left prop er mo del catego ry , it is clea r ly left pro pe r. (Note that, by [24], we could hav e used a simplicial version of the theo ry to o btain a prop er mo del of strict algebras , but this mo del is more diﬃcult to us e for our purp oses.) This left prop ernes s of s C ∞ is necess a ry for the further lo calization we use to deﬁne the category of homo topy sheav es of C ∞ -rings. The use of ho motopy sheaves is one of the key wa ys in which o ur theory diﬀers from the synthetic diﬀer en tia l geometry liter ature. Prop ositio n 5.3. Supp ose that φ : F → G is a morphism of ﬁbr ant C ∞ -rings. Then φ is a we ak e qu ivalenc e if and only if φ ( R ) : F ( R ) → G ( R ) is a we ak e quiva- lenc e of simplicial sets. Pr o of. F ollows from basic facts ab out Bousﬁeld lo calization.  Note that if F ∈ s C ∞ is ﬁbra nt a nd if for all i ∈ N the simplicial set F ( R i ) is a discrete set, then F is a C ∞ -ring in the classica l sens e – that is, it is a strictly pro duct pre s erving functor from E to Sets . Lemma 5 .4. The fun ct or π 0 : sSets E → Sets E sends ﬁbr ant obje cts in s C ∞ to C ∞ -rings (in the classic al sen s e). It sends obje ct-wise ﬁbr ant homotopy pushout diagr ams in s C ∞ to pushouts of C ∞ -rings. Pr o of. If F ∈ s C ∞ is ﬁbrant then Map( p i,j , F ) is a weak equiv alence of s implicia l sets for each p i,j ∈ P . Hence π 0 Map( p i,j , F ) is a bijection of s e ts . Since ea ch p i,j is a map betw e e n discr e te ob jects in s C ∞ , we ha ve π 0 Map( p i,j , F ) = Hom( p i,j , π 0 F ), so π 0 F is indeed a C ∞ -ring. T o pr ov e the second assertion, supp os e Ψ = ( A f ← − B g − → C ) is a diag ram of ﬁbrant ob jects in s C ∞ . F actor f as a coﬁbration B → A ′ follow ed by an acyclic ﬁbration A ′ → A . The homoto p y colimit o f Ψ is given by the usual colimit o f the diag ram Ψ ′ of functors A ′ ← B → C . Applying π 0 commutes with taking co limits. Since A ′ and A a re b oth ﬁbrant and weakly eq uiv alen t in s C ∞ , they ar e weakly equiv alent in sSets E , so π 0 A ′ → π 0 A is an isomorphism in Sets E . Hence we have π 0 ho colim(Ψ) ∼ = π 0 colim(Ψ ′ ) ∼ = colim( π 0 Ψ ′ ) ∼ = colim( π 0 Ψ) , completing the pro o f. DERIVED SMOOTH MANIFOLDS 25  The following lemma is per haps unnecessary , but w e include it to give the reader more of an idea of how classica l (i.e. discr ete) C ∞ -rings work. Lemma 5.5. L et F b e a discr ete ﬁbr ant C ∞ -ring. Then t he set F ( R ) n atur al ly has the st ructur e of a ring. Pr o of. Let R = F ( R ) ∈ Sets a nd note that F ( R i ) ∼ = R i , and in particular F ( R 0 ) = {∗} is a s ingleton set. L et 0 , 1 : R 0 → R denote the a dditive and multiplicativ e units, let + , " : R 2 → R denote the addition a nd m ultiplicatio n functions , and let ι : R → R denote the a dditive inv er se. All of these ar e smo oth functions and a re hence mor phisms in E . Applying F , we obtain elements F (0) , F (1) : {∗ } → R , tw o binary functions F (+) , F ( " ) : R × R → R , and a function F ( ι ) : R → R . Since all of the ring axioms can b e written as the commutativit y condition on a dia g ram, and since F is a functor and pres erves co mmutative diagrams, one s ees that the op erations F (0 ) , F (1) , F (+) , F ( " ), and F ( ι ) satisfy all the axioms for R to b e a ring.  If F ∈ s C ∞ is ﬁbrant but no t necessarily discrete, then the ring axioms hold up to homotopy . Note, howev er, that even in the discr ete cas e , R = F ( R ) ha s muc h more structure than just that of a ring. Any smo o th function R n → R m gives rise to a function R n → R m , satisfying all appr o priate c ommu ta tiv e diag rams (e.g . the functions 3 a + b and 3 a 3 b taking R 2 → R ar e equal, so they ar e sent to equal v er tices of the mapping space Ma p( R 2 , R )). Example 5.6 . Let M b e a manifold. Let H M : E → sSets be deﬁned by H M ( R i ) := Hom Man ( M , R i ). O ne checks ea sily that H M is a discre te ﬁbra n t C ∞ -ring. W e usually denote H M by C ∞ ( M ), though this notation tends to o bscure the role of H M as a functor, instead highlighting the v alue of H M on R , the set of C ∞ -functions M → R . Since s C ∞ is a mo del ca teg ory , it is closed under homotop y colimits. The homotopy colimit o f the diag ram C ∞ ( R 0 ) ← C ∞ ( R ) → C ∞ ( R 0 ), induced by the diagram o f manifolds R 0 0 − → R 0 ← − R 0 , is an example of a non-discr ete C ∞ -ring. Deﬁnition 5.7. Let E alg denote the sub categ ory of E whose ob jects are the Eu- clidean spaces R i , but in which we take as morphisms only those maps R i → R j which a re given by j polynomials in the i c o ordinate functions on R i . F or ea ch i, j ∈ N , let H i ( R j ) = Hom E -alg ( R i , R j ) and let p i,j : H i ∐ H j → H i + j be as ab ove (Equation 5.0.1). Let s R denote the lo calization of the injective mo del c a tegory sSets E alg at the set { p i,j | i, j ∈ N } . W e call s R the categ ory of (lax) simplicial R -algebr as . The functor E alg → E induces a functor U : s C ∞ → s R , whic h we refer to as the underlying R -algebr a functor . It is a right Quillen functor, and as such pr eserves ﬁbrant ob jects. Similarly , the functor s C ∞ → sSets given by F 7→ F ( R ) is a right Quillen functor, whic h we refer to as t he underlying simplicia l set functor . B o th of these right Quillen functors preserve c oﬁbr ations as well b ecause coﬁbr a tions in a ll three mo del categ o ries are monomorphisms. Corollary 5.8. The functor U : s C ∞ → s R pr eserves and r eﬂe cts we ak e qu iva- lenc es b etwe en ﬁbr ant obje cts. 26 DA VID I. SPIV AK Pr o of. Since U is a right Quille n functor, it preser ves ﬁbrant ob jects. The result now follows fro m Pr o po sition 5.3 and the co rresp onding fact ab out s R .  W e do not need the following pro po sition, but include it for the r eader’s co nv e- nience and ediﬁcation. F o r example, it may help orient the reader to Section 7 on cotangent complexes. Prop ositio n 5.9. The mo del c ate gory s R of simplicia l R -algebr as is Q uil len e quiva- lent to the mo del c ate gory of c onne ctive c ommutative diﬀer ential gr ade d R -algebr as. Pr o of. In [26, 1.1 (3)], Sch wede and Shipley prov e that the mo del category of con- nective comm utative diﬀerential grade R - a lgebras is Quillen equiv alent to the model category of strict simplicial commutativ e R - algebras . This is in turn Q uillen equiv- alent to the categor y of la x simplicia l commutativ e R -algebr as b y [19, 15.3].  Deﬁnition 5.1 0. A C ∞ -ring F is ca lle d lo c al if its underlying discrete R -algebra π 0 ( U ( F )) is loca l in the usual sens e , and a morphism φ : F → G of lo cal C ∞ -rings is called lo c al if its underlying morphism of discr e te R -a lg ebras π 0 ( U ( φ )) is lo cal in the us ual sens e . W e will giv e a more in tuitive v ers io n of the locality co nditio n in Prop osition 9.10 . R emark 5.11 . W e reg r et the ov e r use of the word lo c al; when w e “lo ca lize” a mo del category to add weak eq uiv alences, when w e demand a “lo cality condition” on the stalks of a ringed space, and later when we talk ab out derived manifolds as having “lo cal mo dels” (as the lo ca l mo dels for manifolds are Euclidean spaces), we a re using the word lo cal in thr ee diﬀerent w ays. But ea ch is in line with typical usage, so it seems that ov erloading the w o rd co uld not b e a voided. 6. Local C ∞ -ringed sp a ces and derived manifolds In this section we deﬁne the categor y of (la x simplicial) C ∞ -ringed spaces , a sub c ategory called the categor y of lo ca l C ∞ -ringed spa ces, and a full sub c a tegory of that called the category of deriv ed manifolds. Thes e deﬁnitions resemble those of ringed spaces, loc a l ringed spaces, and schemes, from algebraic geometr y . Ordinary C ∞ -ringed spaces a nd C ∞ -schemes hav e been studied for quite a while: See [17],[9], and [22]. Let X b e a top o logical space, and let Op( X ) denote the category of op en inclu- sions in X . A homotopy sheaf of simplicial se ts on X is a functor F : Op( X ) op → sSets w hich is ﬁbra n t as a n ob ject of sSe ts Op( X ) op and whic h satisﬁes a “homotopy descent condition.” Roughly , the descent condition says that given open sets U and V , a sectio n of F o ver each o ne, and a choic e of homotopy b etw een the restricted sections on U ∩ V , ther e is a homotopically uniq ue s ection ov e r U ∪ V whic h r e- stricts to the given sections. Ja rdine showed in [14] that there is a model categor y structure on sSe ts Op( X ) op in which the ﬁbrant ob jects a r e homotopy sheaves on X ; we denote this mo del category Sh v ( X, sSets ). In [10], the authors s how that Shv ( X , sSets ) is a lo ca lization of the injective mo del structur e on sSets Op( X ) op at a certa in set of mor phisms (calle d the h yp er - cov ers) to obtain Sh v ( X , sSets ). F rom this, o ne deduces that Sh v ( X , sSets ) is a left pro per , coﬁbrantly generated simplicial mo del category in which all ob jects DERIVED SMOOTH MANIFOLDS 27 are coﬁbrant. A weak equiv alence b etw een ﬁbrant ob j ec ts o f Sh v ( X , sSets ) is a morphism w hich res tr icts to a weak eq uiv alence on every op en subset of X . W e wish to ﬁnd a s uitable categ ory of homotopy s heav es of C ∞ -rings. This will be obta ined as a lo caliza tion of the injective model structur e on sSets Op( X ) op × E . Prop ositio n 6.1. L et A and B b e c ate gories, and let P b e a set of morphisms in sSets A and Q a set of morphisms in sSets B . Ther e exists a lo c alization of sSets A × B , c al le d the facto r-wise lo caliza tion and denote d M , in which an obje ct F ∈ M is ﬁbr ant if and only if (1) F is ﬁbr ant as an obje ct of sSets A × B (2) for e ach a ∈ A , the induc e d obje ct F ( a, − ) ∈ sSets B is Q -lo c al, and (3) for e ach b ∈ B , the induc e d obje ct F ( − , b ) ∈ sSets A is P - lo c al. Pr o of. The pro jection A × B → A induces a Quillen pair sSets A L A / / sSets A × B R A o o (see Conv ention 4 .1). Similarly , there is a Quillen pair ( L B , R B ). The union of the image of P under L A and the image o f Q under L B is a set o f mo rphisms in sSets A × B , which w e denote R = L A ( P ) ∐ L B ( Q ). Let M deno te the lo caliza tion of sSets A × B at R . The result follows from [1 1, 3.1.1 2].  Deﬁnition 6.2. Let X be a top ological space. Let Q b e the s et of hypercovers in sSets Op( X ) op and let P = { p i,j | i, j ∈ N } denote the set of ma ps from Deﬁnition 5.1. W e denote by Shv ( X, s C ∞ ) the factor-wise loca lization of sSets Op( X ) op × E with resp ect to Q and P , a nd refer to it as the mo del c ate gory of she aves of C ∞ - rings on X . W e refer to F ∈ Sh v ( X , s C ∞ ) as a she af of C ∞ -rings on X if F is ﬁbrant; otherwise we s imply refer to it a s an ob ject o f Shv ( X , s C ∞ ). Similarly , one deﬁnes the mo de l categ ory of sheav es of simplicial R -a lgebras, de- noted Sh v ( X , s R ), as the factor- wise loca lization of sSets Op( X ) op × E alg with resp ect to the same sets, Q and P . Given a morphism o f to p olo gical spaces f : X → Y , one has push-forward a nd pullback functors f ∗ : Sh v ( X , s C ∞ ) → Shv ( Y , s C ∞ ) and f − 1 : Sh v ( Y , s C ∞ ) → Sh v ( X , s C ∞ ) as usual. The functor f − 1 is left adjoint to f ∗ , a nd this adjunction is a Quillen adjunction. Recall that when we sp eak of sheav es on X , w e alwa ys mean ﬁ- brant o b jects in Sh v ( X ). T o that end, we write f ∗ : Sh v ( Y , s C ∞ ) → Sh v ( X , s C ∞ ) to denote the comp osition of f − 1 with the ﬁbrant replac e men t functor . Deﬁnition 6. 3. Let X ∈ CG be a top olo gical space . An ob ject F ∈ Sh v ( X , s C ∞ ) is ca lle d a she af of lo c al C ∞ -rings on X if (1) F is ﬁbra n t, and (2) for every p oint p : {∗} → X , the s talk p ∗ F is a lo ca l C ∞ -ring. In this case, the pa ir ( X , F ) is called a lo c al C ∞ -ringe d sp ac e . Let F and G b e sheav es o f lo cal C ∞ -rings on X . A morphis m a : F → G is called a lo c al morphism if, for every p oint p : {∗} → X , the indu c e d mor phism on stalks p ∗ ( a ) : p ∗ ( F ) → p ∗ ( G ) is a lo cal morphis m (see Deﬁnition 5.10). W e 28 DA VID I. SPIV AK denote b y Map lo c ( F, G ) the simplicia l subset of Ma p( F, G ) spanned b y the vertices a ∈ Map( F, G ) 0 which represent lo cal mor phisms a : F → G . Let ( X, O X ) and ( Y , O Y ) denote lo cal C ∞ -ringed spaces. A morphism of lo c al C ∞ -ringe d sp ac es ( f , f ♯ ) : ( X, O X ) → ( Y , O Y ) consists of a map of topo logical spaces f : X → Y and a morphism f ♯ : f ∗ O Y → O X , such that f ♯ is a lo cal morphism of sheaves of C ∞ -rings on X . More generally , we deﬁne a simplicial ca tegory LRS whose ob jects are the lo cal ringed spaces and whose ma pping spaces hav e as vertices the mor phisms of lo cal C ∞ -ringed spaces . Prec is ely , we deﬁne for lo cal C ∞ -ringed spaces ( X , O X ) and ( Y , O Y ) the mapping space Map LRS (( X, O X ) , ( Y , O Y )) := a f ∈ Hom CG ( X,Y ) Map lo c ( f ∗ O Y , O X ) . Example 6.4 . Giv en a lo cal C ∞ -ringed space ( X , O X ), any subspace is a ls o a lo cal C ∞ -ringed space. F or exa mple, a manifold with b oundar y is a lo cal C ∞ -ringed space. W e do not deﬁne derived ma nifolds with b oundar y in this pap er, but we could do so inside the categ ory of lo cal C ∞ -ringed spa ces. In fact, that was done in the author’s dissertation [27]. If i : U ⊂ X is the inclus io n of an op en subset, then we let O U = i ∗ O X be the restricted sheaf, and we refer to the lo cal C ∞ -ringed space ( U, O U ) as the op en sub obje ct of X over U ⊂ X . Deﬁnition 6.5. A map ( f , f ♯ ) : ( X , O X ) → ( Y , O Y ) is an e quivalenc e of lo c al C ∞ -ringe d sp ac es if f : X → Y is a ho meo morphism of topologica l spa ces and f ♯ : f ∗ O Y → O X is a weak equiv alence in Shv ( X , s C ∞ ). R emark 6 .6 . The relatio n which we called e quiv alence of loc a l C ∞ -ringed spa c e s in Deﬁnition 6 .5 is clearly reﬂexive a nd transitive. Since the sheaves O X and O Y are assumed coﬁbrant-ﬁbran t, it is symmetric as well (see for exa mple [11, 7.5.10 ]). R emark 6.7 . Note that the notion o f equiv alence in Deﬁnition 6 .5 is quite strong. If ( f , f ♯ ) is a n equiv alence, the under lying map f of spaces is a homeomor phism, and the mor phis m π 0 ( f ♯ ) is an isomorphism of the underlying s heaves of C ∞ -rings. Therefore eq uiv alence in this sense should not be thought of as a generalizatio n of homotopy equiv alence, but instead a g e ne r alization o f diﬀeomorphism. In the next lemma, we will need to us e the mapping cylinder constr uction for a morphism f : A → B in a mo del ca tegory . T o deﬁne it, factor the morphism f ∐ id B : A ∐ B → B as a co ﬁbration fo llow ed by an acyclic ﬁbration, A ∐ B / / / / C y l ( f ) ≃ / / / / B ; the intermediate ob ject C y l ( f ) is called the mapping cylinder. Note tha t if f is a weak equiv alence, then so are the induced coﬁbrations A → Cyl ( f ) a nd B → Cyl ( f ); co nsequently , A and B ar e strong deformation r etracts of Cyl ( f ). Lemma 6.8. Su pp ose that X = ( X, O X ) and Y = ( Y , O Y ) ar e lo c al C ∞ -ringe d sp ac es and that U = ( U, O U ) and U ′ = ( U ′ , O U ′ ) ar e op en sub obje cts of X and Y r esp e ctively, and supp ose t hat U and U ′ ar e e quivalent. Th en in p articular ther e is a home omorphism U ∼ = U ′ . DERIVED SMOOTH MANIFOLDS 29 If the union X ∐ U Y of u nderlying top olo gic al sp ac es is Hausdorﬀ, t hen ther e is a lo c al C ∞ -ringe d sp ac e denote d X ∪ Y with underlying sp ac e X ∐ U Y and structur e she af O , su ch that t he diagr am U / /   k " " F F F F F F F F F Y j   X i / / X ∪ Y (6.8.1) c ommutes (up to homotopy), and the natu r al maps i ∗ O → O X , j ∗ O → O Y , and k ∗ O → O U ar e we ak e quivalenc es. Pr o of. W e deﬁne the structure shea f O as follows. First, let V = U , let O V denote the mapping cy linder for the equiv alence O U → O U ′ , and let V = ( V , O V ). Then the natural maps U , U ′ → V ar e equiv alences, and we have natural maps V → X and V → Y which ar e equiv alent to the o riginal sub ob jects U → X and U ′ → Y . Redeﬁne k to b e the natural map k : V → X ∐ V Y . W e ta ke O to b e the homotopy limit in Sh v ( X ∐ V Y , s C ∞ ) g iven by the diagr am O / /   ⋄ j ∗ O Y   i ∗ O X / / k ∗ O V . On the op en set X ⊂ X ∪ Y , the structure sheaf O restricts to O X ( X ) × O V ( V ) O Y ( V ). Since V is an o pen sub ob ject o f Y , we in par ticular hav e an equiv alence O Y ( V ) ≃ O V ( V ). This implies that O ( X ) → O X ( X ) is a weak equiv alence. The same holds for any o pen subset o f X , so we hav e a weak equiv alence i ∗ O ∼ = O X . Symmetric r easoning implies that j ∗ O → O Y is als o a weak equiv alence. The prop erty of b eing a loca l shea f is lo cal on X ∐ V Y , and the op en sets in X and in Y tog ether form a basis for the top olog y on X ∐ V Y . Thus, O is a lo cal sheaf o n X ∐ V Y .  Prop ositio n 6.9. L et X , Y , U and U ′ b e as in L emma 6.8 . Then Diagr am 6.8.1 is a homotopy c olimit diagr am. Pr o of. Let Z ∈ LRS denote a lo cal C ∞ -ringed space, and let V ≃ U ≃ U ′ , X ∪ Y = ( X ∐ V Y , O ), a nd k : V → X ∐ V Y b e as in the pr o of of Lemma 6.8. W e m ust show that the natural map Map( X ∪ Y , Z ) → Map( X , Z ) × Map( V , Z ) Map( Y , Z ) (6.9.1) is a weak equiv alence. Recall that for lo cal C ∞ -ringed spaces A , B , the mapping space is deﬁned as Map LRS ( A , B ) = a f : A → B Map lo c ( f ∗ O B , O A ) , i.e. it is a disjoint union of spaces, indexe d b y maps of underlying top ologica l spaces A → B . Since X ∐ V Y is the colimit of X ← V → Y in CG , the indexing set of Map LRS ( X ∪ Y , Z ) is the ﬁb er pr o duct of the indexing sets for the mapping spaces on the rig ht hand side of E q uation 6 .9.1. Thus, we may apply Lemma 9.12 30 DA VID I. SPIV AK to reduce to the case of a single index f : X ∐ V Y → Z . That is, we must show that the natural map Map loc ( f ∗ O Z , O ) → Map loc ( i ∗ f ∗ O Z , O X ) × Map loc ( k ∗ f ∗ O Z , O V ) Map loc ( j ∗ f ∗ O Z , O Y ) , (6.9.2) indexed by f , is a weak equiv alence. By deﬁnition o f O , we have the ﬁr st weak equiv alence in the following display Map loc ( f ∗ O Z , O ) ≃ Map loc ( f ∗ O Z , i ∗ O X × k ∗ O V j ∗ O Y ) (6.8.3) ≃ Ma p loc ( f ∗ O Z , i ∗ O X ) × Map loc ( f ∗ O Z ,k ∗ O V ) Map loc ( f ∗ O Z , j ∗ O Y ) , and the second weak equiv alence follows from the fact that the pro per t y of a map being lo cal is itself a lo cal prop erty . Note that i, j, and k are o pe n inclusions . W e ha ve reduced to the follo wing : given an o pen inclusion o f top olog ic al spa c es ℓ : A ⊂ B a nd given sheav es F on A and G on B , we hav e a solid- a rrow dia gram Map lo c ( F , ℓ ∗ G ) / / _ _ _   Map lo c ( ℓ ∗ F , G )   Map( F , ℓ ∗ G ) ∼ = / / Map( ℓ ∗ F , G ) coming from the adjointness of ℓ ∗ and ℓ ∗ . W e must s how that the dotted map ex is ts and is an isomorphism. Recall tha t a simplex is in Map lo c if a ll of its vertices ar e lo cal morphisms. So it suﬃces to show that lo ca l morphis ms F → ℓ ∗ G ar e in bijective co rresp ondence with lo cal morphisms ℓ ∗ F → G . This is easily ch e cked b y ta k ing stalk s : one simply uses the fact that for any p oint a ∈ A , a basis of o pen neighborho o ds o f a are s e n t under ℓ to a ba s is of op en neigh b or ho o ds of ℓ ( a ), and vice versa. Now w e can see that the rig h t-ha nd sides of Equa tio ns 6.9 .2 and 6.8.3 are e q uiv- alent, whic h shows that Equation 6.9 .1 is indeed a weak equiv alence, as desir ed.  Deﬁnition 6.10. Let X , Y , U and U ′ be a s in Lemma 6.8. W e refer to X ∪ Y as the u nion of X and Y along the eq uiv alent lo ca l C ∞ -ringed spaces U ≃ U ′ . Prop ositio n 6.1 1. T her e is a ful ly faithful fun ctor i : Man → LRS . Pr o of. Giv en a manifold M , let i ( M ) denote the lo ca l C ∞ -ringed space ( M , C ∞ M ) whose underlying spa ce is that of M , and such that for any op en set U ⊂ M , the discrete C ∞ -ring C ∞ M ( U ) is the functor E → Sets given b y C ∞ M ( U )( R n ) = Hom Man ( U, R n ) . It is easy to check that C ∞ M is a ﬁbra n t ob ject in Sh v ( M , s C ∞ ). It sa tisﬁes the lo cality condition in the sense of Deﬁnition 6.3 beca use, for every point p in M , the smo oth real-v alued functions whic h are deﬁned on a neighborho o d of p but are not inv ertible in any neighborho o d of p are exactly thos e functions that v a nish at p . W e need to show that i takes morphisms of manifolds to morphisms of lo cal C ∞ -ringed spaces. If f : M → N is a smo oth map, p ∈ M is a point, and o ne ha s a smo oth map g : N → R m such that g ha s a ro ot in every suﬃciently small op en neighborho o d of f ( p ), then g ◦ f has a roo t on ev ery suﬃciently s mall neighborho o d of p . DERIVED SMOOTH MANIFOLDS 31 It only rema ins to show that i is fully faithful. It is clear by deﬁnition that i is injectiv e on morphisms, so we must show that any lo ca l morphism i ( M ) → i ( N ) in LRS co mes from a morphism of ma nifolds. Suppose that ( f , f ♯ ) : ( M , C ∞ M ) → ( N , C ∞ N ) is a lo cal morphis m; w e must show that f : M → N is smo oth. This do es not even use the lo cality co ndition: for every chart V ⊂ N with c : V ∼ = R n , a smo oth map f ♯ ( c ) : f − 1 ( V ) → R n is de ter mined, and these are compa tible with op en inclusions .  Notation 6.12. Supp ose ( X , O X ) is a C ∞ -ringed space. Recall that O X is a ﬁbrant sheaf of (simplicial) C ∞ -rings on X , so that for any ope n subset U ⊂ X , the ob ject O X ( U ) is a (w ea k ly) pro duct preser ving functor from E to sSets . The v alue which “ma tter s most” is |O X ( U ) | := O X ( U )( R ) , bec ause every other v a lue O X ( U )( R n ) is weakly equiv alent to an n - fold pro duct o f it. W e similarly denote | F | := F ( R ) for a C ∞ -ring F . Lemmas 5.3 and 5.5 further demonstrate the imp ortance of | F | . The theo rem below states that for lo cal C ∞ -ringed spaces X = ( X , O X ), the sheaf O X holds the answer to the question “what are the r e a l v alued functions o n X ?” Theorem 6. 13. L et ( X , O X ) b e a lo c al C ∞ -ringe d s p ac e, and let ( R , C ∞ R ) denote (the image u n der i of ) the manifold R ∈ Man . Ther e is a homotopy e quivalenc e Map LRS (( X, O X ) , ( R , C ∞ R )) ∼ = − → |O X ( X ) | . W e prov e Theor em 6.13 in Sec tio n 8 as Theorem 9.11. As in Algebraic Geometry , there is a “ prime sp ectrum” functor ta k ing C ∞ -rings to lo c a l C ∞ -ringed spaces. W e will no t use this constr uction in any essential way , so we pres en t it as a remark w itho ut pr o of. R emark 6.14 . The global sections functor Γ : LRS → s C ∞ , given by Γ( X, O X ) := O X ( X ), has a right adjoint, denoted Sp ec (see [9]). Given a C ∞ -ring R , let us brieﬂy explain Sp ec R = (Sp ec R, O ). The p oints of the underlying space of Sp ec R are the maxima l ideals in π 0 R , and a closed set in the top ology on Sp e c R is a se t o f po int s on which some element o f π 0 R v anishes. The sheaf O assigns to a n o pen set U ⊂ Spec R the C ∞ -ring R [ χ − 1 U ], where χ U is a ny element of R that v anishes on the c omplement o f U . The unit o f the (Γ , Spec ) adjunction is a natural transformation, η X : ( X, O X ) → Spec O X ( X ). If X is a manifold then η X is an eq uiv alence of lo cal C ∞ -ringed spaces (see [22, 2.8]). The c ategory of lo c al C ∞ -ringed spaces contains a full sub catego ry , ca lled the category of derived smo oth manifolds, which we now de ﬁne. Basica lly , it is the full sub c ategory of LRS s pa nned by the lo ca l C ∞ -ringed spac e s that ca n b e cov ere d by aﬃne der ived manifo lds , wher e an aﬃne derived ma nifold is the v anishing s et of a smo oth function o n aﬃne space. See also Axiom 5 of Deﬁnition 2.1. 32 DA VID I. SPIV AK Deﬁnition 6.15. An aﬃne derive d manifold is a pair X = ( X, O X ) ∈ LRS , wher e O X ∈ s C ∞ is ﬁbrant, and which can b e obtained as the homotopy limit in a diagram o f the form ( X, O X ) / / g   ⋄ R 0 0   R n f / / R k . W e sometimes refer to an a ﬃne derived ma nifold as a lo c al mo del for derive d man- ifolds. W e refer to the map g : X → R n as the c anonic al inclusion of the zer oset . A d erive d smo oth manifold (or derive d manifold ) is a loca l C ∞ -ringed s pace ( X, O X ) ∈ LRS , wher e O X ∈ s C ∞ is ﬁbra nt , and for which there exists an op en cov ering ∪ i U i = X , such that ea ch ( U i , O X | U i ) is a n aﬃne derived ma nifold. W e denote by dMan the full sub catego ry of LRS spanned b y th e derived smoo th manifolds, and refer to it a s the (simplicial) category of deriv ed manifolds. A morphism o f derived manifolds is called an e quivalenc e of derive d manifolds if it is an equiv alence of lo cal C ∞ -ringed spaces. Manifolds ca no nically hav e the structure of derived manifolds; more precis e ly , we ha ve the following lemma. Lemma 6.16. The functor i : Man → LRS factors thr ough dMan . Pr o of. Euclidean space ( R n , C ∞ R n ) is a principle derived manifold.  An imbedding of derived manifolds is a map g : X → Y that is lo cally a zeroset. Precisely , the de ﬁnitio n is g iven b y Deﬁnition 2.3, with the s implicial catego r y dM replaced b y dMan . Note that if g is an im b edding, then the morphism g ♯ : g ∗ O Y → O X is surjective on π 0 , becaus e it is a pushout of such a morphism. 7. Cot angent Complexes The idea be hind cota ngent complexes is as follows. Given a manifold M , the cotangent bundle T ∗ ( M ) is a vector bundle o n M , which is dual to the tangent bundle. A smo oth map f : M → N induces a map on vector bundles T ∗ ( f ) : T ∗ ( N ) → T ∗ ( M ) . Its cokernel, co ker ( T ∗ ( f )), is not necessa rily a vector bundle, and is thus genera lly not disc us sed in the basic theory of smooth manifolds. How ever, it is a sheaf of mo dules o n M and do es hav e geometric meaning: its sections mea sure tange nt vectors in the ﬁb ers of the map f . Int e r estingly , if f : M → N is an em b edding, then T ∗ ( f ) is a surjection and the cokernel is zer o. In this case its kernel , ker( T ∗ ( f )), has mea ning instea d – it is dual to the norma l bundle to the imbedding f . In general the cotangent complex of f : M → N is a s heaf on M which enco des all of the a bove information (and more) ab out f . It is in some sense the “universal linearization” o f f . The co nstruction of the cotangent complex L f asso ciated to a ny morphism f : A → B of commutativ e rings was worked on b y several p eople, including Andr ´ e [1], Quillen [23], Lic hten baum and Schlessinger [1 8], and Illus ie [13]. La ter, Sc hw ede int r o duced cotangent complexes in more genera lit y [25], by intro ducing sp ectra in DERIVED SMOOTH MANIFOLDS 33 mo del categor ie s and showing that L f emerges in a cano nic a l w ay when one studies the s tabilization of the mo del c ategory of A -algebr as ov er B . The co tangent complex fo r a morphism o f C ∞ -ringed spaces c a n b e obtained using the pr o cess given in Sch wede’s pa per [25]. Let us call this the C ∞ -cotangent complex. I t has a ll of the usual for mal prop erties of cotangent complexes (a nal- ogous to those given in Theorem 7.1). Unfortunately , to adequately present this notion requires a goo d bit of setup, na mely the constr uction of spectra in the mode l category o f sheaves of la x simplicia l C ∞ -rings, following [25]. Mo reov er , the C ∞ - cotangent complex is unfamiliar and quite technical, a nd it is diﬃcult to co mpute with. Underlying a morphism of C ∞ -ringed spaces is a mor phism of ringed s pa ces. It to o has an asso ciated cota ngent complex, whic h we call th e ring - theoretic c otangent complex. Cle arly , the C ∞ -cotangent complex is more canonica l than the ring- theoretic cota ngent complex in the setting of C ∞ -ringed spaces. How ever, for the reasons g iven in the above paragra ph, w e opt to us e the ring-theoretic version instead. This version do es not have quite as many useful formal pro p e r ties as do es the C ∞ -version (P rop erty (4) is weakened), but it has all the pro p er ties we will need. In [13], the cotangent complex for a morphism A → B of simplicial commutativ e rings is a s implicia l B -mo dule. The mo del ca teg ory of simplicial mo dules over B is equiv alent to the mo del catego ry of non-negatively gr aded chain complex es ov er B , by the Dold-Kan cor resp ondence. W e use the la tter appr oach for simplicity . Let us b egin with the prop erties that we will use ab out r ing-theoretic cotang ent complexes. Theorem 7.1. L et X b e a top olo gic al sp ac e. Given a morphism f : R → S of she aves of simplicial c ommut ative rings on X , ther e exists a c omplex of she aves of S -mo dules, denote d L f or L S/R , c al le d the cotangent complex asso ciated to f , with the fol lowing pr op erties. (1) L et Ω 1 f = H 0 L f b e the 0 th homolo gy gr oup. Then, as a she af on X , one has that Ω 1 f is the usual S - mo dule of K¨ ahler diﬀer entials of S over R . (2) The c otangent c omplex is functorial ly re late d to f in the sense t hat a mor- phism of arr ows i : f → f ′ , i.e. a c ommu tative diagr am of she aves in s R R / / f   R ′ f ′   S / / S ′ , induc es a morphism of S -mo dules L i : L f → L f ′ . (3) If i : f → f ′ is a we ak e quivalenc e (i.e. if b oth the top and b ottom maps in the ab ove squar e ar e we ak e quivalenc es of simplicial c ommutative rings), then L i : L f → L f ′ is a we ak e quivalenc e of S - mo dules, and its ad joint L f ⊗ S S ′ → L f ′ is a we ak e quivalenc e of S ′ -mo dules. (4) If the diagr am fr om (2) is a homotopy pushout of she aves of simplicial c ommutative rings, then the induc e d morphsim L i : L f → L f ′ is a we ak e quivalenc e of mo dules. 34 DA VID I. SPIV AK (5) T o a p air of c omp osable arr ows R f − → S g − → T , one c an fun ctorial ly assign an exact triangle in the homotopy c ate gory of T -mo dules, L S/R ⊗ S T → L T /R → L T /S → L S/R ⊗ S T [ − 1 ] . Pr o of. The a bove ﬁve prop erties are pr oved in Chapter 2 o f [13], as Pro po sition 1.2.4.2, Statemen t 1 .2.3, Prop o sition 1.2.6.2, Prop osition 2.2.1, and Prop osition 2.1.2, r esp e ctively .  All o f the a bove prop erties o f the co tangent complex fo r morphisms of s he aves hav e co ntrav ar iant c o rollar ies for morphisms of r inged s paces, and almo s t all of them hav e contra v aria nt corolla ries for maps of C ∞ -ringed space s . F or example, a pair of comp osable ar rows of C ∞ -ringed spac e s X f ← − Y g ← − Z induces an exact triangle in the homotopy categor y of O Z -mo dules, g ∗ L Y / X → L Z / X → L Z / Y → g ∗ L Y / X [ − 1] . It is in this sense t ha t P rop erties (1), (2), (3), and (5) hav e con trav ariant coro llaries. The exception is Pro pe rty (4); o ne asks, do es a homotopy pullback in the cat- egory of C ∞ -ringed spaces induce a w ea k equiv alence of c o tangent co mplexes? In general, the answer is no beca use the ringed space underlying a homo topy pullbac k of C ∞ -ringed spaces is not the ho motopy pullback of the underlying ringed spaces. In other words the “underlying simplicial R -algebra ” functor U : s C ∞ → s R do es not preserve homotopy colimits. How e ver, there a r e ce rtain t y pes o f homotopy colimits tha t U do es preser ve. In particular , if X → Z ← Y is a dia g ram of C ∞ - ringed spaces in which o ne of the tw o ma ps is a n imb e dding of derived manifolds, then ta k ing underlying rings do es c ommut e with tak ing the homoto p y colimit (this follows from Cor ollary 9 .6), and the cotangent complexes will sa tisfy Prop erty (4) of Theor em 7.1. This is the only ca se in which cotangent co mplexes will b e nec- essary for our work. Th us, we restrict our attention to the ring -theoretic version rather than the C ∞ -version of cotangent co mplexes. Here is our analog ue of Theo rem 7 .1. Corollary 7.2. Given a lo c al C ∞ -ringe d sp ac e X = ( X , O X ) , let U ( O X ) denote the underlying she af of simplicial c ommutative rings on X . Given a morphism of lo c al C ∞ -ringe d sp ac es f : X → Y , ther e exists a c omplex of she aves of U ( O X ) - mo dules on X , denote d L f or L X / Y , c al le d the ring-theor etic cotang ent complex asso ciated to f (or jus t the cotangent complex for f ), with the fol lowing pr op ert ies. (1) L et Ω 1 f = H 0 L f b e the 0 th homolo gy gr oup. The n Ω 1 f is the usual O X - mo dule of K¨ ahler diﬀer entials of X over Y . (2) The c otangent c omplex is c ontr avariantly re late d to f in the sense that a morphism of arr ows i = ( i 0 , i 1 ) : f → f ′ , i.e. a c ommutative diagr am in LRS Y i 1 / / Y ′ X f O O i 0 / / X ′ , f ′ O O induc es a morphism of U ( O X ) -mo dules L i : i ∗ 0 L f ′ → L f . DERIVED SMOOTH MANIFOLDS 35 (3) If i : f → f ′ is an e quivalenc e (i.e. it induc es two e quivalenc es of she aves), then L i : i ∗ 0 L f ′ → L f is a we ak e quivalenc e of U ( O X ) -mo dules. (4) If t he diagr am fr om (2) is a homotopy pul lb ack in LRS and either f ′ or i 1 is an immersion, then the induc e d morphism L i : L f ′ → L f is a we ak e quivalenc e of U ( O X ) -mo dules. (5) T o a p air of c omp osable arr ows X f − → Y g − → Z , one c an functorial ly assign an exact triangle in the homotopy c ate gory of U ( O X ) -mo dules, f ∗ L Y / Z → L X / Z → L X / Y → f ∗ L Y / Z [ − 1] . Pr o of. The map f : X → Y induces U ( f ♯ ) : U ( f ∗ O Y ) → U ( O X ), a nd w e set L f := L U ( f ♯ ) , whic h is a sheaf of U ( O X )-mo dules. Let h : X → Y b e a morphism of top olog ical spa ces, and let F → G b e a map of sheav es o f simplicia l commutativ e rings on Y . By [13, 1.2.3.5], s ince inductive limit s commute with the cotangent complex functor for simplicial commutativ e rings , one has a n isomo rphism h ∗ ( L G / F ) → L h ∗ G /h ∗ F . Prop erties (1), (2), and (5) now follow from Theorem 7 .1. Sheav es on lo ca l C ∞ -ringed spaces are assumed coﬁbr ant-ﬁbran t, and weak equiv alences b etw een ﬁbrant o b jects ar e pres e r ved b y U (Corollary 5.8). Prop erty (3) follows fro m Theorem 7.1. Finally , if f ′ or i 1 is an immersion, then tak ing homotopy pullback commutes with taking underlying ring e d spa ces, by Corolla ry 9.6, and so Pro pe r ty (4) also follows fr om Theor em 7.1.  If X is a lo cal C ∞ -ringed spa ce, we wr ite L X to denote the cotangent complex asso ciated to the unique map t : X → R 0 . It is called the absolute c otangent c omplex asso ciated to X . Corollary 7.3. L et t : R n → R 0 b e the unique map. Then the c otangent c omplex L t is 0-trun c ate d, and its 0 th homolo gy gr oup Ω 1 t ∼ = C ∞ R n h dx 1 , . . . , dx n i is a fr e e C ∞ R n mo dule of r ank n . L et p : R 0 → R n b e any p oint. Then the c otangent c omplex L p on R 0 has homol- o gy c onc entr ate d in de gr e e 1, and H 1 ( L p ) is an n -dimensional r e al ve ctor sp ac e. Pr o of. The ﬁrst statement follows fr o m Prop erty (1) of Theorem 7.1. The second statement follows from the exact triangle , g iven by P rop erty (5), for the co mpos able arrows R 0 p − → R n t − → R 0 .  Let X = ( X, O X ) be a derived ma nifold a nd L X its cotangent complex. F or any po int x ∈ X , let L X ,x denote the stalk of L X at x ; it is a ch a in complex ov er the ﬁeld R so its ho mology g roups are vector spaces . Let e ( x ) denote the alterna ting sum of the dimensions of these vector spaces. As deﬁned, e : X → Z is a just function b etw een sets. Corollary 7.4. L et X = ( X, O X ) b e a derive d manifold, L X its c otangent c omplex, and e : X → Z the p ointwise Euler char acteristic of L X deﬁne d ab ove. Then e is c ontinuous (i.e. lo c al ly c onstant ) , and for al l i ≥ 2 we have H i ( L X ) = 0 . 36 DA VID I. SPIV AK Pr o of. W e c a n a ssume that X is a n aﬃne derived manifold; i.e. that there is a homotopy limit squa r e of the form X t / / i   ⋄ R 0 0   R n f / / R k . By Pr op erty (4), the map L i : L t → L f is a weak equiv alence o f sheaves. Recall that L X is sho rthand for L t , so it suﬃces to show that the Euler character is tic o f L f is co ns tant o n X . The compo sable pair of morphisms R n f − → R k → R 0 induces an exa c t triangle f ∗ L R k → L R n → L f → f ∗ L R k [ − 1] . By Co rollary 7.3, this reduces to an exact sequence o f real vector spaces 0 → H 1 ( L f ) → R k → R n → H 0 ( L f ) → 0 . (7.4.1) Note also that for all i ≥ 2 w e hav e H i ( L f ) = 0, proving the se c ond assertio n. The ﬁrst a ssertion follows fro m the exactness o f (7.4 .1), b ecaus e rank( H 0 ( L f )) − rank( H 1 ( L f )) = n − k at all p oints in X .  Deﬁnition 7 .5. Let X = ( X , O X ) b e a derived manifold, and e : X → Z the function deﬁned in Corolla ry 7.4. F or any p oint x ∈ X , the v alue e ( x ) ∈ Z is called the virtual dimension , or just the dimension , of X at x , and denoted dim x X . Corollary 7.6. Supp ose t hat X is a derive d manifold and M is a smo oth mani- fold. If i : X → M is an imb e dding, then the c otangent c omplex L i has homolo gy c onc entr ate d in de gr e e 1. The ﬁrst homolo gy gr oup H 1 ( L i ) is a ve ctor bu n d le on X , c al le d the co normal bundle of i and denote d N i or N X / M . The r ank of N i at a p oint x ∈ X is given by the formula rank x N i = dim i ( x ) M − dim x X . In c ase X ∼ = P is a smo oth manifold, t he bund le N P / M is the dual to the us u al normal bu nd le for t he imb e dding. In p articular, if i : P → E is t he zer o se ction of a ve ctor bund le E → P , then N P /E is c anonic al ly isomorphic to the dual E ∨ of E . Pr o of. All but the ﬁnal a ssertion can esta blished lo cally o n X , and we pr o ceed as follows. Imbeddings i of der ived manifolds ar e lo cally of the form X / / i   ⋄ R 0 z   R n f / / R k . By P rop erty (4), w e hav e a quasi- isomorphism L i ≃ L z . The claim that L i is lo cally free and has homology concentrated in degree 1 now fo llows from Cor ollary 7.3. Note that the cono rmal bundle N i = H 1 ( L i ) ha s ra nk k . DERIVED SMOOTH MANIFOLDS 37 The exact triangle for the comp osable morphisms X i − → R n f − → R k implies that the E uler characteristic of L X is n − k , and the s e cond as sertion follows. F or the ﬁnal ass ertion, we use the exac t sequence 0 → N P / M → i ∗ Ω 1 M → Ω 1 P → 0 .  7.1. Other calculations. In this subs ection we prov e some r esults which will b e useful later. Lemma 7.7. Supp ose given a diagr am X g / / f   X ′ f ′ ~ ~ } } } } } } } } Y of lo c al smo oth-ringe d sp ac es such that g , f , and f ′ ar e close d immersions, the u n- derlying map of top olo gic al s p ac es g : X → X ′ is a home omorphi sm , and the induc e d map g ∗ L f ′ → L f is a qu asi-isomorph ism. Then g is an e quivalenc e. Pr o of. It suﬃces to prov e this on stalks; thus we may assume that X and X ′ are po int s . Let U X and U X ′ denote the lo cal s implicial commutativ e rings underlying O X and O X ′ , and let U g : U X ′ → U X denote the map induced by g . B y Corollary 5.8, it suﬃces to show that U g is a weak equiv alence. Since g is a clo sed immersio n, U g is surjective; let I b e the kernel of U g . It is prov ed in Corollary [13, I II.1 .2.8.1] that the co normal bundle H 1 ( L U g ) is isomorphic to I /I 2 . By the distinguished triangle asso cia ted to the comp o sition f ′ ◦ g = f , we ﬁnd that L U g = 0, so I /I 2 = 0. Thus, by Na k a yama’s lemma, I =0 , so U g is indeed a weak equiv alence.  Prop ositio n 7. 8. Supp ose that p : E → M is a ve ctor bund le. S upp ose t hat s : M → E is a se ction of p such that the diagr am ( X, O X ) f / / f   M s   M z / / E (7.8.1) c ommutes, and that the diagr am of sp ac es underlying (7.8 .1 ) is a pul lb ack; i.e. X = M × E M in CG . The diagr am induc es a morphism of ve ctor bund les λ s : f ∗ ( E ) ∨ → N f , which is an isomorphism if and only if Diagr am 7.8.1 is a pu l lb ack in LRS . Pr o of. On any op en subset U of X , we hav e a co mmutative square O E ( U ) / /   O M ( U )   O M ( U ) / / O X ( U ) 38 DA VID I. SPIV AK of sheav es on X . By Pr o pe rty (2) of cotangent complexes, this square induces a morphism f ∗ L z → L f . By Corolla ry 7.6, we may identify H 1 ( f ∗ L z ) with f ∗ ( E ) ∨ , and H 1 ( L f ) with N f , and we let λ s : f ∗ ( E ) ∨ → N f denote the induced map. If Diagra m 7 .8.1 is a pullback then λ s is an isomo rphism b y P rop erty (4), so w e hav e only to prov e the conv er s e. Suppo se that λ s is an isomo rphism, let X ′ be the ﬁb er pro duct in the diagram X ′ f ′ / / f ′   ⋄ M s   M z / / E , and let g : X → X ′ be the induced ma p. Note that on underlying s paces g is a homeomorphism. Since the co mpo sition X g − → X ′ f ′ − → M , namely f , is a closed immersion, so is g . By a s sumption and Pro per ty (4), g induces an isomor phism o n cotangent complexes g ∗ L f ′ ∼ = − → f ∗ L z ∼ = − → L f . The result follows from Lemma 7.7.  Here is a kind of linea rity r esult. Prop ositio n 7 .9. S u pp ose t hat M is a manifold and X 1 and X 2 ar e derive d man- ifolds which ar e deﬁne d as pul lb acks X 1 j 1 / / j 1   ⋄ M z 1   X 2 j 2 / / j 2   ⋄ M z 2   M s 1 / / E 1 M s 2 / / E 2 , wher e E i → M is a ve ctor bu nd le, s i is a se ct ion, and z i is the zer o se ction, for i = 1 , 2 . Then X 1 and X 2 ar e e quivalent as derive d manifolds over M if and only if t her e exists an op en neighb orho o d U ⊂ M c ontaining b oth X 1 and X 2 , and an isomorphi sm σ : E 1 → E 2 over U , such that s 2 | U = σ ◦ s 1 | U . Pr o of. Suppose ﬁrs t U ⊂ M is an op en neighbo rho o d of b oth X 1 and X 2 , and deno te E i | U and s i | U by E i and s i , resp ectively for i = 1 , 2. Suppose that σ : E 1 → E 2 is an iso morphism. Supp ose that s 2 = σ ◦ s 1 . Co ns ider the diagr am X 1 ⋄ i / / i   U z 1   U z 2   U s 1 / / E 1 σ / / E 2 , where the left-hand square is Car tesian. In fact, the right-hand squar e is Cartesian as well, b ecause σ is an isomorphism of vector bundles, and in particular ﬁxes the zero sectio n. Thus w e see that X 1 and X 2 are equiv alent as derived manifolds ov er U . F or the co n verse, s uppo se that we hav e an equiv alence f : X 2 → X 1 such that j 2 = j 1 ◦ f . In particular , on underly ing top ologica l spaces we hav e X 1 = X 2 =: X , and on cotangent complexe s we hav e a qua si-isomor phism f ∗ L j 1 ∼ = − → L j 2 . In particular, this implies that H 1 ( L z 1 ) is is omorphic to H 1 ( L z 2 ), so E 1 and E 2 are DERIVED SMOOTH MANIFOLDS 39 isomorphic v e c tor bundles on U by Prop osition 7.8. W e can write E to denote bo th bundles, and adjust the sections a s necess ary . Consider the dia g ram X 1 j 1 / / j 1   M z 1   } } } } } } } } } } X 2 f = = | | | | | j 2 / / j 2   M z 2   M z z z z z z z z z z s 1 / / E M s 2 / / E By the commutativit y of the left-hand square of the diagr am and by Prop erty (4) of cota ngent complexes, o ne has a chain of quasi- isomorphisms j ∗ 2 L s 1 = f ∗ j ∗ 1 L s 1 ∼ = − → f ∗ L j 1 ∼ = − → L j 2 ∼ = − → j ∗ 2 L s 2 . W e write pa rt of the long-ex a ct sequences coming from the c o mpo sable ar rows U s 1 − → E → R 0 and U s 2 − → E → R 0 and furnish some morphisms to obtain the solid-arr ow diagram: 0 / / H 1 ( L s 1 ) / / ∼ =   Ω 1 E s ∗ 1 / / τ   Ω 1 U / / 0 0 / / H 1 ( L s 2 ) / / Ω 1 E s ∗ 2 / / Ω 1 U / / 0 . By the 5-lemma, there is an isomorphism τ : Ω 1 E → Ω 1 E making the diag rams com- m ute. Now every section s : U → E of a v ecto r bundle E → U induces a pullback map s ∗ : Ω 1 E → Ω 1 U , and t wo sections induce equiv alent pullback ma ps if a nd only if they diﬀer by an auto mo rphism o f E ﬁxing U . Since s 1 and s 2 induce equiv alent pullback maps, there is an a utomorphism σ : E → E with s 2 = σ ◦ s 1 .  8. Pr oo fs of technical resul ts Recall from Sec tio n 5 that H R i ∈ s C ∞ is the disc rete C ∞ -ring corepres ent ing R i ∈ E . A s mo o th ma p f : R i → R j (contra v aria ntly) induces a morphism of C ∞ - rings H R j → H R i , whic h we often denote by f for co n venience. Rec all also that in s C ∞ , there is a canonical weak equiv alence H R i + j ≃ − → H R i ∐ H R j . Let U : s C ∞ → s R denote the “underlying simplicial comm uta tive ring ” functor from Co rollary 5.8. Lemma 8. 1 . L et m, n, p ∈ N with p ≤ m . L et x : R n + m → R m denote the pr oje c- tion onto the ﬁrst m c o or dinates, let g : R p → R m denote t he inclusion of a p -plane 40 DA VID I. SPIV AK in R m , and let Ψ b e the diagr am H R m x / / g   H R n + m H R p of C ∞ -rings. The homotopy c olimit of Ψ is we akly e quivalent t o H R n + p . Mor e over, the applic ation of U : s C ∞ → s R c ommutes with taking homotopy c olimit of Ψ in t he sense t hat the natura l map ho colim( U Ψ) → U ho colim(Ψ) is a we ak e quivalenc e of simplicial c ommut ative R -algebr as. Pr o of. W e may ass ume that g sends the origin to the origin. F or now, we a ssume that p = 0 and m = 1. Consider the a ll-Cartesian diagr am o f manifolds R n / /   p R n +1 / / x   p R n   R 0 g / / R / / R 0 . Apply H : Man op → s C ∞ to obtain the diagra m H R n H R n +1 o o H R n o o H R 0 O O H R g o o x O O H R 0 O O o o Since H sends pro ducts in E to homotopy pushouts in s C ∞ , the right-hand squar e and the big rectangle are homotopy pushouts. Hence the left square is as well, proving the ﬁr st ass ertion (in case p = 0 a nd m = 1 ). F or notationa l reas ons, let U i denote U ( H R i ), so that U i is the (discr ete s im- plicial) commutativ e ring whose elemen ts are s mo o th maps R i → R . Deﬁne a simplicial co mm utative R -a lgebra D as the homoto p y colimit in the diagra m U 1 x / / g   ⋄ U n +1   U 0 / / D . W e m us t show that the natural map D → U n is a weak eq uiv alence of simplicial commutativ e rings. Recall that the homoto p y g roups of a ho motopy pusho ut of simplicial commutativ e r ings are the T or groups of the corres po nding tensor pro duct of chain complexes . Since x is a nonzero divisor in the ring U n +1 , the homo to p y groups of D are π 0 ( D ) = U n +1 ⊗ U 1 U 0 ; and π i ( D ) = 0 , i > 0 . Thu s , π 0 ( D ) can be iden tiﬁed with the equiv alence classes of smoo th functions f : R n +1 → R , where f ∼ g if f − g is a multiple of x . DERIVED SMOOTH MANIFOLDS 41 On the o ther hand, w e ca n ident ify e lemen ts of the ring U n of smo oth functions on R n with the equiv alence classes o f functions f : R n +1 → R , where f ∼ g if f (0 , x 2 , x 3 , . . . x n +1 ) = g (0 , x 2 , x 3 , . . . , x n +1 ) . Thu s to prov e that t he map D → U n is a w eak equiv alence, we must o nly show t ha t if a smo oth function f ( x 1 , . . . , x n +1 ) : R n +1 → R v anishes whenever x 1 = 0, then x 1 divides f . This is called Hadamar d’s lemma, and it follows fro m the deﬁnition of smo o th functions. Indeed, given a smo oth function f ( x 1 , . . . , x n +1 ) : R n +1 → R which v anis he s whenever x 1 = 0, deﬁne a function g : R n +1 → R b y the fo rmula g ( x 1 , . . . , x n +1 ) = lim a → x 1 f ( a, x 2 , x 3 , . . . , x n +1 ) a . It is clear that g is smo oth, and xg = f . W e have now prov ed both assertions in the case that p = 0 and m = 1; w e contin ue to assume p = 0 and prov e the result for genera l m + 1 by induction. T he inductive step follows from the all-Cartesian diagra m b elow, in whic h ea ch vertical arrow is a n inclusion of a plane and each horizontal arrow is a coo rdinate pro j ec tio n: R n / /   p R 0   R m + n / /   p R m / /   p R 0   R n + m +1 / / R m +1 / / R . Apply H to this dia gram. The ar guments ab ov e implies that the tw o as sertions hold for the right squar e and b ottom rectangle; th us they hold for the bo ttom left square. The inductive hypothesis implies that the tw o asser tions hold for the top square, s o they hold for the left rectangle, a s desir ed. F or the case of general p , let k : R m → R m − p denote the pro jection orthogo nal to g . Applying H to the all-Car tesian diagr am R n + p / /   p R p / / g   p R 0   R n + m x / / R m k / / R m − p , the result holds for the right square and the big rectangle (for b oth of which p = 0), so it holds for the left square a s well.  R emark 8.2 . Note that (the non-for mal part of ) Lemma 9 .1 relies heavily on the fact that we ar e dea ling with smo oth maps. It is this le mma which fails in the setting o f top olog ical manifolds, piecewise linea r manifolds , e tc. Let M b e a mo del categ o ry , and let ∆ denote the simplicial indexing categor y . Recall that a s implicial resolution of an ob ject X ∈ M consists of a s implicial diagram X ′  : ∆ op → M and an augment a tion map X ′  → X , such tha t the induced map ho colim( X ′ ) → X is a w ea k equiv alence in M . Conv ersely , the geometric realizatio n of a simplicial ob ject Y  : ∆ op → M is the ob ject in M obtained b y taking the homo topy colimit of the diagram Y  . 42 DA VID I. SPIV AK Prop ositio n 8.3. The funct ors U : s C ∞ → s R and − ( R ) : s C ∞ → sSe ts e ach pr eserve ge ometric re alizations. Pr o of. By [19, A.1], the geometric realiz ation of a n y simplicial ob ject in a n y of the mo del ca tegories s C ∞ , s R , and s Sets is equiv alent to the diagona l of the co rre- sp onding bis implicia l ob ject. Since b oth U and − ( R ) are functors which pres e r ve the dia gonal, they each commute with geometric rea lization.  Let − ( R ) : s C ∞ → sSets deno te the functor F 7→ F ( R ). It is easy to s ee that − ( R ) is a right Quillen functor. Its left adjoin t is ( − ⊗ H R ) : K 7→ K ⊗ H R (although note that K ⊗ H R is not g enerally ﬁbrant in s C ∞ ). W e call a C ∞ -ring fr e e if it is in the essent ia l image of this functor − ⊗ H R , and similarly , we call a morphism o f free C ∞ -rings fr e e if it is in the imag e of this functor. Thu s , a morphism d ′ : H R k → H R ℓ is free if and only if it is induced by a function d : { 1 , . . . , k } → { 1 , . . . , ℓ } . T o ma ke d ′ explicit, w e just need to provide a dua l map R ℓ → R k ; for each 1 ≤ i ≤ k , we supply the map R ℓ → R given b y pro j ec tion onto the d ( i ) co or dinate. If K is a simplicial set suc h that eac h K i is a ﬁnite set with cardinality | K i | = n i , then X := K ⊗ H R is the simplicial C ∞ -ring with X i = H R n i , and for a map [ ℓ ] → [ k ] in ∆ , the structure map X k → X ℓ is the free map deﬁned b y K k → K ℓ . Lemma 8.4 . L et X b e a simplicia l C ∞ -ring. Ther e exist s a funct orial simplicial r esolution X ′  → X in which X ′ n is a fr e e simplicial C ∞ -ring for e ach n . Mor e over, if f : X → Y is a morphism of C ∞ -rings and f ′  : X ′  → Y ′  is the induc e d map on simplicial r esolutions, then for e ach n ∈ N , the map f ′ n : X ′ n → Y ′ n is a m orphism of fr e e C ∞ -rings. Pr o of. Let R denote the underly ing simplicial se t functor − ( R ) : s C ∞ → sSets , and let F deno te its left adjoint. The como nad F R gives rise to an aug men ted simplicial C ∞ -ring, Φ = · · · / / / / / / F RF R ( X ) / / / / F R ( X ) / / X . By [3 1, 8.6.1 0], the induced a ugmented simplicial set R Φ = · · · / / / / / / RF R F R ( X ) / / / / RF R ( X ) / / R ( X ) is a weak equiv alence. By Pr op ositions 5.8 and 9.3 , Φ is a s implicia l res o lution. The second as sertion is clear b y construction.  In the following theor em, we use a basic fact ab out simplicia l sets: if g : F → H is a ﬁbration of simplicial sets and π 0 g is a surjection of sets, then for each n ∈ N the function g n : F n → H n is surjective. This is proved using the left lifting pr o p e r ty for the cone po int inclusion ∆ 0 → ∆ n +1 . Theorem 8.5. L et Ψ b e a diagr am F f / / g   G H DERIVED SMOOTH MANIFOLDS 43 of c oﬁbr ant-ﬁ br ant C ∞ -rings. Assu me that π 0 g : π 0 F → π 0 H is a su rje ction. Then applic ation of U : s C ∞ → s R c ommutes with taking the homotopy c olimit of Ψ in the sense t hat the n atur al map ho colim( U Ψ) → U ho colim(Ψ) is a we ak e quivalenc e of simplicial c ommut ative rings. Pr o of. W e prov e the result by using simplicial r esolutions to reduce to the case prov ed in Lemma 9.1. W e b egin with a series of replacements and simpliﬁcations of the diagra m Ψ, each o f which preser ves b oth ho colim( U Ψ ) and U hoco lim(Ψ). First, re pla ce g with a ﬁbr ation a nd f with a coﬁbra tio n. Note that sinc e g is surjective on π 0 and is a ﬁbration, it is sur jectiv e in each degree; note also that f is injective in every degree. Next, replace the diagra m b y the simplicia l re solution giv en b y Lemma 9.4. This is a dia gram H ′  g ′  ← − F ′  f ′  − → G ′  , which has the s a me ho mo topy c o limit. W e ca n compute this ho motopy co limit b y ﬁrst c omputing the ho mo topy co limits ho colim( H ′ n g ′ n ← − F ′ n f ′ n − → G ′ n ) in each degree, and then taking the geo metric realiza- tion. Since U preserves ge o metric re alization (Lemma 9 .3), we may a ssume that Ψ is a diagra m H g ← − F f − → G , in which F , G, and H a re free simplicia l C ∞ -rings, and in which g is surjective and f is injective. By p erfo r ming another simplicial resolution, we may assume further that F, G, and H ar e discr e te. A fre e C ∞ -ring is the ﬁltered colimit of its ﬁnitely g enerated (f r ee) sub- C ∞ -rings; hence we ma y ass ume that F , G, and H a re ﬁnitely genera ted. In other w o rds, each is of the form S ⊗ H R , where S is a ﬁnite set. W e are reduced to the cas e in which Ψ is the diagram H R m f / / g   H R n H R p , where a gain g is surjective and f is injective. Since f and g are free maps, they are induced by maps of s ets f 1 : { 1 , . . . m } ֒ → { 1 , . . . , n } and g 1 : { 1 , . . . , m } → { 1 , . . . , p } . The map R n → R m underlying f is g iven b y ( a 1 , . . . , a n ) 7→ ( a f (1) , . . . , a f ( m ) ) , which is a pro jection onto a coor dinate plane through the origin, and we may arrang e that it is a pro jection on to the ﬁrst m co ordina tes . The result now follows from Lemma 9.1.  Corollary 8.6. Supp ose that the diagr am ( A, O A ) G / / F   ⋄ ( Y , O Y ) f   ( X, O X ) g / / ( Z, O Z ) is a homotopy pul lb ack of lo c al C ∞ -ringe d sp ac es. If g is an imb e ddi n g then t he underlying diagr am of ringe d sp ac es is also a homotopy pul lb ack. 44 DA VID I. SPIV AK Pr o of. In b oth the co nt ex t of lo ca l C ∞ -ringed spaces and lo cal r inged spaces, the space A is the pullback of the diagra m X → Z ← Y . The s heaf on A is the homotopy colimit o f the diagram F ∗ O X F ∗ ( g ♯ ) ← − − − − F ∗ g ∗ O Z G ∗ ( f ♯ ) − − − − → G ∗ O Y , either as a sheaf of C ∞ -rings or as a shea f o f simplicial commutativ e rings . Note that taking in verse-image she aves commutes with tak ing underlying simplicial com- m uta tive ring s. Since g is an imbedding, we hav e seen that g ♯ : g ∗ O Z → O X is sur jectiv e on π 0 , and thus so is its pullback F ∗ ( g ♯ ). The result now follows from Theor em 9.5.  W e will now give another way o f viewing the “locality condition” on ringed spaces. Co ns idering sections o f the structure sheaf as functions to aﬃne space, a ringed space is lo cal if these functions pull cov ers bac k to cov er s. This p oint of view can b e found in [2] and [7], for exa mple. Recall the notation | F | = F ( R ) for a C ∞ -ring F ; see Notatio n 6.12. Recall als o that C ∞ ( R ) deno tes the free (discrete) C ∞ -ring on one generato r . Deﬁnition 8.7. Let X b e a top olog ical s pace and F a sheaf of C ∞ -rings on X . Given an op e n set U ⊂ R , let χ U : R → R denote a c ha racteristic function o f U (i.e. χ U v anishes precisely on R − U ). Let f ∈ |F ( X ) | 0 denote a global section. W e will say tha t a n op en subset V ⊂ X is c ont aine d in the pr eimage under f of U if there exists a dotted arrow making the diagr am of C ∞ -rings C ∞ ( R ) f / /   F ( X ) ρ X,V   C ∞ ( R )[ χ − 1 U ] / / F ( V ) (8.7.1) commute up to ho motopy . W e say that V is t he pr eimage under f of U , a nd by abuse of notation write V = f − 1 ( U ), if it is maximal with resp ect to b eing contained in the preimage . Note that these notions are independent o f the choice of characteristic function χ U for a given U ⊂ R . No te also that since lo ca lization is an epimor phism of C ∞ -rings (as it is for ordinar y commutativ e r ings; see [2 2, 2.2,2.6]), the dotted arr ow is unique if it exists. If f , g ∈ |F ( X ) | 0 are homo topic vertices, then for any op en subset U ⊂ R , one has f − 1 ( U ) = g − 1 ( U ) ⊂ X . Therefore , this preima g e functor is well-deﬁned on the set of connec ted co mp onents π 0 |F ( X ) | . F urthermor e, if f ∈ |F ( X ) | n is any simplex, all of its vertices ar e found in the same connected comp onent, roug hly denoted π 0 ( f ) ∈ π 0 |F ( X ) | , so we can write f − 1 ( U ) to deno te π 0 ( f ) − 1 ( U ). Example 8.8 . The ab ove deﬁnition can b e understo o d fro m the v ie w p oint of al- gebraic geometry . Given a scheme ( X , O X ) and a g lobal sec tio n f ∈ O X ( X ) = π 0 O X ( X ), one can co nsider f as a scheme mor phism fro m X to the aﬃne line A 1 . Given a principle op en subset U = Sp ec ( k [ x ][ g − 1 ]) ⊂ A 1 , we are interested in its preimag e in X . This preimage will b e the la rgest V ⊂ X on w hich the map k [ x ] f − → O X ( X ) can be lifted to a map k [ x ][ g − 1 ] → O X ( V ). This is the conten t of Diagram 9 .7.1. DERIVED SMOOTH MANIFOLDS 45 Up next, we will give an alter nate for m ula tio n of the condition that a s heaf of C ∞ -rings be a lo cal in terms of these preimages . In the algebro- geometric setting, it comes down to the fact that a sheaf of rings F is a shea f of lo c al rings on X if and only if, for ev er y global section f ∈ F ( X ), the pr eimages under f of a cover of Spec ( k [ x ]) form a c over of X . Lemma 8.9. Supp ose that U ⊂ R is an op en subset of R , and let χ U denote a char acteristic function for U . Then C ∞ ( U ) = C ∞ ( R )[ χ − 1 U ] . Ther e is a n atur al bije ct ion b etwe en the s et of p oints p ∈ R and the set of C ∞ - functions A p : C ∞ ( R ) → C ∞ ( R 0 ) . U nder t his c orr esp ondenc e, p is in U if and only if A p factors thr ough C ∞ ( R )[ χ − 1 U ] . Pr o of. This follows from [22, 1.5 and 1.6].  Recall that F ∈ s C ∞ is s a id to b e a lo ca l C ∞ -ring if the co mm utative ring underlying π 0 F is a lo ca l ring (see Deﬁnition 5 .10). Prop ositio n 8 .10. L et X b e a sp ac e and F a she af of ﬁnitely pr esente d C ∞ -rings on X . Then F is lo c al if and only if, for any c over of R by op en subsets R = S i U i and for any op en V ⊂ X a n d lo c al se ction f ∈ π 0 |F ( V ) | , the pr eimages f − 1 ( U i ) form a c over of V . Pr o of. Choose a repr esentativ e for f ∈ π 0 | F | , call it f ∈ | F | 0 for simplicity , and recall that it can b e identiﬁ e d with a map f : C ∞ ( R ) → F , which is unique up to homotopy . Both the pro pe r ty of b eing a sheaf o f lo c a l C ∞ -rings a nd the ab ov e “preimage of a covering is a covering” pr op erty is lo ca l on X . Thus we may assume that X is a p oint. W e are reduced to proving that a C ∞ -ring F is a loca l C ∞ -ring if a nd only if, for any cov er of R by op en sets R = S i U i and for any element f ∈ | F | 0 , there ex is ts an index i and a dotted a rrow mak ing the diagra m C ∞ ( R )   f / / F C ∞ ( R )[ χ − 1 U i ] : : commute up to homotopy . Suppo se ﬁr st that for any cov er of R by op en subsets R = S i U i and for any element f ∈ F 0 , ther e exists a lift as ab ov e. Let U 1 = ( −∞ , 1 2 ) a nd let U 2 = (0 , ∞ ). By as sumption, either f factor s through C ∞ ( U 1 ) o r through C ∞ ( U 2 ), and 1 − f factors through the other by Lemma 9.9. It is easy to show that any element of π 0 F which factor s throug h C ∞ ( U 2 ) is inv ertible (using the fact that 0 6∈ U 2 ). Hence, either f or 1 − f is inv er tible in π 0 F , so π 0 F is a lo c al C ∞ -ring. Now supp ose that F is lo cal, i.e. that it has a unique maximal idea l, and suppose R = S i U i is an op en cover. Choo se f ∈ | F | 0 , consider ed as a map of C ∞ -rings f : C ∞ ( R ) → F . By [22, 3 .8], π 0 F ha s a unique p oint F → π 0 F → C ∞ ( R 0 ). By Lemma 9.9, there e x ists i such that the co mpo s ition C ∞ ( R ) f − → F → C ∞ ( R 0 ) 46 DA VID I. SPIV AK factors thro ugh C ∞ ( R )[ χ − 1 U i ], giving the solid arrow square C ∞ ( R ) f / /   F   C ∞ ( R )[ χ − 1 U i ] / / 8 8 C ∞ ( R 0 ) . Since χ U i ∈ C ∞ ( R ) is not sent to 0 ∈ C ∞ ( R 0 ), its image f ( χ U i ) is not contained in the maximal ideal of π 0 F , so a dotted ar row lift exists ma king the diagr a m commute up to homotopy . This pr ov es the prop osition.  In the following theor e m, we use the notation | A | to denote the simplicia l s et A ( R ) = Map s C ∞ ( C ∞ ( R ) , A ) underlying a simplicial C ∞ -ring A ∈ s C ∞ . Theorem 8.11. L et X = ( X , O X ) b e a lo c al C ∞ -ringe d sp ac e, and let i R = ( R , C ∞ R ) denote the (image u nder i of the) r e al line. Ther e is a natu ra l homotopy e quivalenc e of simplicial sets Map LRS (( X, O X ) , ( R , C ∞ R )) ≃ − → |O X ( X ) | . Pr o of. W e will construct morphisms K : Map LRS ( X , i R ) → |O X ( X ) | , a nd L : |O X ( X ) | → Map LRS ( X , i R ) , and show that they are homo topy inv er ses. F or the reader ’s conv enience, we recall the de ﬁnitio n Map LRS ( X , R ) = a f : X → R Map lo c ( f ∗ C ∞ R , O X ) . The map K is fair ly eas y and ca n b e deﬁned without use of the loca lit y condition. Suppo se that φ : X → R is a map o f top ologica l spaces. The r estriction of K to the corres p onding summand of Map LRS ( X , i R ) is given b y taking global s ections Map lo c ( φ ∗ C ∞ R , O X ) → Map( C ∞ ( R ) , O X ( X )) ∼ = |O X ( X ) | . T o deﬁne L is a bit harder a nd dep ends heavily on the a ssumption that O X is a lo ca l s heaf o n X . Fir s t, given a n n - simplex g ∈ |O X ( X ) | n we need to deﬁne a map of topo logical spaces L ( g ) : X → R . Let g 0 ∈ π 0 |O X ( X ) | denote the connected comp onent con taining g . By Prop osition 9.10, g 0 gives rise to a function from op en cov ers o f R to op en covers of X , and this function commutes with re ﬁnement of op en covers. Since R is Hausdor ﬀ, there is a unique map o f top olo g ical spac e s X → R , which we take as L ( g ), consis ten t with such a function. Denote L ( g ) by G , fo r ease of nota tion. Now w e need to de ﬁne a map o f sheaves o f C ∞ -rings G ♭ : C ∞ R ⊗ ∆ n → G ∗ ( O X ) . On global sections, we hav e s uch a function alrea dy , since g ∈ |O X ( X ) | n can b e considered a s a ma p g : C ∞ R ( R ) → |O X ( X ) | n . Let V ⊂ R denote an op en subset and g − 1 ( V ) ⊂ X its preimage under g . The map ρ : C ∞ R ( R ) → C ∞ R ( V ) is a loc a lization; hence, it is an epimo rphism of C ∞ -rings. DERIVED SMOOTH MANIFOLDS 47 T o deﬁne G ♭ , w e will need to show that there exists a unique dotted arrow making the diagra m C ∞ R ( R ) ⊗ ∆ n g / / ρ ⊗ ∆ n   |O X ( X ) |   C ∞ R ( V ) ⊗ ∆ n / / |O X ( g − 1 ( V )) | (8.11.1) commute. Such an ar row exists by deﬁnition of g − 1 . It is unique bec ause ρ is an epimorphism of C ∞ -rings, so ρ ⊗ ∆ n is a s well. W e hav e now deﬁned G ♭ , and we take G ♯ : G ∗ C ∞ R → O X to b e the left adjoint o f G ♭ . W e m ust sho w that f o r ev ery g ∈ |O X ( X ) | 0 , the ma p G ♯ : G ∗ C ∞ R → O X provided by L is lo c a l. (W e ca n choos e g to be a vertex b eca use, by deﬁnition, a simplex in Map( G ∗ C ∞ R , O X ) is lo cal if a ll of its vertices ar e lo ca l.) T o prove this, we may take X to b e a p oint, F = O X ( X ) a lo cal C ∞ -ring, and x = G ( X ) ∈ R the image po int o f X . W e hav e a morphis m of C ∞ -rings G ♯ : ( C ∞ R ) x → F , in w hich b oth the domain a nd co domain are lo c a l. It is a lo c al ring homomorphis m beca use all prime ideals in the lo cal ring ( C ∞ R ) x are maximal. The maps K a nd L hav e now b een deﬁned, and they a re homotopy inv erses by construction.  The following is a technical lemma that allows us to take homotopy limits comp onent-wise in the mo del category of simplicial sets. Lemma 8.12. L et sSets denote the c ate gory of simplicial sets. L et I , J, and K denote sets and let A = a i ∈ I A i , B = a j ∈ J B j , and C = a k ∈ K C k denote c opr o ducts in sSets indexe d by I , J, and K . Supp ose that f : I → J and g : K → J ar e functions and that F : A → B and G : C → B ar e maps in sSets that r esp e ct f and g in the sense that F ( A i ) ⊂ B f ( i ) and G ( C k ) ⊂ B g ( k ) for al l i ∈ I and k ∈ K . L et I × J K = { ( i, j, k ) ∈ I × J × K | f ( i ) = j = g ( k ) } denote the ﬁb er pr o duct of sets. F or typ o gr aphi c al r e asons, we use × h to denote homotopy limit in sSets , and × to denote the 1-c ate goric al limit. Then t he natur al map   a ( i,j,k ) ∈ I × J K A i × h B j C k   − → A × h B C is a we ak e quivalenc e in sSets . Pr o of. If w e r eplace F by a ﬁbration, then each co mpo nen t F i := F | A i : A i → B f ( i ) is a ﬁbration. W e re duce to showing that the map   a ( i,j,k ) ∈ I × J K A i × B j C k   − → A × B C (8.12.1) 48 DA VID I. SPIV AK is a n isomor phism of s implicial sets . Res tricting to the n -simplicies of bo th sides, we may a ssume that A, B , and C are (discrete simplicial) sets. It is an easy exercis e to show that the map in (9.12.1) is injective and surjective, i.e. an isomor phism in Sets .  Prop ositio n 8.1 3. Supp ose that a : M 0 → M and b : M 1 → M ar e morphisms of manifolds, and supp ose that a ﬁb er pr o duct N exists in the c ate gory of m anifold s . If X = ( X , O X ) is the ﬁb er pr o duct X / /   ⋄ M 0 a   M 1 b / / M taken in t he c ate gory of derive d manifolds, then the natur al m ap g : N → X is an e quivalenc e if and only if a and b ar e t ra ns verse. Pr o of. Since limits taken in M an and in dMan co mm ute with tak ing underlying top ological s paces, the ma p N → X is a homeomorphis m. W e hav e a commutative diagram N g / / f   X f ′ z z u u u u u u u u u u M 0 × M 1 , in which f and f ′ are closed immersions (pullbacks of the diagona l M → M × M ). If a and b are not transverse, one shows easily that the ﬁrst homolo gy gr oup H 1 L X 6 = 0 of the cotangent complex for X is nonzer o, wher eas H 1 L N = 0 becaus e N is a manifold; hence X is not equiv alent to N . If a and b ar e transverse, then one can s how that g induces a quasi- is omorphism g ∗ L f ′ → L f . By Lemma 7.7, the map g is a n equiv alence o f der ived manifolds.  Prop ositio n 8.14. The simplici al c ate gory LRS of lo c al C ∞ -ringe d sp ac es is close d under t aking ﬁnite homotopy limits. Pr o of. The lo cal C ∞ -ringed space ( R 0 , C ∞ ( R 0 )) is a homotopy terminal ob ject in LRS . Hence it s uﬃces to show that a ho motopy limit ex ists for any diagra m ( X, O X ) F ← − ( Y , O Y ) G − → ( Z, O Z ) in LRS . W e ﬁrst describ e the appro priate candidate for this homoto p y limit. The underlying s pace of the candidate is X × Y Z , a nd we lab el the maps as in the dia gram X × Y Z h # # H H H H H H H H H g / / f   Z G   X F / / Y The structur e sheaf on the ca ndidate is the homotopy colimit of pullback sheav es O X × Y Z := ( g ∗ O Z ) ⊗ ( h ∗ O Y ) ( f ∗ O X ) . (8.14.1) DERIVED SMOOTH MANIFOLDS 49 T o show tha t O X × Y Z is a shea f of lo c a l C ∞ -rings, we take the stalk at a p oint, apply π 0 , and show that it is a lo ca l C ∞ -ring. The ho motopy colimit written in Equation (9.1 4.1) beco mes the C ∞ -tensor pro duct of po in ted lo ca l C ∞ -rings. By [22, 3.1 2 ], the result is indee d a lo cal C ∞ -ring. One shows that ( X × Y Z, O X × Y Z ) is the homotopy limit of the diagram in the usual way . W e do no t prove it her e, but refer the reader to [27, 2.3.21] o r, for a m uch more genera l result, to [21, 2.4.2 1].  Theorem 8.15. L et M b e a manifold, let X and Y b e derive d manifolds, and let f : X → M and g : Y → M b e morphisms of derive d manifolds. Then a ﬁb er pr o duct X × M Y exists in the c ate gory of derive d manifolds. Pr o of. W e show ed in Prop ositio n 9.1 4 that X × M Y exists as a lo cal C ∞ -ringd space. T o show that it is a derived ma nifold, we m ust only show that it is lo ca lly an aﬃne derived manifold. This is a lo cal prop erty , so it suﬃces to lo ok lo ca lly on M , X , and Y . W e will pr ov e the r esult b y ﬁr s t showing that aﬃne derived manifolds are closed under taking pr o ducts, then tha t they are clos ed under solving equations, and ﬁnally that these t wo facts combine to pr ov e the result. Given aﬃne derived manifolds R n a =0 and R m b =0 , it follows formally that R n + m ( f ,g )=0 is their pro duct, and it is an aﬃne der ived manifold. Now le t X = R n a =0 , wher e a : R n → R m , and supp ose that b : X → R k is a morphism. By Theorem 9.11, we can consider b as an element of O X ( X )( R k ). By Lemma 5.4, it is ho motopic to a comp osite X → R n b ′ − → R k , wher e X → R n is the ca nonical imbedding. Now we can re alize X b =0 as the homo to p y limit in the all-Cartes ia n diagram X b =0 / /   ⋄ R 0   X b / /   ⋄ R k / /   ⋄ R 0   R n ( b ′ ,a ) / / R k × R m / / R m . Therefore, X b =0 = R n ( b ′ ,a )=0 is aﬃne. Finally , suppo se that X and Y are a ﬃne and that M = R p . Let − : R p × R p → R p denote the co ordinate- wise s ubtr action map. Then ther e is an all-Ca rtesian diagram X × R p Y / /   ⋄ R p diag   / / ⋄ R 0 0   X × Y / / R p × R p − / / R p , where di a g : R p → R p × R p is the diag onal map. W e hav e s e en tha t X × Y is aﬃne, so since X × R p Y is the s olution to an equation on a n a ﬃne derived manifold, it to o is aﬃne. This co mpletes the pro of.  50 DA VID I. SPIV AK R emark 8.16 . No te that Theorem 9.15 do es not s ay that the ca tegory dMan is closed under a rbitrary ﬁb er pro ducts. Indeed, if M is not a ssumed to b e a smo oth manifold, then the ﬁbe r pro duct o f derived manifolds o ver M need not be a derived manifold in our sense. The cotangent complex of any derived manifold ha s homology concentrated in degrees 0 a nd 1 (see Corolla ry 7.4), whereas a ﬁber pr o duct of derived ma nifolds (o ver a non-smo oth base) would not hav e that prop erty . Of co ur se, using the s pec tr um functor Sp ec, deﬁned in Remark 6.14, o ne could deﬁne a mor e genera l category C of “ derived manifolds” in the us ual scheme- theoretic way . Then our dMan would form a full sub categor y o f C , whic h o ne might call the sub categ ory of quasi-smo oth ob jects (see [27]). The r e ason we did not in tr o duce this catego ry C is that it does not have the gener al c up pro duct formula in cob or dism; i.e. Theorem 1.8 do es not a pply to C . 9. Derived manif ol ds are good for do ing intersection theor y Recall from Sec tio n 5 that H R i ∈ s C ∞ is the disc rete C ∞ -ring corepres ent ing R i ∈ E . A s mo o th ma p f : R i → R j (contra v aria ntly) induces a morphism of C ∞ - rings H R j → H R i , whic h we often denote by f for co n venience. Rec all also that in s C ∞ , there is a canonical weak equiv alence H R i + j ≃ − → H R i ∐ H R j . Let U : s C ∞ → s R denote the “underlying simplicial comm uta tive ring ” functor from Co rollary 5.8. Lemma 9. 1 . L et m, n, p ∈ N with p ≤ m . L et x : R n + m → R m denote the pr oje c- tion onto the ﬁrst m c o or dinates, let g : R p → R m denote t he inclusion of a p -plane in R m , and let Ψ b e the diagr am H R m x / / g   H R n + m H R p of C ∞ -rings. The homotopy c olimit of Ψ is we akly e quivalent t o H R n + p . Mor e over, the applic ation of U : s C ∞ → s R c ommutes with taking homotopy c olimit of Ψ in t he sense t hat the natura l map ho colim( U Ψ) → U ho colim(Ψ) is a we ak e quivalenc e of simplicial c ommut ative R -algebr as. Pr o of. W e may ass ume that g sends the origin to the origin. F or now, we a ssume that p = 0 and m = 1. Consider the a ll-Cartesian diagr am o f manifolds R n / /   p R n +1 / / x   p R n   R 0 g / / R / / R 0 . Apply H : Man op → s C ∞ to obtain the diagra m H R n H R n +1 o o H R n o o H R 0 O O H R g o o x O O H R 0 O O o o DERIVED SMOOTH MANIFOLDS 51 Since H sends pro ducts in E to homotopy pushouts in s C ∞ , the right-hand squar e and the big rectangle are homotopy pushouts. Hence the left square is as well, proving the ﬁr st ass ertion (in case p = 0 a nd m = 1 ). F or notationa l reas ons, let U i denote U ( H R i ), so that U i is the (discr ete s im- plicial) commutativ e ring whose elemen ts are s mo o th maps R i → R . Deﬁne a simplicial co mm utative R -a lgebra D as the homoto p y colimit in the diagra m U 1 x / / g   ⋄ U n +1   U 0 / / D . W e m us t show that the natural map D → U n is a weak eq uiv alence of simplicial commutativ e rings. Recall that the homoto p y g roups of a ho motopy pusho ut of simplicial commutativ e r ings are the T or groups of the corres po nding tensor pro duct of chain complexes . Since x is a nonzero divisor in the ring U n +1 , the homo to p y groups of D are π 0 ( D ) = U n +1 ⊗ U 1 U 0 ; and π i ( D ) = 0 , i > 0 . Thu s , π 0 ( D ) can be iden tiﬁed with the equiv alence classes of smoo th functions f : R n +1 → R , where f ∼ g if f − g is a multiple of x . On the o ther hand, w e ca n ident ify e lemen ts of the ring U n of smo oth functions on R n with the equiv alence classes o f functions f : R n +1 → R , where f ∼ g if f (0 , x 2 , x 3 , . . . x n +1 ) = g (0 , x 2 , x 3 , . . . , x n +1 ) . Thu s to prov e that t he map D → U n is a w eak equiv alence, we must o nly show t ha t if a smo oth function f ( x 1 , . . . , x n +1 ) : R n +1 → R v anishes whenever x 1 = 0, then x 1 divides f . This is called Hadamar d’s lemma, and it follows fro m the deﬁnition of smo o th functions. Indeed, given a smo oth function f ( x 1 , . . . , x n +1 ) : R n +1 → R which v anis he s whenever x 1 = 0, deﬁne a function g : R n +1 → R b y the fo rmula g ( x 1 , . . . , x n +1 ) = lim a → x 1 f ( a, x 2 , x 3 , . . . , x n +1 ) a . It is clear that g is smo oth, and xg = f . W e have now prov ed both assertions in the case that p = 0 and m = 1; w e contin ue to assume p = 0 and prov e the result for genera l m + 1 by induction. T he inductive step follows from the all-Cartesian diagra m b elow, in whic h ea ch vertical arrow is a n inclusion of a plane and each horizontal arrow is a coo rdinate pro j ec tio n: R n / /   p R 0   R m + n / /   p R m / /   p R 0   R n + m +1 / / R m +1 / / R . Apply H to this dia gram. The ar guments ab ov e implies that the tw o as sertions hold for the right squar e and b ottom rectangle; th us they hold for the bo ttom left square. The inductive hypothesis implies that the tw o asser tions hold for the top square, s o they hold for the left rectangle, a s desir ed. 52 DA VID I. SPIV AK F or the case of general p , let k : R m → R m − p denote the pro jection orthogo nal to g . Applying H to the all-Car tesian diagr am R n + p / /   p R p / / g   p R 0   R n + m x / / R m k / / R m − p , the result holds for the right square and the big rectangle (for b oth of which p = 0), so it holds for the left square a s well.  R emark 9.2 . Note that (the non-for mal part of ) Lemma 9 .1 relies heavily on the fact that we ar e dea ling with smo oth maps. It is this le mma which fails in the setting o f top olog ical manifolds, piecewise linea r manifolds , e tc. Let M b e a mo del categ o ry , and let ∆ denote the simplicial indexing categor y . Recall that a s implicial resolution of an ob ject X ∈ M consists of a s implicial diagram X ′  : ∆ op → M and an augment a tion map X ′  → X , such tha t the induced map ho colim( X ′ ) → X is a w ea k equiv alence in M . Conv ersely , the geometric realizatio n of a simplicial ob ject Y  : ∆ op → M is the ob ject in M obtained b y taking the homo topy colimit of the diagram Y  . Prop ositio n 9.3. The funct ors U : s C ∞ → s R and − ( R ) : s C ∞ → sSe ts e ach pr eserve ge ometric re alizations. Pr o of. By [19, A.1], the geometric realiz ation of a n y simplicial ob ject in a n y of the mo del ca tegories s C ∞ , s R , and s Sets is equiv alent to the diagona l of the co rre- sp onding bis implicia l ob ject. Since b oth U and − ( R ) are functors which pres e r ve the dia gonal, they each commute with geometric rea lization.  Let − ( R ) : s C ∞ → sSets deno te the functor F 7→ F ( R ). It is easy to s ee that − ( R ) is a right Quillen functor. Its left adjoin t is ( − ⊗ H R ) : K 7→ K ⊗ H R (although note that K ⊗ H R is not g enerally ﬁbrant in s C ∞ ). W e call a C ∞ -ring fr e e if it is in the essent ia l image of this functor − ⊗ H R , and similarly , we call a morphism o f free C ∞ -rings fr e e if it is in the imag e of this functor. Thu s , a morphism d ′ : H R k → H R ℓ is free if and only if it is induced by a function d : { 1 , . . . , k } → { 1 , . . . , ℓ } . T o ma ke d ′ explicit, w e just need to provide a dua l map R ℓ → R k ; for each 1 ≤ i ≤ k , we supply the map R ℓ → R given b y pro j ec tion onto the d ( i ) co or dinate. If K is a simplicial set suc h that eac h K i is a ﬁnite set with cardinality | K i | = n i , then X := K ⊗ H R is the simplicial C ∞ -ring with X i = H R n i , and for a map [ ℓ ] → [ k ] in ∆ , the structure map X k → X ℓ is the free map deﬁned b y K k → K ℓ . Lemma 9.4 . L et X b e a simplicia l C ∞ -ring. Ther e exist s a funct orial simplicial r esolution X ′  → X in which X ′ n is a fr e e simplicial C ∞ -ring for e ach n . Mor e over, if f : X → Y is a morphism of C ∞ -rings and f ′  : X ′  → Y ′  is the induc e d map on simplicial r esolutions, then for e ach n ∈ N , the map f ′ n : X ′ n → Y ′ n is a m orphism of fr e e C ∞ -rings. DERIVED SMOOTH MANIFOLDS 53 Pr o of. Let R denote the underly ing simplicial se t functor − ( R ) : s C ∞ → sSets , and let F deno te its left adjoint. The como nad F R gives rise to an aug men ted simplicial C ∞ -ring, Φ = · · · / / / / / / F RF R ( X ) / / / / F R ( X ) / / X . By [3 1, 8.6.1 0], the induced a ugmented simplicial set R Φ = · · · / / / / / / RF R F R ( X ) / / / / RF R ( X ) / / R ( X ) is a weak equiv alence. By Pr op ositions 5.8 and 9.3 , Φ is a s implicia l res o lution. The second as sertion is clear b y construction.  In the following theor em, we use a basic fact ab out simplicia l sets: if g : F → H is a ﬁbration of simplicial sets and π 0 g is a surjection of sets, then for each n ∈ N the function g n : F n → H n is surjective. This is proved using the left lifting pr o p e r ty for the cone po int inclusion ∆ 0 → ∆ n +1 . Theorem 9.5. L et Ψ b e a diagr am F f / / g   G H of c oﬁbr ant-ﬁ br ant C ∞ -rings. Assu me that π 0 g : π 0 F → π 0 H is a su rje ction. Then applic ation of U : s C ∞ → s R c ommutes with taking the homotopy c olimit of Ψ in the sense t hat the n atur al map ho colim( U Ψ) → U ho colim(Ψ) is a we ak e quivalenc e of simplicial c ommut ative rings. Pr o of. W e prov e the result by using simplicial r esolutions to reduce to the case prov ed in Lemma 9.1. W e b egin with a series of replacements and simpliﬁcations of the diagra m Ψ, each o f which preser ves b oth ho colim( U Ψ ) and U hoco lim(Ψ). First, re pla ce g with a ﬁbr ation a nd f with a coﬁbra tio n. Note that sinc e g is surjective on π 0 and is a ﬁbration, it is sur jectiv e in each degree; note also that f is injective in every degree. Next, replace the diagra m b y the simplicia l re solution giv en b y Lemma 9.4. This is a dia gram H ′  g ′  ← − F ′  f ′  − → G ′  , which has the s a me ho mo topy c o limit. W e ca n compute this ho motopy co limit b y ﬁrst c omputing the ho mo topy co limits ho colim( H ′ n g ′ n ← − F ′ n f ′ n − → G ′ n ) in each degree, and then taking the geo metric realiza- tion. Since U preserves ge o metric re alization (Lemma 9 .3), we may a ssume that Ψ is a diagra m H g ← − F f − → G , in which F , G, and H a re free simplicia l C ∞ -rings, and in which g is surjective and f is injective. By p erfo r ming another simplicial resolution, we may assume further that F, G, and H ar e discr e te. A fre e C ∞ -ring is the ﬁltered colimit of its ﬁnitely g enerated (f r ee) sub- C ∞ -rings; hence we ma y ass ume that F , G, and H a re ﬁnitely genera ted. In other w o rds, each is of the form S ⊗ H R , where S is a ﬁnite set. W e are reduced to the cas e in which 54 DA VID I. SPIV AK Ψ is the diagram H R m f / / g   H R n H R p , where a gain g is surjective and f is injective. Since f and g are free maps, they are induced by maps of s ets f 1 : { 1 , . . . m } ֒ → { 1 , . . . , n } and g 1 : { 1 , . . . , m } → { 1 , . . . , p } . The map R n → R m underlying f is g iven b y ( a 1 , . . . , a n ) 7→ ( a f (1) , . . . , a f ( m ) ) , which is a pro jection onto a coor dinate plane through the origin, and we may arrang e that it is a pro jection on to the ﬁrst m co ordina tes . The result now follows from Lemma 9.1.  Corollary 9.6. Supp ose that the diagr am ( A, O A ) G / / F   ⋄ ( Y , O Y ) f   ( X, O X ) g / / ( Z, O Z ) is a homotopy pul lb ack of lo c al C ∞ -ringe d sp ac es. If g is an imb e ddi n g then t he underlying diagr am of ringe d sp ac es is also a homotopy pul lb ack. Pr o of. In b oth the co nt ex t of lo ca l C ∞ -ringed spaces and lo cal r inged spaces, the space A is the pullback of the diagra m X → Z ← Y . The s heaf on A is the homotopy colimit o f the diagram F ∗ O X F ∗ ( g ♯ ) ← − − − − F ∗ g ∗ O Z G ∗ ( f ♯ ) − − − − → G ∗ O Y , either as a sheaf of C ∞ -rings or as a shea f o f simplicial commutativ e rings . Note that taking in verse-image she aves commutes with tak ing underlying simplicial com- m uta tive ring s. Since g is an imbedding, we hav e seen that g ♯ : g ∗ O Z → O X is sur jectiv e on π 0 , and thus so is its pullback F ∗ ( g ♯ ). The result now follows from Theor em 9.5.  W e will now give another way o f viewing the “locality condition” on ringed spaces. Co ns idering sections o f the structure sheaf as functions to aﬃne space, a ringed space is lo cal if these functions pull cov ers bac k to cov er s. This p oint of view can b e found in [2] and [7], for exa mple. Recall the notation | F | = F ( R ) for a C ∞ -ring F ; see Notatio n 6.12. Recall als o that C ∞ ( R ) deno tes the free (discrete) C ∞ -ring on one generato r . Deﬁnition 9.7. Let X b e a top olog ical s pace and F a sheaf of C ∞ -rings on X . Given an op e n set U ⊂ R , let χ U : R → R denote a c ha racteristic function o f U (i.e. χ U v anishes precisely on R − U ). Let f ∈ |F ( X ) | 0 denote a global section. W e will DERIVED SMOOTH MANIFOLDS 55 say tha t a n op en subset V ⊂ X is c ont aine d in the pr eimage under f of U if there exists a dotted arrow making the diagr am of C ∞ -rings C ∞ ( R ) f / /   F ( X ) ρ X,V   C ∞ ( R )[ χ − 1 U ] / / F ( V ) (9.7.1) commute up to ho motopy . W e say that V is t he pr eimage under f of U , a nd by abuse of notation write V = f − 1 ( U ), if it is maximal with resp ect to b eing contained in the preimage . Note that these notions are independent o f the choice of characteristic function χ U for a given U ⊂ R . No te also that since lo ca lization is an epimor phism of C ∞ -rings (as it is for ordinar y commutativ e r ings; see [2 2, 2.2,2.6]), the dotted arr ow is unique if it exists. If f , g ∈ |F ( X ) | 0 are homo topic vertices, then for any op en subset U ⊂ R , one has f − 1 ( U ) = g − 1 ( U ) ⊂ X . Therefore , this preima g e functor is well-deﬁned on the set of connec ted co mp onents π 0 |F ( X ) | . F urthermor e, if f ∈ |F ( X ) | n is any simplex, all of its vertices ar e found in the same connected comp onent, roug hly denoted π 0 ( f ) ∈ π 0 |F ( X ) | , so we can write f − 1 ( U ) to deno te π 0 ( f ) − 1 ( U ). Example 9.8 . The ab ove deﬁnition can b e understo o d fro m the v ie w p oint of al- gebraic geometry . Given a scheme ( X , O X ) and a g lobal sec tio n f ∈ O X ( X ) = π 0 O X ( X ), one can co nsider f as a scheme mor phism fro m X to the aﬃne line A 1 . Given a principle op en subset U = Sp ec ( k [ x ][ g − 1 ]) ⊂ A 1 , we are interested in its preimag e in X . This preimage will b e the la rgest V ⊂ X on w hich the map k [ x ] f − → O X ( X ) can be lifted to a map k [ x ][ g − 1 ] → O X ( V ). This is the conten t of Diagram 9 .7.1. Up next, we will give an alter nate for m ula tio n of the condition that a s heaf of C ∞ -rings be a lo cal in terms of these preimages . In the algebro- geometric setting, it comes down to the fact that a sheaf of rings F is a shea f of lo c al rings on X if and only if, for ev er y global section f ∈ F ( X ), the pr eimages under f of a cover of Spec ( k [ x ]) form a c over of X . Lemma 9.9. Supp ose that U ⊂ R is an op en subset of R , and let χ U denote a char acteristic function for U . Then C ∞ ( U ) = C ∞ ( R )[ χ − 1 U ] . Ther e is a n atur al bije ct ion b etwe en the s et of p oints p ∈ R and the set of C ∞ - functions A p : C ∞ ( R ) → C ∞ ( R 0 ) . U nder t his c orr esp ondenc e, p is in U if and only if A p factors thr ough C ∞ ( R )[ χ − 1 U ] . Pr o of. This follows from [22, 1.5 and 1.6].  Recall that F ∈ s C ∞ is s a id to b e a lo ca l C ∞ -ring if the co mm utative ring underlying π 0 F is a lo ca l ring (see Deﬁnition 5 .10). Prop ositio n 9 .10. L et X b e a sp ac e and F a she af of ﬁnitely pr esente d C ∞ -rings on X . Then F is lo c al if and only if, for any c over of R by op en subsets R = S i U i and for any op en V ⊂ X a n d lo c al se ction f ∈ π 0 |F ( V ) | , the pr eimages f − 1 ( U i ) form a c over of V . 56 DA VID I. SPIV AK Pr o of. Choose a repr esentativ e for f ∈ π 0 | F | , call it f ∈ | F | 0 for simplicity , and recall that it can b e identiﬁ e d with a map f : C ∞ ( R ) → F , which is unique up to homotopy . Both the pro pe r ty of b eing a sheaf o f lo c a l C ∞ -rings a nd the ab ov e “preimage of a covering is a covering” pr op erty is lo ca l on X . Thus we may assume that X is a p oint. W e are reduced to proving that a C ∞ -ring F is a loca l C ∞ -ring if a nd only if, for any cov er of R by op en sets R = S i U i and for any element f ∈ | F | 0 , there ex is ts an index i and a dotted a rrow mak ing the diagra m C ∞ ( R )   f / / F C ∞ ( R )[ χ − 1 U i ] : : commute up to homotopy . Suppo se ﬁr st that for any cov er of R by op en subsets R = S i U i and for any element f ∈ F 0 , ther e exists a lift as ab ov e. Let U 1 = ( −∞ , 1 2 ) a nd let U 2 = (0 , ∞ ). By as sumption, either f factor s through C ∞ ( U 1 ) o r through C ∞ ( U 2 ), and 1 − f factors through the other by Lemma 9.9. It is easy to show that any element of π 0 F which factor s throug h C ∞ ( U 2 ) is inv ertible (using the fact that 0 6∈ U 2 ). Hence, either f or 1 − f is inv er tible in π 0 F , so π 0 F is a lo c al C ∞ -ring. Now supp ose that F is lo cal, i.e. that it has a unique maximal idea l, and suppose R = S i U i is an op en cover. Choo se f ∈ | F | 0 , consider ed as a map of C ∞ -rings f : C ∞ ( R ) → F . By [22, 3 .8], π 0 F ha s a unique p oint F → π 0 F → C ∞ ( R 0 ). By Lemma 9.9, there e x ists i such that the co mpo s ition C ∞ ( R ) f − → F → C ∞ ( R 0 ) factors thro ugh C ∞ ( R )[ χ − 1 U i ], giving the solid arrow square C ∞ ( R ) f / /   F   C ∞ ( R )[ χ − 1 U i ] / / 8 8 C ∞ ( R 0 ) . Since χ U i ∈ C ∞ ( R ) is not sent to 0 ∈ C ∞ ( R 0 ), its image f ( χ U i ) is not contained in the maximal ideal of π 0 F , so a dotted ar row lift exists ma king the diagr a m commute up to homotopy . This pr ov es the prop osition.  In the following theor e m, we use the notation | A | to denote the simplicia l s et A ( R ) = Map s C ∞ ( C ∞ ( R ) , A ) underlying a simplicial C ∞ -ring A ∈ s C ∞ . Theorem 9.11. L et X = ( X , O X ) b e a lo c al C ∞ -ringe d sp ac e, and let i R = ( R , C ∞ R ) denote the (image u nder i of the) r e al line. Ther e is a natu ra l homotopy e quivalenc e of simplicial sets Map LRS (( X, O X ) , ( R , C ∞ R )) ≃ − → |O X ( X ) | . Pr o of. W e will construct morphisms K : Map LRS ( X , i R ) → |O X ( X ) | , a nd L : |O X ( X ) | → Map LRS ( X , i R ) , DERIVED SMOOTH MANIFOLDS 57 and show that they are homo topy inv er ses. F or the reader ’s conv enience, we recall the de ﬁnitio n Map LRS ( X , R ) = a f : X → R Map lo c ( f ∗ C ∞ R , O X ) . The map K is fair ly eas y and ca n b e deﬁned without use of the loca lit y condition. Suppo se that φ : X → R is a map o f top ologica l spaces. The r estriction of K to the corres p onding summand of Map LRS ( X , i R ) is given b y taking global s ections Map lo c ( φ ∗ C ∞ R , O X ) → Map( C ∞ ( R ) , O X ( X )) ∼ = |O X ( X ) | . T o deﬁne L is a bit harder a nd dep ends heavily on the a ssumption that O X is a lo ca l s heaf o n X . Fir s t, given a n n - simplex g ∈ |O X ( X ) | n we need to deﬁne a map of topo logical spaces L ( g ) : X → R . Let g 0 ∈ π 0 |O X ( X ) | denote the connected comp onent con taining g . By Prop osition 9.10, g 0 gives rise to a function from op en cov ers o f R to op en covers of X , and this function commutes with re ﬁnement of op en covers. Since R is Hausdor ﬀ, there is a unique map o f top olo g ical spac e s X → R , which we take as L ( g ), consis ten t with such a function. Denote L ( g ) by G , fo r ease of nota tion. Now w e need to de ﬁne a map o f sheaves o f C ∞ -rings G ♭ : C ∞ R ⊗ ∆ n → G ∗ ( O X ) . On global sections, we hav e s uch a function alrea dy , since g ∈ |O X ( X ) | n can b e considered a s a ma p g : C ∞ R ( R ) → |O X ( X ) | n . Let V ⊂ R denote an op en subset and g − 1 ( V ) ⊂ X its preimage under g . The map ρ : C ∞ R ( R ) → C ∞ R ( V ) is a loc a lization; hence, it is an epimo rphism of C ∞ -rings. T o deﬁne G ♭ , w e will need to show that there exists a unique dotted arrow making the diagra m C ∞ R ( R ) ⊗ ∆ n g / / ρ ⊗ ∆ n   |O X ( X ) |   C ∞ R ( V ) ⊗ ∆ n / / |O X ( g − 1 ( V )) | (9.11.1) commute. Such an ar row exists by deﬁnition of g − 1 . It is unique bec ause ρ is an epimorphism of C ∞ -rings, so ρ ⊗ ∆ n is a s well. W e hav e now deﬁned G ♭ , and we take G ♯ : G ∗ C ∞ R → O X to b e the left adjoint o f G ♭ . W e m ust sho w that f o r ev ery g ∈ |O X ( X ) | 0 , the ma p G ♯ : G ∗ C ∞ R → O X provided by L is lo c a l. (W e ca n choos e g to be a vertex b eca use, by deﬁnition, a simplex in Map( G ∗ C ∞ R , O X ) is lo cal if a ll of its vertices ar e lo ca l.) T o prove this, we may take X to b e a p oint, F = O X ( X ) a lo cal C ∞ -ring, and x = G ( X ) ∈ R the image po int o f X . W e hav e a morphis m of C ∞ -rings G ♯ : ( C ∞ R ) x → F , in w hich b oth the domain a nd co domain are lo c a l. It is a lo c al ring homomorphis m beca use all prime ideals in the lo cal ring ( C ∞ R ) x are maximal. The maps K a nd L hav e now b een deﬁned, and they a re homotopy inv erses by construction.  The following is a technical lemma that allows us to take homotopy limits comp onent-wise in the mo del category of simplicial sets. 58 DA VID I. SPIV AK Lemma 9.12. L et sSets denote the c ate gory of simplicial sets. L et I , J, and K denote sets and let A = a i ∈ I A i , B = a j ∈ J B j , and C = a k ∈ K C k denote c opr o ducts in sSets indexe d by I , J, and K . Supp ose that f : I → J and g : K → J ar e functions and that F : A → B and G : C → B ar e maps in sSets that r esp e ct f and g in the sense that F ( A i ) ⊂ B f ( i ) and G ( C k ) ⊂ B g ( k ) for al l i ∈ I and k ∈ K . L et I × J K = { ( i, j, k ) ∈ I × J × K | f ( i ) = j = g ( k ) } denote the ﬁb er pr o duct of sets. F or typ o gr aphi c al r e asons, we use × h to denote homotopy limit in sSets , and × to denote the 1-c ate goric al limit. Then t he natur al map   a ( i,j,k ) ∈ I × J K A i × h B j C k   − → A × h B C is a we ak e quivalenc e in sSets . Pr o of. If w e r eplace F by a ﬁbration, then each co mpo nen t F i := F | A i : A i → B f ( i ) is a ﬁbration. W e re duce to showing that the map   a ( i,j,k ) ∈ I × J K A i × B j C k   − → A × B C (9.12.1) is a n isomor phism of s implicial sets . Res tricting to the n -simplicies of bo th sides, we may a ssume that A, B , and C are (discrete simplicial) sets. It is an easy exercis e to show that the map in (9.12.1) is injective and surjective, i.e. an isomor phism in Sets .  Prop ositio n 9.1 3. Supp ose that a : M 0 → M and b : M 1 → M ar e morphisms of manifolds, and supp ose that a ﬁb er pr o duct N exists in the c ate gory of m anifold s . If X = ( X , O X ) is the ﬁb er pr o duct X / /   ⋄ M 0 a   M 1 b / / M taken in t he c ate gory of derive d manifolds, then the natur al m ap g : N → X is an e quivalenc e if and only if a and b ar e t ra ns verse. Pr o of. Since limits taken in M an and in dMan co mm ute with tak ing underlying top ological s paces, the ma p N → X is a homeomorphis m. W e hav e a commutative diagram N g / / f   X f ′ z z u u u u u u u u u u M 0 × M 1 , DERIVED SMOOTH MANIFOLDS 59 in which f and f ′ are closed immersions (pullbacks of the diagona l M → M × M ). If a and b are not transverse, one shows easily that the ﬁrst homolo gy gr oup H 1 L X 6 = 0 of the cotangent complex for X is nonzer o, wher eas H 1 L N = 0 becaus e N is a manifold; hence X is not equiv alent to N . If a and b ar e transverse, then one can s how that g induces a quasi- is omorphism g ∗ L f ′ → L f . By Lemma 7.7, the map g is a n equiv alence o f der ived manifolds.  Prop ositio n 9.14. The simplici al c ate gory LRS of lo c al C ∞ -ringe d sp ac es is close d under t aking ﬁnite homotopy limits. Pr o of. The lo cal C ∞ -ringed space ( R 0 , C ∞ ( R 0 )) is a homotopy terminal ob ject in LRS . Hence it s uﬃces to show that a ho motopy limit ex ists for any diagra m ( X, O X ) F ← − ( Y , O Y ) G − → ( Z, O Z ) in LRS . W e ﬁrst describ e the appro priate candidate for this homoto p y limit. The underlying s pace of the candidate is X × Y Z , a nd we lab el the maps as in the dia gram X × Y Z h # # H H H H H H H H H g / / f   Z G   X F / / Y The structur e sheaf on the ca ndidate is the homotopy colimit of pullback sheav es O X × Y Z := ( g ∗ O Z ) ⊗ ( h ∗ O Y ) ( f ∗ O X ) . (9.14.1) T o show tha t O X × Y Z is a shea f of lo c a l C ∞ -rings, we take the stalk at a p oint, apply π 0 , and show that it is a lo ca l C ∞ -ring. The ho motopy colimit written in Equation (9.1 4.1) beco mes the C ∞ -tensor pro duct of po in ted lo ca l C ∞ -rings. By [22, 3.1 2 ], the result is indee d a lo cal C ∞ -ring. One shows that ( X × Y Z, O X × Y Z ) is the homotopy limit of the diagram in the usual way . W e do no t prove it her e, but refer the reader to [27, 2.3.21] o r, for a m uch more genera l result, to [21, 2.4.2 1].  Theorem 9.15. L et M b e a manifold, let X and Y b e derive d manifolds, and let f : X → M and g : Y → M b e morphisms of derive d manifolds. Then a ﬁb er pr o duct X × M Y exists in the c ate gory of derive d manifolds. Pr o of. W e show ed in Prop ositio n 9.1 4 that X × M Y exists as a lo cal C ∞ -ringd space. T o show that it is a derived ma nifold, we m ust only show that it is lo ca lly an aﬃne derived manifold. This is a lo cal prop erty , so it suﬃces to lo ok lo ca lly on M , X , and Y . W e will pr ov e the r esult b y ﬁr s t showing that aﬃne derived manifolds are closed under taking pr o ducts, then tha t they are clos ed under solving equations, and ﬁnally that these t wo facts combine to pr ov e the result. Given aﬃne derived manifolds R n a =0 and R m b =0 , it follows formally that R n + m ( f ,g )=0 is their pro duct, and it is an aﬃne der ived manifold. Now le t X = R n a =0 , wher e a : R n → R m , and supp ose that b : X → R k is a morphism. By Theorem 9.11, we can consider b as an element of O X ( X )( R k ). By Lemma 5.4, it is ho motopic to a comp osite X → R n b ′ − → R k , wher e X → R n is 60 DA VID I. SPIV AK the ca nonical imbedding. Now we can re alize X b =0 as the homo to p y limit in the all-Cartes ia n diagram X b =0 / /   ⋄ R 0   X b / /   ⋄ R k / /   ⋄ R 0   R n ( b ′ ,a ) / / R k × R m / / R m . Therefore, X b =0 = R n ( b ′ ,a )=0 is aﬃne. Finally , suppo se that X and Y are a ﬃne and that M = R p . Let − : R p × R p → R p denote the co ordinate- wise s ubtr action map. Then ther e is an all-Ca rtesian diagram X × R p Y / /   ⋄ R p diag   / / ⋄ R 0 0   X × Y / / R p × R p − / / R p , where di a g : R p → R p × R p is the diag onal map. W e hav e s e en tha t X × Y is aﬃne, so since X × R p Y is the s olution to an equation on a n a ﬃne derived manifold, it to o is aﬃne. This co mpletes the pro of.  R emark 9.16 . No te that Theorem 9.15 do es not s ay that the ca tegory dMan is closed under a rbitrary ﬁb er pro ducts. Indeed, if M is not a ssumed to b e a smo oth manifold, then the ﬁbe r pro duct o f derived manifolds o ver M need not be a derived manifold in our sense. The cotangent complex of any derived manifold ha s homology concentrated in degrees 0 a nd 1 (see Corolla ry 7.4), whereas a ﬁber pr o duct of derived ma nifolds (o ver a non-smo oth base) would not hav e that prop erty . Of co ur se, using the s pec tr um functor Sp ec, deﬁned in Remark 6.14, o ne could deﬁne a mor e genera l category C of “ derived manifolds” in the us ual scheme- theoretic way . Then our dMan would form a full sub categor y o f C , whic h o ne might call the sub categ ory of quasi-smo oth ob jects (see [27]). The r e ason we did not in tr o duce this catego ry C is that it does not have the gener al c up pro duct formula in cob or dism; i.e. Theorem 1.8 do es not a pply to C . 10. Rela tionship to simil ar work In this section we discuss o ther resea rch which is in so me wa y rela ted to the present pap er. Most relev ant is Section 10.1, in whic h we discus s the relationship to Lurie’s work on derived algebraic geometry , and in par ticular to structured spaces. The other sections discuss manifolds with singularities , Chen spaces and diﬀeolog i- cal spaces, synthetic diﬀerential geometry , a nd a catch-all section to c oncisely s tate how one might orient our work within the canon. 10.1. Lurie’s Structured spaces. There is a version o f the ab ove work on de- rived manifolds, pre s ent ed in the author’s PhD disser ta tion [27], which v er y closely follows Jacob Lurie’s theor y of structured spac e s, as pr esented in [21]. Similar in DERIVED SMOOTH MANIFOLDS 61 spirit is T o en a nd V ezzosi’s work [2 9], [3 0] on homo topical alg e braic geometry . In this section w e attempt to orient the reader to L ur ie’s theory o f structure d spaces. In order to deﬁne structured spaces, Lur ie b egins with the deﬁnition of a ge om- etry . A g e o metry is an ∞ -catego ry , equipp ed with a given choice of “admissible” morphisms that generate a Grothendieck top ology , and sa tisfying certain condi- tions. F or exa mple, there is a ge o metry who se underly ing categ ory is the category of aﬃne schemes Sp ec R , with a dmissible morphisms given by principal op en s ets Spec R [ a − 1 ] → Sp ec R , and with the usual Grothendieck top ology of op en cover- ings. Given a geometry G and a top ological spa c e X , a G -structure on X is roug hly a functor O X : G → Shv ( X ) which preser ves ﬁnite limits and sends cov ering s ieves on G to eﬀectiv e epimorphisms in Sh v ( X ). One should visualize the ob jects of G as space s and the admissible mor phisms in G as op en inclusions . In this vis ua lization, a G -structure on X provides, for each “space” g ∈ G , a shea f O X ( g ), whose s ections ar e s een as “ma ps” from X to g . Since a map from X to a limit o f g ’s is a limit of maps, o ne sees immediately why we req uir e O X to b e left exact. Given a covering sieve in G , we wan t to be able to say that to give a map fr om X to the union of the co ver is ac complished by g iving lo cal maps to the piec es of the cov er, such that these maps agree on ov erlaps . This is the cov er ing sieve condition in the deﬁnition of G -str ucture. Our a pproach follows Lurie’s in spirit, but not in practice. The issue is in his deﬁnition [21, 1.2.1 ] of a dmissibilit y str uc tur e, which we recall here. Deﬁnition 10.1. Let G b e an ∞ -category . An admissibility structu r e on G cons is ts of the following data: (1) A sub categor y G ad ⊂ G , containing every ob ject of G . Morphisms of G which belo ng to G ad will be called admissible morphisms in G . (2) A Grothendieck top olog y on G ad . These da ta are required to sa tisfy the following conditions: (i) Let f : U → X b e an admissible morphism in G , and g : X ′ → X a n y mor- phism. Then there exists a pullback diag ram U ′ / / f ′   U f   X ′ g / / X , where f ′ is admissible. (ii) Suppo se given a commutativ e triangle Y g   @ @ @ @ @ @ @ X h / / f > > ~ ~ ~ ~ ~ ~ ~ Z in G , where g and h are admissible. Then f is admissible. (iii) Every retract o f an admissible mo rphism of G is a dmissible. In our ca se, the role of G is play ed by E , the ca tegory of Euclidean spaces , and the r ole o f G ad is pla yed by op en inclusions R n ֒ → R n (see Sections 5 and 6 ). But, as 62 DA VID I. SPIV AK such, G ad is not a n admissibility structure on G b ecause it do es not satisfy condition (i): the pullback of a Euclidea n op en subset is no t necessa rily Euclidean. How ever such a pullback is lo c al ly Euc lidea n. This should b e enoug h to deﬁne something like a “pre-a dmissibility structure,” which do es the same job a s an ad- missibility str ucture. In pr iv ate corres po ndence, Lurie told me that such a notion would be useful – p er haps this issue will be reso lved in a la ter version o f [21]. In the author’s disser ta tion, how ever, we did not use C ∞ -rings as our basic algebraic ob jects. Instead, we used something calle d “smo oth rings.” A smo o th ring is a functor M an → sSets which preser ves pullbacks along submersions (see [27, Deﬁnition 2.1.3]). In this ca s e, the ro le of G is played by M an , and the r o le of G ad is play ed by submersions. This is a n admissibility structure in Lurie’s sense, and it should not b e ha r d to prov e that the categ ory of structured spaces o ne obta ins in this case is equiv alen t to our catego ry of loca l C ∞ -ringed spaces. 10.2. Manifolds wi th singul arities. A common misconception about derived manifolds is that every sing ular homolog y cla ss a ∈ H ∗ ( M , Z ) of a manifold M should b e representable b y an or ient ed der ived manifold. The misco nception seems to arise fr o m the idea that manifolds with singula rities, ob jects obtained by “coning oﬀ a s ubmanifold of M ,” s hould b e examples of de r ived manifolds. This is not the case. By the phr a se “coning oﬀ a submanifold A ⊂ M ,” one means taking the colimit of a diagram {∗} ← A → M . Derived manifolds are not clo s ed under taking arbitrar y colimits, e.g. quo tients. In particular , one ca nnot naturally obtain a derived manifold structure on a ma nifold with sing ularities. Instead, one obtains derived manifolds from taking the zer o -set o f a se c tio n o f a smo oth vector bundle: see Example 2.7. The c o llection of derived manifolds is quite larg e, but it do es not include manifolds with sing ularities in a natural wa y . In order to obtain ar bitrary colimits, per haps one should consider stacks on derived manifolds, but we have not work ed out this idea. 10.3. Chen spaces, diﬀ eological s paces. Another common generalization o f the category of manifolds was in vented b y Kuo Tsai Che n in [8]. Let C onv denote the category whos e o b jects ar e convex subs ets of R ∞ , and whos e morphisms are smo o th maps. With the Gr o thendieck top olo gy in whic h cov er ings a re given by open covers in the usual se ns e, w e can deﬁne the t o po s Sh v ( C onv ). The category of Chen spa c es is ro ughly this top os , the diﬀerence b eing that p oints are g iven mor e impo rtance in Chen s pa ces tha n in Shv ( C onv ), in a sense known a s c oncr eteness (see [3] for a precise account). Diﬀeolog ical spa ces ar e s imilar – they are deﬁned ro ughly as sheav es on the category of op en subsets o f Euclidea n spaces. The diﬀer ence b etw een these a pproaches and our own is tha t Chen spaces a r e based o n “maps in” to the o b ject in question (the Chen space) whereas our ob jects carry informatio n a bo ut “ maps out” of the ob ject in q uestion (the der ived mani- fold). In other words, the simplest questio n one can ask ab out a Chen spa ce X is “what are the maps from R n to X ?” Since X is a she a f on a site in which R n is an ob ject, the answer is simply X ( R n ). In the ca se of derived manifolds, the simplest question o ne can ask is “what ar e the maps from X to R n ?” By the s tructure theo- rem, Theorem 9.11, information ab out maps fr om X to Euclidean spaces is carried by the struc tur e sheaf O X – the answer to the question is O X ( X ) n . DERIVED SMOOTH MANIFOLDS 63 If o ne is in ter ested in generalizing manifolds to b etter study maps in to them, one should pr obably use Chen spaces or diﬀeologica l s paces. In o ur case, we were int er ested in cohomologic a l pro p er ties (intersection theory and cup pro duct); since elements o f co ho mology on X are determined by maps out of X , we constr ucted our generalized manifolds to be well-behaved with r esp ect to maps out. It may b e po ssible to generalize further and talk ab out “derived Chen spaces,” but we hav e not yet pur s ued this idea. 10.4. Syn thetic di ﬀ eren tial geo metry. In the bo o k [22], Mo er dijk a nd Reyes discuss yet another generaliza tion of manifolds, called smo oth functors. These are functors from the ca tegory of (discrete) C ∞ -rings to sets that satisfy a descen t condition (see [2 2, 3.1.1 ]). A smo o th functor can be considere d a s a patching o f lo cal neig hborho o ds, each of which is a formal C ∞ -v ariety . Neither o ur setup nor theirs is mor e general than the other. While bo th are based on C ∞ -rings, o ur approa ch uses ho motopical ideas , whereas theirs do es not; their approach gives a top os, whereas our s does not. It cer tainly may b e p ossible to combine these ideas into “derived smo oth functors,” but we have no t pursued this idea either . The non-ho mo topical approach do es not seem adequate for a g eneral cup pro duct formula in the sense of Deﬁnition 1 .7. 10.5. Other s imilar w o rk. There ha s be e n far to o muc h written a bo ut the in- tersection theory of manifolds for us to list here. In our work, we achiev e an int er section pairing at the level of spaces: the in tersection of tw o s ubmanifolds is still a geo metric ob ject (i.e. an o b ject in a ge o metric categor y in the sens e of Deﬁnition 1 .3), and this g e o metric o b ject ha s an appropriate fundamental c lass in cohomolog y (see Deﬁnition 1.7). No “gener a l po sition” requirements are necessary . W e hope that this is enough to distinguish o ur results from previous o nes. References [1] M. Andr ´ e. M´ e t ho de simpliciale en alg` ebr e homolo gique e t alg` ebr e c ommutative . Lectu r e Notes in Mathematics, V ol. 32. Springer-V erlag, Berlin, 1967. [2] M. Artin, A. Grothendiec k, and J.- L. V er dier. Sminair e de Gomtrie Algbrique du Bois Marie - Thorie des top os et c ohomolo gie t ale des schmas - (SGA 4) , volume 269. Spr inger-V erl ag, New Y ork, 1963-64. [3] J. Baez and A. Hoﬀn ung. Con venien t cate gori es of smooth spaces. ePrin t a v ailable: h ttp://front.math.ucda vis.edu/0807.1704 , 2008. [4] C. Barw i c k. On (enric hed) left bousﬁeld lo calizations of mo del categories. ePri nt av ailable: h ttp://arxiv.org/p df/0708.2067.pdf , 2007. [5] F. Borceux. Handb o ok of c ate goric al algeb r a. 2 , v olume 51 of Encyclop e dia of Mathematics and its Applic ations . Cambridge Universit y Press, Cambridge, 1994. Categories and struc- tures. [6] G. E. Bredon. T op olo gy and ge ometry , volume 139 of Gr aduate T exts in Mathematics . Springer-V erlag, New Y ork, 1997. Corr ected third printing of the 1993 original. [7] M. Bunge a nd E. J. Dubu c. Arc him edian local C ∞ -rings and models of synthetic diﬀeren tial geometry . Cahiers T op olo g i e G ´ eom. Diﬀ´ er entiel le Cat´ e g. , 27(3):3–22, 1986. [8] K. T. Chen. Iterated path inte gr al s. Bul l. Amer. Math. So c. , 83(5):831–87 9, 1977. [9] E. J. Dubuc. C ∞ -sche mes. Amer. J. Math. , 103(4):683–690 , 1981. [10] D . Dugger, S. Hollander, and D. C. Isaksen. Hyp ercov ers and simpl i cial presheav es. Math. Pr o c. Cambridge Philos. So c. , 136(1):9–51, 2004. [11] P . S. Hir sc hhorn. Mo del ca tegories and t heir lo c alizations , v olume 99 of Mathematic al Surveys and Mono g r aphs . A merican Mathematical So ciet y , Pr o vidence, RI, 2003. [12] M . Ho vey . Mo del c ate gories , volume 63 of Mathematic al Surve ys and Mono gr aphs . Ameri can Mathematical Society , Pro vidence, RI, 1999. 64 DA VID I. SPIV AK [13] L. Illusie. Complexe c otangent et d ´ eformations. I . Springer- V erlag, Berlin, 1971. Lecture Notes in Mathematics, V ol. 239. [14] J. F. Jardine. Simpli ci al presheav es. J. Pur e Appl. A lgeb ra , 47(1):35–87 , 1987. [15] M . Kont sevich. Enumeration of rational curves via torus actions. In The mo duli sp ac e of curves (Texel Island, 1994) , v olume 129 of Pr o gr. Math. , pages 335–3 68. Bi rkh¨ auser Boston, Boston, MA, 1995. [16] F. W. La wvere. Algebraic th eori es, algebraic cat egories, and algebraic funct ors . In The ory of Mo dels (Pr o c. 1963 Internat. Symp os. Berkeley) , pages 413–418. North-Holland, Amsterdam, 1965. [17] F. W. La wvere. Cat egorical dynamics. In T op os the or etic metho ds in ge ometry , volume 30 of V arious Publ. Ser. , pages 1–28. Aarh us Univ., Aarhus, 1979. [18] S. Lic hten baum and M . Schlessinger. The cotange nt complex of a morphism. T r ans. Amer. Math. So c. , 128:41–70, 1967. [19] J. Lurie. Deriv ed alge br aic ge ometry 1. ePrint av ail able: h ttp://arxiv.org/p df/math/0608228 v1.p df , 2006. [20] J. Lurie. Higher top os theory . ePrint av ailable: ht tp://front.math.ucda vis.edu/0608.5040 , 2006. [21] J. Lurie. Derived algebraic geo m etry 5. ePrint a v ailable, htt p: //www- math.mit.edu/ ∼ lurie/, 2008. [22] I. Mo erdijk and G. E. Reyes. Mo dels for smo oth inﬁnitesimal analysis . S pr inger-V erl ag, N ew Y ork, 1991. [23] D . Quill en. On the (co-) homology of commutat i ve rings. In A pplic ations of Cate goric al Algebr a (Pr o c. Symp os. Pur e Math., V ol. XVII, New Y ork, 196 8) , pages 65–87. A mer. Math. Soc., P r o vi dence, R.I., 1970. [24] C . Rezk. Every homotop y theory of simplicial algebras admits a prop er mo del. T op olo gy Appl. , 119(1):65–94 , 2002. [25] S. Sc h wede. Sp ectra in model c ategories and applications to t he algebraic cotange nt complex. J. Pur e Appl. A lgeb r a , 120(1):77–104 , 1997. [26] S. Sc h wede and B. Shipley . Equiv alences of monoidal model categories. Algebr. Ge om. T op ol. , 3:287–334 (electronic), 2003. [27] D . I. Spiv ak. Q uasi-smo oth Derived Manifolds . PhD thesis, Univ er s it y of Calif ornia, Berk eley , 2007. [28] R . E. Stong. Notes on c ob or dism the ory . Mathematical notes. Princeton Universit y Press, Princeton, N.J., 1968. [29] B . T o¨ en and G. V ezzosi. Homotopical algebraic geometry . I. Topos theory . A dv. Math. , 193(2):257 –372, 2005. [30] B . T o¨ en and G. V ezzosi. Homotopical algebraic geometry . I I. Geometric stacks and applica- tions. Mem. A mer. Math. So c. , 193(902 ):x+224, 2008. [31] C . A. W eib el. An int r o duction to homolo gica l algebr a , vo l ume 38 of Cambridge Studies in A dvanc e d Mathematics . Cambridge Universit y Pr ess, Cam bri dge, 1994.

Derived Smooth Manifolds

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