A homotopy theory for enrichment in simplicial modules

A HOMOTOPY THEOR Y F OR ENRICH M ENT IN SIMPLICIAL MODULES ALEXANDRU E. ST ANCULESCU Abstract. W e put a Quil len model structure on the catego ry of small cate - gories enriched in s implicial k - modules and non-negativ ely gr aded ch ain com- plexes of k -mo dules, where k is a comm utativ e ri ng. The mo del structure i s obtained b y transfer f rom the mo del structure on si mplicial categories due to J. Bergner. 1. Introduction: DK-equiv alences a n d DK-fibra tions 1.1. Let Cat the ca tegory of sma ll categorie s. It has a natur al mo del structure in whic h a co fibration is a functor monic on ob jects, a weak eq uiv alence is an equiv ale nc e of categ ories a nd a fibration is an isofibr a tion [5]. The fibration weak equiv ale nc e s a re the eq uiv alences sur jectiv e on ob jects. Let V b e a monoida l mo del categ ory [8] with unit I . W e denote by W the cla ss of weak equiv alences of V , b y Fib the class of fibrations and by Cof the cla ss of cofibrations. The small V -categorie s tog ether with the V -functors betw een them fo r m a ca t- egory wr itten V Cat . Let M be a clas s of maps of V . W e say that a V -functor f : A → B is lo c al ly in M if for each pair x, y ∈ A o f ob jects, the map f x,y : A → B is in M . W e hav e a functor [ ] V : V Cat → Cat obtained b y change o f base along the (symmetric monoidal) comp osite functor V γ / / H o ( V ) H om H o ( V ) ( I , ) / / S et. Definition 1.1. L et f : A → B b e a morphism in V Cat . 1. The morphi sm f is homotopy ess e n tially surjective if the induc e d functor [ f ] V : [ A ] V → [ B ] V is essential ly surje ctive. 2. The morphism f is a DK − equiv ale nce if it is homo topy essent ial ly surje ctive and lo c al ly in W . 3. The morphism f is a DK − fibration if it satisfies the fol lowing two c ondi- tions. (a) f is lo c al ly in Fib . (b) F or any x ∈ A , and any isomorphi sm v : [ f ] V ( x ) → y ′ in [ B ] V , ther e exists an isomorphism u : x → y in [ A ] V such t hat [ f ] V ( u ) = v . That is, if [ f ] V is an isofibr ation. One can e asily see that a morphism f is a DK-equiv alence and a DK-fibration iff f is s urjective o n ob jects a nd lo cally in W ∩ F ib . The class of maps having the left lifting pro per t y with resp ect to the V -functors surjective o n ob ject a nd lo ca lly in W ∩ Fib is gener ated by the map u : ∅ → I , where I is the V -categ ory with a single ob ject ∗ and I ( ∗ , ∗ ) = I , together with the ma ps ¯ 2 i : ¯ 2 A → ¯ 2 B , Date : Decem b er 6, 2007. 1 2 ALEXANDRU E. ST ANCULESCU where i is a genera ting cofibration of V . Here the V -categor y ¯ 2 A has o b jects 0 and 1, with ¯ 2 A (0 , 0) = ¯ 2 A (1 , 1) = I , ¯ 2 A (0 , 1) = A and ¯ 2 A (1 , 0) = ∅ . 1.2. Let k b e a commutativ e ring. W e denote by SMo d k the category of simplicial k -mo dules and by C h + ( k ) the ca teg ory of non-negatively graded c hain complexes of k -mo dules . The purpose of this note is to prove the following theorem. Theorem 1.2. L et V b e one of the c ate gories SMo d k or C h + ( k ) . Then V Cat admits a mo del stru ctur e in which the we ak e quivalenc es ar e the DK-e quivalenc es and t he fibr ations ar e t he DK - fibr ations. T o prov e this r e s ult we use the (similar) mo del structure on simplicial categorie s [2] and Quillen’s path ob ject arg umen t ([7], Lemma 2.3(2) and [1 ], 2.6). An explicit description of a cofibra tion of V Cat can b e given [9]. 1.3. The pro of of theo rem 1.2 relies dec is ively on the cons tr uction of path o b jects for dg-catego ries due to G. T abuada ([11 ], 4.1). In fact, our attempt to understand his constructio n led us to the pro o f o f our r esult. 1.4. In [1 2], B . T o¨ en character ised the maps in the homotopy ca tegory of dg- categorie s, where the ca tegory of dg-categor ies has a mo del str ucture in which the weak equiv alenc e s are the DK- equiv alenc e s and the fibr ations a re the DK -fibrations. One ca n show tha t his r esults ( lo c. cit. , Thm. 4.2 a nd 6.1 ) hold for V Cat , wher e V is SMo d k or C h + ( k ). Note add e d in pr o of. After the completion o f this w ork we learned about the existence of a pap er b y G. T abuada [10], which tr eats the s ame sub ject matter, and more, but differently . O ne can see that the mo del structure prop os e d in theorem 1.2 coincides with the o ne in [10 ], a lthough the cla sses of fibra tions and cofibr a tions are not explicitly identified in lo c. cit. On the o ther ha nd, T abua da shows that the mo del structur es on SMo d k Cat a nd C h + ( k ) Cat are Q uillen equiv alent, an issue that we have initially neglected. One can easily g ive a pro of of this fac t, adapted to our context, using se ction 2.2 below and the general results of [9]. 2. Ca tegories enriched in SMo d k and C h + ( k ) 2.1. The category SM o d k is a clos ed symmetric monoidal ca tegory with tensor pro duct defined p oint wise and unit ck , where ( ck ) n = k for all n ≥ 0. A model structure on SMo d k is obta ined b y transfer from the categor y S of simplicia l sets, regar ded as having the cla ssical model structure, via the free-for g etful a djunction k : S ⇄ SM o d k : U. All ob jects a r e fibrant and the mo del s tr ucture is simplicial. The functor k is str ong symmetric monoidal (and it pr eserves the unit), hence SMo d k is a monoida l mo del category . The adjunction ( k, U ) induces an adjunction k ′ : SCat ⇄ SMo d k Cat : U ′ . W e claim that a map f o f SMo d k Cat is a DK-equiv alence (resp. DK-fibration) iff U ′ ( f ) is a weak equiv alence (resp. fibr a tion) in the B ergner mo del structur e on SCat [2]. C le arly , f is lo cally in W (resp. Fib ) iff U ′ ( f ) is lo cally in W (res p. Fib ). In the induced adjoint pair Lk : H o ( S ) ⇄ H o ( SMo d k ) : RU, the functor Lk is strong symmetric monoidal and preserves the unit o b ject, hence one has a natural isomorphis m o f functor s η : [ ] SMo d k ∼ = [ ] S U ′ : SMo d k Cat → C at such that for all A ∈ SMo d k Cat , η A is the identit y o n ob jects. The r est of the claim follows from this obs e r v atio n. A HOMOTOPY THEOR Y FOR ENRICHMENT IN SIMPLI CIAL MODULES 3 2.2. Co nsider the nor malized ch ain co mplex functor N : SMo d k → C h + ( k ). It was shown in ([8], 4 .3) that N is part of a weak monoidal Quillen equiv alence N : SM o d k ⇄ C h + ( k ) : Γ in which b oth functors preser ve the unit ob jects. Therefore the comp osite adjunc- tion N k : S ⇄ C h + ( k ) : U Γ is a w eak monoidal Quillen pair with N k preserving the unit ob ject. The functor U Γ induces a functor ( U Γ) ′ : C h + ( k ) Cat → SCat which has a left adjoint F defined ” fibrewise”. W e claim that a map f in C h + ( k ) Cat is a DK-equiv alence (resp. DK-fibration) iff ( U Γ) ′ ( f ) is a weak equiv alence (resp. fibration) in the Bergner model structure on SCat . F or this, it is e no ugh to remark that in the induced adjunction L ( N k ) : H o ( S ) ⇄ H o ( C h + ( k )) : R ( U Γ) , the functor L ( N k ) is strong monoidal and preserves the unit ob ject, and then conclude as in 2.1. 2.3. In o rder to pr ov e theor em 1.2, it suffices to apply Quillen’s path ob ject argument to the adjunctions ( k ′ , U ′ ) a nd ( F, ( U Γ) ′ ). This will b e ac hieved in the next section. 3. Coca tegor y object structure on the inter v al 3.1. Let V b e a mo noidal mo del ca tegory with cofibr a nt unit I and all o b jects fibrant. W e wr ite Y X for the internal hom o f t wo ob jects X , Y of V . W e say that V has a c o c ate gory interval if there is a co ca tegory ob ject s tructure I d 1 / / d 0 / / I [1] p o o i 0 / / c / / i 1 / / I [2] such that I ⊔ I ▽ / / d 0 ⊔ d 1 " " F F F F F F F F I I [1] p > > } } } } } } } } is a cylinder ob ject for I . The map c denotes the co comp osition. Examples . ( a ) The sta nda rd exa mple is when V = Cat as in 1.1. Here I [1] is the ”free-living” iso morphism and I [2] is the g roup oid with three ob jects and one isomorphism b etw een any tw o ob jects. W e leav e to the reader the task to identify all the maps inv olved. ( b ) The case whic h in terests us is when V = C h + ( k ). The interv al I [1] is well known to be ... → 0 → k e ∂ → k a ⊕ k b , where ∂ ( e ) = b − a . The maps d 0 and d 1 are the inclusions, and the map p is a, b 7→ 1 . The ob ject I [2] is ... → 0 → k e 1 ⊕ k e 2 ∂ → k a 0 ⊕ k a 1 ⊕ k a 2 , where ∂ ( e 1 ) = a 1 − a 0 and ∂ ( e 2 ) = a 2 − a 1 . The co co mpos ition c is given by e 7→ e 1 + e 2 , a 7→ a 0 and b 7→ a 2 . The ma p i 0 is given by e 7→ e 1 , a 7→ a 0 and b 7→ a 1 ; the map i 1 is given by e 7→ e 2 , a 7→ a 1 and b 7→ a 2 . ( c ) Since the functor Γ from 2.2 pr eserves the unit ob ject and is an equiv alence of categories, we obtain that SMo d k has a co categor y interv al. ( d ) Let k b e a field and let H be a finite dimensiona l co co mm utative Hopf algebr a ov er k . W e le t V = H M od , the categor y o f left H -mo dules , a nd we view V as having 4 ALEXANDRU E. ST ANCULESCU the stable mo del str ucture [4]. Let u (resp. ǫ ) b e the unit (r e sp. counit) of H . A cylinder ob ject for k is k ⊕ k ▽ / / id ⊕ u $ $ I I I I I I I I I k k ⊕ H p :=( id,ǫ ) < < y y y y y y y y y The maps d 0 and d 1 are given by d 0 (1) = (1 , 0) and d 1 (1) = (0 , 1 H ). W e set I [2] = k ⊕ k ⊕ H and c ( α, h ) = ( α, 0 , h ). The map i 0 is ( α, h ) 7→ ( α, ǫ ( h ) , 0) a nd the ma p i 1 is ( α, h ) 7→ (0 , α, h ). One can chec k that the res ulting gadget is a co categor y interv al with I [1] = k ⊕ H . It is easy to see that I [1] is an ”interv al with a coa sso ciative and co commutativ e comultiplication” in the sens e o f ([1], pag e 813). 3.2. W e shall now construct (DK-)path o b jects for V -catego ries, where V is as in 3.1. In the case of dg - categorie s , the cons truction is due to G. T abuada ([1 1], 4.1). Let A ∈ V Cat . W e first construct a factorisa tion A △ / / i 0 ! ! C C C C C C C C A × A P 0 A ( s,t ) : : v v v v v v v v v such that i 0 is lo cally in W and ( s, t ) is lo cally in Fib . An ob ject of P 0 A is a map f : a → b of [ A ] V . If f 0 : a 0 → b 0 and f 1 : a 1 → b 1 are t wo ob jects of P 0 A , we define P 0 A ( f 0 , f 1 ) to be the limit of the diagr am A ( a 0 , a 1 ) f 1 ∗ & & L L L L L L L L L L A ( a 0 , b 1 ) I [1] t x x p p p p p p p p p p p s & & N N N N N N N N N N N A ( b 0 , b 1 ) f ∗ 0 y y r r r r r r r r r r A ( a 0 , b 1 ) A ( a 0 , b 1 ) The unit of P 0 A ( f , f ) is induced by the adjoint tr ansp ose of I [1] p → I f → A ( a, b ). Let f i : a i → b i ( i = 0 , 2) b e three o b jects of P 0 A and let A i = P 0 A ( f i , f i +1 ) ( i = 0 , 1). W e denote by p i (resp. q i ) the canonical map A i → A ( a i , a i +1 ) (r e s p. A i → A ( b i , b i +1 )) ( i = 0 , 1). The pa ir ( p i , q i ) gives rise to a commutative diagra m A i p i / / j 1 ,i   A ( a i , a i +1 ) f i +1 ∗   I [1] ⊗ A i H i / / A ( a i , b i +1 ) A i j 0 ,i O O q i / / A ( b i , b i +1 ) f ∗ i O O where j k,i = d k ⊗ A i ( k = 0 , 1). Observe that in or der to define a map A 0 ⊗ A 1 → P 0 A ( f 0 , f 2 ) it suffices to find a map G : A 0 ⊗ A 1 → A ( a 0 , b 2 ) I [1] which mak es A HOMOTOPY THEOR Y FOR ENRICHMENT IN SIMPLI CIAL MODULES 5 commutativ e the diagram A ( a 0 , b 2 ) A 0 ⊗ A 1 G / / f 2 ∗ ( p 0 ⊗ p 1 ) & & M M M M M M M M M M M f ∗ 0 ( q 0 ⊗ q 1 ) 8 8 q q q q q q q q q q q A ( a 0 , b 2 ) I [1] t   s O O A ( a 0 , b 2 ) W e define the map G 1 as the comp osite I [1] ⊗ A 0 ⊗ A 1 H 0 ⊗ A 1 − → A ( a 0 , b 1 ) ⊗ A 1 id ⊗ q 1 − → A ( a 0 , b 1 ) ⊗ A ( b 1 , b 2 ) → A ( a 0 , b 2 ) . Then G 1 is a ”homotopy” b etw een f ∗ 0 ( q 0 ⊗ q 1 ) a nd p 0 ⊗ q 1 . W e define the map G 2 as the comp osite I [1] ⊗ A 0 ⊗ A 1 A 0 ⊗ H 1 − → A 0 ⊗ A ( a 1 , b 2 ) p 0 ⊗ id − → A ( a 0 , a 1 ) ⊗ A ( a 1 , b 2 ) → A ( a 0 , b 2 ) . Then G 2 is a ”homo topy” be tw een p 0 ⊗ q 1 and f 2 ∗ ( p 0 ⊗ p 1 ). The tw o homoto pies induce a map A 0 ⊗ A 1 → A ( a 0 , b 2 ) I [1] × A ( a 0 ,b 2 ) A ( a 0 , b 2 ) I [1] such that the diagr am A ( a 0 , b 2 ) A 0 ⊗ A 1 / / ¯ G 2 * * T T T T T T T T T T T T T T T T T ¯ G 1 4 4 j j j j j j j j j j j j j j j j j A ( a 0 , b 2 ) I [1] × A ( a 0 ,b 2 ) A ( a 0 , b 2 ) I [1]   O O A ( a 0 , b 2 ) commutes, wher e ¯ G i is the adjoint tra nspo se of G i ( i = 0 , 1). Since A ( a 0 , b 2 ) I [1] is a category ob ject in V , we hav e then a map A 0 ⊗ A 1 → A ( a 0 , b 2 ) I [1] × A ( a 0 ,b 2 ) A ( a 0 , b 2 ) I [1] m − → A ( a 0 , b 2 ) I [1] which is the req uir ed map G . In this wa y P 0 A b ecomes a V -catego ry . The as s o ciation i 0 : O b ( A ) → Ob ( P 0 A ), a 7→ ( id a : a → a ), ( i 0 ) a,b = A ( a, b ) p , is a V -functor A → P 0 A . By co nstruction, the ma ps s, t : P 0 A → A , s ( f 0 : a 0 → b 0 ) = a 0 , t ( f 0 : a 0 → b 0 ) = b 0 , s f 0 ,f 1 = p 0 and t f 0 ,f 1 = q 0 , a re V -functors . O ne clearly has ( s, t ) i 0 = △ . Mor eov er, ( s, t ) is lo cally in Fib since ( s, t ) f 0 ,f 1 : A 0 → A ( a 0 , a 1 ) × A ( b 0 , b 1 ) is the pullback A ( a 0 , a 1 ) × A ( b 0 , b 1 ) f 1 ∗ × f ∗ 0   A ( a 0 , b 1 ) I [1] ( s,t ) / / A ( a 0 , b 1 ) × A ( a 0 , b 1 ) . Next, let P A b e the full sub- V - category o f P 0 A whose ob jects consist of isomor- phisms f : a → b of [ A ] V . Then i 0 factors thro ugh P A . The resulting factorisation A △ / / i ! ! C C C C C C C C A × A P A ( s,t ) : : v v v v v v v v v 6 ALEXANDRU E. ST ANCULESCU is the desired (DK- )path o b ject: a leng th y but straig ht forward computatio n shows that i is homotopy esse ntially surjective and that [( s, t )] V is an isofibra tio n. Ac kno wledgments. I w ould lik e to thank Pr ofessor Andr´ e Joy al for man y useful discussions ab out the sub ject and for prono uncing the w ord ”co catego ry”. References [1] C. Berger, I. Mo erdijk, Axiomatic homotopy t he ory for op e ra ds , Comment . Math. Hel v. 78 (2003), no. 4, 805–831. 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Sch wede, B. E. Shipley , Equivalenc es of monoidal mo del c ate gories , Algebr. Geom. T op ol. 3 (2003), 287–334 (electronic). [9] A. E. Stanculescu, Bifib r ations and we ak factorisation systems , Pr eprin t, May 2007. [10] G. T abuada, D iffer ential gr ade d versus Simplicial c ategories , Preprint [math.KT], Nov em b er 2007. [11] G. T abuada, A new Q uil len mo del for the Morita homotopy the ory of DG c ate gories , Prepri n t \protect \vrule width0pt\prot ect\href{http://arxiv.org/abs/math/0701205}{math.KT/0701205} , F ebruary 2007. [12] B . T o ¨ en, The homotopy the ory of dg-c ate gories and deriv e d Morita the ory , Preprint \protect \vrule width0pt\prot ect\href{http://arxiv.org/abs/math/0408337}{math.AG/0408337} , Septem ber 2006. Dep a r tmen t of Ma thema tics and S t at istics, McGill University, 805 Sherbrooke Str. West, Montr ´ eal, Qu ´ ebec, Canada, H3A 2K6 E-mail addr ess : stanculescu@ma th.mcgill.ca