Abelian Groups in omega-categories

Ab elian groups in ω -categories Brett Milburn ∗ Abstract W e study ab elian group ob jects in ω -categories and d iscuss the wel l-known Dold-Kan corresp ondence from the p ersp ectiv e of ω - categories as a mo del for strict ∞ -categories. The first part of the p aper is in tended to compile results from t he existing literature and to fill some gaps th erein. W e go on to consider a p arameteri zed Dold-Kan corresp ondence, i.e. a Dold-Kan corresponden ce for preshea ves of ω -categories. The main result is to describe the descen t or sheaf condition in terms of a gl ueing condition that is familiar for 1 and 2-stacks. Con ten ts 1 In troductio n 2 2 ω -categories 3 2.1 Equiv a lences o f ω -categ ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Picard ω -categories 5 3.1 Picard ω -Categ ories and Chain Complexe s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Equiv a lences o f P icard ω -ca tegories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Useful F acts for Picard ω -catego ries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 I-categor ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 ω -categor ies in a categor y C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 The Do ld-Kan Corresp ondence 13 4.1 ω -categor ies and quasicateg ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.1 Ba sics of Parity Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.2 The Nerve of an ω -Catego ry and the Street-Ro berts Conjecture . . . . . . . . . . . . . 15 4.2 The Dold-Kan T riangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 The Do ld-Kan Corresp ondence for Shea v es 16 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Mo del Structures and Derived Dold- K an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6 Omega Descen t 18 6.1 Lo op a nd P ath F unctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.2 Equiv a lence o f Descent Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 ∞ -torsors 25 7.1 Compariso n with o ther Appro ac hes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∗ milburn@math.utexas.edu 1 8 App endix 29 8.1 Deligne’s Theor em . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.2 Nerve and Path F unctors for ω -categor ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8.3 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.3.1 Homo topies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.3.2 Homo topies fro m the Persp ective of ω -Categor ies . . . . . . . . . . . . . . . . . . . . . 34 1 In tro duction Our g oal is to inv estigate ab elian gro up ob jects in ∞ - c a tegories. There are many notions o f ∞ - c ategory . Lurie [31] a nd Leinster [30] provide go od– tho ugh not exhaustiv e–surveys of v arious definitions. O ther models such as complete Segal spaces [32] and cross ed complexe s [8] also a ppear in the litera ture. The a pproach taken in this pa p er is to consider ω -categ ories as strict ∞ - categories. Roughly , a n ω -category coincides with the intuitiv e des cription of an ∞ -ca tegory . It ha s ob jects a nd n-morphisms for n ≥ 1. One can comp ose n-morphisms and tak e the k-th source or target of an n-morphism to get a k- morphism. In co ntrast, quasica tegories (simplicia l sets which admit fillers for inner horns) are also a mo del for ∞ -categ o ries. While simplicia l sets ar e useful fro m the p erspec tiv e of homotopy theory , they are not endow e d with all of the desired structure that one would like for a n ∞ -categ o ry . Namely , there is no natura l choice for identit y morphisms or comp osition. F rom this pe r spective, it is useful to consider ω -categ ories, which have the adv an tage of not having the same deficits. F urthermore, ω -categor ies enjoy ma n y nice prop erties. F or example, n-catego ries ar e ea sily defined. Howev er, ω -catego ries ar e stric t ∞ -categor ies in the sens e that comp osition is ass ociative on the nose, and they do not contain the coher- ence da ta that o ne might desire for weak ∞ -categ ories (although weak ω -ca tegories have also b een studied [46, 30]). P resent ly our int erest is in ab elian group ob jects in ∞ -ca tegories. W e will see that a belian g roup ob jects in simplicia l sets a re in fa c t strict ∞ -categorie s , and we may therefore in terpret them as ω -ca tegories. Sections 1-4 are prima rily exp ository . W e be gin by de fining ω -ca tegories in § 2 and int ro ducing the no- tion of equiv ale nce of ω -categor ies. In § 3 , we formulate and prove the well-kno wn statement that ab elian groups ob jects in ∞ -ca tegories are the s a me as chain complexes o f ab elian groups in non-negative degrees. F urthermore, we show that this equiv ale nc e induces a derived equiv alence. Two gener alizations are pur sued. Firstly , w e introduce the notio n of an I -categor y for a n y partially or dered set I . Of particular interest ar e the Z -categ ories. Abelia n group ob jects in Z -categories are eq uiv alent to c hain complexes o f abelia n groups. The analo gue of the co rresp ondence b et w een ω -catego ries and simplicial sets is now betw een Z -categ o ries and combinatorial spe c tr a (cf. [24]), though this is not made precise her e. Similarly , the relationship b et w een chain co mplexes and Z -ca tegories is als o analog ous to the Quillen eq uiv alence b etw een chain co mplexes o f ab elian groups and H Z -mo dule sp ectra describ ed by Sch w ede and Shipley [38, 39]. Sec o ndly , we observe that since an ω -ca tegory is determined by a set e quipped with so me structure ma ps, we may think of a n ω -categor y as an ω -category in S ets . This c an b e extended to define an ω - category in an arbitrary ca tegory with fib ered pro ducts. W e generalize the equiv alence b et w een Chain co mplexes C h + ( Ab ) of a belian g roups in non-neg a tiv e deg r ee a nd P ic ω , a belian group o b jects in ω - categories. F or any ab elian categor y C with countable direct sums, we show that C h + ( C ) is equiv alen t to C ω , ω -catego ries in C . Simplicial a b elian groups, denoted sAb , ca n also b e tho ugh t of a s ab elian group o b jects in ∞ -catego ries. The Dold-K an corresp ondence [13] states that there is a n equiv alence C h + ( Ab ) ≃ sAb . In section § 4 w e cite a r ecen t result of Brown, Higgins, a nd Sivera [10] that relates the Dold-Kan corr espondence to the eq uiv a- lence C h + ( Ab ) ≃ P ic ω . T o put it succinctly , there is a nerve functor, due to Street [43], N : ω C at − → sS e t from ω -c ategories to s implicia l sets, which when restricted to P ic ω gives N : P ic ω − → sA b . The Dold- Kan equiv ale nce C h + ( Ab ) − → sAb is , up to isomorphism, the comp osition o f C h + ( Ab ) − → P ic ω with the nerve functor. Each of C h + ( Ab ), P ic ω , and sAb a re na turally equiv alent not just as categ ories but also as mo del 2 categorie s. The c ore of the pap er is in § 6 , where we cons ider descent for pres heaves of ω -categories . Desce n t for ω -categor ies has been co nsidered by Street in [44, 4 6], and V erity sho wed [48] that Street’s definition o f descent is equiv ale nt to the standa r d notion of desce nt for presheaves of s implicia l sets, wher e the tw o ar e related b y the nerve functor . The Dold-Kan corres pondence ex tends to pres hea v es with v alues in C h + ( Ab ), P ic ω , or sAb . W e co ns ider several mo del structures o n the preshea f ca tegories, in pa r ticular one wher e the fibrant o b jects a re pr ecisely the sheaves (i.e. those satisfying descent with resp ect to all hypercovers) and one wher e the fibra n t ob jects are those s a tisfying ˇ Cech desc en t (i.e. descent with r e s pect to op en cov ers). W e show that the homo top y categor y of simplicial sheaves of ab elian groups o n a s pace X satisfying de- scent is e q uiv alent to the derived catego ry in no n-negative degrees D ≥ 0 ( A b ) of sheav es of abelia n gro ups on X . The key result is Theorem 6.4 which states that a presheaf of simplicial ab elian gr oups on site S satisfies ˇ Cech descent if and o nly if it satisfies a more co ncrete glueing co ndition, which ca n b e expla ined ro ug hly as being able to g lue ob jects and n-mor phism fro m lo cal sections. In more detail, a pr esheaf A o f simplicia l ab elian g roups, A satisfies ˇ Cech desce n t if and only if fo r every X ∈ S a nd op en cov er U = { U i } i ∈ I of X , 1. given lo cal ob jects x i ∈ A ( U i ) 0 which are glued toge ther by 1-mor phisms and hig her deg r ee morphisms in a coher en t w a y , there exists a global o b ject x ∈ A ( X ), unique up to is omorphism, which glues the x i , and 2. for any n-morphisms x, y ∈ A ( X ) n , the pre s heaf H om A ( x, y ) who se ob jects are the ( n + 1)-morphisms from x to y satis fie s the a bov e glueing co ndition for ob jects. This ca n b e interpreted a s providing a computational to ol for determining whether a presheaf satisfies ˇ Cech descent or a wa y of constructing a sheafification of a giv en preshea f. W e hop e that this has applica tions in the study of n-ge r bes. Let G b e an abelia n g roup and X a top ological space. With the appropria te notion of torsor, one may view an n-gerb e for G on X as a torsor for a presheaf o f s implicia l abelian gro ups , namely it is generated by G in degree n and 0 elsewhere. Given the computational descent condition, it should bec ome appare nt that isomorphis m classes of n - gerbe s for G are g iv en b y ˇ H n ( X, G ). In this wa y , this pape r is a step tow a rds viewing s hea v es of ∞ -categories a s geometric realizations o f cohomolo gy classes. This po in t of view is further e x plained in the final section, where we draw on the insights of Fiorenza , Sati, Schreiber, and Stasheff [15, 37, 41] to descr ibe torso r s for sheaves v alued in ∞ -gr oups. 2 ω -categories W e beg in b y de fining ω -catego ries, which are a mo del for strict ∞ -categ o ries. Definition 1 . The data for an ω - c ate gory is a se t A with maps s i , t i : A − → A fo r i ∈ N and ma ps ∗ i : A × A A − → A , where A × A A − → A is the fib ered pr oduct, given maps s i : A − → A and t i : A − → A . Let ρ i , σ i ∈ { s i , t i } denote any sourc e or targ et map. ( A, s i , t i , ∗ i ) i ∈ N is said to b e an ω -category if the following 3 conditio ns ar e sa tisfied: 1. F or all i ∈ N , ( A, s i , t i , ∗ i ) is a catego ry . In other words, (a) ρ i σ i = σ i (b) a ∗ i s i ( a ) = t i ( a ) ∗ i a = a (c) ( a ∗ i b ) ∗ i c = a ∗ i ( b ∗ i c ) (d) s i ( a ∗ i b ) = s i b , and t i ( a ∗ i b ) = t i a 2. F or all i < j ,( A i , A j ) is a strict 2 -category . That is, 3 (a) ρ j σ i = σ i (b) σ i ρ j = σ i (c) ρ j ( a ∗ i b ) = ρ j a ∗ i ρ j b (d) ( a ∗ j b ) ∗ i ( α ∗ j β ) = ( a ∗ i α ) ∗ j ( b ∗ i β ) whene ver both s ides are defined. 3. F or all a ∈ A , there is some i ∈ N such that s i a = t i a = a . Definition 2. 1. F or an ω -categor y A and i ∈ N , i-obje ct s in A are A i := s i A , and strict i-obje ct s are A i \ A i − 1 . In c o nforming to c on v en tion, w e a lso refer to i-ob jects as i-m orphisms . 2. Let ω Cat denote the category w ho se ob jects are ω - c a tegories and morphisms are functor s b et w een ω -categor ies, meaning maps of sets which preser v e all structures s i , t i , ∗ i for all i . 3. W e write Ob : ωC a t − → S ets for the forg etful functor which sends an ω -categor y to its underlying set. 4. F or A ∈ ω C at and a, b ∈ A i , let H om i A ( a, b ) := { x ∈ A | s i x = a, and t i x = b } . Remark 2.1. ω Cat is a symmetric monoida l categor y . F o r A , B ∈ ∞ - cat, The pro duct A × B is just the cartesian pro duct as sets, and sour ce, target, and comp osition maps are defined comp onent wise. Definition 3. W e say that A ∈ ω C at is a gr oup oid if every n-morphism n ≥ 1 is an isomorphism, meaning that if x ∈ A n , there exists for every j < n a y ∈ A s uc h that x ∗ j y = t j x and y ∗ j x = s j x . Remark 2.2. It is a w ell known result of Br own and Higgins [9] that there is an equiv alence b et w een ω -group oids and c rossed co mplexes. 2.1 Equiv alences of ω -categories Definition 4. 1. F or any ω - category A , tw o i-ob jects a, a ′ ∈ A i are said to b e isomorphic if there exists u ∈ H om i +1 ( a, a ′ ) and v ∈ H om i +1 ( a ′ , a ) such that u ∗ i v = a ′ and v ∗ i u = a . (In the la nguage of Street [43], a and a ′ are 1-equiv alen t.) 2. Let F : A − → B b e a functor of ω - categories. W e say that F is a n equiv alence of ω -categor ies if (a) Any 0-ob ject, b ∈ B is isomor phic to F a for some 0-ob ject a in A , (b) for any i ≥ 0 and a, a ′ ∈ A i such that s i − 1 a = s i − 1 a ′ , t i − 1 a = t i − 1 a ′ and ψ ∈ H om i B ( F a, F a ′ ), there exists φ ∈ H om i A ( a, a ′ ) and an isomor phism β ∈ H om i +1 B ( F φ, ψ ), and (c) fo r i-o b jects a, a ′ ∈ A , if F a is iso morphic to F a ′ in B , then a is isomorphic to a ′ in A Conditions 2a, 2 b, and 2c are the hig her-categor ic a l analogues of being ess en tially surjective, full, and faithful r espectively . The meaning of this definition is roughly that F A i should b e the same as B i , up to (i+1)-isomor phism. In the notation of definition 1, the fir st tw o conditions can be restated as: (a) F or all y ∈ B 0 , there exists x ∈ A 0 and isomor phism f ∈ B 1 such that s 0 f = F x and t 0 f = y . (b) If a, a ′ ∈ A i such that s i − 1 a = s i − 1 a ′ , t i − 1 a = t i − 1 a ′ , a nd ψ ∈ B i +1 such that s i ψ = F a and t i ψ = F a ′ , then there exis ts φ ∈ A i +1 and an isomor phism β ∈ B i +2 such that s i φ = a , t i φ = a ′ , s i +1 β = F φ , and t i +1 β = ψ . Remark 2.3. Note that when A a nd B are gro upoids, then F : A − → B automatically satisfies condition 2c of Definition 4 if it satisfies 2a and 2 b. Th us, the thir d co ndition is s uperfluous when w e are dealing with group oids. F urthermor e , when A and B ar e group oids, condition 2 a c a n b e viewed as a sp ecial case of condition 2b if we a dd a po in t {∗ } = A − 1 = B − 1 in degree − 1. Equiv a lences of ω -catego ries are in fact w eak equiv alences in a co fibran tly gener ated mo del structure on ω C at [29]. Ara and M´ etay er show ed in [1] that this mo del str uctures r estricts to one on ω -g roupo ids in a wa y that is compa tible with Brown a nd Go lansi´ nski’s mo del structure on crossed complexe s [7]. 4 3 Picard ω -categories Definition 5. A Pic ar d ω -c ate gory is an a b elian gr oup ob ject in ω Ca t. W e let P ic ω denote the ca tegory of Picar d ω -categor ies, where H om P ic ω ( A, B ) = { F ∈ H om ω cat ( A, B ) | F ◦ + = + ◦ ( F × F ) } . Remark 3. 1 . F or a Picar d ω -category A , the fact that + : A × A − → A is a functor implies that each A i is a subgroup. Also, we observe that if A is a Pica r d ω -catego ry , then O b ( A ) is an ab elian g roup. Prop osition 3. 2. 1. An ω -c ate gory A su ch that O b ( A ) is endowe d with the stru ctur e of an ab elian gr oup is a Pic ar d ω -c ate gory if and only if (a) + : A × A − → A is a fun ctor of ω -c ate gories and (b) x ∗ i y = x + y − s i ( x ) whenever t he left hand side is define d. 2. P i c ω is an ab elian c ate gory. Pr o of. 1. Fir s t supp ose tha t A ∈ P ic ω . Then + is a functor. F urther more, it is clear tha t O b ( A ) must be an ab elian gr oup ob ject in S et . W e only need to verify that x ∗ i y = x + y − s i ( x ) whenever the left-hand side is defined. Since + is a functor, ( x + y ) ∗ n ( x ′ + y ′ ) = ( x ∗ n x ′ ) + ( y ∗ n y ′ ) if the right side is defined. Suppose that s i x = t i y . Then x ∗ i y = ( x + 0) ∗ i ( s i x + ( y − t i y )) = x ∗ i s i x + 0 ∗ i ( y − t i y ) = x + ( t i y − t i y ) ∗ i ( y − t i y ) = x + t i y ∗ i y + ( − t i y ∗ i − t i y ) = x + y + − t i y = x + y − s i x . Now supp ose that A ∈ ω C at such that O b ( A ) is an ab elian group and conditions 1a and 1b are satisfied. W e wish to show that A ∈ P i c ω . Since Ob is faithful and Ob A is an a belian group o b ject in S et , A is an ab elian gro up ob ject in ω C at provided that a dditio n + and inv erse ι : A − → A a re functor s of ω -categor ies. By co ndition 1 a, + is a functor, so it o nly remains to see that ι is a functor . Since 0 = ρ n 0 = ρ n ( x + x − 1 ) = ρ n x + ρ n x − 1 , ρ n x − 1 = ( ρ n x ) − 1 for ρ n ∈ { s n , t n } so that ι resp ects source and target maps. Also, since ( x ∗ n y ) − 1 = ( x + y − s n x ) − 1 = x − 1 + y − 1 − s n x − 1 = x − 1 ∗ n y − 1 , ι resp ects compo s itions. W e conclude that A is a gro up ob ject in ω Cat. 2. F or A , B ∈ P i c ω , H om ( A, B ) is an ab elian group. The sum φ + ψ o f t w o functors preser v es all source and ta r get maps and also pres erv es comp osition b ecause + is a functor: ( φ + ψ ) x ∗ n y = φx ∗ n φy + ψ x ∗ n ψ y = ( φx + ψx ) ∗ n ( φy + ψ y ) = ( φ + ψ ) x ∗ n ( φ + ψ ) y . Direct sums, kernels, and cokernels are g otten by taking each on the level o f ab elian groups, e.g. O b ( K er φ ) = K er Ob ( φ ). It is clear how to define so urce, targ ets and comp ositions on direct sums and kernels. F or c ok ernels, since functors resp e c t source and tar get maps, there is no difficulty in defining source and targ et maps on a cokernel. Comp osition in a cokernel is defined b y letting x ∗ n y = x + y − s n x . Remark 3.3. The forge tful functors F Ab , F S et = Ob , F ω taking v alues in Ab , S et , a nd ω Cat resp ectively are all faithful functors. It follows from Pr opos ition 3.2b that if A , B ∈ P ic ω and g ∈ H om Ab ( F Ab A, F Ab B ) resp ects all source and target maps, then g = F Ab f for some f ∈ H om P ic ω ( A, B ). 3.1 Picard ω -Categories and Chain Complexes Notation. Let C h + ( Ab ) denote the categor y of complexes of ab elian groups in non-neg a tiv e degrees . Let P ic denote the categor y of Picard categories in the se nse of Deligne [12]. That is, a Picard categor y C is a quadruple ( C , + , σ, τ ), where + : C × C − →C is a functor such that for all ob jects x ∈ C , x + : C − →C is a n e quiv alence, and addition is commutative and asso ciative up to is omorphisms τ and σ . Deligne made the obse r v ation that o ne can assign to a co mplex A 1 − → A 0 of a b elian gr oups a Picard categ ory P A , whose ob jects are A 0 and H om P A ( a, b ) = { f ∈ A 1 | d f = b − a } . He go es on to s how that P : C h 0 , 1 ( Ab ) − → P i c is 5 an equiv alence and also induces an equiv a lence betw een D 0 , 1 ( Ab ) and P ic mo dulo na tural is omorphism. Define P ic 1 ω = { A ∈ P ic ω | A = A 1 } . The r elationship b etw een P ic 1 ω and P ic is explained in Pr opos itio n 3.4, the pro of o f which is found in section 8 .1 of the app endix. F urthermor e , when restricted to s hort complexes in degr e e s 1 and 0 only , Theor em 3.7 is a strictifica tion theor em which states that P ic 1 ω and P ic are equiv a len t. Prop osition 3.4. P ic 1 strict c onsists of al l smal l Pic ar d c ate gories in P ic such that + is strictly asso ciative and c ommutative (i.e. τ and σ ar e identities) and for e ach x ∈ ob ( C ) , x + : C − →C is an isomorphism, not just an e quivalenc e. Deligne’s cor respo ndence extends to longer complexes. The cor respo ndence in Pr opositio n 3.5 was con- sidered by Bourn, Steiner, Brown, Higgins [4, 42, 8], et a l. Prop osition 3. 5. A c o mplex A of ab elian gr oups defines a Pic ar d ω -c ate gory P ( A ) . Th is assignment P : C h + ( Ab ) − → P i c ω is a fun ctor. Pr o of. Let P = P ( A ) consist of sequences x = (( x − 0 , x + 0 ) , ( x − 1 , x + 1 ) , ... ) with x α i ∈ A i such that dx α i = x + i − 1 − x − i − 1 for all i ≥ 1, α ∈ { + , −} . W e define so urce a nd ta r get maps by s i x = (( x − 0 , x + 0 ) , ( x − i − 1 , x + i − 1 ) , ( x − i , x − i ) , (0 , 0) , ... ), and t i x = (( x − 0 , x + 0 ) , ( x − i − 1 , x + i − 1 ) , ( x + i , x + i ) , (0 , 0) , ... ). If s i x = t i y , define x ∗ i y = (( x − 0 , x + 0 ) , ..., ( x − i − 1 , x + i − 1 ) , ( y − i , x + i ) , ( x − i +1 + y − i +1 , x + i +1 + y + i +1 ) , ... ) W e need to check that x ∗ i y is an elemen t of P A . Firstly , dx + i = dx − i = dy + i = dy − i since s i x = t i y . Seco ndly , d ( x + i +1 + y + i +1 ) = d ( x − i +1 + y − i +1 ) = ( x + i − x − i ) + ( y + i − y − i ) = x + i − y − i since x − i = y + i . Finally , for j > i + 1, it is obvious that d ( x ∗ i y ) α j = d (( x ∗ i y ) + j − 1 − ( x ∗ i y ) − j − 1 ). Hence, ( x ∗ i y ) ∈ P . It is easily chec k ed that P A is an ω -catego ry . W e now define an op eration P × P − → P which makes P into a P icard ω -ca tegory . This is the obvious op eration x + y = (( x − 0 + y − 0 , x + 0 + y + 0 ) , ( x − 1 + y − 1 , x + 1 + y + 1 ) , ... ) , which ob viously sa tisfies x + y − s n x = x ∗ n y when comp osition is defined. T o see that + is a functor, let x, y , a, b ∈ P such that s n x = t n a and s n y = t n b . Then +(( x, y ) ∗ i ( a, b )) = ( x ∗ i a ) + ( y ∗ i b ) = (( x − 0 , x + 0 ) , ..., ( a − i , x + i ) , ( a − i +1 + x − i +1 , a + i +1 + x + i +1 ) , ... ) +(( y − 0 , y + 0 ) , ..., ( b − i , y + i ) , ( b − i +1 + y − i +1 , b + i +1 + y + i +1 ) , ... ) = (( x − 0 + y − 0 , x + 0 + y + 0 ) , ..., ( a − i + b − i , x + i + y + i ) , ( a − i +1 + x − i +1 + b − i +1 + y − i +1 , a + i +1 + x + i +1 + b + i +1 + y + i +1 ) , ... ) = ( x + y ) ∗ i ( a + b ) , so b y Pro position 3.2, P a Picard ω -categ ory . The assig nmen t P : C ≤ 0 ( Ab ) − → P i c ω is o bviously a functor; for a morphism f : A − → B in C h + ( Ab ), P f is g iv en by ( P f x ) α i = f ( x α i ) for α ∈ { + , −} . Henceforth, we shall denote a sequence x = (( x − 0 , x + 0 ) , ( x − 1 , x + 1 ) , ... ) b y ( x i ) i ∈ N or simply ( x i ), wher e x i = ( x − i , x + i ). Lemma 3. 6. A Pic ar d ω -c ate gory A defines a chain c omplex Q ( A ) ∈ C h + ( Ab ) in such a way t hat Q : P i c ω − → C h + ( Ab ) is a functor. Pr o of. Since + A × A − → A is a functor, it r espects all op erators ∗ i , s i , t i . Hence, + A i × A i − → A i , so A i is a subgro up of A . Therefore it makes sense to define Q i := ( QA ) i := A i / A i − 1 for i > 0 and Q 0 = A 0 . W e 6 define, for ea ch i > 0 an op e ration d : Q i − → Q i − 1 by first defining a homomo r phism d 0 : A i − → A i − 1 . Define d 0 = t i − 1 − s i − 1 . W e must check that this is a homomorphism. If x, y ∈ A i , d 0 ( x + y ) = t i − 1 ( x + y ) − s i − 1 ( x + y ) = t i − 1 x + t i − 1 y − ( s i − 1 x + s i − 1 y ) = t i − 1 x − s i − 1 x + ( t i − 1 y − s i − 1 y ) = d 0 x + d 0 y . If x ∈ A i − 1 , then t i − 1 x = x = s i − 1 x , so d 0 ( A i − 1 ) = 0. Therefore, d 0 is a group homomorphism such that A i − 1 ⊂ K er d 0 . This determines a homomor phism d : Q i − → Q i − 1 . T o see that d 2 = 0, for x ∈ Q i , d 2 x = d ( t i − 1 x − s i − 1 x ) = t i − 2 ( t i − 1 x − s i − 1 x ) − s i − 2 ( t i − 1 x − s i − 1 x ) = t i − 2 x − t i − 2 x − ( s i − 2 x − s i − 2 x ) = 0. Hence, Q is a complex of a belian gro ups. W e have co ns tructed Q from A , whic h gives a map Q : P ic ω − → C ≤ 0 ( Ab ). If F : A − → B is a map of ω -categor ies, F : A i − → B i for each i ≥ 0 , so F descends to a map Q ( F ) : A i / A i − 1 − → B i /B i − 1 , which is easily seen to b e a map of complexes. This makes Q is a functor of 1-categ ories. Theorem 3. 7. ([4]) Q ◦ P ≃ id and P ◦ Q ≃ id . Ther efor e, Q and P ar e e quivalenc es of c ate gories. Pr o of. F rom a complex A ∈ C h + ( Ab ), we get a complex Q = Q ( P A ), where Q i = ( P A ) i / ( P A ) i − 1 . First define a map A i − → ( P A ) i , x 7→ ˆ x , in the following wa y: ( ˆ x ) j =    (0 , 0 ) if j < i − 1 (0 , dx ) if j = i − 1 ( x, x ) if j = i . Now define a map h : A i − → Q i by h ( x ) = [ ˆ x ] ∈ P A i /P A i − 1 . Observe that fo r y ∈ ( P A ) i , [ y ] = [ y − s i − 1 y ] = [ ˆ y + i ] = h ( y + i ). Therefor e, h is surjective. It is clear that h is a homomorphis m and that it is injective. But h must also b e a map of complexes. If x ∈ A i , dh ( x ) = d [ ˆ x ] = [ t i − 1 ˆ x − s i − 1 ˆ x ], where ( t i − 1 ˆ x − s i − 1 ˆ x ) j =  (0 , 0 ) if j 6 = i − 1 ( dx, dx ) if j = i − 1 Since d 2 = 0, this is ob viously the class of ˆ dx . Hence, h is a morphism o f chain complexes, and we conclude by injectivity and surjectivity that h A : A − → QP ( A ) is an isomor phism. How ev er, to b e an iso- morphism of functors , QP ˜ → 1, these maps must satisfy h B ◦ f = (( QP ( f )) ◦ h A for any map of co mplexes f : A − → B . F o r x ∈ A i , h B ( f ( x )) = [ ˆ ( f ( x )) ] ∈ ( P B ) i / ( P B ) i − 1 . Applying the r igh t-hand side to x , we get (( QP ( f )) ◦ h A x = Q ( P ( f ))[ ˆ x ] = [ P ( f ) ˆ x ] = [ ˆ ( f ( x ))]. So h do es in fact define an isomor phism b et w een endofunctors QP and 1 of C h + ( Ab ). Now we wish to sho w that for A ∈ P ic ω , there is an isomorphism ϕ : A − → P QA in P ic ω . F or the r est of the pro of, for x ∈ A i , let [ x ] be its image in Q i = A i / A i − 1 , and for a n y x ∈ A , let µ ( x ) := min { m ∈ N | s m x = x } . Now, for x ∈ A , define ϕx = { ( ϕx ) i } i ∈ N by: ( ϕx ) i =  ([ s i x ] , [ t i x ]) if i ≤ µ ( x ) (0 , 0 ) if i > µ ( x ) . It is clea r that ϕx ∈ P QA , but we still must chec k that ϕ is a functor. F o r ρ i ∈ { s i , t i } , it must b e shown that ϕ ( ρ i x ) = ρ i ϕx . If i > µ ( x ), this is obvious. If i < µ ( x ), ( ϕ ( ρ i x )) j =    ([ s i x ] , [ t i x ]) if i < j ([ ρ i x ] , [ ρ i x ]) if i = j (0 , 0 ) if j > i, 7 which is exactly the same formula for ( ρ i ( ϕx )) j . Additionally , ϕ must be a homomorphis m of ab elian groups, but this follows ea sily . F o r s implicity , assume µ ( x ) ≤ µ ( y ). ( ϕ ( x + y )) i =    ([ s i ( x + y )] , [ t i ( x + y )]) if i ≤ µ ( x + y ) = µ ( y ) ([ x + y ] , [ x + y ]) if i = µ ( y ) (0 , 0 ) if i > µ ( y ) =    ([ s i x ] + [ s i y ] , [ t i x ] + [ t i y ]) if i ≤ µ ( x + y ) = µ ( y ) ([ x ] + [ y ] , [ x ] + [ y ]) if i = µ ( y ) (0 , 0 ) if i > µ ( y ) = ( ϕ ( x ) + ϕ ( y )) i . By Remark 3.3, ϕ is a morphism o f Pica rd ω -catego ries. T o show that ϕ is an iso morphism o f ω - categories, w e simply s ho w that ϕ is bijection o f sets. Let P = P Q ( A ). First w e show that ϕ is surjectiv e. W e prov e b y inductio n that ( P ) n is in the image of ϕ for each each n ∈ N . If n = 0, let a = (( a 0 , a 0 ) , (0 , 0) , ... ) ∈ P 0 , so a 0 ∈ A 0 = P 0 , and ϕ ( a 0 ) = a . Now supp ose that P k ⊂ ϕ ( A ). Let x = ( x i ) i ∈ N = (([ a − i ] , [ a + i ])) i ∈ N ∈ P k +1 , a ± i ∈ A i . Then [ a − k +1 ] = [ a + k +1 ], and ϕ ( a + k +1 ) − x ∈ ( P ) k ⊂ ϕ ( A ) by induction hypothesis, so ϕ ( a + k +1 ) − x = ϕ ( z ) for so me z ∈ A k . Hence, x = ϕ ( a + k +1 − z ) ∈ ϕ ( A ). Thus, P k +1 ⊂ ϕ ( A ) and ther efore ϕ is surjective. Now we demonstrate that ϕ is injective. Let x ∈ K er φ and µ = µ ( x ) as a b ov e. Then s n x = x for all n ≥ µ and s k x 6 = x for all k < µ . If φx = 0 , [ s k x ] = 0 for all k , whence s k x ∈ A k − 1 . In pa r ticular, s µ x ∈ A µ − 1 , which implies that s µ − 1 x = x , which contradicts the minimality o f µ . Ther efore to av oid a contradiction, x must be 0 a nd K er φ = 0. Therefore, ϕ is an isomor phism. F o r each A ∈ P i c ω , we’v e pro duced an ϕ A : A ˜ − → P Q ( A ). T o complete the pro of, we simply m ust s e e if this satisfies compatibility with morphisms in P i c ω . F or x ∈ A , we require that ϕ B ◦ f ( x ) = ( P Q ( f )) ◦ ϕ A . W e consider the i-th entry in each sequence and see that the are the same. Since ( ϕ A ( x )) i = ([ s i x ] , [ t i x ]), with [ s i x ] , [ t i x ] ∈ A i / A i − 1 , w e have that ( P ( Qf )( ϕ A ( x )) i = ( Qf )( ϕ A ( x )) i = (( Q ( f ))[ s i x ] , ( Q ( f ))[ t i x ]) = ([ f ( s i x )] , [ f ( t i x )]) = ([ s i f ( x )] , [ t i f ( x )]) = ( ϕ B ◦ f ( x )) i . 3.2 Equiv alences of Picard ω -categories Prop osition 3.8. F or c omplexes A, B ∈ C + ( Ab ) and map of c omplexes f : A − → B , f is a quasi-isomorphism if and only if P f : P A − → P B is an e quiva lenc e of ω -c ate gories. Pr o of. This pro of will use the descr iption in the first part o f Definition 4, a s it simplifies the no tation. F or an ob ject a ∈ A n satisfying da = 0, we denote its image in H n ( A ) by [ a ], and for a map f : A − → B , by abuse of notatio n, let f also denote the induced ma p on cohomolo gy . F o r e ase of notation, let F : A − → B denote P f : P A − → P B . Let A f − → B b e a q ua si-isomorphism o f co mplex es A B ∈ C h + ( Ab ) . If y ∈ P B 0 = B 0 , there exists x ∈ A 0 such that [ f x ] = [ y ], so there exists some z ∈ B 1 such that dz = f x − y . Th us, (( y , f x ) , ( z , z ) , 0 , ... ) ∈ H om 1 ( f x, y ) is an iso morphism as req uired. This prov es condition (1 ) of Definition 4 . T o prov e con- dition (2), let ψ ∈ H om n P B ( F x, F y ) with x , y ∈ P A n such that s n − 1 x = s n − 1 y and t n − 1 x = t n − 1 y . Then ψ = (( ψ − 0 , ψ + 0 ) , ..., ( ψ − n +1 , ψ + n +1 ) , 0 , ... ) with s n ψ = F x = (( f x − 0 , f x + 0 ) , ..., ( f x − n , f x + n ) , 0 , ... ) and t n ψ = F x = (( f y − 0 , f y + 0 ) , ..., ( f y − n , f y + n ) , 0 , ... ), so ψ − n = f x − n = f x + n and ψ + n = f y − n = f y + n . This means that dψ ± n +1 = f y ± n − f x ± n so that [ f x + n − f y + n ] = 0 and therefore [ x + n − y + n ] = 0 b ecause f is a quasi-is o morphism . Hence, dφ = y + n − x + n for some φ ∈ A n +1 . Since σ i x = σ i y for i < n , σ i ∈ { s i , t i } , x ± i = y ± i . W e conclude that (( x − 0 , x + 0 ) , ..., , ( x − n − 1 , x + n − 1 ) , ( x + n , y + n ) , ( φ, φ ) , 0 , ... ) is an ( n + 1)- morphism from x to y a s required. Since P A and P B ar e group oids, condition (2 ) entails co ndition (3). Therefor e, F : A − → B is an eq uiv alence of 8 ω -categor ies. Now supp ose that F is an equiv alence. W e will show that f : H n ( A ) − → H n ( B ) is an is omorphism for each n . Let x ∈ K er ( d : A n − → A n − 1 ). If [ f x ] = 0, then there is some ψ such that dψ = f x . W e see that (0 , ..., (0 , 0) , (0 , f x ) , ( ψ , ψ ) , 0 ... ) is an ( n +1 )-isomorphism from 0 to ˆ f x = (0 , ..., (0 , 0) , ( f x, f x ) , 0 ... ). By condi- tion (3) of Definition 4, ˆ x is isomorphic to 0. Such an isomo r phism is of the form (0 , ..., (0 , 0) , (0 , x ) , ( φ, φ ) , 0 ... ) for some φ sa tisfying dφ = x , so we see tha t [ x ] = 0 and therefore f is injective. T o see that f is sur jectiv e, let [ y ] ∈ H n ( B ) for n > 0 and consider ˆ y = (0 , ..., 0 , (0 , 0) , ( y , y ) , 0 , ... ), which is an n - isomorphism from 0 = F (0) to itself. By co ndition (2), there exists an n - isomorphism ˆ x = (0 , ..., 0 , ( x, x ) , 0 ... ) in A together with an ( n + 1 )-morphism (0 , ..., 0 , ( f x, y ) , ( z , z ) , 0 ... ) fro m F ˆ x to ˆ y . Hence, dz = y − f x so that f [ x ] = [ y ] and f is surjective. F o r n = 0 , let y ∈ H 0 ( B ), w e use the sa me argument, except this time in v oking condition (1) to ensure the existence of s uch an x ∈ A 0 . This shows that f is a quasi- isomorphism . Let H o ( P ic ω ) denote P i c ω lo calized at the equiv ale nc e s . W e now hav e the following coro lla ry . Corollary 3. 9. The derive d c ate gory D ≤ 0 ( Ab ) of ab elian gr ou ps in de gr e es ≤ 0 is e quivalent t o the homotopy c ate gory H o ( P ic ω ) . 3.3 Useful F acts for Picard ω -categories Lemma 3.10 shows that if A is a sub categor y of B , and B can b e extended to an ω -categor y C , then A can be extended to an ω -s ub- category of C . Lemma 3.10 . L et A and B b e 1-c ate gories with A a sub c ate gory of B . If ther e is an ω -c ate gory C such that ( C 1 , s 0 , t 0 , ∗ 0 ) = B , then ther e is an ω - sub c ate gory C ′ of C such that ( C ′ 1 , s 0 , t 0 , ∗ 0 ) = A . Pr o of. Define O b ( C ′ ) = { c ∈ O b ( C ) | s 1 c , t 1 c ∈ C ′ 1 } . W e show that C ′ is an ω -catego r y by first showing that it is s table under all sour ce and tar g et maps and then s ho wing that it is close d under comp osition. If x ∈ O b ( C ′ ) a nd j > 1 , ρ 1 σ j x = ρ 1 x ∈ C ′ 1 . If j = 1, ρ 1 σ j x = σ j x ∈ C ′ 1 . Finally , if j = 0, ρ 1 σ 0 x = σ 0 x = σ 0 ρ 1 x . Since A is a categor y , σ 0 ρ 1 x ∈ A 0 = C ′ 0 ⊂ C ′ 1 . This shows that C ′ is stable under maps σ j ∈ { s j , t j } . Now supp ose x, y ∈ C ′ . If i ≥ 1, s 1 ( x ∗ i y ) = s 1 y ∈ C ′ , a nd t 1 ( x ∗ i y ) = t 1 x ∈ C ′ . If i = 0, ρ 1 ( x ∗ i y ) = ρ 1 x ∗ 0 ρ 1 y ∈ A ⊂ C ′ since A is a catego ry . This shows that C ′ is stable under all co mp ositions ∗ i . Clea rly C ′ is an ω -catego ry be cause all o ther nece s sary pr operties a r e inherited fro m C . In [43], Street constructs a n ω -category C at ∞ such that (( C at ∞ ) 1 , s 0 , t 0 , ∗ 0 ) = ω C at so that ω -Cat is a category enriched over itself. The construction is natural since it comes from an inner hom in ω Cat. Lemma 3.10 can b e applied to P ic ω : there is an ω -catego ry C such that ( C 1 , s 0 , t 0 , ∗ 0 ) = P ic ω . In fact, we see in sections 8.3 and 3.4 that there is more than one ω - category which has P ic ω as its 0-ob jects and 1-morphisms. W e now mak e the obse r v ation that for A ∈ P ic ω , since there is a section of A i − → A i / A i − 1 sending [ x ] to x − s i − 1 x . It follows that A i ≃ A i / A i − 1 ⊕ A i − 1 as ab elian groups. Mo r eo ver, A ≃ A 0 ⊕ ∞ M i =1 A i / A i − 1 With this identification, for x ∈ A k / A k − 1 , s n x =  x if n ≥ k 0 if n < k t n x =    x if n ≥ k dx if n = k − 1 0 if n < k − 1 9 This can be written explicitly as s n : ( x 0 , ..., x m , ... ) 7→ ( x 0 , ...x n , 0 , 0 , ... ) and t n : ( x 0 , ..., x m , ... ) 7→ ( x 0 , ..., x n − 1 , x n + dx n +1 , 0 , 0 , ... ), wher e d = t n − s n . There ar e several p ossible iden tifications of A with L ∞ i =0 A i / A i − 1 . It is p erhaps most transpare nt if A = P ( C ) for C ∈ C h + ( Ab ), so C i ≃ A i / A i − 1 , and L ∞ i =0 C i − → A is given by ( x 0 , x 1 , x 2 , ..., x n , 0 , 0 , ... ) 7→ (( x 0 , x 0 + dx 1 ) , ( x 1 , x 1 + dx 2 ) , ..., ( x n − 1 , x n − 1 + dx n ) , ( x n , x n ) , (0 , 0) , ... ). Prop osition 3. 11. Ther e is a functor [ − 1] : P ic ω − → P ic ω such that for C ∈ C h + ( Ab ) , P ( C [ − 1 ]) = ( P ( C ))[ − 1] , wher e C [ − 1] denotes the c omplex ... − → C 1 − → C 0 − → 0 ∈ C h + ( Ab ) with C [ − 1] n = C n − 1 for n > 0 and C [ − 1 ] 0 = 0 . F or A ∈ P ic ω , we define A [ − 1 ] by lett ing O b ( A [ − 1]) = O b ( A ) , ( ∗ [ − 1] n = ∗ n − 1 , s [ − 1] n = s n − 1 , t [ − 1] n = t n − 1 ) for al l n ≥ 1 , s [ − 1] 0 = t [ − 1] 0 = 0 , and x ∗ 0 y = x + y . Pr o of. That A [ − 1] ∈ P ic ω is e v iden t. If A = P ( C ), ther e is a bijection betw een O b ( P C [ − 1 ]) a nd O b (( P C )[ − 1]). F o r (( x − 0 , x + 0 ) , ( x − 1 , x + 1 ) , ..., ( x − n − 1 , x + n − 1 )) ∈ ( P C )[ − 1] n maps to ((0 , 0) , ( x − 0 , x + 0 ) , ( x − 1 , x + 1 ) , ... ). The following prop osition is a genera lization of a lemma found in [8] Prop osition 3.12. L et A b e any ab elian gr oup. If A admits Z -line ar maps s n , t n : A − → A for n ∈ N satisfying c onditions 1a, 2a, and 2b of definition 1, then t her e is a unique ω -c ate gory st ructur e on A such that A ∈ P i c ω . Pr o of. If a, b ∈ A suc h that s n a = t n b , then we define a ∗ n b = a + b − s n a and chec k that this ma k es A in to a Picard ω -categ ory . If comp ositions satisfy all ω Cat axio ms, then it is easily see n that + is a functor, i.e. ( a + b ) ∗ n ( a ′ + b ′ ) = ( a ∗ n a ′ ) + ( b ∗ n b ′ ), whenever the right-hand side is defined. W e can easily chec k that ( a + b ) ∗ n ( a ′ + b ′ ) = a + b + a ′ + b ′ − s n ( a + b ) = a + b + a ′ + b ′ − s n a − s n b = ( a ∗ n a ′ ) + ( b ∗ n b ′ ). Thus, it suffices to chec k that A sa tisfies all ω -ca tegory axio ms of Definition 1. (1b) a ∗ n s n ( a ) = a + s n a − s n a = a , and t n a ∗ n a = t n a + a − s n t n a = t n a + a − t n a = a . (1d) s n ( a ∗ n b ) = s n ( a + b − s n a ) = s n a + s n b − s n a = s n b , and s imilarly , t n ( a ∗ n b ) = t n ( a + b − s n a ) = t n a + t n b − s n a = t n a since t n b = s n a . (1c) ( a ∗ n b ) ∗ n c = ( a + b − s n a ) + c − s n ( a ∗ n b ) = a + b + c − s n a − s n b , whereas a ∗ n ( b ∗ n c ) = a + ( b + c − s n b ) = s n a . (2c) ρ j ( a ∗ i b ) = ρ j ( a + b − s i a ) = ρ j a + ρ j b − ρ j s i a = ρ j a + ρ j b − s i ( ρ j a ) = ( ρ j a ) ∗ i ( ρ j b ) since j > i . (2d) ( a ∗ j b ) ∗ i ( α ∗ j β ) = ( a + b − s j a ) + ( α + β − s j α ) − s i ( a + b − s j a ) = a + b + α + β − s j a − s j α − s i a − s i b + s i a = a + b + α + β − s j a − s j α − s i b . On the r igh t-hand side we hav e ( a ∗ i α ) ∗ j ( b ∗ i β ) = ( a + α − s i a ) + ( b + β − s i b ) − s j ( a ∗ i α ) = a + b + α + β − s i a − s i b − ( s j a ∗ i s j α ) = a + b + α + β − s i a − s i b − ( s j a + s j α − s i a ) = a + b + α + β − s j a − s j α − s i b . F o r a set A , we let Z [ A ] denote the free ab elian gro up on the set A . If A ∈ ω C at , we can extend all s o urce and target maps Z -linearly . Prop osition 3 .12 implies that Z [ A ] ∈ P ic ω . Lemma 3.1 3 . The functor Z : ω C at − → P ic ω sending A to the fr e e ab elian gr oup gener ate d by A is left- adjoint to the for getful funct or F ω : P ic ω − → ω C at . Pr o of. Let A ∈ ω C at and B ∈ P ic ω . Any ϕ ∈ H om ω C at ( A, F ω ( B )) can b e extended Z -linea r ly to a map ˆ ϕ ∈ H om Ab ( Z [ A ] , B )) of ab elian groups. The fact that all s n and t n are Z -linear mea ns that ˆ ϕ commutes with all s o urce and ta rget maps . But since compos ition ∗ n in a Pica rd ω -catego ry is determined by all +, s n , t n , ˆ ϕ a lso res pects comp ositions ∗ n . The r efore ˆ ϕ ∈ H om P ic ω ( Z [ A ] , B ). It is clear that the function ϕ 7→ ˆ ϕ is injective. T o s ee that it is surjective, any ψ ∈ H om P ic ω ( Z [ A ] , B ) is also a map of ab elian groups , so it comes from some ϕ ∈ H om S ets ( Ob ( A ) , Ob ( F ( B ))). Of course since Ob ( A ) ⊂ O b ( Z [ A ]), ϕ ( a ) = ψ ( a ) and so ϕ is actually a map of ω -c a tegories and ˆ ϕ = ψ . The inv erse map is ψ 7→ ψ | A . 10 T o check that G and F ar e adjoints, w e m ust a ls o se e that we ha ve a map of functors H om P ic ω ( Z [ − ] , − ) ˜ − → H om ω C at ( − , F − ). In other words, for f ∈ H om ω C at ( A, A ′ ) and g ∈ H om P ic ω ( B , B ′ ), then for ψ ∈ H om P ic ω ( Z [ A ′ ] , B ), the maps ( g ◦ ψ ◦ Z [ − ]) | A = F ( g ) ◦ ( ψ | A ′ ◦ f ∈ H om ω C at ( A, F ( B ′ )). It is easy to see that these maps ag ree at the level of sets. In [43] it is shown that O b : ω C at − → S et is repr esen ted by an o b ject 2 ω ∈ ω C a t . Corollary 3. 1 4. Z [2 ω ] is a c or epr esentative for the functor O b : P ic ω − → S e ts . Pr o of. H om P ic ω ( Z [2 ω ] , B ) = H om ω C at (2 ω , F ω ( B )) = O b ( F ω ( B )) = O b ( B ). 3.4 I-categories Since the structure maps for an ω -catego ry are indexed by natura l num bers , we may think of an ω -c ategory a s an N - c ategory . The ω -catego ry axioms dep ended only on N b eing a partially ordered set. W e may therefor e extend the definition and define an I -categor y for any partially ordered set I . In particular, w e ar e in terested in Z -categor ies and show that “nice” abelia n group ob jects in Z -categor ies a r e the same as un bounded chain complexes of ab elian groups. Definition 6. Let I be a linearly order ed set. 1. An I-c ate gory is a set qua druple ( A, s i , t i , ∗ i ) i ∈ I as in Definition 1 except that instead of N , we hav e I . Let I -Cat denote the ca tegory o f all I-categ ories. 2. Let P ic I denote ab elian group ob jects in I -cat, a nd let P ic 0 I denote the full s ub categor y , ca lled P icard I-categor ies, the ob jects o f which ar e A ∈ P i c I such that for all x ∈ A , there exists n ∈ I such tha t s n x = 0 and there exists m ∈ I for w hich s m x = x . W e can e xtend some of the r esults ab out ω -ca tegories to Z -ca tegories. Theorem 3.7, for instance, can b e extended to Z -categor ies. Definition 7. A functor F : A − → B of Z -catego ries is an eq uiv alence if conditions 2b and 2c of Definition 4 are met and for each b ∈ B , there exists n such tha t s n b is isomor phic to some F ( a ). It follows directly from the definition of P ic 0 Z that any map of Picar d Z -categor ies which satisfies co ndi- tions 2b a nd 2c of Definition 4 is an equiv alence. In fac t, since Pica rd Z -catego ries are group oids, condition 2b of Definition 4 is sufficient. Theorem 3.15 . The c ate gory C h ( A b ) of chain c omplexes of ab elian gr o ups is e quivalent t o the c ate gory P i c 0 Z of Pic ar d Z -c ate gories. Pr o of. The pr o ofs o f Prop o sition 3.2, Lemma 3.5, 3.6, and Theor em 3 .7 extend naturally to Z -catego ries with only a few mo difications. First we define P ( A ) to consist of sequences ( .... ( x − i , x + i ) , ... ) as b efore but require that only finitely many x ± i are nonzero . Seco ndly , in lemma 3.6, to show the sur jectivit y of φ : A − → P Q ( A ), we c ho ose y ∈ P Q ( A ) and such that y ± i = 0 for i ≤ n ∈ Z . T o s ho w that y is in the image of φ , we star t the induction at n instead of 0 . Also , w e note that µ ( x ) of Lemma 3 .6 is w ell defined except for when x = 0 . Letting H o ( P ic 0 Z ) denote Picard Z -ca teg ories lo calized at equiv a lences, we ar riv e at the following corolla ry , the pro of of which is identical to the pr oof of P ropo sition 3.8. Corollary 3. 16. The derive d c ate gory DAb of ab elia n gr oups is e quivalent to the homotopy c ate gory of P ic 0 Z . The following results ab out ω -cats also extend to Z -catego ries: Prop 3.2, P ropo sition 3.8. Also, the equiv ale nce P : C h ( Ab ) − → P ic 0 Z is, just a s for Picar d ω -categ ories, isomor phic to the o ne that sends A ∈ C h ( Ab ) to L n ∈ Z A n ∈ P ic 0 Z . The followin g prop osition is patent. 11 Prop osition 3 .17. P ic 0 Z is a triangulate d c ate gory with shift functor given as in Pr op ositio n 3.11. This gives D P i c 0 Z the stru ct ur e of a triangulate d c ate gory. The mapping c one of f : A − → B is ( B [ − 1] × A, s n = s B n − 1 × s A n , t n = t B n − 1 × ( f + t A n )) . The t-stru ctur e c oming fr om the standar d t -structur e on C h ( Ab ) is D ≥ 0 = H o ( P ic ω ) , D ≤ 0 = { A ∈ P ic 0 Z | s n = id, t n = id for al l n > 0 } . Its he art is { A ∈ P ic ω | al l σ n = id } ≃ Ab . Remark 3.18 . The ω - category str ucture o n C h ( Ab ) induced from Theore m 3.15 is given in the following wa y . 1-ob jects ar e maps of complexes. Strict 2-ob jects a re maps b et w een ma ps of complexes F , G : A − → B , i.e. φ : F = ⇒ G is a map φ ∈ H om ( A [ − 1] , B ) such that dφ = G − F . etc. Remark 3.19 . Prop osition 3.1 7 is just ano ther wa y to say that the categ ory of ab elian gro up sp ectra is equiv ale nt to C h ( Ab ). The derived version also ho lds fro m this po in t of view, as was shown by Shipley [40]. 3.5 ω -categories in a category C Just a s an ω - category can be viewed as an ω -categ ory in sets and P ic ω consists of ω -ca tegories in ab elian groups, we can define ω -catego ries in any categ ory C with fib ered pro ducts, and when C is ab elian, we generalize Theorem 3.7. Definition 8. Le t C be any categ ory with fib ered pro ducts. An ω -categ ory in C is an ob ject X of C with maps s n , t n : X − → X for n ∈ N a nd comp ositions X × X X ∗ n − → X sa tisfying the axio ms in definition 1 (where X × X X is the fib ered pro duct with resp ect to t n and s n ). Morphisms b et w een ω -catego r ies in C are simply mor phisms in C commuting with all source, ta r get, and co mposition maps. W e denote the ca tegory of ω -catego ries in C b y C ω . Lemma 3.20. F or any ab eli an c ate gory C with infin ite dir e ct su ms, C h + ( C ) and C ω ar e e quivalent. Pr o of. W e sketch a pro o f and leav e the details to the rea der. First, w e define a functor P : C h + ( C ) − →C ω as follows. Let A b e a complex in C h + ( C ), a nd let P ( A ) = L ∞ n =0 A i . W e must show that B = P ( A ) is an ω -categ ory in C . W e define s ource a nd tar get maps s n , t n : L ∞ n =0 A i − → L ∞ n =0 A i as follows. First, let s i,j n , t i,j n : A i − → A j be given by s i,j n :=  1 if n ≥ i = j 0 otherwise t i,j n :=    1 if n ≥ i = j d if n = j = i − 1 0 otherwise . Now, let s i n be the s um ov er j of the comp ositions A i s i,j n − → A j − → L ∞ k =0 A k and similarly for t i n . The mo r- phisms s i n , t i n : A i − → P A , i ≥ 0, deter mine s n , t n . Defining comp osition a s ∗ n = π 2 + π 1 − s n π 1 , one may verify that ( ∗ n , t n , s n ) n ≥ 0 satisfies conditions of Definition 1. F or condition (2d) of Definition 1, the statemen t for ω -categor ies in C should rea d: ∗ i ( ∗ j π 1 × ∗ j π 2 ) = ∗ j ( ∗ i π 1 × ∗ i π 2 )(( π 1 π 1 × π 1 π 2 ) × ( π 2 π 1 × π 2 π 2 )) when re- stricted to the appr opriate subo b ject o f ( B × B ) × ( B × B ). Our definition of compositio n ∗ n can b e extended to B × B − → B , a nd one ma y chec k that ∗ i ( ∗ j π 1 × ∗ j π 2 ) agrees with ∗ j ( ∗ i π 1 × ∗ i π 2 )(( π 1 π 1 × π 1 π 2 ) × ( π 2 π 1 × π 2 π 2 )) on ( B × B ) × ( B × B ). Hence, they also agree on the appropr iate fib ered pro duct. Therefor e , B is an ω - category in C . Let B = P ( A ), D = P ( C ) for A, C ∈ C h + ( C ). The fact that H om C ω ( B , D ) ≃ H om C h + ( C ) ( A, C ) follows easily from a few o bserv ations. Let f be a morphis m from B to D . First, since f s n = s n f , an inductive pro of shows that f ( A i ) − → D factors thro ugh C i − → D . The fact that f commutes with all t n shows that the induced maps A i − → C i commute with the differentials A i d − → A i − 1 and C i d − → C i − 1 . W e conclude that a morphism f : B − → D is equiv a len t to a mor phism fro m A to C in C h + ( C ). Therefo r e, P is fully faithful. 12 W e now show that P is essentially sur jectiv e. Define Q : C ω − → C h + ( C ) as follows. Given B ∈ C ω , let A = Q ( B ) b e given by letting A n be the co k ernel B n /B n − 1 of the monomo rphism B n − 1 − → B n , where B n is the image o f s n : B − → B . The morphism t n − 1 − s n − 1 : B n − → B n − 1 induces a morphism from B n /B n − 1 − → B n − 1 , and we denote the comp osition with B n − 1 − → B n − 1 /B n − 2 by d = A n − → A n − 1 . Let us see that P QB ≃ B . Let f : B n − → B n be f = 1 − s n − 1 . Then f induces a morphism f : B n /B n − 1 − → B n such that π f = 1. Hence, B n ≃ B n /B n − 1 ⊕ B n − 1 . It is now clea r that P QB ≃ B . 4 The Dold-Kan Corresp ondence W e b egin by laying out basic definitions and notations which ca n b e found in an y standar d text on the sub ject, such as [18]. Let ∆ denote the categ ory o f ordinals, with ob jects [ n ] = { 0 , 1 , ..., n } for n ∈ N and morphisms the (non-strictly) increasing set morphisms b et w een them. A simplicial set is a functor X : ∆ op − → S e t . W e define r-simplic es in a simplicial s et X to b e the set X r := X ([ r ]). W e let ∆ n denote the simplicial set H om ∆ ( − , [ n ]) a nd denote an r -simplex α : [ r ] − → [ n ] by listing ( α (0) , α (1) , ..., ( α ( r )). F or a s implicial set X , we let d i , s i denote the fac e and degener acy maps X ( ∂ i ) and X ( σ i ) resp ectiv ely , wher e [ n − 1] ∂ i − → [ n ] is the mor phism whic h skips only i , and σ i : [ n ] − → [ n − 1] is the morphism whic h rep eats only i . F o r a category C , a simplici al obje ct in C is a functor from ∆ op to C , and the categor y of simplicial o b jects in C is denoted simply by s C . A simplicial ab elia n gr oup is a simplicial ob ject in the categ o ry Ab o f abelian gro ups. The Dold- Kan co r respo ndence was discov ered indep e nden tly b y Dold and Kan and can b e found orig inally in [13] as well as a num ber of other references such a s [1 8], [49]. Theorem 4. 1. If C is an ab elia n c ate gory, ther e is an e quivale nc e K : s C − → C h + ( C ) . K : s C − → C h + ( C ) is given by K ( A ) n = T n − 1 i =0 K er d i , and the differential d : K ( A ) n − → K ( A ) n − 1 is d = ( − 1) n d n . W e will be pa rticularly interested in the case when C = Ab or sheaves of ab elian g roups o n some site. 4.1 ω -categories and quasicategories T o extend the idea of the nerve of an or dinary categor y , Street defines in [43] the nerve of an ω -c a tegory , which defines a functor N : ω C a t − → sS et . W e firs t review some ba c kground on par it y complexes a nd the Street-Rob erts conjecture. The results prese nted in this section are a s ummary o f so me of the r esults in [4 7]. The origina l nerve constructio n is can b e found in [4 3], and the ideas were streamlined using the language of parity complexes in [4 4, 4 5]. 4.1.1 Basics of Par it y Complexes Definition 9. A pr e-p arity is a graded set C = F ∞ n =0 C n and a pair of op erations sending x ∈ C n to x − ⊂ C n − 1 and x + ⊂ C n − 1 , called negative a nd p ositiv e faces of x resp ectively . If x ∈ C 0 , we take x − = x + = ∅ b y conv en tion. W e also say that for x ∈ C n , a face a ∈ x − has parity 1 (o dd) a nd a ∈ x + has parity 0 (even). E lemen ts in C n are said to b e n-dimensional. A Parity c omplex is a pre -parit y complex s a tisfying some additional axio ms delineated in [47, 4 4]. The additional tec hnical assumptions do not conern us because the pre-parity complexes which we dea l with here are all parity co mplexes. F o r a parity complex C and S ⊂ C , let | S | n = S n k =0 S k , where S k = S ∩ C k . F or S ⊂ C and ξ ∈ { + , −} , let S ξ = S x ∈ S x ξ , and let S ∓ = S − \ S + and S ± = S + \ S − . If C is a graded set, we le t N ( C ) denote the ω -category with under lying set { ( M , P ) | M , P are finite s ubsets of C } . Source, target and comp ositions are given by 13 • s n ( M , P ) = ( | M | n , M n ∪ | P | n − 1 ) • t n ( M , P ) = ( | M | n − 1 ∪ P n , | P | n ) • ( N , Q ) ∗ n ( M , P ) = ( M ∪ ( N \ N n ) , Q ∪ ( P \ P n ). There is a nother ω -categ ory O ( C ) attained fro m a par it y complex C , which we will now descr ibe. F or a parity complex C and subsets S, T ⊂ C , w e say that S ⊥ T if ( S + ∩ T + ) ∪ ( S − ∩ T − ) = ∅ . Ano ther wa y to ex - press this is to say that S ⊥ T if S and T hav e no common faces of the same parity . A subset S of C is called wel l-forme d if it has at mos t o ne 0-dimensional element and for distinct elements x, y ∈ S , x ⊥ y . Define O ( C ) to b e the subcateg ory of N ( C ) co nsisting of all ( M , P ) ∈ N ( C ) such that M and P a re b oth non-empt y , well-formed subsets of C , P = ( M ∪ M + ) \ M − = ( M ∪ P + ) \ P − , and M = ( P ∪ M − ) \ M + = ( P ∪ P − ) \ P + . It is not immediately c lear that O ( C ) is a n ω -category . How ev er, the work in [4 3, 44] demonstrates tha t it is. F o r a par it y complex C , there are dis tinguished elements of O ( C ). Let x ∈ C n . W e inductively define subsets π ( x ) , µ ( x ) ⊂ C . Let π ( x ) m = µ ( x ) m = ∅ for m > n , let π ( x ) m = µ ( x ) m = { x } for m = n , and let µ ( x ) m = µ ( x ) ∓ m +1 and π ( x ) m = π ( x ) ± m +1 for 0 ≤ m < n . Then the element < x > := ( µ ( x ) , π ( x )) ∈ O ( C ) is called an atom . Let < C > = { < x > | x ∈ C } . Street pr o ved [43, 44] that < C > freely generates O ( C ) in the sense defined below. First we introduce so me notation. F or n ∈ N and B ∈ ω C at , let | B | n denote the n -categor y ( s n B , ∗ i , s i , t i ) 0 ≤ i ≤ n . Definition 10. Let A be an ω -categor y and G a subset of its elements, with grading G n = G ∩ A n . 1. A is fr e ely gener ate d by G if for all ω -ca tegories B , all functors f : | A | n − → B of ω -categ ories and maps of sets g : G n +1 − → B such tha t s n g ( x ) = f ( s n x ) and t n g ( x ) = f ( t n x ) for a ll x ∈ G n +1 , there exists a unique functor ˆ f : | A | n +1 − → B of ω -categor ies such that ˆ f || A | n = f and f | G n +1 = g . 2. A is gener ate d by G if for ea c h n ≥ 0, | A | n +1 is the smallest sub- ω -categ ory of A containing | A | n ∪ G n +1 . If A is freely generated by G , then A is gener a ted by G ([47]). F o r Parity complexes C , D , a ma p of sets f : C − → D which resp ects the g rading induces a morphism N ( f ) : N ( C ) − →N ( D ), sending ( M , P ) to ( f ( M ) , f ( P )). Let us co nsider only graded maps of sets f : C − → D such that • for a ll x ∈ C 0 , f ( x ) ⊂ D 0 is a singleton set, and • for all n ≥ 0 and x ∈ C n +1 , f ( x ) is well for med, f ( x + ) = ( f ( x − ) ∪ f ( x ) + ) \ f ( x ) − , and f ( x − ) = ( f ( x + ) ∪ f ( x ) − ) \ f ( x ) + . Parit y complex es to gether with graded set maps f : C − → D with these tw o pr oper ties for m a category P ar ity of par it y complexes. The tw o co nditions are chosen so that the functor N : Gr aded S ets − → ω C at restricts to a functor O : P ar ity − → ω C at . Of particular interest a re the parity co mplexes ˜ ∆ n , which we now define. r-dimensiona l elements of ˜ ∆ n are s ubsets v = { v 0 < v 1 < ... < v r } of [ n ] := { 0 , 1 , ...n } ⊂ N of size r + 1. W e will often deno te such a v ∈ ˜ ∆ n r by ( v 0 v 1 ...v r ). The i-th face o f v ∈ ˜ ∆ n r , deno ted δ i v = { v 0 , ..v i − 1 , ˆ v i , v i +1 , ..., v r } ∈ ˜ ∆ n r +1 , where ˆ v i denotes omission of v i . Now define the face o p erator s v ξ = { δ i v | i ∈ [ r ] and i is of par it y ξ } for ξ ∈ { + , −} . A morphism [ n ] α − → [ n ] in ∆ induces a mor phism ˜ ∆( α ) : ˜ ∆ m − → ˜ ∆ m sending v ∈ ˜ ∆ m r to ∅ if αv i = αv i +1 for some i and to αv = { αv 0 , ..., αv r } o therwise. Thus, ˜ ∆ is a functor from ∆ to P a rity , and we obtain the co mposition ∆ ˜ ∆ − → P ar ity O − → ω C at . The ω -categor y O ( ˜ ∆ n ) is ca lled the n-th o rien tal and has a uniq ue non-identit y n-mo rphism h (01 ..n ) i . F or a morphism [ m ] α − → [ n ] in ∆, O ( ˜ ∆( α )) maps h v i to h αv i . The pro duct of parity complexes was shown in [44] to b e a pa rit y co mplex. F o r parity complexes C , D , let ( C × D ) n = S p + q = n C p × D q , and for ξ ∈ { + , −} , ( x, y ) ξ = x ξ × { y } ∪ { x } × y ξ ( p , where ξ ( p ) = ξ if p is even and has the o pposite parity of p is o dd. 14 4.1.2 The N erv e of an ω -Category and the Street-Rob erts Co njecture In [43], Street defines the nerve functor N : ωC at − → sS et with left adjoin t F ω . The nerve of a n ω -categor y A consists of co mposing O ˜ ∆ with the Y oneda embedding ω C at − → S et . More explicitly , The n-simplices of N A are H om ω C at ( O ( ˜ ∆ n ) , A ). F or a n n-simplex x : O ( ˜ ∆ n ) − → A and morphism α : [ m ] − → [ n ] in ∆, α ∗ x ∈ N A m is the comp osition of x with O ( ˜ ∆( α )). The left-adjoint F ω : sS et − → ω C at is the left Kan extension of O ◦ ˜ ∆ : ∆ − → ω C at a long the Y oneda embedding Y : ∆ − → sS et . F or a simplicial set X , F ω ( X ) is charac- terized by the following pro perty . F o r each n-simplex x ∈ X n , ther e is a a functor ι x : O ( ˜ ∆ n ) − → F ω ( X ) of ω -categ ories such that for any morphism α : [ m ] − → [ n ] in ∆, ι α ∗ x = ι x ◦ O ( ˜ ∆( α )), a nd for an y other ω -categor y A with such a family of maps j x : O ( ˜ ∆ n ) − → A , n ∈ N , x ∈ X n , j factors through ι . F o r X ∈ sS e t and x ∈ X n , let [ [ x ] ] = ι x ( h 01 ..n i ). T o get an idea of what the nerve of an ω -category lo oks like, a n n- simplex of N A lo oks like a draw- ing of an n-simplex in the ω catego ry A , mea ning a n n-simplex lab eled with an n- morphism in A a nd k-dimensional faces a re la b eled with k-morphisms in A . It is an easy e x ercise to c hec k that N A 0 = A 0 , N A 1 = A 1 . A 2 -simplex x ∈ N A 2 is a functor of ω -ca tegories x : O ( ˜ ∆ 2 ) − → A , which co nsists of a 0- ob jects x ( h 0 i ) , x ( h 1 i ) , x ( h 2 i ) ∈ A 0 , 1-mo rphisms x ( h i i ) x ( h ij i ) − → x ( h j i ) in A 1 for i, j ∈ [2], and a 2-mor phism x ( h 012 i ) ∈ A 2 such that s 1 x ( h 012 i ) = x ( h 02 i ) and t 1 x ( h 012 i ) = x ( h 12 i ) ∗ 0 x ( h 01 i ). When we restrict to P ic ω , Theor em 4.5 g uarantees tha t N : P ic ω − → sAb is an equiv alence. In g eneral, how ev er, N : ω C at − → sS et is not an equiv alence. The problem is that viewing an ω -categ ory as a simplicial set by taking its nerve loses so me informatio n. The simplicial set no lo nger remember s which n-s implices represent identit y morphisms and so it fo r gets how to comp ose morphisms . T o remedy this situation, in [43, 36], Stree t and Ro berts mo dify the mo dify the nerve construction to take v alues in the ca tegory C s o f “complicial sets.” A complicial set is a simplicial set X to gether w ith a co llection of simplices tX called thin simplices which sa tisfy certa in axio ms. T o name a few, • No 0- simplex o f X is in tX , • the o nly 1 - simplices in tX ar e dege nerate 1-s implices, • the de g enerate simplices o f X a re in tX , • a nd for each ( n − 1)-dimensio nal k- horn for n ≥ 2, 0 < k < n has a unique thin fille r . The other pro perties ca n b e found in [47]. A mor phism o f co mplicia l s ets f : ( X, tX ) − → ( Y , tY ) is a morphism f : X − → Y of s implicial sets such that f ( tX ) ⊂ tY . Remark 4.2. Co mplicial sets is a full sub categor y of a larg er catego ry S tr at of str a tifie d sets who se ob jects are pairs ( X , tX ) but which a re not re q uired to satisfy a ll of the a x ioms listed ab ov e for complicial sets . Morhphisms, of c ourse, ar e simply mor phis ms of simplicial se ts which pres e r v e thin simplices. There is a natural wa y of taking the pro duct ⊗ of tw o stra tified sets, wher e the underlying simplicial set of X ⊗ Y is X × Y . F or instance, the thin r-s implices in ∆ n ⊗ ∆ 1 are the simplice s ( x, y ) ∈ ∆ n r × ∆ 1 r such that x is degenerate at some 0 ≤ j < r and y is dege nerate at s o me k ≥ j . The enhanced nerve construction N : ω C at − → C s sends A to ( N A, tN A ), where the thin n-simplices in N A a re the simplices x : O ( ˜ ∆ n ) − → A such that x ( h 01 ...n i ) is an ( n − 1)-morphism. Co mp osing N with the forgetful functor C s − → sS et (( X, tX ) 7→ X ) gives the or iginal nerve construction. The nerve N has a left adjoint F ω so that F ω (( X, tX )) is attained from F ω ( X ) by “collaps ing ” mo rphisms co rresp onding to thin simplices, a pro cess des cribed in detail in [4 7]. Theor em 4.3, known as the Street-Rob erts conjectur e, was prov en by V erity in [47]. Theorem 4. 3. N : ω C at − → C s is an e quivalenc e of c ate gories. Remark 4.4. Mo re recently , in [3 3], Nik olaus defines a model categ ory o f algebr aic Kan c omplexes similar to the categ ory C s , whic h spe c ifies a distinguished filler for each horn. He shows that algebraic Kan co mplexes is Quillen equiv alen t to simplicia l sets. 15 4.2 The Dold-Kan T riangle The Dold-Ka n co rresp ondence [1 3] gives an equiv alence betw een C h + ( Ab ) and sAb , simplicial o b jects in ab elian groups (or e q uiv alently , a belian g r oup ob jects in sS et ). F ur thermore, sAb and C h + ( Ab ) hav e model structures. T he mo del structure o n sAb is induced by the forg etful functor U : sAb − → sS et . Sp ecifically , sAb inherits the w eak equiv alences and fibrations fr o m sS et ; f is a weak equiv alenc e in sAb if and o nly if U ( f ) is a weak e quiv alence in sS et , and f is a fibration in sAb if and only if U f is a fibration in sS et . The mo del structure on C h + ( Ab ) has qua si-isomorphisms as the weak equiv alences, degree- wise e pimorphisms (in po sitiv e degree) as the fibrations, and degree-wis e monomo r phisms with pro jective cok ernels as cofibratio ns . Additionally , P ic ω inherits a model structure from C h + ( Ab ) via the equiv a lence C h + ( Ab ) − → P i c ω . The weak-equiv alences in P ic ω are morphisms which are e quiv alences of the underlying ω - categories. The Dold- Kan corresp ondence is in fact an equiv alence of mo del categories, as is explained in [38]. W e hav e seen that P i c ω ≃ C h + ( Ab ) ≃ sAb as model categories, but also the following theor em of Brown relates these tw o corres p ondences in the following wa y . Theorem 4.5. ([10], [34]) The c omp osition N ◦ P : C h + ( Ab ) − → sAb is t he same as the Dold-Kan c orr e- sp ondenc e. In other wor ds, the fol lowing diagr am c ommu tes u p to isomorphism. C h + ( Ab ) P − − − − → P ic ω   y D   y N sAb sAb wher e D denotes the Dold-Kan c orr esp onde nc e. Since the Dold-Kan corre s pondence sends X ∈ sAb to A • , where A n = L n − 1 i =0 K er d i , it is now clear from the comments in sectio n 3.3 that N − 1 : sAb − → P ic ω satisfies N − 1 ( X ) ≃ L ∞ n =0 ∩ n − 1 i =0 K er d i . In the Dold-Kan co rresp ondence q uasi-isomorphisms corr espond to weak eq uiv alences in sS et , and by Prop osition 3.8, quasi- isomorphisms corres pond to equiv alences of P ic ard ω -categ ories. Under the equiv a- lence N : P ic ω − → sAb , equiv alence s of ω -categorie s co rresp ond to weak e q uiv alences in sAb . Moreover, upon taking the geometric realiza tio n : | · | : sAb − → T op , the equiv alences of ω -categories a re iden tified with weak equiv ale nce s of top ological spaces . Lo calizing with r espect to weak equiv alences, we hav e an embedding of H o ( P ic ω ) into the homotopy ca tegory o f to polog ic al spaces. 5 The Dold-Kan Corresp ondence for S h ea v es Throughout the next tw o sections, fix an essentially small site S with enough p oint s equipp ed with a Grothenieck top ology , such as the ca tegory of manifolds with the Etale to p ology o r o p en sets o n a fixe d manifold X . Henceforth, let “prehseaf ” mean a preshea f on S , i.e. a functor from S op int o some category . 5.1 Definitions Definition 11. F or a pre sheaf F with v a lues in mo del categor y M , an ob ject X of S a nd an op en cover U = { U i } of X , let ˇ F U denote the cosimplicia l diag ram Y F ( U i ) ⇒ Y F ( U ij ) ⇛ F ( U ij k ) ... in M . W e write ˇ F = ˇ F U when the op en cov er is unders too d. W e say that F satisfies ˇ Ce ch desc ent with r esp e ct t o U if the natura l map F ( X ) − → hol im ˇ F U is a w eak equiv alence in M . W e say that F satisfie s ˇ Ce ch desc ent if F s atisfies ˇ Cech descent with resp ect to all ob jects X ∈ S a nd all op en covers U of X . 16 Let X be an ob ject of S , which we think of as a discrete pr esheaf o f simplicia l sets. The conce pt of hyp er c over U − → X is defined precisely in [14]. Informally , we may think of it a s a resolution of a ˇ Cech co v er of X . Notice that for a hyper co ver U − → X , and pr esheaf F with v a lues in mo del categor y M , F ( U ) is a cosimplicial diagr am in M since U is a simplicial diagram in S , and we have a morphism F ( X ) − → F ( U ) in M ∆ , where X is co ns idered as a consta nt dia gram. Definition 12. Let U − → X b e a hypercover and F b e a preshea f with v alues in mo del categ ory M . W e say that F satisfies desc ent with r esp e ct t o U − → X if F ( X ) − → hol imF ( U ) is a weak equiv alence in M . W e say that F satisfies desc ent if it satisfies descent with resp ect to all hypercovers. By “simplicial presheaf ” w e mean a presheaf with v alues in sS et . Let ˜ P r e sS et denote simplicial pr esheav es which ar e levelwise sheaves of sets (i.e. simplicial o b jects in sheav es of sets). In gener al, fo r a categ ory C , we denote pres hea v es on S with v alues in C by P r e C , and we let ˇ S h C denote those presheaves whic h s a tisfy ˇ Cech descent and S h C denote thos e satisfying descent, provided that C is a mo del ca tegory . F or sho rthand we write P r e ω for P r e ω C at and P re ω Ab for P r e P ic ω . Remark 5 .1. It was shown in [14] that , ˇ S h sS et are the presheav es which satisfy de s cen t for all b ounded hypercov ers and that there exist presheav es satisfying ˇ Cech desce n t but not desc e n t for all hyp e r co vers. Prop osition 5.2. F or a pr eshe af F of s impli cial ab elia n gr oups, F sat isfies ( ˇ Ce ch ) desc ent if and only if U F : S op − → sS et satisfies ( ˇ Ce ch) desc ent. Pr o of. Let D : ∆ op − →S b e a simplicial diagr am a s socia ted to a C ¸ ech complex ˇ C U − → X (i.e. the ˇ Cech nerve of an op en cov er of some X ∈ S ). W e know that F ( X ) − → holi m ( F D ) is a weak equiv alence if and only if U F ( X ) − → U hol im ( F D ) is a weak equiv alenc e . W e would like to show that F ( X ) − → hol imF D is a weak equiv ale nce in sAb if and only if U F ( X ) − → hol im ( U F D ) is a weak equiv a lence in sS et . By the t w o out of three pr operty of weak eq uiv alences, this is true provided that U h olim ( F D ) − → holim ( U F D ) is a weak equiv ale nce in sS et . Thus, to c o mplete the pro of, it suffices to show that U ( hol imF D ) − → holim ( U F D ) is a weak equiv alence. Since U : sAb − → sS et is the right a djoin t in a Quillen pair ( Z [ − ] , U ), it naturally follows that for any diag ram G in sAb , U ( hol imG ) − → holi m ( U G ) is a weak equiv a le nc e . Remark 5.3. Since the category A b of shea ves of ab elian groups on S is ab elian, the Dold-Ka n corr e- sp ondence provides an equiv alences C h + ( A b ) − → ˜ P r e sAb bec ause simplicial ob jects in A b are the same as ˜ P r e sAb . Prop osition 5. 4. Neither ˇ S h sAb nor ˜ P r e sAb is c ontaine d in the other. Pr o of. T o s ee this, consider the following example of a presheaf F ∈ P r e sS et which satisfies ˇ Cech descent but which is not a levelwise preshea f. Let S = O p ( X ), op en sets on a manifold X , and let A 0 be any non-zero ab elian group. Consider the pr esheaf of abelia n groups A suc h that A ( X ) = A 0 and A ( U ) = 0 if U 6 = X and the complex A ∗ ∈ C + ( A b ) which is A in each degr ee and whos e differential is the ident it y map o n A . The corres p onding presheaf of simplicial a b elian gr oups satisfies ˇ Cech descent but levelwise is not a shea f o f s ets. On the other hand, take any sheaf of abelia n groups A and consider the complex A − → 0 in degrees 1 and 0. The corr esponding presheaf P ( A − → 0) of ω -catego ries and in fact a pr esheaf o f 1 -categories. How ever, it do es not satisfy desce nt fo r sta cks. Ho lla nder shows in [20] that a stack satisfies descent if and only if its nerve satisfies ˇ Cech descent as a simplicia l presheaf. Hence, N P ( A − → 0) is a preshea f of simplicia l ab elian groups which is levelwise a sheaf but do es no t satisfy ˇ Cech descent, showing tha t neither condition implies the other. 5.2 Mo del Structures and Deriv ed Dold-Kan There are several mo del structur es on simplicial presheaves. There are, o f course, the pr o jective and injec- tive model structures [3 1, 19], whic h we denote by P r e pro j sS et , P r e inj sS et resp ectiv ely . W ea k equiv a lences are the sectionwise weak equiv a lences. In the pr o jective mo del str ucture, fibra tions are the sectionwise fibratio ns, 17 and in the injective mo del structure, the cofibra tio ns are the sectionwise cofibra tions. F o r each of these, one can take the left B ousfield lo calizatio n P r e loc,inj sS et and P r e loc,pr oj sS et at the hyperc o v ers. The existence of the lo calalization P r e loc,inj sS et follows from the work of Jardine [2 5], and the co nstruction of the lo cal pr o jective mo del structure is due to Blander [3]. The weak equiv alences in P r e loc,pr oj sS et and P r e loc,inj sS et are the stalkwise weak equiv alences o f s implicial sets since S ha s enoug h p oint s [26]. The imp ortant feature of the lo cal mo del structures is that in P re loc,inj sS et , the fibra n t ob jects a re the presheav es which are fibran t in P r e inj sS et and sa tisfy descent for all hyper co v ers. Fibr an t ob jects in P r e loc,pr oj sS et are the ones which are sectionwise Kan complexes and satisfy descent for a ll hyperc overs. Jardine shows the existence o f a model structure on P r e sAb such that a mo r phism is a w eak eq uiv alence or fibration if and only if it is a weak equiv alence o r fibration in P r e loc,inj sS et [24]. F ro m this, the next tw o results follow easily . First we see that for any presheaf of simplicia l abelia n groups, there exists a shea fifica tion (i.e. lo cal fibr an t replacemen t) which is also a preshea f o f simplicial ab elian groups. The second result states that there is a derived Dold- Kan corresp ondence. Lemma 5.5. L et U : P r e sAb − → P r e sS et denote the for getfu l funct or. F or every X ∈ P r e sAb , ther e is a map X f − → Y i n P r e sAb such t hat U f is a we ak e quiva lenc e and U Y is a fibr ant obje ct in P r e loc,inj sS et and P r e loc,pr oj sS et . Pr o of. If X ∈ P r e sAb , take a fibrant r eplacemen t X − → Y for X in P re sAb in the mo del structure of [24] describ ed above. U Y ∈ P r e loc,inj sS et is fibrant since U : P r e sAb − → P r e loc,inj sS et preserves fibra tions. Additionally , since U Y is fibrant in P re loc,inj sS et , it sa tisfies descent for all hypercovers. Since it is a preshea f taking v alues in sAb , it is sectionwise fibra n t. Ther efore, U Y is also fibrant in P r e loc,pr oj sS et . Prop osition 5.6. L et P ′ denote the ful l sub c ate gory of P re sAb sp anne d by obje ct s satisfying desc ent for hyp er c overs. L o c alizing at lo c al we ak e qu iva lenc es, we c an form the ho motopy c ate ge gory H o ( P ′ ) , and H o ( P ′ ) is e quivale nt to D + ( A b ) , the derive d c ate gory of chain c omplexes of she aves of ab elia n gr oups in non- ne gative de gr e es. Pr o of. First observe that the inclusion ˜ P r e sAb ⇆ P re sAb and the lev elwise sheafifica tion functors descend to equiv ale nce s of homotopy categor ies since weak equiv alence s in Jar dine’s model structure o n P r e sAb are the lo cal weak equiv alences . Beca use C h + ( A b ) is equiv alen t to ˜ P r e sAb , we need only show tha t H o ( P re sAb ) is equiv ale nt to H o ( P ′ ) to complete the pr o of. Since the forgetful functor U : P r e sAb − → P r e pro j,loc sS et preserves fibrant o b jects, as was noted in the pro of of Lemma 5.5, the full subc a tegory P cf of cofibrant fibr an t ob jects is contained in P ′ . It is easy to chec k that since P cf ⊂ P ′ ⊂ P re sAb , H o ( P ′ ) ex is ts and is equiv a len t to H o ( P re sAb ). Remark 5.7. Co nsider the case of a simplicia l presheaf o n a top ological space X . In general it is a str onger requirement on a simplicial pr esheaf o n X to satisfy descent for all hypercovers than it is to satisfy ˇ Cech descent. Lurie explains in [31] that if X has finite covering dimension, then the tw o co nditions are the same. How ev er, we are interested in presheaves o n manifolds, all o f which hav e finite cov ering dimensio n. If we consider sheaves on the site of all differentiable manifo lds , then the result is unchanged since we a re only considering the hypercovers of [14] rather than the most gener al hypercovers. 6 Omega Descen t The standard definition of desce n t, used in [20, 25, 14], is given in Definitions 11 and 12. In this section, how ev er, we descr ibe a g lue ing co ndition for presheav es of ω - categories a nd show that in the ca se when the presheaf takes v alues in P ic ω , it c o incides with the homotopy limit descent condition. This is an a ttempt to expand on the work of Hollander [20], who show ed that descent for 1-sta cks can b e describ ed in a homo - topy theoretic wa y whic h is co ns isten t with descent for simplicial presheaves. Our definition o f the glueing 18 condition is motiv ated by B r een’s description o f descent for 2-stacks [6]. The idea is that a sheaf A of ω - categorie s on S satisfies the glueing condition if o ne ca n glue 0-ob jects, 1- ob jects, and k- ob jects for any k ≥ 0. Informally , glue ing o f 0-ob jects ha s the following meaning. Given a n op en cov er U = { U i } i ∈ I of X ∈ S the data for g lue ing of 0 -ob jects cons ists o f 0 -ob jects a i ∈ A ( U i ) 0 which ar e identified on int ersections via 1-morphisms a ij ∈ A ( U ij ) 1 , a ij : ( a i ) | ij − → ( a j ) | ij , the 1- morphisms a ij are identified o n triple in tersections via 2-mo rphisms a ij k : a j k ∗ 0 a ij = ⇒ a ik and so on. The g lueing condition for 0-o b jects s ta tes that for any such sy stem ( { a i } i ∈ I , { a ij } i,j ∈ I , { a ij k } , ... ), there exists an 0-ob ject x ∈ A ( X )– uniq ue up to isomo rphism– with isomorphisms x | U i − → a i which fit together in a consisten t way . In other words, the sys tem can b e glued to a global 0-ob ject in an essentially unique way . F o r A to satisfy the glueing condition, it m ust satisfy the glueing co ndition for 0-ob jects, a nd for each pair of sections x, y ∈ A ( X ) k , the presheaf o f ω -ca tegories H om k A ( x, y ) satisfies the g lueing condition for 0-ob jects. T o ma k e this descriptio n for mal, Let Y : ∆ − → s S et b e the Y oneda embedding. A system ( a i ∈ A ( U i ) 0 , a ij ∈ A ( U ij ) 1 , ... ) as ab o ve c an b e thought of as a morphism a ∈ H om sS et ∆ ( Y , ˇ N A ). Here, N is the nerve functor so that N A is a simplicia l presheaf, a nd ˇ N A is the diag ram fr om Definition 11. There is a restric tio n map of the consta n t diagr am ρ : N A ( X ) const − → ˇ N A . I n o rder to co mpare N A ( X ) 0 with, H om sS et ∆ ( Y , ˇ N A ), we identify N A ( X ) 0 with { F ∈ H om sS et ∆ ( Y , N A ( X ) const ) | F (∆ n ) = F (∆ 0 ) for all n } , so N A ( X ) 0 is a subset of H om sS et ∆ ( Y , N A ( X ) const ) and ρ maps N A ( X ) 0 to H om sS et ∆ ( Y , ˇ N A ). Remark 6.1. In Street’s definition of the descent [46], he defines a descent o b ject D esc ( ˇ N A ) ∈ ω C at ob ject of A ∈ P r e ω with res pect to the ˇ Cech co mplex for U . Using the adjunction, F ω : sS et ⇆ ω C at : N , H om sS et ∆ ( Y , ˇ N A ) is ident ified with the 0-o b jects o f D esc ( ˇ N A ). Remark 6.2. The c ategory of cosimplicia l ob jects in sS et is a simplicial mo del catego r y [18]. F or X , Y ∈ sS e t ∆ , the enrichmen t ov er simplicia l sets is given by hom ( X, Y ) n = H om sS et ∆ ( X × ∆ n , Y ), where hom ( X , Y ) ∈ sS e t . F or K ∈ sS et , X ∈ sS e t ∆ , X × K ∈ sS et ∆ is ( X × K ) n = X n × K , and for any Y ∈ sS e t ∆ , H om sS et ∆ ( X × K , Y ) ≃ H om sS et ( K, hom ( X , Y )). The homotopy of Definition 13 is really a homotopy H : ∆ 1 − → hom ( Y , ˇ N A ) be tween 0-simplices F a nd ρG in hom ( Y , ˇ N A ). The simplicial se t h om ( Y , ˇ N A ) is commonly called the total s pa ce T ot ( ˇ N A ) o f ˇ N A . Definition 13. Let A b e a preshea f of ω -c ategories on S , and let U = { U i } i ∈ I be an op en cov er o f X ∈ S . W e say that A sa tisfies 0-glueing with respe c t to U if for all F ∈ H om sS et ∆ ( Y , ˇ N A ), there exists a homoto py H : H om sS et ∆ ( Y × ∆ 1 , ˇ N A ) fr o m F to ρG for s ome G ∈ N A ( X ) 0 . W e say tha t A a lso satisfies unique 0-glueing with r e s pect to U if tw o 0-ob jects a, b ∈ A ( X ) 0 = N A ( X ) 0 are iso morphic in A ( X ) whenever ρa and ρb are isomorphic in hom ( Y , ˇ N A ). The intuitiv e meaning of the uniqueness of g lueing is that if a, b ∈ A ( X ) 0 are lo cally isomor phic in a consistent wa y , then a, b are is omorphic. Hence if G a nd G ′ ∈ A ( X ) 0 glue F ∈ hom ( Y , ˇ N A ) in the notation of Definition 13, then G a nd G ′ are isomor phic in A ( X ). T o describ e k -glueing for k > 0, first o bserve that ther e is a shift functor [1 ] : ω C at − → ω C at , where if A = ( O b ( A ) , s k , t k , ∗ k ) k ∈ N , A [1] is the ω -catego ry ( O b ( A ) , s [1] k = s k +1 , t [1 ] k = t k +1 , ∗ [1 ] k = ∗ k +1 ) k ∈ N . F o r x, y ∈ A 0 , we ar e esp ecially interested in sub- ω -catego ries A [1] x,y = H om A ( x, y ) := { a ∈ A [1] | s 0 a = x , t 0 a = y } . More ge nerally , for k ≥ 1, x, y ∈ A k − 1 , le t A [ k ] x,y = { a ∈ A [ k ] | s k − 1 a = x , t k − 1 a = y } . Observe that ( A [ k ])[1] = A [ k + 1]. Remark 6.3. A [ k ] x,y = H om k A ( x, y ) from Definition 2. Definition 14. Let A be a presheaf of ω -ca tegories, X an ob ject in S and U an ope n cov er of X . 1. Supp ose k > 0. Then A satisfies the (unique) k-glueing c ondition with r esp e ct to U if for x, y ∈ A ( X ) k − 1 , A [ k ] x,y satisfies (unique) 0-glueing whenev er s k − 2 x = s k − 2 y and t k − 2 x = t k − 2 y . W e use the con ven tion that s − 1 x = t − 1 x = 0 for all x . 19 2. W e say that A satisfies ω -desc ent with r esp e ct to U if for all k ≥ 0, A satisfies the unique k-glueing condition. 3. W e s a y that A satisfies ω -desc ent for lo ops if for all x ∈ A ( X ) 0 , each A [ k ] x,x satisfies the unique 0-glueing condition. Definition 15. Let A b e a preshea f of Pic a rd ω - categories. W e say that A satisfies ω -desc ent , k-glueing, etc. if it sa tisfies ω -descent with resp ect to U , k-glueing with resp ect to U , et cetera, resp ectiv ely for all ob jects X of S a nd op en cov ers U of X . Our go a l for the r emainder of this section is prov e the following theorem, which relates tw o notions of descent. Theorem 6.4. L et A b e a pr eshe af site S with values in P ic ω . Then A satisfies ω -desc ent if and only if it satisfies ˇ Ce ch desc ent. It is imp ortant to note that since N : P ic ω − → sAb is an equiv alence o f mo del categorie s, a preshea f A ∈ P re ω satisfies ( ˇ Cech) descent if and only if its nerve N A ∈ P re sS et satisfies ( ˇ Cech) descent. T o prov e Theor em 6.4, we will show that for each op en cov er U , a pr esheaf A of P icard ω -categorie s on S satisfies the unique glueing c ondition with r espect to U on X if and o nly if it satisfies ˇ Cech desce n t with resp ect to U . F or the rest of the section, fix an o b ject X ∈ O b ( S ) a nd op en cov er U = { U i } i ∈ I of X . 6.1 Lo op and Pa th F unctor s W e define a pair of adjoint functors [ − 1] : C h + ( Ab ) ⇆ C h + ( Ab ) : Ω, w hich form a Quillen pair for the mo del categ ory C h + ( Ab ). The functor [ − 1] is defined as follows. Let B ∈ C h + ( Ab ). Then ([ − 1 ] B ) i = B i − 1 for i > 0, and ([ − 1 ] B ) 0 = 0. Thus, [ − 1] B = ... − → B 1 − → B 0 − → 0, wher e B 0 is in deg ree 1. O n the other hand, Ω is defined by letting (Ω A ) i = A i +1 for i > 0, and (Ω A ) 0 = K er ( A 1 d − → A 0 ). Thus, ([1] 0 , 0 A ) = ... − → A 3 − → A 2 − → K er ( d ), with K er ( d ) ⊂ A 1 in deg ree 0 . Clearly , H om ch + ( Ab ) ([ − 1] B , A ) ≃ H om ch + ( Ab ) ( B , Ω A ), so [ − 1] is left adjoint to Ω. Since [ − 1] preserves weak equiv alences and c ofibrations in C h + ( Ab ), ([ − 1] , Ω) is a Quillen pair (Pr op 8.5.3 in [1 9]). Lemma 6.5. Given the e quivalenc es P ic ω ≃ C h + ( Ab ) ≃ sAb , t he fol lowing p airs of en dofunctors c orr esp ond to e ach other: 1. ([ − 1] , Ω) on C h + ( Ab ) define d ab ove, 2. ([ − 1] , [1] 0 , 0 ) on P ic ω , wher e [1] 0 , 0 is define d after Definition 13 and [ − 1] is define d in Pr op ositio n 3.11, and 3. ( M , L ) , wher e L is given by L ( X ) n = K er ( d X 0 : X n +1 − → X n ) ∩ K er ( d n 1 ) and d L ( X ) n = − d X n +1 , s L ( X ) n = − s X n +1 . One c an describ e L ( X ) n as the ( n + 1) -simplic es in X such that the 0-th fac e and 0-t h vertex of x is 0 . On t he other hand, M ( X ) n = X n − 1 for n > 0 and M 0 = 0 . The structur e maps ar e d M ( X ) i = d X i − 1 for i > 0 and d 0 = 0 . Pr o of. Using the equiv a lences P : C h + ( Ab ) − → P i c ω and K : sAb − → C h + ( sAb ), the result follows easily . Lemma 6 .5 suggests tha t we should think of Ω( A ) a s lo ops in A based at 0. W e can also cons ide r path functors. Lemma 6.6 . Given the e quivalenc es, P i c ω ≃ C h + ( Ab ) ≃ sAb , the fol lowing functors c orr esp ond to e ach other. 1. In P ic ω the p ath functor [1] : P ic ω − → P i c ω is the r estriction of [1] : ω C a t − → ωC at define d imme diately after D efinition 13. 20 2. Π : C h + ( Ab ) − → C h + ( Ab ) define d by Π( A ) = ... − → A 3 − → A 2 − → A 1 ⊕ A 0 , with A 1 ⊕ A 0 in de gr e e 0 and differ ential d ⊕ 0 : A 2 − → A 1 ⊕ A 0 . 3. P ath : sAb − → s Ab defin e d by P ath ( X ) = ˆ S ( X ) ⊕ X 0 , wher e X 0 is a discr ete simplicial set with X 0 in de gr e e 0 and ˆ S : sAb − → sAb is given by ˆ S ( X ) n := K er ( d X 0 : X n +1 − → X n ) and d ˆ S ( X ) n = − d X n +1 , s ˆ S ( X ) n = − s X n +1 . Pr o of. T o mak e the iden tification o f Π with [1 ], let P denote the equiv alence P : C h + ( Ab ) − → P i c ω . Identify- ing (( a + n +1 , a − n +1 ) , ..., ( a + 2 , a − 2 ) , ( a + 1 + a + 0 , a − 1 + a − 0 )) ∈ P (Π( A )) n with (( a + n +1 , a − n +1 ) , ..., ( a + 2 , a − 2 ) , ( a + 1 , a − 1 ) , ( a + 0 , a + 0 − da + 1 )) ∈ ( P A ) n +1 = ( P A )[1] n establishes that [1] ◦ P = P ◦ Π. T o make the identification of P ath with Π, let S : C h + ( Ab ) − → C h + ( Ab ) denote the functor A 7→ ( ... − → A 3 − → A 2 − → A 1 ). It is easily verified using the Dold-Kan cor respo ndence that ˆ S cor respo nds with S . Observe that Π( A ) = S ( A ) ⊕ ( ... 0 − → A 0 ) so that K Π( A ) = K ( S ( A )) ⊕ K ( A 0 ), where K ( A 0 ) is the discrete simplicial ab elian group with A 0 in degree 0. F o r A ∈ P i c ω and 0-o b jects a, b ∈ A 0 , the sub- ω -categ ory A [1] a,b of A [1] is not a Pica rd ω -ca tegory . How ev er, w e ca n still des c ribe its nerve as a sub-ob ject N A [1] a,b of N A [1] in sS et . Denote the pa th functor sAb − → sS et corresp onding to [1] a,b : P ic ω − → ω C at b y P ath a,b . First we a rgue that the n-simplices of N A [1] can b e viewed as the ( n + 1 )-simplices of N A for which the 0th face is s n +1 0 b for s ome vertex b ∈ N A 0 . By Lemma 6 .6, N A [1] n = { y ∈ ( N A ) n +1 | d 0 y = 0 } ⊕ A 0 , and there is an inclusion N A [1] n ֒ → N A n +1 given b y ( y , b ) 7→ y + s n +1 0 b . Lemma 6. 7 . L et A b e a Pic ar d ω - c ate go ry. Then N A [1] a,b n c onsists of simplic es x ∈ N A n +1 for which t he 0-th vertex v 0 x = ( d 1 ) n +1 x = a and d 0 x = b . Pr o of. W e pro v e by induction that fo r an n-simplex in x ∈ N A [1] n ⊂ N A n +1 such that v 0 x = a and d 0 x = b , x ∈ N A [1] a,b . F or n = 0, let x b e a 0-simplex in N A [1] such that v 0 x = a and d 0 x = b . Since ( N A ) 1 = A 1 , we can think of the 1-simplex x in N A as a 1- morphism in A , wher e the so urce of x is s 0 x = d 1 x = v 0 x = a and the target of x is t 0 x = d 0 x = b . Hence, x ∈ N [1] a,b 0 . Now let x be an n-simplex in N A [1] n ⊂ N A n +1 such that v 0 x = a a nd d 0 x = b . Observe that v 0 x = v 0 ( d i x ) = a and d 0 x = d 0 d i x = b for every 0 < i ≤ n + 1, and in fact, v 0 w = a and d 0 w = b for every r-dimensiona l fa c e w of x . W e ma ke use of the notation defined in § 4.1.1 for the remainder of the pro of. By constr uc tio n, an n-simplex in N A [1] is a morphism x ∈ H om ω C at ( O ( ˜ ∆ n ) , A [1]), where O ( ˜ ∆ n ) is Str eet’s n- th oriental, defined in § 4.1 .1 . F o r an ω -categor y B = ( O b ( B ) , ∗ i , s i , t i ) i ∈ N , and m ≥ 0 , let | B | m = ( B m , ∗ i , s i , t i ) 0 ≤ i ≤ m − 1 denote the m - c ategory for med by taking all k -mor phis ms for k ≤ m . In [43], Street shows that a choice of morphism x ∈ H om ω C at ( O ( ˜ ∆ n ) , A [1]) is equiv alen t to a morphism g ∈ H om ω C at ( |O ( ˜ ∆ n ) | n − 1 , A [1]) and α ∈ A [1] n such that s [1 ] n − 1 α = g ( s n − 1 h (01 ...n ) i ) and t [1 ] n − 1 α = g ( t n − 1 h (01 ...n ) i ), wher e h (01 ...n ) i is the unique non- trivial n-morphism in O ( ˜ ∆ n ). Hence, the 0 -source and 0-targ e t of α in A are determined b y g because s 0 α = s 0 s [1] n − 1 α = s 0 g ( s n − 1 h (0 ...n ) i ) = g ( s 0 s n − 1 h (0 ...n ) i ) = g ( s 0 h (0 ...n ) i ), and similarly , t 0 α = g ( t 0 h (0 ...n ) i ). Street shows that g ( s n − 1 h (0 ...n i ) and g ( t n − 1 h (0 ...n i ) a re comp ositions of of g applied to h β i for so me β : ∆ r ֒ → ∆ n with r < n . Using prop erties (1d) a nd (2b) of Definition 1, tak ing s 0 or t 0 of a comp osition s imply applies s 0 or t 0 to the last morphism in the chain of comp ositions. Th us, s 0 α = s 0 g ( h β i ) a nd t 0 α = t 0 g ( h β i ) for some β : ∆ r ֒ → ∆ n with r < n . But O ( ˜ ∆ r ) β ∗ − → O ( ˜ ∆ n ) x − → A [1] is simply one of the r-faces of the n-simplex x . Since xβ ∗ is an r- face of x , a = v 0 x = v 0 ( xβ ∗ ) and b = d 0 ( xβ ∗ ). By induction hypothesis, s 0 ( g h β i ) = a and t 0 ( g h β i ) = b , so s 0 α = a and t 0 α = b . Also, for r < n , every r-dimensio nal face xγ ∗ for γ : ∆ r ֒ → ∆ n , v 0 ( xγ ∗ ) = v 0 x = a and d 0 ( xγ ∗ ) = d 0 x = b , whence s 0 ( g h γ i ) = a and t 0 ( g h γ i ) = b b y induction hypothesis. Thus, for r ≤ n , every r-face w of x , the r-morphis m g ( h γ w i )in A [1] representing w has a as its 0- s ource and b as its 0- ta rget. Therefore, x ∈ N A [1] a,b . 21 Remark 6.8. F or X ∈ sAb , P ath a,b X (the paths in P ath ( X ) from a to b ) is isomor phic to the left mapping space H om L X ( a, b ) of Lurie [31]. 6.2 Equiv alence of Descen t Conditions Lemma 6 .9. L et A b e a pr eshe af of ω -c ate gories. The u nique 0-glueing c ondition is e qu iva lent to t he c ondition t hat π 0 ( N A ( X )) ≃ π 0 ( holi m ˇ N A ) . Pr o of. In [5] ch.10-11, it was prov ed that for G ∈ sAb ∆ , hom ( Y , G ) − → hol imG is a w eak e q uiv alence. Ther e- fore, π 0 ( hom ( Y , ˇ N A )) ≃ π 0 hol im ˇ N A . W e conclude the pr oof b y arguing that π 0 ( N A ( X )) ≃ π 0 ( hom ( Y , ˇ N A )) if and only if A sa tis fie s the uniq ue 0- glueing condition. The 0-glueing condition states that for all f ∈ H om sS et ∆ ( Y , ˇ N A ), there exists H ∈ H om sS et ∆ ( Y × ∆ 1 , ˇ N A ) such that H |Y ×{ 0 } = f and H |Y ×{ 1 } = ρg for some g ∈ A ( X ) 0 . Since H om sS et ∆ ( Y × ∆ 1 , ˇ N A ) = hom ( Y , ˇ N A ) 1 , this is eq uiv alent to asking that for every vertex f ∈ hom ( Y , ˇ N A ) 0 , there exists H ∈ hom ( Y , ˇ N A ) 1 such that d 1 H = f a nd d 0 H = ρg for some g ∈ N A ( X ) 0 . In o ther words, The 0-glueing condition s tates tha t ρ ∗ : π 0 ( N A ( X )) − → π 0 ( hom ( Y , ˇ N A )) is a surjection. The uniqueness part o f the unique 0-glueing condition states that ρ ∗ is also injective. Lemma 6.1 0 . L et A b e a pr eshe af of Pic ar d ω -c ate gories. Then A s atisfi es the unique glueing c onditio n for al l lo ops if and only if it satisfies t he unique glueing c ondition for lo ops at 0 . F urthermor e, for any a ∈ A ( X ) 0 , A [1 ] 0 , 0 ≃ A [1] a,a in P r e ω . Pr o of. First w e sho w tha t for A ∈ P i c ω and a ∈ A 0 , addition by a , p a : A− →A (mapping x 7→ x + a ) defines an isomorphism of ω -catego ries, thoug h not of Pica rd ω -catego ries. T o see that p a is a mor phis m of ω - categorie s, let σ = s or t and n ≥ 0. Then σ n p a ( x ) = σ n ( a + x ) = σ n ( a ) + σ n ( x ) = a + σ n ( x ) = p a ( σ n x ) since a is a 0- ob ject. Co mposition is also preserved; p a ( x ∗ n y ) = a + x ∗ n y = a + x + y − s n x = a + x + y + a − ( s n x + a ) = a + x + y + a − s n ( x + a ) = ( x + a ) ∗ n ( y + a ) = p a x ∗ n p a y . Therefor e, p a is a morphism of ω - categories. Since p a has inv erse p − a , it is an isomor phism. If A is a presheaf of ω -ca tegories and a ∈ A ( X ) 0 , then p a : A− →A is an isomor phism of presheaves of ω -categor ies sending basep oin t 0 to basep oin t a . Lemma 6.11. Le t G ∈ sAb and L : sAb − → sAb b e the functor describ e d ab ove, c orr esp onding to [1] 0 , 0 : P i c ω − → P i c ω . Then π n +1 L k ( G ) ≃ π n L k +1 G for al l k , n ≥ 0 . Pr o of. It follows fro m the definition o f L that H om sS et (∆ n , L G ) ≃ { f ∈ H om sS et (∆ n +1 , G ) | (0) 7→ 0 and f d 0 (012 ...n ) = 0 } . Therefore, { f ∈ H om sS et (∆ n , L G ) | ∂ ∆ n − → 0 } ≃ { f ∈ H om sS et (∆ n +1 , G ) | ∂ ∆ n +1 − → 0 } . Not only are they isomor o phic as sets, but homotopies in { f ∈ H om sS et (∆ n , L G ) | ∂ ∆ n − → 0 } co incide with those in { f ∈ H om sS et (∆ n +1 , G ) | ∂ ∆ n +1 − → 0 } . T o see this, we will show tha t f and g are homoto pic in { f ∈ H om sS et (∆ n , L G ) | ∂ ∆ n − → 0 } if and only if the corr esponding simplices in { f ∈ H om sS et (∆ n +1 , G ) | ∂ ∆ n +1 − → 0 } are homotopic. How ev er, since G and LG are simplicial ab elian groups, it suffice s to show that f b eing ho- motopic to 0 is the sa me in bo th sets. It is a standard fact, found in [22], that fo r f ∈ G n with d i f = 0 for all i , f is homotopic to 0 if and only if and only if there is some h ∈ G n +1 such that d n +1 h = f and d i h = 0 for a ll i ≤ n . Clearly then, f : ∆ n − → LG is homotopic to 0 if a nd only if the corresp onding map ∆ n +1 − → G is homotopic to 0. Prop osition 6.12 . L et A b e a pr eshe af of Pic ar d ω -c ate gories. Then A is a satisfies ˇ Ce ch desc ent if and only if A satisfies the unique glueing c ondition for lo ops. Pr o of. Let G = N A b e the corr esponding pr esheaf of simplicial ab elian groups . Then G satisfies descent for all h yper co vers if and only if G ( X ) − → hol im ˇ G is a weak equiv alence, i.e. π n G ( X ) − → π n hol im ˇ G is an isomorphism for each n ≥ 0 . By le mma 6 .11, π n G ( X ) − → π n hol im ˇ G is an iso morphism for each n ≥ 0 if and only if π 0 ( L n G ( X )) ≃ π 0 ( L n ( holi m G )) for all n ≥ 0 . Since L is right Q uillen, it preser v es ho motop y limits. Ther e fore, π 0 ( L n ( holi m G )) ≃ π 0 ( holi m ( L n G )). In summary , G satisfies ˇ Cech descent if and only if π 0 ( L n G ( X )) ≃ π 0 ( holi m ( L n ˇ G )) fo r a ll n ≥ 0 . B y Lemma 6.9, we conclude that G sa tisfies ˇ Cech descent if 22 and only if L n G sa tisfies the unique 0-g lueing co nditio n for each n. T o say that L n G sa tisfies the unique 0-glueing condition is just to say that G sa tisfies the unique n-glueing condition. Ther efore, G satisfies ˇ Cech descent if and o nly if G satisfies the unique g lue ing conditio n for lo ops based at 0. How ev er, Lemma 6.10 implies that this is equiv alen t to the unique glueing co nditio n for all lo ops. Corollary 6.13. L et b e a A is a pr eshe af of Pic ar d ω -c ate gories. If A satisfies ω -desc ent, then it satisfies ˇ Ce ch desc ent. T o complete the pro of of Theorem 6 .4, w e now sho w that if a preshea f of simplicial ab elian gro ups sa tisfies ˇ Cech descent for all h yper co v ers, it sa tisfies the unique glueing condition, not just for lo ops. Lemma 6 .14. L et A b e a pr eshe af of Pic a r d ω -c ate gories, X ∈ S , and a, b ∈ A ( X ) 0 . If ther e exists f ∈ A ( X ) 1 such that s 0 f = a , t 0 f = b , t hen A [1] a,b ≃ A [1] 0 , 0 in P r e ω , whenc e A [1] a,b is a pr eshe af of Pic ar d ω -c ate gories. Pr o of. There is an isomorphism A [1] a,b − →A [1] a,a sending a section y to x − 1 ∗ 0 y = y + ( x − 1 − b ). First, let us see that for y ∈ A [1] a,b , x − 1 ∗ 0 y ∈ A [1] a,a . W e easily see that s 0 ( y + x − 1 − b ) = s 0 y + s 0 x − 1 − s 0 b = a + b − b = a and t o ( y + x − 1 − b ) = t 0 y + t 0 x − 1 − t 0 b = b + a − b = a so that x − 1 ∗ 0 y ∈ A [1] a,a . Since x − 1 − b ∈ A [1]( X ) 0 , addition by x − 1 − b is a n isomorphism (The pro of is identical to that of Lemma 6.10). Rec all from Lemma 6.10 that A [1] a,a ≃ A [1] 0 , 0 . Lemma 6.1 5 . L et A b e a she af of ω -c ate gories and a, b ∈ A ( X ) 0 . Then A [1] a,b ( X ) − → hol im ˇ A [1] a,b is a we ak e quivalenc e if and only if A [1] a,b ( X ) − → hom sS et ∆ ( Y , ˇ A [1] a,b ) is a we ak e quivalenc e. Pr o of. Let B = N A [1] a,b ∈ P re sAb . If s ome B ( U i 0 ...i n ) = ∅ , then ˇ B = ∅ , and B ( X ) = ∅ , so b oth A [1] a,b ( X ) − → hol im ˇ A [1] a,b and A [1] a,b ( X ) − → hom sS et ∆ ( Y , A [1] a,b ) ar e weak equiv alences . Now supp ose that ea c h B ( U i 0 ...i n ) 6 = ∅ . By the tw o o ut of three axiom for weak eq uiv alences, it suffices to show that hom sS et ∆ ( Y , A [1 ] a,b ) − → hol im ˇ B is a weak equiv alence. By Ch. 11, § 4 in [5 ], the result will follow if we show that ˇ B is a fibrant ob ject in sS et ∆ with Bousfield and Ka n’s mo del structure. F o r X ∈ s S et ∆ , let the n-th matching spac e M n X = { ( x 0 , ...x n ) ∈ X n × X n × ... × X n | s i x j = s j − 1 x i if 0 ≤ i < j ≤ n } , where s i denotes the i-th c o face map X ( σ i ). There is a map X n +1 − → M n X in sS et given by x 7→ ( s 0 x, ..., s n x ). By definition (Ch. 10 § 4 in [5]), X is fibrant if and only if ea c h X n +1 − → M n X , n ≥ 0 and X 0 − →∗ are fibrations in sS et . W e now pro ceed to s how that ˇ B is fibrant. First w e show that w e can endo w each each ˇ B n with the structure of a s implicial abelian g roup s uc h that all s i : ˇ B n +1 − → ˇ B n is a morphism in sAb . F or the op en cov er U = { U i } i ∈ I , we a re a ssuming that for each n ≥ 0 and ea c h α ∈ I [ n ] , B ( U α ) 6 = ∅ , so by Lemma 6.14, we can choo se a group str uc tur e on ˇ B n by choosing a 1-simplex f = { f α } α ∈ I [ n ] with d 1 f = a , d 0 f = b . The goal is to c hoo se a gr oup structure o n e a c h ˇ B s o tha t the coface maps are all morphisms of simplicial ab elian groups. Fir st declar e an equiv alence rela tionship on I [ n ] by setting α ∼ β if U α = U β , and let I [ n ] denote the set o f equiv alence classes. Observe that the coface maps s m : ˇ B n +1 − → ˇ B n are given by ( s m ( { x α } α ∈ I [ n +1] )) β = x σ ∗ m β , where σ m : [ n + 1] − → [ n ] is the monotonic map whic h rep eats only m . T o choose the gro up stuctures on ˇ B n , w e star t with n = 0. Cho osing any group s tr ucture f = { f α } α ∈ I [0] such that f α = f β for α ∼ β . Now, having chosen gr oup s tr uctures for all ˇ B k for k ≤ n such that all coface maps ar e morphisms of simplicial ab elian groups, choose any g r oup structure f = { f α } α ∈ I [ n +1] such that f α = f β if α ∼ β and if α = σ ∗ m β for some β ∈ I [ n ] , f α = f β . T o s e e that such a choice exists, we s imply observe that if α ∼ γ such that α = σ ∗ m β a nd γ = σ ∗ l δ , then U β = U σ ∗ m β = U α = U γ = U σ ∗ l δ = U δ , so γ ∼ δ and f γ = f δ . The coface maps ( s m ( { x α } α ∈ I [ n +1] ) β = x σ ∗ m β are mor phisms of simplicial a belian g roups bec ause for β ∈ I [ n ] , π β s m is the comp osition of the identit y ma p B ( U σ ∗ m β ) − →B ( U β ) with the pro jection π σ ∗ m β : ˇ B n +1 − →B ( U σ ∗ m β ), b o th of which ar e group ma ps since the gro up stuctures f σ ∗ m β and f β were chosen to coincide. 23 W e now demo nstrate that each ˇ B n +1 − → M n ˇ B is surjective. It will follow that these maps ar e levelwise epimorphisms of ab elian groups, hence fibra tions in sS et . Cho ose any n ≥ 0. The pr oof that ˇ B n +1 − → M n ˇ B is surjective is very muc h the same as the pro of in the previous paragra ph. F or x 0 ... x 0 ∈ ˇ B n ( x i = { x α i } α ∈ I [ n ] ), ( x 0 , ..., x m ) ∈ M n ˇ B if and only if for all 0 ≤ l < m ≤ n and α ∈ I [ n − 1] , x σ ∗ l α m = x σ ∗ m − 1 α l . Cho ose a n y y = { y α } α ∈ I [ n +1] such that for all 0 ≤ m ≤ n , β ∈ I [ n ] , y σ ∗ β = x β m . Hence the map ˇ B n +1 − → M n ˇ B sends y to ( x 0 , ...x n ). How ever, we need to chec k that such a choice exists. W e need to show that if σ ∗ m β = σ ∗ l γ , then x β m = x γ l . Supp ose that σ ∗ m β = σ ∗ l γ . Clear ly α = σ ∗ m σ ∗ l δ for some δ , s o α = ( σ l σ m ) ∗ δ = ( σ m − 1 σ l ) ∗ δ = σ ∗ l σ ∗ m − 1 γ . In general, if σ ∗ m τ = σ ∗ m µ , then τ = µ . Hence, β = σ ∗ l δ and γ = σ ∗ m − 1 γ , so x β m = x σ ∗ l δ m = x σ ∗ m − 1 δ l = x γ l . Ther efore, y α is w ell-defined. Lemma 6.16. L et A b e a pr eshe af of Pic ar d ω -c ate gories t hat satisfies ˇ Ce ch desc ent, and let a , b ∈ A ( X ) 0 . If H om sS et ∆ ( Y , ˇ N A [1] a,b ) is non-empty, then ther e exists a p ath x ∈ A ( X ) 1 fr om a to b , and A [1] a,b ≃ A [1] a,a ≃ A [1] 0 , 0 as she aves of ω -c ate gories. It fol lows that A [1] a,b ∈ P r e ω Ab . Pr o of. Cho ose any F ∈ H om sS et ∆ ( Y , ˇ N A [1] a,b ). Let Y (1) denote the ob ject in sS et ∆ for which Y (1) ([ n ]) = ∆ n +1 , Y (1) ([ n ] ∂ i − → [ n + 1]) = ∆ n +1 ∂ i +1 − → ∆ n +2 , and Y (1) ([ n ] σ i − → [ n − 1]) = ∆ n +1 σ i +1 − → ∆ n . First we obs erv e that H om sS et ∆ ( Y , ˇ N A [1] a,b ) is isomor phic to H om sS et ∆ ( Y (1) , ˇ N A ) a,b := { f ∈ H om sS et ∆ ( Y (1) , ˇ N A ) | f n : ∆ n +1 − → ˇ N A n satisfies f ((0)) = a, d 0 f (01 ...n + 1) = b } . W e will define a morphism p : ∆ n × ∆ 1 in sS et which sends ∆ n × { 0 } to a a nd ∆ n × { 1 } to b . First observe that ∆ n × ∆ 1 has non-degenera te ( n + 1)-simplices z k := (01 2 .., k − 1 , k , k , k + 1 , ...n, s k 0 (01) ∈ ∆ n n +1 × ∆ 1 n +1 , 0 ≤ k ≤ n , which satisfy relatio ns d k z k = d k z k − 1 , for k ≥ 1 . F or any X ∈ sS e t , to give a morphism f ∈ H om sS et (∆ n × ∆ 1 , X ), it is neces sary and sufficient to give ( n + 1)-simplices y k = f ( z k ) ∈ X n +1 such that d k y k = d k y k − 1 . Let p : ∆ n × ∆ 1 − → ∆ n +1 be the mor phism given by y k = s k 0 d k 1 (012 ...n + 1). One may verify that ∆ n × { 0 }− → s n 0 (0) and ∆ n × { 1 }− → d 0 (01 ...n + 1). An eas y but tedious ca lculation shows that p extends to a morphism p : Y × ∆ 1 − →Y (1) in sS et ∆ . Now, F ∈ H om sS et ∆ ( Y , ˇ N A [1] a,b ) c o rresp onds to so me f ∈ H om sS et ∆ ( Y (1) , ˇ N A ) a,b . W e ca n form the comp osition f p ∈ H om sS et ∆ ( Y × ∆ 1 , ˇ N A [1]) = hom ( Y , ˇ N A ) 1 , which sends Y × { 0 } to a and Y × { 1 } to b . Thus, f p ∈ hom ( Y , ˇ N A ) is a 1- s implex in hom ( Y , ˇ N A ) from a to b . Since A satisfies ˇ Cech descent, ρ : N A ( X ) − → hom ( Y , ˇ N A ) is a weak equiv a lence of fibrant simplicial sets. Therefor e , there e x ists a mor phism G : hom ( Y , ˇ N A ) − → N A ( X ) such tha t Gρ is homotopic to the identit y . Since N A ( X ) is fibrant, one ca n also find a 1 -simplex in N A ( X ) from a to b , i.e. a path in A ( X ) 1 from a to b . The result now follows from Lemma 6.14. Remark 6.17. F or a, b ∈ A ( X ) 0 as in Lemma 6.16, the (presheaf of ) ab elian group s tructure of A [1] a,b is not the one endow ed from b eing a sub- ω - category . Lemma 6.18. L et A b e a pr esh e af of Pic ar d ω -c ate gories wh ich satisfies ˇ Ce ch desc ent. Supp ose a , b ∈ A ( X ) k ar e such that s k − 1 a = s k − 1 b and t k − 1 a = t k − 1 b . Then A [ k + 1] a,b satisfies ˇ Ce ch desc ent. A dditionally, if ther e ex ists a ( k + 1) -morphi sm x fr om a to b , then A [ k + 1] a,b is a pr eshe af of Pic ar d ω -c ate gories. Pr o of. W e prov e the statement by induction on k . First let k = 0. If H om sS et ∆ ( Y , ˇ N A [1] a,b ) is non-empty , then there exists a path x ∈ A ( X ) 1 from a to b and A [1] a,b ≃ A [1] a,a ≃ A [1] 0 , 0 ∈ P r e ω Ab . Since A satisfies ˇ Cech descent, it satisfies the unique k-glueing condition for loops at 0, and since ( A [1] 0 , 0 )[ k ] 0 , 0 = A [ k + 1] 0 , 0 , A [1] 0 , 0 satisfies the unique k-g lueing condition for loo ps at 0. Therefore, A [1] 0 , 0 is a pres he a f of Picard ω -categor ies which sa tisfies ˇ Cech descent. It follows that s ince A [1 ] 0 , 0 ≃ A [1] a,b , A [1] a,b is a presheaf of Picard ω -categor ies satisfying ˇ Cech descent. If, o n the other ha nd, H om sS et ∆ ( Y , ˇ N A [1] a,b ) = ∅ , then the simplicial set hom sS et ∆ ( Y , ˇ N A [1] a,b ) = ∅ . But H om sS et ∆ ( Y , ˇ N A [1] a,b ) = ∅ also implies tha t N A [1] a,b ( X ) 0 = ∅ , whence N A [1] a,b ( X ) = ∅ , so A [1] a,b trivially satisfies ˇ Cech descent. This proves the base case ( k = 0). 24 Now supp ose tha t a, b ∈ A ( X ) k are such that s k − 1 a = s k − 1 b and t k − 1 a = t k − 1 b . Then for any op en U ⊂ X , as a set, A [ k + 1 ] a,b ( U ) = { x ∈ O b ( A ( U )) | s k x = a, t k x = b } = { x ∈ Ob ( A ( U )) | s k x = a, t k x = b, s k − 1 x = s k − 1 a, t k − 1 x = t k − 1 b } = { x ∈ Ob ( A [ k ] s k − 1 a,t k − 1 b ( U )) | s k x = a, t k x = b } = { x ∈ Ob ( A [ k ] s k − 1 a,t k − 1 b ( U )) | s [ k ] 0 x = a, t [ k ] 0 x = b } = ( A [ k ] s k − 1 a,t k − 1 b )[1] a,b ( U ) . Hence, A [ k + 1 ] a,b = ( A [ k ] s k − 1 a,t k − 1 b )[1] a,b . But since a is a k -mor phism from s k − 1 a to t k − 1 b , A [ k ] s k − 1 a,t k − 1 b is a prehs e a f of Picard ω - categories satisfying ˇ Cech descent, b y induction hypothesis. By the base case, A [ k + 1 ] a,b = ( A [ k ] s k − 1 a,t k − 1 b )[1] a,b satisfies ˇ Cech descent, and if ther e exists a ( k + 1)-mor phism x ∈ A ( X ) k +1 from a to b , then A [ k + 1] a,b ∈ P r e ω Ab . Theorem 6.4 now follows directly from Lemma 6.18. Pr o of. Supp ose that A ∈ P r e ω Ab and satisfies ˇ Cech desce nt. Then by Lemma 6 .9, A satis fie s the unique 0-glueing condition. Lemma 6.1 8 ensur es that each A [ k ] a,b satisfies ˇ Cech descent hence the unique 0-glueing prop erty by Lemma 6.9. Therefore A satisfies the unique k -glueing prop erty for eac h k , i.e. satisfies ω - descent. 7 ∞ -torsors W e hav e added this section for completeness, as using ω -descent as a way to make ∞ -torsor s accessible was a prima ry mo tiv ation for establishing the equiv alence of the tw o des cen t conditions. The ce ntral ideas in this section (Definition 1 6 and P r opo s itions 7.1 and 7.4) ar e due to Fiorenza, Sati, Schreiber, a nd Stasheff [15, 37, 41]. W e simply for m ulate them in a slightly different wa y , prefering to define ob jects up to homo top y . Definition 16 . Let G b e a presheaf of simplicial abe lia n gr oups on a site C . L e t B G denote any delo oping ob ject of G in the homotopy categor y o f P re pro j,loc sS et ( C ). This means that B G is an ob ject with a p oint ∗− → B G , and G is the homo top y pullba ck of the diagram ∗   y ∗ − − − − → B G W e define T ors G = hom ( − , g B G ), where hom denotes simplicia l enrichm ent in P re sS et ( C ), a nd g B G denotes a sheafification (i.e. fibrant re pla cemen t) of B G [15]. Remark 7.1. No te that T or s G is w ell-definied up to w eak equiv alence due to the uniqueness up to homotopy of lo oping and delo oping functor s [18, 2 1]. F urthermore, Lemma 7.2 shows that for t w o different choices T 1 , T 2 for T or s G , T 1 ( X ) is weakly e q uiv alent to T 2 ( X ) for any X ∈ C . Lemma 7.2. L et F , G ∈ P r e pro j,loc sS et b e fibr ant obje cts which ar e we akly e quivalent. Then F or any X ∈ C , F ( X ) and G ( X ) ar e we akly e quivalent. Pr o of. Let V − → X b e a hypercover o f X , and let V ′ be a cofibrant replacement of V . Then By [14] Lemma 4.4, F ( X ) ≃ hom ( X , F ) − → hom ( V ′ , F ) and G ( X ) − → hom ( V ′ , G ) ar e w eak equiv alences. If there is a weak equiv ale nce F − → G , then it is a g eneral fact [19] Corolla ry 9 .3.3 that in a simplicial mo del catego r y with cofibrant V ′ and a weak equiv ale nce of fibrant ob jects F − → G then h om ( V ′ , F ) − → hom ( V ′ , G ) is a weak equiv ale nce of simplicial sets. Ther efore, F ( X ) is weakly equiv ale n t to G ( X ). If there is a zigz a g of weak equiv ale nce s fro m F to G , then the r esult is the same: F ( X ) and G ( X ) a re weakly equiv alent. 25 The ce n tral obs erv ations of this section ar e Prop ositions 7.1 and 7 .4. The pro of Pr opositio n 7 .1 is a mo dification of the pro of o f Pro position 3 .2.17 in [15], which is for complexes concentrated in o ne degree only . First w e ma ke use of the following fa ct ab out homotopy limits for pr eshea ves. A homotopy pullback is simply the homotopy limit in the mo del categ ory P r e pro j,loc sS et . How e ver, s ince every sectionwise weak equiv a le nc e is a lo cal weak equiv alence, the ident it y map i : P re pro j sS et − → P r e pro j,loc sS et preserves weak equiv a lence a nd is adjoint to itself. It is a general fact that if a functor U b et w een mo del categorie s is a right adjoint and preserves w eak equiv alences, then U preserves homotopy limits. Hence, to compute the ho mo top y limit in the lo cal mo del structure, it is enough to co mpute it in the glo bal pro jective mo del structure. Given a co mplex A ∈ C h + ( A b ) o f pre s hea v es of ab elian gro ups concentrated in non-neg ativ e de g rees, recall that A [1] is A shifted up one degree so that A [1] n = A n − 1 . F o r a preshea f o f simplicial gr oups G , let G [1] b e the pre sheaf of simplicia l gr o ups co r respo nding to the shift functor in C h + ( A b ) by the Dold- Kan corres p ondence. Prop osition 7. 3. L et G b e a pr eshe af of simplicial gr oups. Then G [1] is a delo oping obje ct of G . Pr o of. W e use the Dold-Ka n corr espondence be t w een P r e sAb ( C ) and C h + ( A b ). Given a complex A ∈ C h + ( A b ) of pr eshea ves o f ab elian groups concentrated in non-neg ativ e degrees , reca ll that A [1] is A shifted up one deg ree so that A [1] n = A n − 1 . Define B = B ( A ) as follows. L et B n = A n × A n − 1 , and let d : B n − → B n − 1 be d ( x, y ) = ( da + ( − 1) n b, db ). Clea rly , d 2 = 0 so that B ∈ C h + ( A b ). W e define f : B − → A [1] as the ob vious map: f n : A n − → A n − 1 − → A n − 1 is just π 2 . It is o b vious tha t this is a chain map. F urthermor e, B is a cyclic in the sense that each homolo gy class H n B = 0. Using the pr e ceding facts ab out ho motop y pullbacks, since B − → A [1] is a fibra tio n, the pullback of the diagra m B   y 0 − − − − → A [1] is in fact the homotopy pullback. The pullback P has the pr operty that any g : C − → B suc h that f g = 0 factors thro ugh P . Fir st we see that there is a map h : A − → B given by in degree n by x 7→ ( x, 0). and that f h = 0. I t is easy to s e e that a map g such that f g = 0 is precisely a map C − → A − → B . Hence , A is the pullback. Since there is a weak equiv ale nc e fro m the diag ram B   y 0 − − − − → A [1] to 0   y 0 − − − − → A [1] , A is the homotopy pullback of the latter diagr am, whence A [1] is a delo oping of A . Since G [1] is a delo oping of G , it follows tha t T or s G ( X ) = hom( X , g G [1]) ≃ g G [1]( X ), whence T or s G = g G [1]. Having defined T or s G , we define T or s n G , which is well defined up to lo cal w eak equiv alence. Let T ors n G := g G [ n ], where the fibrant replac emen t g G [ n ] is chosen to b e a s he a f of simplicia l ab elian gr o ups. It was shown in Lemma 5.5 that it is p ossible to make such a c hoice. 26 Prop osition 7. 4. L et G b e a pr eshe af of simplicial ab elian gr oups. Then T ors n G = g G [ n ] = T or s T or s n − 1 G . Pr o of. This follows by induction. By de finitio n, T or s T or s n − 1 G = T ors ^ G [ n − 1] = (( G [ n − 1]) ∼ [1]) ∼ . Since the Dold-Kan corr espondence comm utes with taking stalks , which is to say that the diagr am C h + ( p A b ) − − − − → P re sAb   y   y C h + ( Ab ) − − − − → sAb commutes if the vertical arrows represent ta king stalks at a po in t x . Therefore, for a loc a l w eak eq uiv alence A − → B in P re sAb , the induced map A [1] − → B [1] is a lo cal weak equiv alence. Ther efore, given A ∈ P re sAb with sheafifica tion ˜ A , the lo cal weak equiv alence A − → ˜ A induces a lo cal weak e quiv alence A [1] − → ˜ A [1]. Hence, g A [1] is weakly equiv alent to ˜ A [1]. F r om here it is easy to see that g G [ n ] is weakly equiv alent to (( G [ n − 1]) ∼ [1]) ∼ . The result follows. 7.1 Comparison with other Approaches Jardine and Luo hav e taken a different a pproach to principal bundles [2 6, 27], and we now co mpare their formulations with those o f [15]. Here we consider only the case where C is the site of op en sets on a space X so that X is the terminal ob ject in C . Throughout this section, fix a space X a nd a sheaf G of simplicial groups on X . Let G − sP r e ( C ) denote the simplicial presheaves on X with G -action. Lemma 7.5 . Ther e is a c ofibr antly gener ate d close d mo del struct u r e on t he c ate gory G − sP r e ( C ) of sim- plicial G-pr eshe aves, wher e a map is a fibr ation (r esp. we ak e quivale nc e) if the underlying map of simplicial pr eshe aves is a glob al fibr atio n (r esp. lo c al we ak e qu ival enc e). Definition 17. 1. L e t G − T ors b e the categor y of cofibra n t fibr an t simplicial G-presheav es P such tha t P /G − →∗ is a hyperc o v er (i.e. lo cal trivial fibr ation). W e call the ob jects in G − T or s G-princip al bund les . A G-principal bundle P is called a principa l bundle ov er X = P /G . 2. Cho ose a factor ization ∅− → E G − → X in G − sP re ( C ) where the first map is a cofibr ation and the second is a trivial fibration. Let B G = E G/G . Note that E G and B G are defined up to weak homoto p y equiv alence in G − sP r e ( C ). Lemma 7.6. Any B G is a delo oping obje ct of G . Pr o of. W e would like to see that G is a ho motop y pullback o f ∗   y ∗ − − − − → B G in (some/any) mo del structure where the weak equiv ale nc e s are the lo cal weak equiv alences. In Remark 4 of [2 7], the follo wing observ a tion is made. Let W G and W G b e defined sectionwise. That is to say , ( W G )( U ) = W ( G ( U )) and ( W G )( U ) = W ( G ( U )), wher e W and W a re the standard co nstructions [18, 11, 2 8]. By ta king co fibran t r eplacemen ts a nd factorizing maps, we can find ob jects E G a nd ˜ W G in G − sP re ( C ) tog ether with maps W G p ← − ˜ W G j − → E G s uch that p is a trivial fibra tion in G − s P re ( C ), j is a trivial cofibratio n in G − sP re ( C ), and ˜ W G is cofibrant. Since p is a weak equiv alence of co fibran t simplicial G-spaces, it induces a weak equiv a lence ˜ W G/G − → W G/G = W G . Similar ly , j induces a weak equiv alence ˜ W G/G − → E G/G = B G . W e therefore hav e a seque nc e of weak equiv alences in the dia gram ca tegory: fro m ˜ W G   y ∗ − − − − → ˜ W G/G 27 to W G   y ∗ − − − − → W G and also to E G   y ∗ − − − − → B G, which maps to ∗   y ∗ − − − − → B G. by a weak equiv alence. Replacing an ob ject in the diagr a m categ ory by a weakly e q uiv alent one do es not change the homotopy pullbac k. Hence, we need only show that G is the homotopy pullback of W G   y ∗ − − − − → W G. First we s how that the pullback of ∗ − → W G ← − W G is in fact a ho motop y pullback. The homotopy pullback is simply the homotopy limit in the mo del catego ry P re pro j,loc sS et . F rom the par a graph prec eding Prop osition , we know that to co mpute the homotopy limit in the lo cal mo del structur e , it is enoug h to compute it in the glo bal pr o jective model structure. W e know that P r e pro j sS et is prop er [14]. It is well known that in a right pro per mo del categor y , the pullback o f a diagr am X − → Z ← − Y is the homotopy pullba c k provided that one o f the mor phisms X − → Z o r Y − → Z is a fibration. In the g lobal pro jective mo del struc- ture, W G − → W G is a fibra tion, so the ho motop y pullbac k of the origina l diagra m is simply the pullback o f ∗− → W G ← − W G . The pullback can b e taken sectionwise, and the rea de r can consult [18] for an exp osition of the fact that sectionwise, G is iso morphic to the pullback of W G   y ∗ − − − − → W G. Lemma 7.7. Any delo oping obje ct B G is we akly e quivalent to a classifying sp ac e. Pr o of. T ake a cla ssifying space B G . By Lemma 7 .6, B G is a delo oping of G . Howev er, delo oping is well- defined up to weak equiv alence, so s inc e B G a nd B G are b oth delo opings, they are is omorphic in the homotopy categor y . Alternately , a s traight forward calculation shows that G [1] is w eakly equiv alent to W G and hence B G . 28 8 App endix 8.1 Deligne’s Theorem Let P ic 1 ω = { A ∈ P ic ω | A = A 1 } , and let P ic denote the categor y of Picard categories in the s ense of Deligne [12]. That is, a Picard catego ry C is a quadr uple ( C , + , σ, τ ), whe r e + : C × C − →C is a functor such that for all ob jects x ∈ C , x + : C − →C is an equiv alence, a nd addition is commutativ e and asso ciative up to isomorphisms τ and σ . In this section we explain in detail the relations hip b et w een P ic 1 ω and P ic . Lemma 8.1. L et C ∈ P i c . F or any obje cts x , y ∈ C , id x + id y = id x + y whenever x is isomorphi c t o y . Pr o of. T ake any g ∈ H om C ( x, y ). Then ( g ◦ id x ) + ( g − 1 ◦ id y ) = ( g + g − 1 ) ◦ ( id x + id y ) g + g − 1 = ( g + g − 1 )( id x + id y ) id x + y = id x + id y Lemma 8. 2. Assume C ∈ P ic such that + is strictly c ommutative and asso cia tive. 1. F or any f ∈ H om C ( x, y ) in C , f + f − 1 = id x + y . 2. F or any f ∈ H C ( a, b ) , g ∈ H om C ( b, c ) , id b + g ◦ f = g + f . 3. Assume that for every x ∈ Ob ( C ) , x + : C − →C is actual ly an isomorphism. Then O b ( C ) is an ab elian gr oup. F or al l f ∈ H om C ( x, y ) , f + i d 0 = f , and ther e ex ist s a unique h ∈ H om C ( − x, − y ) such that f + h = id 0 . Henc e, H om C = ∪ x,y ∈ ob ( C ) ( x, y ) is also an ab elia n gr oup. Pr o of. 1. i d x + y = i d x + id y = i d x + f ◦ f − 1 = ( id x ◦ id x ) + ( g ◦ g − 1 ) = ( id x + g ) ◦ ( id x + g − 1 ) = ( id x + g ) ◦ ( g − 1 + id x ) = ( id x ◦ g − 1 ) + ( g ◦ id x ) = g − 1 + g 2. g ◦ f + id b = ( g ◦ f ) + ( g − 1 ◦ g ) = ( g + g − 1 ) ◦ ( f + g ) = id x + y ◦ ( f + g ) = f + g 3. The firs t cla im is obvious. Since 0 + 0 = 0, id 0 + id 0 = id 0 . Therefore, for f ∈ H om C ( x, y ), id 0 + ( id 0 + f ) = id 0 + f . How ev er, since 0+ : C − →C is an equiv alence of categ ories, it gives a bijection H om C ( x, y ) − → H om C ( x, y ) sending f 7→ id 0 + f . Since this is injective, id 0 + f = f . W e show ed tha t f + f − 1 = id x + y , so f + ( f − 1 + id − x − y ) = id x + y + id − x − y = id 0 . W e hav e show ed the existence of an additive in verse h = f − 1 + id − x − y to f . T o show uniqueness. If f + g = id 0 = f + h , g + i d 0 = g + ( f + h ) = ( g + f ) + h = id 0 + h . Again, since 0+ is an equiv alence , h = g . Prop osition 8.3. P ic 1 strict c onsists of al l smal l Pic ar d c ate gories in P ic such that + is strictly asso ciative and c ommutative (i.e. τ and σ ar e identities) and for e ach x ∈ ob ( C ) , x + : C − →C is an isomorphism, not just an e quivalenc e. Pr o of. Clear ly any 1- category in P i c 1 strict ⊂ P ic is a small Picar d catego r y sa tisfying thes e pr o perties. On the other ha nd, if a small Picar d category C ∈ P ic satisfies these prop erties, lemma 8.2 shows that C 1 = H om C is actually an ab elian gro up, and sending the inclusion ob ( C ) ֒ → C 1 ( x 7→ id x ) is an inclusio n of abe lia n groups. F ur thermore, lemma 8.2 demonstrates that the second prop erty f ◦ g = f + g − s 0 f is satisfied. 29 8.2 Nerv e and P ath F unctors for ω -categories W e wish to see that ω -des c e n t a nd ˇ Cech desc en t are e quiv alent for ω -gro up oids gener ally , no t just Picard ω -categor ies. In tegral to our pro of for Picar d ω -categ ories was a descriptio n of the ner v e of A [1] a,b . As a step tow ards extending the pro of to ω -gro upoids, we give a characteriza tion of the nerve of A [1] a,b for any A ∈ ω C at and a, b ∈ A 0 . Prop osition 8. 4. L et A b e an ω -c ate gory. Then H om ω C at ( O ( ˜ ∆ n )) , A [1] a,b ) ≃ { f ∈ H om ω C at ( O ( ˜ ∆ n × ˜ ∆ 1 ) , A ) | f ( h u , 0 i ) = a, f ( h u, 1 i ) = b for al l u ∈ ˜ ∆ n × ˜ ∆ 1 } . Pr o of. Let C = ˜ ∆ n × ˜ ∆ 1 . F or ( M , P ) ∈ N ( C ), let Θ(( M , P )) = ( M ∩ ˜ ∆ n × ˜ ∆ 1 1 , P ∩ ˜ ∆ n × ˜ ∆ 1 1 ) ∈ N ( C ). Notice that ˜ ∆ n × ˜ ∆ 1 1 = ˜ ∆ n × { (01) } . Recall fro m § 4.1 .1 that O ( C ) is gener ated by atoms h c i for c ∈ C . This implies that every element in α ∈ O ( C ) is a gotten by a comp osition of a to ms α = h c 1 i ∗ k 1 c 2 ∗ k 2 ... ∗ k t − 1 h c t i for c 1 , ...c t ∈ C . W e omit parentheses in such comp ositions for gener alit y and ease of exp osition. Viewing Θ as a map of sets Θ : O ( C ) − →N ( C ), for α ∈ O ( C ), let α deno te the set Θ − 1 (Θ α ). W e will prove the following statements: 1. Θ( s n ( M , P )) = s n (Θ( M , P )), a nd Θ( t n ( M , P )) = t n (Θ( M , P )). 2. If N , Q ⊂ ˜ ∆ n × ˜ ∆ 1 0 , then Θ neglects comp osition with ( N , Q ), i.e . Θ(( M , P ) ∗ k ( N , Q )) = Θ ( M , P ) and Θ (( N , Q ) ∗ k ( M , P )) = Θ( M , P ) whenever the comp ositions a re defined. More gener a lly , for any ( N , Q ), ( M , P ), Θ(( N , Q ) ∗ k ( M , P )) = Θ( N , Q ) ∗ k Θ( M , P ) 3. F or u ∈ ˜ ∆ n , Θ h u, (0 1 ) i = h u i × { (01) } , and s n ( h u i × { (01) } ) = s n − 1 h u i × { (01) } (similar ly for t n ). 4. If h u , (01 ) i ∈ ( h x, (01) i ), h v , (01) i ∈ ( h y , (01 ) i ) and we ar e a ble to comp ose h x, (01) i ∗ k h y , (01) i , then one can fo r m the comp osition h u i ∗ k − 1 h v i in O ( ˜ ∆ n ). In this case, Θ( h x, (01) i ∗ k h y , (01) i ) = h u i ∗ k − 1 h v i × { (01) } . The pr oo fs o f (1)-(3), are es sen tially based on the obser v ation that the oper ations inv olv ed in taking the source and tar get a nd compo s ition resp ect the decomp osition of a s ubset S n ⊂ C n = F p + q = n ˜ ∆ n p × ˜ ∆ 1 q int o S n = F p + q = n S n ∩ ( ˜ ∆ n p × ˜ ∆ 1 q ). T o b e mor e precise, the op erations c o nsist of repla cing a set M by M n , | M | n , or ( M \ M n ) as w ell a s tak ing unions . Let C ∗ , 1 = ˜ ∆ n × ˜ ∆ 1 1 . Clearly , ( M ∩ C ∗ , 1 ) n = M n ∩ C ∗ , 1 . Also, ( P ∪ M ) ∩ C ∗ , 1 = ( P ∩ C ∗ , 1 ) ∪ ( M ∩ C ∗ , 1 ). Tha t | M ∩ C ∗ , 1 | n = | M | n ∩ C ∗ , 1 follows from the pre v ious tw o prop erties tog ether with the fact that | M | n = S n k =0 M k . Finally , the first prop erty shows that intersecting with C ∗ , 1 resp ects gr ading so tha t ( M \ M n ) ∩ C ∗ , 1 = ( M ∩ C ∗ , 1 ) \ ( M n ∩ C ∗ , 1 ) = ( M ∩ C ∗ , 1 ) \ ( M ∩ C ∗ , 1 ) n . Statement s (1) - (3) now follow easily . 1. Θ( s n ( M , P )) = Θ( | M | n , M n ∪ | P | n − 1 ) = ( | M | n ∩ ˜ ∆ n × ˜ ∆ 1 1 , ( M n ∪ | P | n − 1 ) ∩ ˜ ∆ n × ˜ ∆ 1 1 ) = ( | M ∩ ˜ ∆ n × ˜ ∆ 1 1 | n , ( M n ∩ ˜ ∆ n × ˜ ∆ 1 1 ) ∪ | P ∩ ˜ ∆ n × ˜ ∆ 1 1 | n − 1 ) = s n (Θ( M , P )). The pr oof for t n is similar. 2. First supp ose that we can for m the co mposition ( N , Q ) ∗ k ( M , P ) = ( M ∪ ( N \ N n ) , ( P \ P n ) ∪ Q ). It follows from the observ ations in the pr e v ious para graph together with the fact that Θ ( N , Q ) = ( ∅ , ∅ ) that Θ(( N , Q ) ∗ k ( M , P )) = ( M ∩ C ∗ , 1 , ( P ∩ C ∗ , 1 ) \ ( P n ∩ C ∗ , 1 ). In order fo r ( N , Q ) and ( M , P ) to b e comp osable, w e r equire that s n ( N , Q ) = t n ( M , P ), which implies that P n = N n , whence P n ∩ C ∗ , 1 = ∅ . Therefore, Θ (( N , Q ) ∗ k ( M , P ) = ( M ∩ C ∗ , 1 , P ∩ C ∗ , 1 ) = Θ( M , P ). Sho wing that Θ( M , P ) ∗ n ( N , Q ) = Θ( M , P ) follows similarly . More g enerally , for a n y ( M , P ), ( N , Q ), Θ(( N , Q ) ∗ k ( M , P )) = ( M ∩ C ∗ , 1 , ( P ∩ C ∗ , 1 ) \ ( P n ∩ C ∗ , 1 ) = (( M ∪ ( N \ N n )) ∩ C ∗ , 1 , ( Q ∪ ( P \ P n )) ∩ C ∗ , 1 = (( M ∩ C ∗ , 1 ) ∪ ( N \ N n ) ∩ C ∗ , 1 , ( Q ∩ C ∗ , 1 ) ∪ ( P \ P n ) ∩ C ∗ , 1 ) = (( M ∩ C ∗ , 1 ) ∪ ( N ∩ C ∗ , 1 \ N n ∩ C ∗ , 1 ) , ( Q ∩ C ∗ , 1 ) ∪ ( P ∩ C ∗ , 1 \ P n ∩ C ∗ , 1 )) = ( N ∩ C ∗ , 1 , Q ∩ C ∗ , 1 ) ∗ k ( M ∩ C ∗ , 1 , P ∩ C ∗ , 1 ) = Θ( N , Q ) ∗ k Θ( M , P ) 30 3. First we verify the c la im that for u ∈ ˜ ∆ n r and ( M , P ) = Θ( h u, (01) i ) ∈ N ( C ), M n − k = µ ( u ) n − k − 1 × { (01) } a nd P n − k = π ( u ) n − k − 1 × { (01) } for 0 ≤ k ≤ n , wher e µ ( z ), and π ( z ) ar e as in § 4 .1.1. Let us prov e the statement by induction on k . Let z = ( u , (01 )) so that < z > = ( µ ( z ) , π ( z )). F or k = 0, µ ( z ) n = { ( u, (01) } , so Θ( µ ( z )) n = { u, (01) } . No w suppose that the cla im is true up for all i ≤ k . Then µ ( z ) n − k = µ ( u ) n − k − 1 × { (01 ) } ∪ S k × { (0) } ∪ T k × { (1) } for some subsets S k , T k ⊂ ˜ ∆ n n − k . Hence, µ ( z ) n − ( k +1) = µ ( z ) − n − k \ µ ( z ) + n − k = ( µ ( u ) − n − k × { (01 ) } ∪ µ ( u ) n − k × { (01 ) } ± ∪ S − k × { (0) } ∪ T − k × { (1) } ) \ ( µ ( u ) + n − k × { (01 ) } ∪ µ ( u ) n − k × { (01 ) } ± ∪ S + k × { (0) } ∪ T + k × { (1) } ) = µ ( u ) n − k − 1 × { (01 ) } ∪ S k +1 × { (0) } ∪ T k +1 × { (1) } for so me sets T k +1 , S k +1 and where ± means, that it c an b e + o r − de p ending on the parity of n and k . Ther efore, Θ( µ ( z )) n − ( k +1) = µ ( u ) n − k − 1 × { (01) } . It follows by induction that Θ( µ ( z )) n − k = µ ( u ) n − k − 1 × { (01 ) } for all 0 ≤ k ≤ n . A similar shows that Θ( π ( z )) n − k = π ( u ) n − k − 1 × { (01 ) } for all 0 ≤ k ≤ n . W e conclude that Θ( µ ( z ) , π ( z )) = h u i × { (01) } . Since ( h u i × { (01) } ) n = h u i n − 1 × { (01) } , an easy calculation sho ws tha t s n ( h u i × { (01) } ) = ( s n − 1 h u i ) × { (01) } and t n ( h u i × { (01) } ) = ( t n − 1 h u i ) × { (01) } 4. Observe that Θ( h u, (0 1) i ) = Θ ( h x, (01) i ) and Θ( h v , (01) i ) = Θ( h y , (01) i ). Since h x, (01) i and h y , (01) i are comp osable, s k h x, (01) i = t k h y , (01) i , so by part (1), s k Θ( h u, (01) i ) = t k Θ( h v , (01) i ) and we can form the comp osition Θ( h u , (01 ) i ) ∗ k Θ( h v , (01) i ). But w e just show ed that Θ( h u, (01) i ) = ( µ ( u ) × { (01) } , π ( u ) × { (01) } ) = h u i × { (01) } and Θ ( h v, (01) i ) = ( µ ( v ) × { (01) } , π ( v ) × { (01) } ) = h v i × { (01) } . It follows that h u i and h u i are compo sable since s k − 1 h u i × { (01) } = s k ( h u i × { (0 1) } ) = s k Θ h u, (01) i = t k Θ h v , (01 ) i = t k ( h v i × { (01) } ) = t k − 1 h v i × { (01) } . Now, Θ( h x , (01 ) i ∗ k h y , (01) i ) = Θ( h x, (01) i ) ∗ k Θ( h y , (01 ) i ) = Θ( h u , (01 ) i ) ∗ k Θ( h v , (01) i ) = ( µ ( v ) × { (01 ) } ∪ ( µ ( u ) \ µ ( u ) k − 1 ) × { (01) } , π ( u ) × { (01) } ∪ ( π ( v ) \ π ( v ) k − 1 ) × { (01) } ) = (( µ ( v ) ∪ ( µ ( u ) \ µ ( u ) k − 1 )) × { (01) } , ( π ( u ) ∪ ( π ( v ) \ π ( v ) k − 1 )) × { (01) } ) = ( h u i ∗ k − 1 h v i ) × { (01) } . W e are now able to prove the main prop osition. W e will use induction to give a bijection H om ω C at ( |O ( ˜ ∆ n ) | k , A [1] a,b ) ˜ − →H n k := { f ∈ H om ( |O ( ˜ ∆ n × ˜ ∆ 1 ) | k +1 , A ) | f ( h u, (0) i ) = a and f ( h u, (1) i ) = a for all u ∈ ˜ ∆ n } . This bijection will send g to the f in H n k such that f ( h u , (0) i ) = a , f ( h u, (0) i ) = a and f ( h u, (01 ) i = g ( h u i ) for a ll u ∈ ˜ ∆ n , and f ∈ H n k corres p onds to g such that g ( h u i ) = f ( h u, (01) i ). W e now pro ceed b y inductio n. F o r k = 0, a functor g ∈ H om ω C at ( |O ( ˜ ∆ n ) | 0 , A [1] a,b ) consists o f a choice o f n + 1 ob jects g (0),..., g ( n ) ∈ A [1] a,b 0 . Equiv alen tly , this is a choice of n + 1 1 - morphisms in A from a to b . Since O ( ˜ ∆ n × ˜ ∆ 1 ) is freely generated by its atoms h c i for c ∈ ˜ ∆ n × ˜ ∆ 1 , an f in H n 1 ⊂ H om ω C at ( |O ( ˜ ∆ n × ˜ ∆ 1 ) | 1 , A ) is freely determined by f ( h c i ) for c ∈ ( ˜ ∆ n × ˜ ∆ 1 ) i , i = 0 , 1 as long a s f is co mpa tible with so ur ce and target ma ps s 0 , t 0 . Since we require that f ( h u , (0) i ) = a a nd f ( h u, (1) i ) = b fo r all u , we ca n freely choo se f ( h u, (01 ) i ) ∈ A 1 as long as s 0 f ( h u , (01 ) i ) = a and t 0 f ( h u, (01) i ) = b for u ∈ ˜ ∆ n 0 . Therefor e, a choice of f ∈ H n 1 is a choice o f ( n + 1) 1-morphisms in A fr o m a to b . It is evident that under this ident ification, fo r u ∈ ˜ ∆ n 0 , g ( h u i ) = f ( h u, (01) i ). Now assume that H om ω C at ( |O ( ˜ ∆ n ) | i , A [1] a,b ) is identified with H n i as ab ov e for all i ≤ k . Since O ( ˜ ∆ n × ˜ ∆ 1 ) is freely g enerated by its atoms, a functor ˆ f ∈ H n k +1 is equiv alen t to a functor f ∈ H n k together with an y choice of ( k +2 )-morphisms ˆ f ( h u , (01 ) i ) for u ∈ ˜ ∆ n k +1 such that s k +1 f ( h u, (01) i ) = f ( s k +1 ( h u, (01 ) i )) and t k +1 f ( h u, (01) i ) = f ( t k +1 ( h u, (01 ) i )). This is b e c ause a ll f ( h u, ( i ) i ) for u ∈ ˜ ∆ n k +2 are for ced to b e a or b . Also, O ( ˜ ∆ n ) is fre e ly ge ne r ated b y its atoms, so a choice of ˆ g ∈ H om ω C at ( |O ( ˜ ∆ n ) | k +1 , A [1] a,b ) is eq uiv - alent to a choice of g ∈ H om ω C at ( |O ( ˜ ∆ n ) | k , A [1] a,b ) to gether with a choice o f ˆ g h u i for u ∈ ˜ ∆ n k +1 with the 31 appropria te k -source and target. Let us then b egin with f ∈ H n k , which cor respo nds to g ∈ H om ω C at ( |O ( ˜ ∆ n | k , A [1] a,b ). Firs t we show that if x ∈ O ( ˜ ∆ n × ˜ ∆ 1 ) is not in the sub- ω -catego ry genera ted by elements of the form h u , ( i ) i , then f ( x ) = g p 1 Θ x , where p 1 denotes pr o jection onto the firs t factor. Cho ose such a n x ∈ O ( ˜ ∆ n × ˜ ∆ 1 ). Since O ( ˜ ∆ n × ˜ ∆ 1 ) is freely g enerated by its ato ms , it is also g enerated by its ato ms , so is a co mposition of a toms: x = h x 1 i ∗ l 1 h x 2 i ∗ l 2 ... ∗ l e − 1 h x e i for some x 1 , ..., x e ∈ ˜ ∆ n × ˜ ∆ 1 (omitting pa ren theses). Since f ( h u , ( i ) i ) is a 0-ob ject in A for i = 0 , 1, the v alue of f is no t affected b y co mposition with elements o f the form h u, ( i ) i . Similarly , Θ also neglects comp osition w ith these elements. Let x i 1 ,... x i r be the o nes of the form x i k = h u i k , (01 ) i , so f ( x ) = f h x 1 i ∗ l 1 f h x 2 i ∗ l 2 ... ∗ l e − 1 h f x e i = f h x i 1 i ∗ l i 1 f h x i 2 i ∗ l i 2 ... ∗ l ir f h x i r i = f h u i 1 , (01 ) i ∗ l i 1 f h u i 2 , (01 ) i ∗ l i 2 ... ∗ l ir f h u i r , (01 ) i , whe reas Θ( x ) = Θ h x 1 i ∗ l 1 θ h x 2 i ∗ l 2 ... ∗ l e − 1 h Θ x e i = Θ h x i 1 i ∗ l i 1 Θ h x i 2 i ∗ l i 2 ... ∗ l ir Θ h x i r i = Θ h u i 1 , (01 ) i ∗ l i 1 Θ h u i 2 , (01 ) i ∗ l i 2 ... ∗ l ir Θ h u i r , (01 ) i = ( h u i 1 i ∗ l i 1 − 1 h u i 2 i ∗ l i 2 − 1 ... ∗ l ir − 1 h u i r i ) × { (01) } . Therefore, g p 1 Θ x = g h u i 1 i ∗ l i 1 − 1 h g u i 2 i ∗ l i 2 − 1 ... ∗ l ir − 1 h g u i r i ), and f ( x ) = f h u i 1 , (01 ) i ∗ l i 1 f h u i 2 , (01 ) i ∗ l i 2 ... ∗ l ir f h u i r , (01 ) i . By inductio n hypo thesis, f h u i j , (01 ) i = g h u i j i for e ac h j , and since the comp ositions app earing in g p 1 Θ x are in A [1] a,b , ∗ [1] n = ∗ n +1 for any n so that the co mpositions coinc ide to o. Now, for u ∈ ˜ ∆ n k +1 , f ( t k +1 ( h u, (01 ) i )) = g p 1 θt k +1 h u, (0 1 ) i = g p 1 t k +1 θ h u , (01) i = g p 1 t k +1 ( h u i × { (01) } = g p 1 t k h u i × { (01) } = g ( t k h u i ) = t [1 ] k g ( h u i ) = t k +1 g ( h u i ). Similar ly , f ( s k +1 ( h u, (01 ) i )) = s k +1 g ( h u i ). In sum- mary , an extensio n ˆ f ∈ H n k +1 of f is equiv alen t to a choice of ( k + 2 )-morphisms ˆ f ( h u , (01 ) i ) for u ∈ ˜ ∆ n k +1 such that s k +1 f h u, (01) i = s k +1 g ( h u i ) and t k +1 f h u, (01) i = s k +1 g ( h u i ). In compar ison, a n e xtension ˆ g of g consists of any choice o f ( k + 1)-morphisms g h u i ∈ A [1] a,b for u ∈ ˜ ∆ n k +1 such that s [1 ] k ˆ g h u i = g s k h u i and t [1 ] k ˆ g h u i = g t k h u i . Since g ( s k h u i ) = s [1 ] k g h u i = s k +1 g h u i a nd g ( t k h u i ) = t k +1 g h u i , an extension ˆ g of g is the sa me as a c hoice of ( k + 2)-mo r phisms ˆ g h u i ∈ A k +2 such that s k +1 ˆ g h u i = s k +1 g h u i and tk + 1 ˆ g h u i = t k +1 g h u i . Note that this implies that s 0 ˆ g h u i = a and t 0 ˆ g h u i = b . It is now evident that the choice for a n extension ˆ f of f is equiv alen t to a c hoice of an extension ˆ g of g . It follows that H n k +2 is in bijection with H om ω C at ( |O ( ˜ ∆ n )) | k +1 , A [1] a,b ), thu s completing our pro of b y inductio n. Corollary 8. 5 . F or A ∈ ωC at , 1. N A [1] a,b n ≃ { f ∈ H om C s (∆ n ⊗ ∆ 1 , N A ) | f | ∆ n × 0 = a, f | ∆ n × 1 = b } . 2. If A is an ω -gr o up oid, ther e is a we ak homotopy e quivalenc e N ( A [1] a,b ) − → H om L N A ( a, b ) . Pr o of. 1. Let ∆ n ⊗ ∆ 1 denote the complicia l set with underly ing simplicial set ∆ n × ∆ 1 and for which the thin r -simplices are ( x, y ) ∈ ∆ n r × ∆ 1 r such that for some i ≤ j in [ r ], x is degener ate at i and y is degener ate a t j . In [47], V erity pr o v es that there is an iso morphism o f ω -categ o ries c n, 1 : F ω (∆ n ⊗ ∆ 1 ) − →O ( ˜ ∆ n × ˜ ∆ 1 ). Since F ω is left adjoint to the nerve functor N : ω C at − → C s , H om ω C at ( O ( ˜ ∆ n × ˜ ∆ 1 ) , A ) ≃ H om ω C at ( F ω (∆ n ⊗ ∆ 1 ) , A ) ≃ H om C s (∆ n ⊗ ∆ 1 , N A ). By Pro position 8 .4, the o nly thing left to pr ove is that { f ∈ H om ω C at ( O ( ˜ ∆ n × ∆ 1 ) , A ) | f ( h u , 0 i ) = a, f ( h u , 1 i ) = b for all u ∈ ˜ ∆ n × ˜ ∆ 1 } ⊂ H om ω C at ( O ( ˜ ∆ n × ∆ 1 ) , A ) co rresp onds to H om C s (∆ n ⊗ ∆ 1 , N A ) | f | ∆ n ×{ 0 } = a, f | ∆ n ×{ 0 } = b } under the isomorphism H om ω C at ( O ( ˜ ∆ n × ˜ ∆ 1 ) , A ) ≃ H om C s (∆ n ⊗ 32 ∆ 1 , N A ). T o describ e this isomo rphism, w e will fir st need the following tw o facts, which can be fo und in [47]. One can tak e pro ducts of maps of parit y complexes, so given morphisms [ r ] φ − → [ n ], [ s ] ψ − → [ m ], there is a mor phism ˜ ∆( φ ) × ˜ ∆( ψ ) : ˜ ∆ r × ˜ ∆ l − → ˜ ∆ n × ˜ ∆ m . In fact, O ( ˜ ∆( φ ) × ˜ ∆( ψ ))( h u , v i ) = h φu, ψ v i . Secondly , there is a morphism ∇ r : ˜ ∆ r − → ˜ ∆ r of pa rit y complexes, which sends v = ( v 0 v 1 ..v r ) to { ( v 0 ...v s , v s ...v r ) ∈ ˜ ∆ r s × ˜ ∆ r r − s | s = 0 , 1 ..., r } . Now, given f ∈ H om ω C at ( O ( ˜ ∆ n × ˜ ∆ 1 ) , A ), we de- scrib e the co r respo nding F ∈ H om C s (∆ n ⊗ ∆ 1 , N A ). Given an r- simplex ( α, β ) ∈ ∆ n r × ∆ 1 r , which we think o f as a pa ir ([ r ] α − → [ n ] , [ r ] β − → [1]), F ( α, β ) is the comp osition O ( ˜ ∆ r ) O ( ∇ r ) − → O ( ˜ ∆ r × ˜ ∆ r ) O ( ˜ ∆( α ) × ˜ ∆( β )) − → O ( ˜ ∆ n × ˜ ∆ 1 ) f − → A (an r -simplex in N A ). By the univ ersal pro perty o f F ω , f ◦ O ( ˜ ∆( α ) × ˜ ∆( β )) ◦ O ( ∇ r ) = f ◦ c n, 1 ◦ ι ( α,β ) . Additionally , F ω (∆ n ⊗ ∆ 1 ) is, by definition [47], a quotient q : F ω (∆ n × ∆ 1 ) − →F ω (∆ n ⊗ ∆ 1 ) of F ω (∆ n × ∆ 1 ), and c n, 1 ◦ q = c n, 1 . Now we will show that f sa tisfies f ( h u , (0) i ) = a and f ( h u, (1) i ) = b for all u ∈ ˜ ∆ n if and only if F ( x, (0)) = a a nd F ( x, (1)) = b for all r and all r-simplices x ∈ ∆ n . F or i ∈ { 0 , 1 } , we wr ite ( i ) as shorthand for the r-simplex ( ii...i ) with i listed r times. Supp ose f satisfies this condition. T o show that F has the desired prop ert y , it is enough to verify the statement for non- degenerate s implices ( x, ( i )) since if it is tr ue for non-degener ate simplices, then F ( σ k ( x, ( i )) = σ k F ( x, ( i )) = σ k a = a , wher e σ k is the k-th degener a cy ma p. Since it is e nough to verify the statement for non-degene r ate s implices, we may also assume that x is non-deg enerate, since x is degenerate if and only if ( x, ( i )) is deg enerate. W e know that F ( α, β ) = f ◦ c n, 1 ◦ ι ( α,β ) . Let α = ( u 0 u 1 ...u r ) non- degenerate and β = ( i ). By definition, ι ( α,β ) ( h 01 ...r i ) = [ [ α, β ] ], so F ( α, β ) h 01 ...r i = f ◦ c n, 1 ([ [ α, β ] ]), which equals f ( h u , ( i ) i ) by Theorem 255 of [4 7]. Hence, F ( α, β ) h 01 ...r i = a if i = 0 or equa ls b if i = 1. F or the gener al ca se, w e show that F ( α, β ) h v i = a or b . A k-dimensional v ∈ ˜ ∆ r k corres p onds to a s trictly increas ing morphism [ k ] φ − → [ r ]. W e know that ι ( α,β ) ◦ O ( ˜ ∆( φ )) = ι φ ∗ ( α,β ) , whence ι ( α,β ) h v i = ι ( α,β ) ◦ O ( ˜ ∆( φ )) h 01 ...k i = ι φ ∗ ( α,β ) h 01 ...k i = [ [ φ ∗ a, φ ∗ β ] ] = [ [ φ ∗ α, ( i )] ]. Then, F ( α, β ) h v i = f ◦ c n, 1 ι ( α,β ) h v i = f ◦ c n, 1 ([ [ φ ∗ α, ( i )] ]) = f ( h u, ( i ) i ) (ag ain by Theorem 25 5 o f [4 7]), which is equa l to a if i = 0 o r b if i = 1. Th us, F is a s desired. This ar gumen t c an be run backw ards to show that if F ( α, (0)) = a , F ( α, (1)) = b for all α ∈ ∆ n , then f ( h u , (0) i ) = a a nd f ( h u, (1) i ) = b for all u ∈ ˜ ∆ n . 2. ∆ n +1 is a retract of ∆ n × ∆ 1 , with the inclusion ∆ n +1 ֒ → ∆ n × ∆ 1 given b y (0 , 1 , ..., n + 1) 7→ (012 ...nn, 0 .... 01 ) and p : ∆ n × ∆ 1 − → ∆ n +1 chosen to s end thin simplices to degenerate ones. 8.3 Homotopies In this section we make some basic obser v ations a bout homoto pies betw een morphisms of chain complexes from the p erspective of ω -categor ies. 8.3.1 Homotopie s Using the eq uiv alence of C h + ( Ab ) with P ic ω , we see that homotopies of maps of complexes co rresp ond to the following notion of ho mo top y b et w een maps F , G : A − → B in P i c ω . A ho motop y consists of maps H n : A n − → B n +1 such that H n ( A n − 1 ) ⊂ B n and ( t n − s n ) H n + H n − 1 ( t n − 1 − s n − 1 ) agrees with G − F when we pass to the quotient A n / A n − 1 − → B n /B n − 1 . Lemma 8.6 . F or c omplexes A , B ∈ C h + ( Ab ) , ther e is a C ∈ P ic ω such that C 0 = { maps of c omplexes A − → B } and H om 1 ( F, G ) = { homotopies fr om F to G } . Pr o of. F or i > 0, let H i consist of all maps from A − → B [ − i ]. By this w e mean h ∈ H i consists of a sequence of maps h k : A k − → B k − i but not necess arily commuting with d . Define H 0 = H om C h + ( Ab ) ( A, B ). Now define a differential ∂ : H i − → H i − 1 by ∂ f = d f + ( − 1) i − 1 f d . This ma kes H ∗ int o a complex of a belian groups. ˜ H is the desired ω -ca teg ory . 33 Lemma 8.6 gives the following corollar y , an obser v ation also made by Street [46]. Corollary 8. 7. Ther e is an ω c at e gory, A su ch that A 0 = C h + ( Ab ) , A 1 = { maps of c omplexes } , and A 1 = { homotopies of m aps of c omplexes } . F o r any ω -catego ry in ab elian groups A , there is a group homomorphism D : A − → A given by D = P n ≥ 0 t n − s n . Since we assume that A is a union o f the A n , this sum makes sense bec a use it is a finite sum on A n . Now we can efficiently define a homtop y betw een tw o functors F , G ∈ H om P ic ω ( A, B ) by reformulating the descr iption at the b eginning o f this sec tio n. Definition 18. Let A, B ∈ P ic ω and F, G ∈ H om P ic ω ( A, B ). W e say that a g r oup homomorphis m H : A − → B is a homotopy if 1. H ( A n ) ⊂ B n +1 , 2. D H + H D = G − F , and 3. s n H ( s n − s n − 1 ) = 0 for a ll n ≥ 0. Lemma 8.8. L et f , g : A − → B b e maps in C h + ( Ab ) . A homotopy h : f − → g is e quivalent to a homotopy H fr om P f t o P g in P ic ω . Pr o of. Since P A = ⊕ K i ≃ A i K i = σ ( A i ) as in § 3.3 a nd P B = ⊕ L i ≃ B i , a homomorphism H : P A − → P B is determined by H i : K i − → P B . And s n H ( s n − s n − 1 ) for all n if and only if H i ( K i ) ⊂ L i +1 . Then for x = ((0 , 0) , ... (0 , dx n ) , ( x n , x n ) , (0 , 0) , ... ) ∈ K n , H x = ((0 , 0) , ... (0 , dy n +1 ) , ( y n +1 , y n +1 , (0 , 0) , ... ) ∈ L n +1 . Defining h : A n − → B n +1 by h = π n +1 H σ gives a bijection betw een maps h : A − → B [1 ] (not necessarily commuting with the differential d ) and ho momorphisms H : P A − → P B such that s n H ( s n − s n − 1 ) and H ( P A n ) ⊂ P B n +1 . A direct c o mputation shows that with x = ((0 , 0) , ... (0 , dx n ) , ( x n , x n ) , (0 , 0) , ... ) ∈ K n , ( D H + H D ) x = ((0 , 0) , ... (0 , ( hd + dh ) dx n ) , (( dh + hd ) x n , ( dh + hd ) x n ) , (0 , 0) , ... ) ∈ L n . Hence H D + D H = P g − P f if and only if hd + dh = g − f . Remark 8.9. P A is a pro jective a belian gro up if and only if each A i is pro jective. Prop osition 8.10. L et P denote the ful l sub c ate gory of P ic ω c onsisting of obje cts which ar e pr oje ctive ab elian gr oups, and let P denote the c ate gory the obje cts of which ar e ob ( P ) and the morphisms of which ar e morphisms in P mo dulo homotopy. Ther e is an e quivalenc e of c ate gories D ≤ 0 ( Ab ) − →P . Pr o of. Let K ≤ 0 ( P r oj ) deno te the homo top y catego ry of complexes of pro jective ab elian gro ups in de g rees ≤ 0. Lemma 8.8 shows tha t H : K ≤ 0 ( P r oj ) − → P is a n equiv a lence of categor ies. The theor em no w follows since K ≤ 0 ( P r oj ) − → D ≤ 0 ( Ab ) is an equiv alence. 8.3.2 Homotopie s from the P ersp ectiv e of ω -Categories The homotopies de s cribed above are alg ebraic in nature, but we ca n still define homotopies for ordinar y ω -categor ies. The following definition extends the concept of ho motop y for from P i c ω to ω -cat. Definition 19. Let f , g : A − → B b e tw o functors of ω -c a tegories. A homotopy H : F − → G is a map H : A − → B with H ( A n ) ⊂ B n +1 , and for x ∈ A n , H x is a n n + 1 iso morphism H t n − 1 x ∗ n − 1 ( H t n − 2 x ∗ n − 2 ( ... ( H t 1 x ∗ 1 ( H t 0 x ∗ 0 f x )) .. ) H x − → n +1 ( ... (( g x ∗ 0 H s 0 x ) ∗ 1 H s 1 x ) ∗ 2 ... ) ∗ n − 1 H s n − 1 x Int uitiv ely , a homotopy H : f − → g lo oks like a “na tural transformatio n” from f to g , except any diagram of n -morphisms which sho uld commute only commut es up to an ( n + 1)-isomorphism spec ified by H . Prop osition 8.11. L et f , g : A − → B b e maps of c omplexes in C h + ( Ab ) . A homotopy h : f − → g defines a homotopy H in the sense of definition 19. 34 Pr o of. Having defined H on ˜ A i for i < n , we define H on ˜ A n . Le t x = (( x − 0 , x + 0 ) , ..., ( x − n − 1 , x + n − 1 ) , ( x n , x n ) , (0 , 0) , ... ) ∈ ˜ A n . 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