aa r X i v : . [ m a t h . C T ] J a n Abelian groups in ω -categories Brett Milburn ∗ Abstract
We study abelian group objects in ω -categories and discuss the well-known Dold-Kan correspondencefrom the perspective of ω -categories as a model for strict ∞ -categories. The first part of the paperis intended to compile results from the existing literature and to fill some gaps therein. We go on toconsider a parameterized Dold-Kan correspondence, i.e. a Dold-Kan correspondence for presheaves of ω -categories. The main result is to describe the descent or sheaf condition in terms of a glueing conditionthat is familiar for 1 and 2-stacks. Contents ω -categories 3 ω -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ω -categories 5 ω -Categories and Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Equivalences of Picard ω -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Useful Facts for Picard ω -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 I-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 ω -categories in a category C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ω -categories and quasicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.1 Basics of Parity Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.2 The Nerve of an ω -Category and the Street-Roberts Conjecture . . . . . . . . . . . . . 154.2 The Dold-Kan Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ∞ -torsors 25 ∗ [email protected] Appendix 29 ω -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3.1 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3.2 Homotopies from the Perspective of ω -Categories . . . . . . . . . . . . . . . . . . . . . 34 Our goal is to investigate abelian group objects in ∞ -categories. There are many notions of ∞ -category.Lurie [31] and Leinster [30] provide good–though not exhaustive–surveys of various definitions. Other modelssuch as complete Segal spaces [32] and crossed complexes [8] also appear in the literature. The approachtaken in this paper is to consider ω -categories as strict ∞ -categories.Roughly, an ω -category coincides with the intuitive description of an ∞ -category. It has objects andn-morphisms for n ≥
1. One can compose n-morphisms and take the k-th source or target of an n-morphismto get a k-morphism. In contrast, quasicategories (simplicial sets which admit fillers for inner horns) arealso a model for ∞ -categories. While simplicial sets are useful from the perspective of homotopy theory,they are not endowed with all of the desired structure that one would like for an ∞ -category. Namely,there is no natural choice for identity morphisms or composition. From this perspective, it is useful toconsider ω -categories, which have the advantage of not having the same deficits. Furthermore, ω -categoriesenjoy many nice properties. For example, n-categories are easily defined. However, ω -categories are strict ∞ -categories in the sense that composition is associative on the nose, and they do not contain the coher-ence data that one might desire for weak ∞ -categories (although weak ω -categories have also been studied[46, 30]). Presently our interest is in abelian group objects in ∞ -categories. We will see that abelian groupobjects in simplicial sets are in fact strict ∞ -categories, and we may therefore interpret them as ω -categories.Sections 1-4 are primarily expository. We begin by defining ω -categories in § ω -categories. In §
3, we formulate and prove the well-known statement that abeliangroups objects in ∞ -categories are the same as chain complexes of abelian groups in non-negative degrees.Furthermore, we show that this equivalence induces a derived equivalence. Two generalizations are pursued.Firstly, we introduce the notion of an I -category for any partially ordered set I . Of particular interest arethe Z -categories. Abelian group objects in Z -categories are equivalent to chain complexes of abelian groups.The analogue of the correspondence between ω -categories and simplicial sets is now between Z -categoriesand combinatorial spectra (cf. [24]), though this is not made precise here. Similarly, the relationship betweenchain complexes and Z -categories is also analogous to the Quillen equivalence between chain complexes ofabelian groups and H Z -module spectra described by Schwede and Shipley [38, 39]. Secondly, we observethat since an ω -category is determined by a set equipped with some structure maps, we may think of an ω -category as an ω -category in Sets . This can be extended to define an ω -category in an arbitrary categorywith fibered products. We generalize the equivalence between Chain complexes Ch + ( Ab ) of abelian groupsin non-negative degree and P ic ω , abelian group objects in ω -categories. For any abelian category C withcountable direct sums, we show that Ch + ( C ) is equivalent to C ω , ω -categories in C .Simplicial abelian groups, denoted sAb , can also be thought of as abelian group objects in ∞ -categories.The Dold-Kan correspondence [13] states that there is an equivalence Ch + ( Ab ) ≃ sAb . In section § Ch + ( Ab ) ≃ P ic ω . To put it succinctly, there is a nerve functor, due to Street [43], N : ωCat −→ sSet from ω -categories to simplicial sets, which when restricted to P ic ω gives N : P ic ω −→ sAb . The Dold-Kanequivalence Ch + ( Ab ) −→ sAb is, up to isomorphism, the composition of Ch + ( Ab ) −→ P ic ω with the nervefunctor. Each of Ch + ( Ab ), P ic ω , and sAb are naturally equivalent not just as categories but also as model2ategories.The core of the paper is in §
6, where we consider descent for presheaves of ω -categories. Descent for ω -categories has been considered by Street in [44, 46], and Verity showed [48] that Street’s definition ofdescent is equivalent to the standard notion of descent for presheaves of simplicial sets, where the two arerelated by the nerve functor. The Dold-Kan correspondence extends to presheaves with values in Ch + ( Ab ), P ic ω , or sAb . We consider several model structures on the presheaf categories, in particular one where thefibrant objects are precisely the sheaves (i.e. those satisfying descent with respect to all hypercovers) andone where the fibrant objects are those satisfying ˇCech descent (i.e. descent with respect to open covers).We show that the homotopy category of simplicial sheaves of abelian groups on a space X satisfying de-scent is equivalent to the derived category in non-negative degrees D ≥ ( A b ) of sheaves of abelian groups on X .The key result is Theorem 6.4 which states that a presheaf of simplicial abelian groups on site S satisfiesˇCech descent if and only if it satisfies a more concrete glueing condition, which can be explained roughly asbeing able to glue objects and n-morphism from local sections. In more detail, a presheaf A of simplicialabelian groups, A satisfies ˇCech descent if and only if for every X ∈ S and open cover U = { U i } i ∈ I of X ,1. given local objects x i ∈ A ( U i ) which are glued together by 1-morphisms and higher degree morphismsin a coherent way, there exists a global object x ∈ A ( X ), unique up to isomorphism, which glues the x i , and2. for any n-morphisms x, y ∈ A ( X ) n , the presheaf Hom A ( x, y ) whose objects are the ( n + 1)-morphismsfrom x to y satisfies the above glueing condition for objects.This can be interpreted as providing a computational tool for determining whether a presheaf satisfiesˇCech descent or a way of constructing a sheafification of a given presheaf. We hope that this has applicationsin the study of n-gerbes. Let G be an abelian group and X a topological space. With the appropriate notionof torsor, one may view an n-gerbe for G on X as a torsor for a presheaf of simplicial abelian groups, namelyit is generated by G in degree n and 0 elsewhere. Given the computational descent condition, it shouldbecome apparent that isomorphism classes of n -gerbes for G are given by ˇ H n ( X, G ). In this way, this paperis a step towards viewing sheaves of ∞ -categories as geometric realizations of cohomology classes. This pointof view is further explained in the final section, where we draw on the insights of Fiorenza, Sati, Schreiber,and Stasheff [15, 37, 41] to describe torsors for sheaves valued in ∞ -groups. ω -categories We begin by defining ω -categories, which are a model for strict ∞ -categories. Definition 1.
The data for an ω -category is a set A with maps s i , t i : A −→ A for i ∈ N and maps ∗ i : A × A A −→ A , where A × A A −→ A is the fibered product, given maps s i : A −→ A and t i : A −→ A . Let ρ i , σ i ∈ { s i , t i } denote any source or target map.( A, s i , t i , ∗ i ) i ∈ N is said to be an ω -category if the following 3 conditions are satisfied:1. For all i ∈ N , ( A, s i , t i , ∗ i ) is a category. 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. For all i < j ,( A i , A j ) is a strict 2-category. That is,3a) 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 ) whenever both sides are defined.3. For all a ∈ A , there is some i ∈ N such that s i a = t i a = a . Definition 2.
1. For an ω -category A and i ∈ N , i-objects in A are A i := s i A , and strict i-objects are A i \ A i − . In conforming to convention, we also refer to i-objects as i-morphisms .2. Let ω Cat denote the category whose objects are ω -categories and morphisms are functors between ω -categories, meaning maps of sets which preserve all structures s i , t i , ∗ i for all i .3. We write Ob : ωCat −→ Sets for the forgetful functor which sends an ω -category to its underlying set.4. For A ∈ ωCat and a, b ∈ A i , let Hom iA ( a, b ) := { x ∈ A | s i x = a, and t i x = b } . Remark 2.1. ω Cat is a symmetric monoidal category. For A , B ∈ ∞ -cat, The product A × B is just thecartesian product as sets, and source, target, and composition maps are defined componentwise. Definition 3.
We say that A ∈ ωCat is a groupoid if every n-morphism n ≥ x ∈ A n , there exists for every j < n a y ∈ A such that x ∗ j y = t j x and y ∗ j x = s j x . Remark 2.2.
It is a well known result of Brown and Higgins [9] that there is an equivalence between ω -groupoids and crossed complexes. ω -categories Definition 4.
1. For any ω -category A , two i-objects a, a ′ ∈ A i are said to be isomorphic if there exists u ∈ Hom i +1 ( a, a ′ ) and v ∈ Hom i +1 ( a ′ , a ) such that u ∗ i v = a ′ and v ∗ i u = a . (In the language ofStreet [43], a and a ′ are 1-equivalent.)2. Let F : A −→ B be a functor of ω -categories. We say that F is an equivalence of ω -categories if(a) Any 0-object, b ∈ B is isomorphic to F a for some 0-object a in A ,(b) for any i ≥ a, a ′ ∈ A i such that s i − a = s i − a ′ , t i − a = t i − a ′ and ψ ∈ Hom iB ( F a, F a ′ ),there exists φ ∈ Hom iA ( a, a ′ ) and an isomorphism β ∈ Hom i +1 B ( F φ, ψ ), and(c) for i-objects a, a ′ ∈ A , if F a is isomorphic to
F a ′ in B , then a is isomorphic to a ′ in A Conditions 2a, 2b, and 2c are the higher-categorical analogues of being essentially surjective, full, andfaithful respectively. The meaning of this definition is roughly that
F A i should be the same as B i , upto (i+1)-isomorphism. In the notation of definition 1, the first two conditions can be restated as:(a) For all y ∈ B , there exists x ∈ A and isomorphism f ∈ B such that s f = F x and t f = y .(b) If a, a ′ ∈ A i such that s i − a = s i − a ′ , t i − a = t i − a ′ , and ψ ∈ B i +1 such that s i ψ = F a and t i ψ = F a ′ , then there exists φ ∈ A i +1 and an isomorphism 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 and B are groupoids, then F : A −→ B automatically satisfies condition2c of Definition 4 if it satisfies 2a and 2b. Thus, the third condition is superfluous when we are dealingwith groupoids. Furthermore, when A and B are groupoids, condition 2a can be viewed as a special case ofcondition 2b if we add a point {∗} = A − = B − in degree − ω -categories are in fact weak equivalences in a cofibrantly generated model structure on ωCat [29]. Ara and M´etayer showed in [1] that this model structures restricts to one on ω -groupoids in away that is compatible with Brown and Golansi´nski’s model structure on crossed complexes [7].4 Picard ω -categories Definition 5. A Picard ω -category is an abelian group object in ω Cat. We let
P ic ω denote the categoryof Picard ω -categories, where Hom
P ic ω ( A, B ) = { F ∈ Hom ωcat ( A, B ) | F ◦ + = + ◦ ( F × F ) } . Remark 3.1.
For a Picard ω -category A , the fact that + : A × A −→ A is a functor implies that each A i isa subgroup. Also, we observe that if A is a Picard ω -category, then Ob ( A ) is an abelian group. Proposition 3.2.
1. An ω -category A such that Ob ( A ) is endowed with the structure of an abelian groupis a Picard ω -category if and only if(a) + : A × A −→ A is a functor of ω -categoriesand(b) x ∗ i y = x + y − s i ( x ) whenever the left hand side is defined.2. P ic ω is an abelian category.Proof.
1. First suppose that A ∈ P ic ω . Then + is a functor. Furthermore, it is clear that Ob ( A ) mustbe an abelian group object in Set . We only need to verify that x ∗ i y = x + y − s i ( x ) whenever theleft-hand side is defined. Since + is a functor, ( x + y ) ∗ n ( x ′ + y ′ ) = ( x ∗ n x ′ ) + ( y ∗ n y ′ ) if the right sideis 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 suppose that A ∈ ωCat such that Ob ( A ) is an abelian group and conditions 1a and 1b are satisfied.We wish to show that A ∈ P ic ω . Since Ob is faithful and ObA is an abelian group object in
Set , A is an abelian group object in ωCat provided that addition + and inverse ι : A −→ A are functors of ω -categories. By condition 1a, + is a functor, so it only remains to see that ι is a functor. Since0 = ρ n ρ n ( x + x − ) = ρ n x + ρ n x − , ρ n x − = ( ρ n x ) − for ρ n ∈ { s n , t n } so that ι respects sourceand target maps. Also, since ( x ∗ n y ) − = ( x + y − s n x ) − = x − + y − − s n x − = x − ∗ n y − , ι respects compositions. We conclude that A is a group object in ω Cat.2. For A , B ∈ P ic ω , Hom ( A, B ) is an abelian group. The sum φ + ψ of two functors preserves allsource and target maps and also preserves composition because + is a functor: ( φ + ψ ) x ∗ n y = φx ∗ n φy + ψx ∗ n ψy = ( φx + ψx ) ∗ n ( φy + ψy ) = ( φ + ψ ) x ∗ n ( φ + ψ ) y . Direct sums, kernels, andcokernels are gotten by taking each on the level of abelian groups, e.g. Ob ( Kerφ ) =
KerOb ( φ ). It isclear how to define source, targets and compositions on direct sums and kernels. For cokernels, sincefunctors respect source and target maps, there is no difficulty in defining source and target maps on acokernel. Composition in a cokernel is defined by letting x ∗ n y = x + y − s n x . Remark 3.3.
The forgetful functors F Ab , F Set = Ob , F ω taking values in Ab , Set , and ω Cat respectivelyare all faithful functors. It follows from Proposition 3.2b that if A , B ∈ P ic ω and g ∈ Hom Ab ( F Ab A, F Ab B )respects all source and target maps, then g = F Ab f for some f ∈ Hom
P ic ω ( A, B ). ω -Categories and Chain Complexes Notation.
Let Ch + ( Ab ) denote the category of complexes of abelian groups in non-negative degrees.Let P ic denote the category of Picard categories in the sense of Deligne [12]. That is, a Picard category C is a quadruple ( C , + , σ, τ ), where + : C × C−→C is a functor such that for all objects x ∈ C , x + : C−→C is an equivalence, and addition is commutative and associative up to isomorphisms τ and σ . Deligne madethe observation that one can assign to a complex A −→ A of abelian groups a Picard category P A , whoseobjects are A and Hom
P A ( a, b ) = { f ∈ A | df = b − a } . He goes on to show that P : Ch , ( Ab ) −→ P ic is5n equivalence and also induces an equivalence between D , ( Ab ) and P ic modulo natural isomorphism.Define
P ic ω = { A ∈ P ic ω | A = A } . The relationship between P ic ω and P ic is explained in Proposition3.4, the proof of which is found in section 8.1 of the appendix. Furthermore, when restricted to shortcomplexes in degrees 1 and 0 only, Theorem 3.7 is a strictification theorem which states that
P ic ω and P ic are equivalent.
Proposition 3.4.
P ic strict consists of all small Picard categories in P ic such that + is strictly associativeand commutative (i.e. τ and σ are identities) and for each x ∈ ob ( C ) , x + : C−→C is an isomorphism, notjust an equivalence.
Deligne’s correspondence extends to longer complexes. The correspondence in Proposition 3.5 was con-sidered by Bourn, Steiner, Brown, Higgins [4, 42, 8], et al.
Proposition 3.5.
A complex A of abelian groups defines a Picard ω -category P ( A ) . This assignment P : Ch + ( Ab ) −→ P ic ω is a functor.Proof. Let P = P ( A ) consist of sequences x = (( x − , x +0 ) , ( x − , x +1 ) , ... ) with x αi ∈ A i such that dx αi = x + i − − x − i − for all i ≥ α ∈ { + , −} . We define source and target maps by s i x = (( x − , x +0 ) , ( x − i − , x + i − ) , ( x − i , x − i ) , (0 , , ... ),and t i x = (( x − , x +0 ) , ( x − i − , x + i − ) , ( x + i , x + i ) , (0 , , ... ). If s i x = t i y , define x ∗ i y = (( x − , x +0 ) , ..., ( x − i − , x + i − ) , ( y − i , x + i ) , ( x − i +1 + y − i +1 , x + i +1 + y + i +1 ) , ... )We need to check that x ∗ i y is an element of P A . Firstly, dx + i = dx − i = dy + i = dy − i since s i x = t i y . Secondly, 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 ) αj = d (( x ∗ i y ) + j − − ( x ∗ i y ) − j − ). Hence, ( x ∗ i y ) ∈ P . It is easily checked that P A is an ω -category.We now define an operation P × P −→ P which makes P into a Picard ω -category . This is the obviousoperation x + y = (( x − + y − , x +0 + y +0 ) , ( x − + y − , x +1 + y +1 ) , ... ) , which obviously satisfies x + y − s n x = x ∗ n y when composition is defined. To 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 − , x +0 ) , ..., ( a − i , x + i ) , ( a − i +1 + x − i +1 , a + i +1 + x + i +1 ) , ... )+(( y − , y +0 ) , ..., ( b − i , y + i ) , ( b − i +1 + y − i +1 , b + i +1 + y + i +1 ) , ... )= (( x − + y − , 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 by Proposition 3.2, P a Picard ω -category . The assignment P : C ≤ ( Ab ) −→ P ic ω is obviously a functor;for a morphism f : A −→ B in Ch + ( Ab ), P f is given by (
P f x ) αi = f ( x αi ) for α ∈ { + , −} .Henceforth, we shall denote a sequence x = (( x − , x +0 ) , ( x − , x +1 ) , ... ) by ( x i ) i ∈ N or simply ( x i ), where x i =( x − i , x + i ). Lemma 3.6.
A Picard ω -category A defines a chain complex Q ( A ) ∈ Ch + ( Ab ) in such a way that Q : P ic ω −→ Ch + ( Ab ) is a functor.Proof. Since + A × A −→ A is a functor, it respects all operators ∗ i , s i , t i . Hence, + A i × A i −→ A i , so A i isa subgroup of A . Therefore it makes sense to define Q i := ( QA ) i := A i /A i − for i > Q = A . We6efine, for each i > d : Q i −→ Q i − by first defining a homomorphism d : A i −→ A i − . Define d = t i − − s i − . We must check that this is a homomorphism. If x, y ∈ A i , d ( x + y ) = t i − ( x + y ) − s i − ( x + y )= t i − x + t i − y − ( s i − x + s i − y )= t i − x − s i − x + ( t i − y − s i − y )= d x + d y. If x ∈ A i − , then t i − x = x = s i − x , so d ( A i − ) = 0. Therefore, d is a group homomorphism suchthat A i − ⊂ Kerd . This determines a homomorphism d : Q i −→ Q i − . To see that d = 0, for x ∈ Q i , d x = d ( t i − x − s i − x ) = t i − ( t i − x − s i − x ) − s i − ( t i − x − s i − x ) = t i − x − t i − x − ( s i − x − s i − x ) = 0.Hence, Q is a complex of abelian groups.We have constructed Q from A , which gives a map Q : P ic ω −→ C ≤ ( Ab ). If F : A −→ B is a map of ω -categories, F : A i −→ B i for each i ≥
0, so F descends to a map Q ( F ) : A i /A i − −→ B i /B i − , which iseasily seen to be a map of complexes. This makes Q is a functor of 1-categories. Theorem 3.7. ([4]) Q ◦ P ≃ id and P ◦ Q ≃ id . Therefore, Q and P are equivalences of categories.Proof. From a complex A ∈ Ch + ( Ab ), we get a complex Q = Q ( P A ), where Q i = ( P A ) i / ( P A ) i − . Firstdefine a map A i −→ ( P A ) i , x ˆ x , in the following way:(ˆ x ) j = (0 ,
0) if j < i − , dx ) if j = i − 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 − . Observe that for y ∈ ( P A ) i , [ y ] = [ y − s i − y ] =[ ˆ y + i ] = h ( y + i ). Therefore, h is surjective. It is clear that h is a homomorphism and that it is injective. But h must also be a map of complexes. If x ∈ A i , dh ( x ) = d [ˆ x ] = [ t i − ˆ x − s i − ˆ x ], where( t i − ˆ x − s i − ˆ x ) j = (cid:26) (0 ,
0) if j = i − dx, dx ) if j = i − d = 0, this is obviously the class of ˆ dx . Hence, h is a morphism of chain complexes, and weconclude by injectivity and surjectivity that h A : A −→ QP ( A ) is an isomorphism. However, to be an iso-morphism of functors, QP ˜ →
1, these maps must satisfy h B ◦ f = (( QP ( f )) ◦ h A for any map of complexes f : A −→ B . For x ∈ A i , h B ( f ( x )) = [ ˆ( f ( x ))] ∈ ( P B ) i / ( P B ) i − . Applying the right-hand side to x , weget (( QP ( f )) ◦ h A x = Q ( P ( f ))[ˆ x ] = [ P ( f )ˆ x ] = [ ˆ( f ( x ))]. So h does in fact define an isomorphism betweenendofunctors QP and 1 of Ch + ( Ab ).Now we wish to show that for A ∈ P ic ω , there is an isomorphism ϕ : A −→ P QA in P ic ω . For the rest of theproof, for x ∈ A i , let [ x ] be its image in Q i = A i /A i − , and for any 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 = (cid:26) ([ s i x ] , [ t i x ]) if i ≤ µ ( x )(0 ,
0) if i > µ ( x ) . It is clear that ϕx ∈ P QA , but we still must check that ϕ is a functor. For ρ i ∈ { s i , t i } , it must be shownthat 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, ρ i ( ϕx )) j . Additionally, ϕ must be a homomorphism of abelian groups,but this follows easily. For simplicity, 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 of Picard ω -categories.To show that ϕ is an isomorphism of ω -categories, we simply show that ϕ is bijection of sets. Let P = P Q ( A ).First we show that ϕ is surjective. We prove by induction that ( P ) n is in the image of ϕ for each each n ∈ N .If n = 0, let a = (( a , a ) , (0 , , ... ) ∈ P , so a ∈ A = P , and ϕ ( a ) = a . Now suppose 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 some z ∈ A k . Hence, x = ϕ ( a + k +1 − z ) ∈ ϕ ( A ). Thus, P k +1 ⊂ ϕ ( A ) and therefore ϕ is surjective.Now we demonstrate that ϕ is injective. Let x ∈ Kerφ and µ = µ ( x ) as above. Then s n x = x forall n ≥ µ and s k x = x for all k < µ . If φx = 0, [ s k x ] = 0 for all k , whence s k x ∈ A k − . In particular, s µ x ∈ A µ − , which implies that s µ − x = x , which contradicts the minimality of µ . Therefore to avoid acontradiction, x must be 0 and Kerφ = 0. Therefore, ϕ is an isomorphism.For each A ∈ P ic ω , we’ve produced an ϕ A : A ˜ −→ P Q ( A ). To complete the proof, we simply must see ifthis satisfies compatibility with morphisms in P ic ω . For x ∈ A , we require that ϕ B ◦ f ( x ) = ( P Q ( f )) ◦ ϕ A .We 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 − , we 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 . ω -categories Proposition 3.8.
For complexes
A, B ∈ C + ( Ab ) and map of complexes f : A −→ B , f is a quasi-isomorphismif and only if P f : P A −→ P B is an equivalence of ω -categories.Proof. This proof will use the description in the first part of Definition 4, as it simplifies the notation. Foran object a ∈ A n satisfying da = 0, we denote its image in H n ( A ) by [ a ], and for a map f : A −→ B , byabuse of notation, let f also denote the induced map on cohomology. For ease of notation, let F : A −→ B denote P f : P A −→ P B .Let A f −→ B be a quasi-isomorphism of complexes A B ∈ Ch + ( Ab ) . If y ∈ P B = B , there exists x ∈ A such that [ f x ] = [ y ], so there exists some z ∈ B such that dz = f x − y . Thus, (( y, f x ) , ( z, z ) , , ... ) ∈ Hom ( f x, y ) is an isomorphism as required. This proves condition (1) of Definition 4. To prove con-dition (2), let ψ ∈ Hom nP B ( F x, F y ) with x , y ∈ P A n such that s n − x = s n − y and t n − x = t n − y .Then ψ = (( ψ − , ψ +0 ) , ..., ( ψ − n +1 , ψ + n +1 ) , , ... ) with s n ψ = F x = (( f x − , f x +0 ) , ..., ( f x − n , f x + n ) , , ... ) and t n ψ = F x = (( f y − , f y +0 ) , ..., ( f y − n , f y + n ) , , ... ), 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 because f is a quasi-isomorphism .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 . We concludethat (( x − , x +0 ) , ..., , ( x − n − , x + n − ) , ( x + n , y + n ) , ( φ, φ ) , , ... ) is an ( n + 1)-morphism from x to y as required. Since P A and
P B are groupoids, condition (2) entails condition (3). Therefore, F : A −→ B is an equivalence of8 -categories.Now suppose that F is an equivalence. We will show that f : H n ( A ) −→ H n ( B ) is an isomorphism foreach n . Let x ∈ Ker ( d : A n −→ A n − ). If [ f x ] = 0, then there is some ψ such that dψ = f x . We see that(0 , ..., (0 , , (0 , f x ) , ( ψ, ψ ) , ... ) is an ( n +1)-isomorphism from 0 to ˆ f x = (0 , ..., (0 , , ( f x, f x ) , ... ). By condi-tion (3) of Definition 4, ˆ x is isomorphic to 0. Such an isomorphism is of the form (0 , ..., (0 , , (0 , x ) , ( φ, φ ) , ... )for some φ satisfying dφ = x , so we see that [ x ] = 0 and therefore f is injective. To see that f is surjective,let [ y ] ∈ H n ( B ) for n > y = (0 , ..., , (0 , , ( y, y ) , , ... ), which is an n -isomorphism from0 = F (0) to itself. By condition (2), there exists an n -isomorphism ˆ x = (0 , ..., , ( x, x ) , ... ) in A togetherwith an ( n + 1)-morphism (0 , ..., , ( f x, y ) , ( z, z ) , ... ) from F ˆ x to ˆ y . Hence, dz = y − f x so that f [ x ] = [ y ]and f is surjective. For n = 0, let y ∈ H ( B ), we use the same argument, except this time invoking condition(1) to ensure the existence of such an x ∈ A . This shows that f is a quasi-isomorphism .Let Ho ( P ic ω ) denote P ic ω localized at the equivalences. We now have the following corollary. Corollary 3.9.
The derived category D ≤ ( Ab ) of abelian groups in degrees ≤ is equivalent to the homotopycategory Ho ( P ic ω ) . ω -categories Lemma 3.10 shows that if A is a subcategory of B , and B can be extended to an ω -category C , then A canbe extended to an ω -sub-category of C . Lemma 3.10.
Let A and B be 1-categories with A a subcategory of B . If there is an ω -category C such that ( C , s , t , ∗ ) = B , then there is an ω -subcategory C ′ of C such that ( C ′ , s , t , ∗ ) = A .Proof. Define Ob ( C ′ ) = { c ∈ Ob ( C ) | s c , t c ∈ C ′ } . We show that C ′ is an ω -category by first showingthat it is stable under all source and target maps and then showing that it is closed under composition. If x ∈ Ob ( C ′ ) and j > ρ σ j x = ρ x ∈ C ′ . If j = 1, ρ σ j x = σ j x ∈ C ′ . Finally, if j = 0, ρ σ x = σ x = σ ρ x .Since A is a category, σ ρ x ∈ A = C ′ ⊂ C ′ . This shows that C ′ is stable under maps σ j ∈ { s j , t j } .Now suppose x, y ∈ C ′ . If i ≥ s ( x ∗ i y ) = s y ∈ C ′ , and t ( x ∗ i y ) = t x ∈ C ′ . If i = 0, ρ ( x ∗ i y ) = ρ x ∗ ρ y ∈ A ⊂ C ′ since A is a category. This shows that C ′ is stable under all compositions ∗ i . Clearly C ′ is an ω -category because all other necessary properties are inherited from C .In [43], Street constructs an ω -category Cat ∞ such that (( Cat ∞ ) , s , t , ∗ ) = ωCat so that ω -Cat is acategory enriched over itself. The construction is natural since it comes from an inner hom in ω Cat. Lemma3.10 can be applied to
P ic ω : there is an ω -category C such that ( C , s , t , ∗ ) = P ic ω . In fact, we see insections 8.3 and 3.4 that there is more than one ω -category which has P ic ω as its 0-objects and 1-morphisms.We now make the observation that for A ∈ P ic ω , since there is a section of A i −→ A i /A i − sending [ x ] to x − s i − x . It follows that A i ≃ A i /A i − ⊕ A i − as abelian groups. Moreover, A ≃ A ⊕ ∞ M i =1 A i /A i − With this identification, for x ∈ A k /A k − , s n x = (cid:26) x if n ≥ k n < kt n x = x if n ≥ kdx if n = k −
10 if n < k − s n : ( x , ..., x m , ... ) ( x , ...x n , , , ... ) and t n : ( x , ..., x m , ... ) ( x , ..., x n − , x n + dx n +1 , , , ... ), where d = t n − s n .There are several possible identifications of A with L ∞ i =0 A i /A i − . It is perhaps most transparent if A = P ( C )for C ∈ Ch + ( Ab ), so C i ≃ A i /A i − , and L ∞ i =0 C i −→ A is given by ( x , x , x , ..., x n , , , ... ) (( x , x + dx ) , ( x , x + dx ) , ..., ( x n − , x n − + dx n ) , ( x n , x n ) , (0 , , ... ). Proposition 3.11.
There is a functor [ −
1] :
P ic ω −→ P ic ω such that for C ∈ Ch + ( Ab ) , P ( C [ − P ( C ))[ − , where C [ − denotes the complex ... −→ C −→ C −→ ∈ Ch + ( Ab ) with C [ − n = C n − for n > and C [ − = 0 . For A ∈ P ic ω , we define A [ − by letting Ob ( A [ − Ob ( A ) , ( ∗ [ − n = ∗ n − , s [ − n = s n − , t [ − n = t n − ) for all n ≥ , s [ − = t [ − = 0 , and x ∗ y = x + y .Proof. That A [ − ∈ P ic ω is evident. If A = P ( C ), there is a bijection between Ob ( P C [ − Ob (( P C )[ − x − , x +0 ) , ( x − , x +1 ) , ..., ( x − n − , x + n − )) ∈ ( P C )[ − n maps to ((0 , , ( x − , x +0 ) , ( x − , x +1 ) , ... ).The following proposition is a generalization of a lemma found in [8] Proposition 3.12.
Let A be any abelian group. If A admits Z -linear maps s n , t n : A −→ A for n ∈ N satisfying conditions 1a, 2a, and 2b of definition 1, then there is a unique ω -category structure on A suchthat A ∈ P ic ω .Proof. If a, b ∈ A such that s n a = t n b , then we define a ∗ n b = a + b − s n a and check that this makes A intoa Picard ω -category. If compositions satisfy all ω Cat axioms, then it is easily seen that + is a functor, i.e.( a + b ) ∗ n ( a ′ + b ′ ) = ( a ∗ n a ′ ) + ( b ∗ n b ′ ), whenever the right-hand side is defined. We can easily check 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, itsuffices to check that A satisfies all ω -category axioms 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 similarly, 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 right-hand side we have ( 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 .For a set A , we let Z [ A ] denote the free abelian group on the set A . If A ∈ ωCat , we can extend all sourceand target maps Z -linearly. Proposition 3.12 implies that Z [ A ] ∈ P ic ω . Lemma 3.13.
The functor Z : ωCat −→ P ic ω sending A to the free abelian group generated by A is left-adjoint to the forgetful functor F ω : P ic ω −→ ωCat .Proof. Let A ∈ ωCat and B ∈ P ic ω . Any ϕ ∈ Hom ωCat ( A, F ω ( B )) can be extended Z -linearly to a mapˆ ϕ ∈ Hom Ab ( Z [ A ] , B )) of abelian groups. The fact that all s n and t n are Z -linear means that ˆ ϕ commuteswith all source and target maps. But since composition ∗ n in a Picard ω -category is determined by all+, s n , t n , ˆ ϕ also respects compositions ∗ n . Therefore ˆ ϕ ∈ Hom
P ic ω ( Z [ A ] , B ). It is clear that the function ϕ ˆ ϕ is injective. To see that it is surjective, any ψ ∈ Hom
P ic ω ( Z [ A ] , B ) is also a map of abelian groups,so it comes from some ϕ ∈ Hom
Sets ( Ob ( A ) , Ob ( F ( B ))). Of course since Ob ( A ) ⊂ Ob ( Z [ A ]), ϕ ( a ) = ψ ( a )and so ϕ is actually a map of ω -categories and ˆ ϕ = ψ . The inverse map is ψ ψ | A .10o check that G and F are adjoints, we must also see that we have a map of functors Hom
P ic ω ( Z [ − ] , − ) ˜ −→ Hom ωCat ( − , F − ). In other words, for f ∈ Hom ωCat ( A, A ′ ) and g ∈ Hom
P ic ω ( B, B ′ ),then for ψ ∈ Hom
P ic ω ( Z [ A ′ ] , B ), the maps ( g ◦ ψ ◦ Z [ − ]) | A = F ( g ) ◦ ( ψ | A ′ ◦ f ∈ Hom ωCat ( A, F ( B ′ )). It iseasy to see that these maps agree at the level of sets.In [43] it is shown that Ob : ωCat −→ Set is represented by an object 2 ω ∈ ωCat . Corollary 3.14. Z [2 ω ] is a corepresentative for the functor Ob : P ic ω −→ Sets .Proof.
Hom
P ic ω ( Z [2 ω ] , B ) = Hom ωCat (2 ω , F ω ( B )) = Ob ( F ω ( B )) = Ob ( B ). Since the structure maps for an ω -category are indexed by natural numbers, we may think of an ω -category asan N -category. The ω -category axioms depended only on N being a partially ordered set. We may thereforeextend the definition and define an I -category for any partially ordered set I . In particular, we are interestedin Z -categories and show that “nice” abelian group objects in Z -categories are the same as unbounded chaincomplexes of abelian groups. Definition 6.
Let I be a linearly ordered set.1. An I-category is a set quadruple (
A, s i , t i , ∗ i ) i ∈ I as in Definition 1 except that instead of N , we have I .Let I -Cat denote the category of all I-categories.2. Let P ic I denote abelian group objects in I -cat, and let P ic I denote the full subcategory, called PicardI-categories, the objects of which are A ∈ P ic I such that for all x ∈ A , there exists n ∈ I such that s n x = 0 and there exists m ∈ I for which s m x = x .We can extend some of the results about ω -categories to Z -categories. Theorem 3.7, for instance, can beextended to Z -categories. Definition 7.
A functor F : A −→ B of Z -categories is an equivalence if conditions 2b and 2c of Definition4 are met and for each b ∈ B , there exists n such that s n b is isomorphic to some F ( a ).It follows directly from the definition of P ic Z that any map of Picard Z -categories which satisfies condi-tions 2b and 2c of Definition 4 is an equivalence. In fact, since Picard Z -categories are groupoids, condition2b of Definition 4 is sufficient. Theorem 3.15.
The category Ch ( A b ) of chain complexes of abelian groups is equivalent to the category P ic Z of Picard Z -categories.Proof. The proofs of Proposition 3.2, Lemma 3.5, 3.6, and Theorem 3.7 extend naturally to Z -categories withonly a few modifications. First we define P ( A ) to consist of sequences ( .... ( x − i , x + i ) , ... ) as before but requirethat only finitely many x ± i are nonzero. Secondly, in lemma 3.6, to show the surjectivity of φ : A −→ P Q ( A ),we choose y ∈ P Q ( A ) and such that y ± i = 0 for i ≤ n ∈ Z . To show that y is in the image of φ , we start theinduction at n instead of 0. Also, we note that µ ( x ) of Lemma 3.6 is well defined except for when x = 0.Letting Ho ( P ic Z ) denote Picard Z -categories localized at equivalences, we arrive at the following corollary,the proof of which is identical to the proof of Proposition 3.8. Corollary 3.16.
The derived category
DAb of abelian groups is equivalent to the homotopy category of
P ic Z . The following results about ω -cats also extend to Z -categories: Prop 3.2, Proposition 3.8. Also, theequivalence P : Ch ( Ab ) −→ P ic Z is, just as for Picard ω -categories, isomorphic to the one that sends A ∈ Ch ( Ab ) to L n ∈ Z A n ∈ P ic Z . The following proposition is patent.11 roposition 3.17. P ic Z is a triangulated category with shift functor given as in Proposition 3.11. Thisgives DP ic Z the structure of a triangulated category. The mapping cone of f : A −→ B is ( B [ − × A, s n = s Bn − × s An , t n = t Bn − × ( f + t An )) . The t-structure coming from the standard t-structure on Ch ( Ab ) is D ≥ = Ho ( P ic ω ) , D ≤ = { A ∈ P ic Z | s n = id, t n = id for all n > } . Its heart is { A ∈ P ic ω | all σ n = id } ≃ Ab . Remark 3.18.
The ω -category structure on Ch ( Ab ) induced from Theorem 3.15 is given in the followingway. 1-objects are maps of complexes. Strict 2-objects are maps between maps of complexes F, G : A −→ B ,i.e. φ : F = ⇒ G is a map φ ∈ Hom ( A [ − , B ) such that dφ = G − F . etc. Remark 3.19.
Proposition 3.17 is just another way to say that the category of abelian group spectra isequivalent to Ch ( Ab ). The derived version also holds from this point of view, as was shown by Shipley [40]. ω -categories in a category C Just as an ω -category can be viewed as an ω -category in sets and P ic ω consists of ω -categories in abeliangroups, we can define ω -categories in any category C with fibered products, and when C is abelian, wegeneralize Theorem 3.7. Definition 8.
Let C be any category with fibered products. An ω -category in C is an object X of C withmaps s n , t n : X −→ X for n ∈ N and compositions X × X X ∗ n −→ X satisfying the axioms in definition 1(where X × X X is the fibered product with respect to t n and s n ). Morphisms between ω -categories in C aresimply morphisms in C commuting with all source, target, and composition maps. We denote the categoryof ω -categories in C by C ω . Lemma 3.20.
For any abelian category C with infinite direct sums, Ch + ( C ) and C ω are equivalent.Proof. We sketch a proof and leave the details to the reader. First, we define a functor P : Ch + ( C ) −→C ω asfollows. Let A be a complex in Ch + ( C ), and let P ( A ) = L ∞ n =0 A i .We must show that B = P ( A ) is an ω -category in C . We define source and target maps s n , t n : L ∞ n =0 A i −→ L ∞ n =0 A i as follows. First, let s i,jn , t i,jn : A i −→ A j be given by s i,jn := (cid:26) n ≥ i = j t i,jn := n ≥ i = jd if n = j = i −
10 otherwise . Now, let s in be the sum over j of the compositions A i s i,jn −→ A j −→ L ∞ k =0 A k and similarly for t in . The mor-phisms s in , t in : A i −→ P A , i ≥
0, determine s n , t n . Defining composition as ∗ n = π + π − s n π , one mayverify that ( ∗ n , t n , s n ) n ≥ satisfies conditions of Definition 1. For condition (2d) of Definition 1, the statementfor ω -categories in C should read: ∗ i ( ∗ j π × ∗ j π ) = ∗ j ( ∗ i π × ∗ i π )(( π π × π π ) × ( π π × π π )) when re-stricted to the appropriate subobject of ( B × B ) × ( B × B ). Our definition of composition ∗ n can be extended to B × B −→ B , and one may check that ∗ i ( ∗ j π ×∗ j π ) agrees with ∗ j ( ∗ i π ×∗ i π )(( π π × π π ) × ( π π × π π ))on ( B × B ) × ( B × B ). Hence, they also agree on the appropriate fibered product. Therefore, B is an ω -category in C .Let B = P ( A ), D = P ( C ) for A, C ∈ Ch + ( C ). The fact that Hom C ω ( B, D ) ≃ Hom Ch + ( C ) ( A, C ) followseasily from a few observations. Let f be a morphism from B to D . First, since f s n = s n f , an inductiveproof shows that f ( A i ) −→ D factors through C i −→ D . The fact that f commutes with all t n shows that theinduced maps A i −→ C i commute with the differentials A i d −→ A i − and C i d −→ C i − . We conclude that amorphism f : B −→ D is equivalent to a morphism from A to C in Ch + ( C ). Therefore, P is fully faithful.12e now show that P is essentially surjective. Define Q : C ω −→ Ch + ( C ) as follows. Given B ∈ C ω ,let A = Q ( B ) be given by letting A n be the cokernel B n /B n − of the monomorphism B n − −→ B n , where B n is the image of s n : B −→ B . The morphism t n − − s n − : B n −→ B n − induces a morphism from B n /B n − −→ B n − , and we denote the composition with B n − −→ B n − /B n − by d = A n −→ A n − . Let ussee that P QB ≃ B . Let f : B n −→ B n be f = 1 − s n − . Then f induces a morphism f : B n /B n − −→ B n such that πf = 1. Hence, B n ≃ B n /B n − ⊕ B n − . It is now clear that P QB ≃ B . We begin by laying out basic definitions and notations which can be found in any standard text on thesubject, such as [18]. Let ∆ denote the category of ordinals, with objects [ n ] = { , , ..., n } for n ∈ N and morphisms the (non-strictly) increasing set morphisms between them. A simplicial set is a functor X : ∆ op −→ Set . We define r-simplices in a simplicial set X to be the set X r := X ([ r ]). We let ∆ n denotethe simplicial set Hom ∆ ( − , [ n ]) and denote an r -simplex α : [ r ] −→ [ n ] by listing ( α (0) , α (1) , ..., ( α ( r )). Fora simplicial set X , we let d i , s i denote the face and degeneracy maps X ( ∂ i ) and X ( σ i ) respectively, where[ n − ∂ i −→ [ n ] is the morphism which skips only i , and σ i : [ n ] −→ [ n −
1] is the morphism which repeats only i .For a category C , a simplicial object in C is a functor from ∆ op to C , and the category of simplicial objects in C is denoted simply by s C . A simplicial abelian group is a simplicial object in the category Ab of abelian groups.The Dold-Kan correspondence was discovered independently by Dold and Kan and can be found originallyin [13] as well as a number of other references such as [18], [49]. Theorem 4.1. If C is an abelian category, there is an equivalence K : s C−→ Ch + ( C ) . K : s C−→ Ch + ( C ) is given by K ( A ) n = T n − i =0 Kerd i , and the differential d : K ( A ) n −→ K ( A ) n − is d = ( − n d n . We will be particularly interested in the case when C = Ab or sheaves of abelian groups onsome site. ω -categories and quasicategories To extend the idea of the nerve of an ordinary category, Street defines in [43] the nerve of an ω -category,which defines a functor N : ωCat −→ sSet . We first review some background on parity complexes and theStreet-Roberts conjecture. The results presented in this section are a summary of some of the results in [47].The original nerve construction is can be found in [43], and the ideas were streamlined using the languageof parity complexes in [44, 45]. A pre-parity is a graded set C = F ∞ n =0 C n and a pair of operations sending x ∈ C n to x − ⊂ C n − and x + ⊂ C n − , called negative and positive faces of x respectively. If x ∈ C , we take x − = x + = ∅ by convention. We also say that for x ∈ C n , a face a ∈ x − has parity 1 (odd) and a ∈ x + hasparity 0 (even). Elements in C n are said to be n-dimensional.A Parity complex is a pre-parity complex satisfying some additional axioms delineated in [47, 44]. Theadditional technical assumptions do not conern us because the pre-parity complexes which we deal with hereare all parity complexes.For a parity complex C and S ⊂ C , let | S | n = S nk =0 S k , where S k = S ∩ C k . For 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 let N ( C ) denote the ω -category with underlying set { ( M, P ) | M, P are finite subsets of C } .Source, target and compositions are given by 13 s n ( M, P ) = ( | M | n , M n ∪ | P | n − ) • t n ( M, P ) = ( | M | n − ∪ P n , | P | n ) • ( N, Q ) ∗ n ( M, P ) = ( M ∪ ( N \ N n ) , Q ∪ ( P \ P n ).There is another ω -category O ( C ) attained from a parity complex C , which we will now describe. For aparity complex C and subsets S, T ⊂ C , we say that S ⊥ T if ( S + ∩ T + ) ∪ ( S − ∩ T − ) = ∅ . Another way to ex-press this is to say that S ⊥ T if S and T have no common faces of the same parity. A subset S of C is called well-formed if it has at most one 0-dimensional element and for distinct elements x, y ∈ S , x ⊥ y . Define O ( C ) to be the subcategory of N ( C ) consisting of all ( M, P ) ∈ N ( C ) such that M and P are both non-empty,well-formed subsets of C , P = ( M ∪ M + ) \ M − = ( M ∪ P + ) \ P − , and M = ( P ∪ M − ) \ M + = ( P ∪ P − ) \ P + .It is not immediately clear that O ( C ) is an ω -category. However, the work in [43, 44] demonstrates that itis. For a parity complex C , there are distinguished elements of O ( C ). Let x ∈ C n . We inductively definesubsets 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 ) iscalled an atom . Let < C > = { < x > | x ∈ C } . Street proved [43, 44] that < C > freely generates O ( C ) inthe sense defined below. First we introduce some notation. For n ∈ N and B ∈ ωCat , let | B | n denote the n -category ( s n B, ∗ i , s i , t i ) ≤ i ≤ n . Definition 10.
Let A be an ω -category and G a subset of its elements, with grading G n = G ∩ A n .1. A is freely generated by G if for all ω -categories B , all functors f : | A | n −→ B of ω -categories and mapsof sets g : G n +1 −→ B such that s n g ( x ) = f ( s n x ) and t n g ( x ) = f ( t n x ) for all x ∈ G n +1 , there exists aunique functor ˆ f : | A | n +1 −→ B of ω -categories such that ˆ f || A | n = f and f | G n +1 = g .2. A is generated by G if for each n ≥ | A | n +1 is the smallest sub- ω -category of A containing | A | n ∪ G n +1 .If A is freely generated by G , then A is generated by G ([47]).For Parity complexes C , D , a map of sets f : C −→ D which respects the grading induces a morphism N ( f ) : N ( C ) −→N ( D ), sending ( M, P ) to ( f ( M ) , f ( P )). Let us consider only graded maps of sets f : C −→ D such that • for all x ∈ C , f ( x ) ⊂ D is a singleton set, and • for all n ≥ x ∈ C n +1 , f ( x ) is well formed, f ( x + ) = ( f ( x − ) ∪ f ( x ) + ) \ f ( x ) − , and f ( x − ) =( f ( x + ) ∪ f ( x ) − ) \ f ( x ) + .Parity complexes together with graded set maps f : C −→ D with these two properties form a category P arity of parity complexes. The two conditions are chosen so that the functor N : Graded Sets −→ ωCat restrictsto a functor O : P arity −→ ωCat .Of particular interest are the parity complexes ˜∆ n , which we now define. r-dimensional elements of˜∆ n are subsets v = { v < v < ... < v r } of [ n ] := { , , ...n } ⊂ N of size r + 1. We will often denotesuch a v ∈ ˜∆ nr by ( v v ...v r ). The i-th face of v ∈ ˜∆ nr , denoted δ i v = { v , ..v i − , ˆ v i , v i +1 , ..., v r } ∈ ˜∆ nr +1 ,where ˆ v i denotes omission of v i . Now define the face operators v ξ = { δ i v | i ∈ [ r ] and i is of parity ξ } for ξ ∈ { + , −} . A morphism [ n ] α −→ [ n ] in ∆ induces a morphism ˜∆( α ) : ˜∆ m −→ ˜∆ m sending v ∈ ˜∆ mr to ∅ if αv i = αv i +1 for some i and to αv = { αv , ..., αv r } otherwise. Thus, ˜∆ is a functor from ∆ to P arity , andwe obtain the composition ∆ ˜∆ −→ P arity O −→ ωCat . The ω -category O ( ˜∆ n ) is called the n-th oriental andhas a unique non-identity n-morphism h (01 ..n ) i . For a morphism [ m ] α −→ [ n ] in ∆, O ( ˜∆( α )) maps h v i to h αv i .The product of parity complexes was shown in [44] to be a parity complex. For 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 iseven and has the opposite parity of p is odd. 14 .1.2 The Nerve of an ω -Category and the Street-Roberts Conjecture In [43], Street defines the nerve functor N : ωCat −→ sSet with left adjoint F ω . The nerve of an ω -category A consists of composing O ˜∆ with the Yoneda embedding ωCat −→ Set . More explicitly, The n-simplices of
N A are
Hom ωCat ( O ( ˜∆ n ) , A ). For an n-simplex x : O ( ˜∆ n ) −→ A and morphism α : [ m ] −→ [ n ] in ∆, α ∗ x ∈ N A m is the composition of x with O ( ˜∆( α )). The left-adjoint F ω : sSet −→ ωCat is the left Kan extension of O ◦ ˜∆ : ∆ −→ ωCat along the Yoneda embedding Y : ∆ −→ sSet . For a simplicial set X , F ω ( X ) is charac-terized by the following property. For each n-simplex x ∈ X n , there is a a functor ι x : O ( ˜∆ n ) −→ F ω ( X )of ω -categories such that for any morphism α : [ m ] −→ [ n ] in ∆, ι α ∗ x = ι x ◦ O ( ˜∆( α )), and for any other ω -category A with such a family of maps j x : O ( ˜∆ n ) −→ A , n ∈ N , x ∈ X n , j factors through ι . For X ∈ sSet and x ∈ X n , let [[ x ]] = ι x ( h ..n i ).To get an idea of what the nerve of an ω -category looks like, an n-simplex of N A looks like a draw-ing of an n-simplex in the ω category A , meaning an n-simplex labeled with an n-morphism in A andk-dimensional faces are labeled with k-morphisms in A . It is an easy exercise to check that N A = A , N A = A . A 2-simplex x ∈ N A is a functor of ω -categories x : O ( ˜∆ ) −→ A , which consists of a 0-objects x ( h i ) , x ( h i ) , x ( h i ) ∈ A , 1-morphisms x ( h i i ) x ( h ij i ) −→ x ( h j i ) in A for i, j ∈ [2], and a 2-morphism x ( h i ) ∈ A such that s x ( h i ) = x ( h i ) and t x ( h i ) = x ( h i ) ∗ x ( h i ).When we restrict to P ic ω , Theorem 4.5 guarantees that N : P ic ω −→ sAb is an equivalence. In general,however, N : ωCat −→ sSet is not an equivalence. The problem is that viewing an ω -category as a simplicialset by taking its nerve loses some information. The simplicial set no longer remembers which n-simplicesrepresent identity morphisms and so it forgets how to compose morphisms. To remedy this situation, in[43, 36], Street and Roberts modify the modify the nerve construction to take values in the category Cs of“complicial sets.” A complicial set is a simplicial set X together with a collection of simplices tX called thinsimplices which satisfy certain axioms. To name a few, • No 0-simplex of X is in tX , • the only 1-simplices in tX are degenerate 1-simplices, • the degenerate simplices of X are in tX , • and for each ( n − n ≥
2, 0 < k < n has a unique thin filler.The other properties can be found in [47]. A morphism of complicial sets f : ( X, tX ) −→ ( Y, tY ) is amorphism f : X −→ Y of simplicial sets such that f ( tX ) ⊂ tY . Remark 4.2.
Complicial sets is a full subcategory of a larger category
Strat of stratified sets whose objectsare pairs ( X, tX ) but which are not required to satisfy all of the axioms listed above for complicial sets.Morhphisms, of course, are simply morphisms of simplicial sets which preserve thin simplices. There is anatural way of taking the product ⊗ of two stratified sets, where the underlying simplicial set of X ⊗ Y is X × Y . For instance, the thin r-simplices in ∆ n ⊗ ∆ are the simplices ( x, y ) ∈ ∆ nr × ∆ r such that x isdegenerate at some 0 ≤ j < r and y is degenerate at some k ≥ j .The enhanced nerve construction N : ωCat −→ Cs sends A to ( N A, tN A ), where the thin n-simplices in
N A are the simplices x : O ( ˜∆ n ) −→ A such that x ( h ...n i ) is an ( n − N with theforgetful functor Cs −→ sSet (( X, tX ) X ) gives the original nerve construction. The nerve N has a leftadjoint F ω so that F ω (( X, tX )) is attained from F ω ( X ) by “collapsing” morphisms corresponding to thinsimplices, a process described in detail in [47]. Theorem 4.3, known as the Street-Roberts conjecture, wasproven by Verity in [47]. Theorem 4.3. N : ωCat −→ Cs is an equivalence of categories. Remark 4.4.
More recently, in [33], Nikolaus defines a model category of algebraic Kan complexes similar tothe category Cs , which specifies a distinguished filler for each horn. He shows that algebraic Kan complexesis Quillen equivalent to simplicial sets. 15 .2 The Dold-Kan Triangle The Dold-Kan correspondence [13] gives an equivalence between Ch + ( Ab ) and sAb , simplicial objects inabelian groups (or equivalently, abelian group objects in sSet ). Furthermore, sAb and Ch + ( Ab ) have modelstructures. The model structure on sAb is induced by the forgetful functor U : sAb −→ sSet . Specifically, sAb inherits the weak equivalences and fibrations from sSet ; f is a weak equivalence in sAb if and only if U ( f ) is a weak equivalence in sSet , and f is a fibration in sAb if and only if U f is a fibration in sSet . Themodel structure on Ch + ( Ab ) has quasi-isomorphisms as the weak equivalences, degree-wise epimorphisms (inpositive degree) as the fibrations, and degree-wise monomorphisms with projective cokernels as cofibrations.Additionally, P ic ω inherits a model structure from Ch + ( Ab ) via the equivalence Ch + ( Ab ) −→ P ic ω . Theweak-equivalences in P ic ω are morphisms which are equivalences of the underlying ω -categories. The Dold-Kan correspondence is in fact an equivalence of model categories, as is explained in [38]. We have seen that P ic ω ≃ Ch + ( Ab ) ≃ sAb as model categories, but also the following theorem of Brown relates these twocorrespondences in the following way. Theorem 4.5. ([10], [34]) The composition N ◦ P : Ch + ( Ab ) −→ sAb is the same as the Dold-Kan corre-spondence. In other words, the following diagram commutes up to isomorphism. Ch + ( Ab ) P −−−−→ P ic ω y D y N sAb sAb where D denotes the Dold-Kan correspondence. Since the Dold-Kan correspondence sends X ∈ sAb to A • , where A n = L n − i =0 Kerd i , it is now clear fromthe comments in section 3.3 that N − : sAb −→ P ic ω satisfies N − ( X ) ≃ L ∞ n =0 ∩ n − i =0 Kerd i .In the Dold-Kan correspondence quasi-isomorphisms correspond to weak equivalences in sSet , and byProposition 3.8, quasi-isomorphisms correspond to equivalences of Picard ω -categories. Under the equiva-lence N : P ic ω −→ sAb , equivalences of ω -categories correspond to weak equivalences in sAb . Moreover, upontaking the geometric realization : | · | : sAb −→ T op , the equivalences of ω -categories are identified with weakequivalences of topological spaces. Localizing with respect to weak equivalences, we have an embedding of Ho ( P ic ω ) into the homotopy category of topological spaces. Throughout the next two sections, fix an essentially small site S with enough points equipped with aGrothenieck topology, such as the category of manifolds with the Etale topology or open sets on a fixedmanifold X . Henceforth, let “prehseaf” mean a presheaf on S , i.e. a functor from S op into some category. Definition 11.
For a presheaf F with values in model category M , an object X of S and an open cover U = { U i } of X , let ˇ F U denote the cosimplicial diagram Y F ( U i ) ⇒ Y F ( U ij ) ⇛ F ( U ijk ) ... in M . We write ˇ F = ˇ F U when the open cover is understood. We say that F satisfies ˇCech descent withrespect to U if the natural map F ( X ) −→ holim ˇ F U is a weak equivalence in M . We say that F satisfies ˇCechdescent if F satisfies ˇCech descent with respect to all objects X ∈ S and all open covers U of X .16et X be an object of S , which we think of as a discrete presheaf of simplicial sets. The concept of hypercover U −→ X is defined precisely in [14]. Informally, we may think of it as a resolution of a ˇCech coverof X . Notice that for a hypercover U −→ X , and presheaf F with values in model category M , F ( U ) is acosimplicial diagram in M since U is a simplicial diagram in S , and we have a morphism F ( X ) −→ F ( U ) in M ∆ , where X is considered as a constant diagram. Definition 12.
Let U −→ X be a hypercover and F be a presheaf with values in model category M . Wesay that F satisfies descent with respect to U −→ X if F ( X ) −→ holimF ( U ) is a weak equivalence in M . Wesay that F satisfies descent if it satisfies descent with respect to all hypercovers.By “simplicial presheaf” we mean a presheaf with values in sSet . Let ˜ P re sSet denote simplicial presheaveswhich are levelwise sheaves of sets (i.e. simplicial objects in sheaves of sets). In general, for a category C ,we denote presheaves on S with values in C by P re C , and we let ˇ Sh C denote those presheaves which satisfyˇCech descent and Sh C denote those satisfying descent, provided that C is a model category. For shorthandwe write P re ω for P re ωCat and
P re ωAb for
P re
P ic ω . Remark 5.1.
It was shown in [14] that , ˇ Sh sSet are the presheaves which satisfy descent for all boundedhypercovers and that there exist presheaves satisfying ˇCech descent but not descent for all hypercovers. Proposition 5.2.
For a presheaf F of simplicial abelian groups, F satisfies ( ˇCech ) descent if and only if U F : S op −→ sSet satisfies ( ˇCech) descent.Proof. Let D : ∆ op −→S be a simplicial diagram associated to a C¸ ech complex ˇ C U −→ X (i.e. the ˇCech nerveof an open cover of some X ∈ S ). We know that F ( X ) −→ holim ( F D ) is a weak equivalence if and only if
U F ( X ) −→ U holim ( F D ) is a weak equivalence. We would like to show that F ( X ) −→ holimF D is a weakequivalence in sAb if and only if U F ( X ) −→ holim ( U F D ) is a weak equivalence in sSet . By the two outof three property of weak equivalences, this is true provided that
U holim ( F D ) −→ holim ( U F D ) is a weakequivalence in sSet . Thus, to complete the proof, it suffices to show that U ( holimF D ) −→ holim ( U F D ) is aweak equivalence. Since U : sAb −→ sSet is the right adjoint in a Quillen pair ( Z [ − ] , U ), it naturally followsthat for any diagram G in sAb , U ( holimG ) −→ holim ( U G ) is a weak equivalence.
Remark 5.3.
Since the category A b of sheaves of abelian groups on S is abelian, the Dold-Kan corre-spondence provides an equivalences Ch + ( A b ) −→ ˜ P re sAb because simplicial objects in A b are the same as˜ P re sAb . Proposition 5.4.
Neither ˇ Sh sAb nor ˜ P re sAb is contained in the other.Proof.
To see this, consider the following example of a presheaf F ∈ P re sSet which satisfies ˇCech descent butwhich is not a levelwise presheaf. Let S = Op ( X ), open sets on a manifold X , and let A be any non-zeroabelian group. Consider the presheaf of abelian groups A such that A ( X ) = A and A ( U ) = 0 if U = X andthe complex A ∗ ∈ C + ( A b ) which is A in each degree and whose differential is the identity map on A . Thecorresponding presheaf of simplicial abelian groups satisfies ˇCech descent but levelwise is not a sheaf of sets.On the other hand, take any sheaf of abelian groups A and consider the complex A −→ P ( A −→
0) of ω -categories and in fact a presheaf of 1-categories. However, itdoes not satisfy descent for stacks. Hollander shows in [20] that a stack satisfies descent if and only if itsnerve satisfies ˇCech descent as a simplicial presheaf. Hence, N P ( A −→
0) is a presheaf of simplicial abeliangroups which is levelwise a sheaf but does not satisfy ˇCech descent, showing that neither condition impliesthe other.
There are several model structures on simplicial presheaves. There are, of course, the projective and injec-tive model structures [31, 19], which we denote by
P re projsSet , P re injsSet respectively. Weak equivalences are thesectionwise weak equivalences. In the projective model structure, fibrations are the sectionwise fibrations,17nd in the injective model structure, the cofibrations are the sectionwise cofibrations. For each of these, onecan take the left Bousfield localization
P re loc,injsSet and
P re loc,projsSet at the hypercovers. The existence of thelocalalization
P re loc,injsSet follows from the work of Jardine [25], and the construction of the local projectivemodel structure is due to Blander [3]. The weak equivalences in
P re loc,projsSet and
P re loc,injsSet are the stalkwiseweak equivalences of simplicial sets since S has enough points [26]. The important feature of the local modelstructures is that in P re loc,injsSet , the fibrant objects are the presheaves which are fibrant in
P re injsSet and satisfydescent for all hypercovers. Fibrant objects in
P re loc,projsSet are the ones which are sectionwise Kan complexesand satisfy descent for all hypercovers.Jardine shows the existence of a model structure on
P re sAb such that a morphism is a weak equivalence orfibration if and only if it is a weak equivalence or fibration in
P re loc,injsSet [24]. From this, the next two resultsfollow easily. First we see that for any presheaf of simplicial abelian groups, there exists a sheafification (i.e.local fibrant replacement) which is also a presheaf of simplicial abelian groups. The second result states thatthere is a derived Dold-Kan correspondence.
Lemma 5.5.
Let U : P re sAb −→ P re sSet denote the forgetful functor. For every X ∈ P re sAb , there is amap X f −→ Y in P re sAb such that
U f is a weak equivalence and
U Y is a fibrant object in
P re loc,injsSet and
P re loc,projsSet .Proof. If X ∈ P re sAb , take a fibrant replacement X −→ Y for X in P re sAb in the model structure of [24]described above.
U Y ∈ P re loc,injsSet is fibrant since U : P re sAb −→ P re loc,injsSet preserves fibrations. Additionally,since
U Y is fibrant in
P re loc,injsSet , it satisfies descent for all hypercovers. Since it is a presheaf taking valuesin sAb , it is sectionwise fibrant. Therefore,
U Y is also fibrant in
P re loc,projsSet . Proposition 5.6.
Let P ′ denote the full subcategory of P re sAb spanned by objects satisfying descent forhypercovers. Localizing at local weak equivalences, we can form the homotopy categegory Ho ( P ′ ) , and Ho ( P ′ ) is equivalent to D + ( A b ) , the derived category of chain complexes of sheaves of abelian groups in non-negativedegrees.Proof. First observe that the inclusion ˜
P re sAb ⇆ P re sAb and the levelwise sheafification functors descend toequivalences of homotopy categories since weak equivalences in Jardine’s model structure on
P re sAb are thelocal weak equivalences. Because Ch + ( A b ) is equivalent to ˜ P re sAb , we need only show that Ho ( P re sAb ) isequivalent to Ho ( P ′ ) to complete the proof. Since the forgetful functor U : P re sAb −→ P re proj,locsSet preservesfibrant objects, as was noted in the proof of Lemma 5.5, the full subcategory P cf of cofibrant fibrant objectsis contained in P ′ . It is easy to check that since P cf ⊂ P ′ ⊂ P re sAb , Ho ( P ′ ) exists and is equivalent to Ho ( P re sAb ). Remark 5.7.
Consider the case of a simplicial presheaf on a topological space X . In general it is a strongerrequirement on a simplicial presheaf on X to satisfy descent for all hypercovers than it is to satisfy ˇCechdescent. Lurie explains in [31] that if X has finite covering dimension, then the two conditions are the same.However, we are interested in presheaves on manifolds, all of which have finite covering dimension. If weconsider sheaves on the site of all differentiable manifolds, then the result is unchanged since we are onlyconsidering the hypercovers of [14] rather than the most general hypercovers. The standard definition of descent, used in [20, 25, 14], is given in Definitions 11 and 12. In this section,however, we describe a glueing condition for presheaves of ω -categories and show that in the case when thepresheaf takes values in P ic ω , it coincides with the homotopy limit descent condition. This is an attemptto expand on the work of Hollander [20], who showed that descent for 1-stacks can be described in a homo-topy theoretic way which is consistent with descent for simplicial presheaves. Our definition of the glueing18ondition is motivated by Breen’s description of descent for 2-stacks [6]. The idea is that a sheaf A of ω -categories on S satisfies the glueing condition if one can glue 0-objects, 1-objects, and k-objects for any k ≥ U = { U i } i ∈ I of X ∈ S the data for glueing of 0-objects consists of 0-objects a i ∈ A ( U i ) which are identified on intersections via1-morphisms a ij ∈ A ( U ij ) , a ij : ( a i ) | ij −→ ( a j ) | ij , the 1-morphisms a ij are identified on triple intersectionsvia 2-morphisms a ijk : a jk ∗ a ij = ⇒ a ik and so on. The glueing condition for 0-objects states that for anysuch system ( { a i } i ∈ I , { a ij } i,j ∈ I , { a ijk } , ... ), there exists an 0-object x ∈ A ( X )–unique up to isomorphism–with isomorphisms x | U i −→ a i which fit together in a consistent way. In other words, the system can be gluedto a global 0-object in an essentially unique way.For A to satisfy the glueing condition, it must satisfy the glueing condition for 0-objects, and for each pairof sections x, y ∈ A ( X ) k , the presheaf of ω -categories Hom k A ( x, y ) satisfies the glueing condition for 0-objects.To make this description formal, Let Y : ∆ −→ sSet be the Yoneda embedding. A system ( a i ∈A ( U i ) , a ij ∈ A ( U ij ) , ... ) as above can be thought of as a morphism a ∈ Hom sSet ∆ ( Y , ˇ N A ). Here, N is the nerve functor so that N A is a simplicial presheaf, and ˇ N A is the diagram from Definition 11. Thereis a restriction map of the constant diagram ρ : N A ( X ) const −→ ˇ N A . In order to compare N A ( X ) with, Hom sSet ∆ ( Y , ˇ N A ), we identify N A ( X ) with { F ∈ Hom sSet ∆ ( Y , N A ( X ) const ) | F (∆ n ) = F (∆ ) for all n } ,so N A ( X ) is a subset of Hom sSet ∆ ( Y , N A ( X ) const ) and ρ maps N A ( X ) to Hom sSet ∆ ( Y , ˇ N A ). Remark 6.1.
In Street’s definition of the descent [46], he defines a descent object
Desc ( ˇ N A ) ∈ ωCat object of A ∈
P re ω with respect to the ˇCech complex for U . Using the adjunction, F ω : sSet ⇆ ωCat : N , Hom sSet ∆ ( Y , ˇ N A ) is identified with the 0-objects of Desc ( ˇ N A ). Remark 6.2.
The category of cosimplicial objects in sSet is a simplicial model category [18]. For
X, Y ∈ sSet ∆ , the enrichment over simplicial sets is given by hom ( X, Y ) n = Hom sSet ∆ ( X × ∆ n , Y ), where hom ( X, Y ) ∈ sSet . For K ∈ sSet , X ∈ sSet ∆ , X × K ∈ sSet ∆ is ( X × K ) n = X n × K , and for any Y ∈ sSet ∆ , Hom sSet ∆ ( X × K, Y ) ≃ Hom sSet ( K, hom ( X, Y )). The homotopy of Definition 13 is really a homotopy H : ∆ −→ hom ( Y , ˇ N A ) between 0-simplices F and ρG in hom ( Y , ˇ N A ). The simplicial set hom ( Y , ˇ N A ) iscommonly called the total space T ot ( ˇ N A ) of ˇ N A . Definition 13.
Let A be a presheaf of ω -categories on S , and let U = { U i } i ∈ I be an open cover of X ∈ S .We say that A satisfies with respect to U if for all F ∈ Hom sSet ∆ ( Y , ˇ N A ), there exists a homotopy H : Hom sSet ∆ ( Y × ∆ , ˇ N A ) from F to ρG for some G ∈ N A ( X ) . We say that A also satisfies unique0-glueing with respect to U if two 0-objects a, b ∈ A ( X ) = N A ( X ) are isomorphic in A ( X ) whenever ρa and ρb are isomorphic in hom ( Y , ˇ N A ).The intuitive meaning of the uniqueness of glueing is that if a, b ∈ A ( X ) are locally isomorphic in aconsistent way, then a, b are isomorphic. Hence if G and G ′ ∈ A ( X ) glue F ∈ hom ( Y , ˇ N A ) in the notationof Definition 13, then G and G ′ are isomorphic in A ( X ).To describe k -glueing for k >
0, first observe that there is a shift functor [1] : ωCat −→ ωCat , whereif A = ( Ob ( A ) , s k , t k , ∗ k ) k ∈ N , A [1] is the ω -category ( Ob ( A ) , s [1] k = s k +1 , t [1] k = t k +1 , ∗ [1] k = ∗ k +1 ) k ∈ N .For x, y ∈ A , we are especially interested in sub- ω -categories A [1] x,y = Hom A ( x, y ) := { a ∈ A [1] | s a = x , t a = y } . More generally, for k ≥ x, y ∈ A k − , let A [ k ] x,y = { a ∈ A [ k ] | s k − a = x , t k − a = y } .Observe that ( A [ k ])[1] = A [ k + 1]. Remark 6.3. A [ k ] x,y = Hom kA ( x, y ) from Definition 2. Definition 14.
Let A be a presheaf of ω -categories, X an object in S and U an open cover of X .1. Suppose k >
0. Then A satisfies the (unique) k-glueing condition with respect to U if for x, y ∈ A ( X ) k − , A [ k ] x,y satisfies (unique) 0-glueing whenever s k − x = s k − y and t k − x = t k − y . We use the conventionthat s − x = t − x = 0 for all x . 19. We say that A satisfies ω -descent with respect to U if for all k ≥ A satisfies the unique k-glueingcondition.3. We say that A satisfies ω -descent for loops if for all x ∈ A ( X ) , each A [ k ] x,x satisfies the unique0-glueing condition. Definition 15.
Let A be a presheaf of Picard ω -categories. We say that A satisfies ω -descent, k-glueing,etc. if it satisfies ω -descent with respect to U , k-glueing with respect to U , et cetera, respectively for allobjects X of S and open covers U of X .Our goal for the remainder of this section is prove the following theorem, which relates two notions ofdescent. Theorem 6.4.
Let A be a presheaf site S with values in P ic ω . Then A satisfies ω -descent if and only if itsatisfies ˇCech descent. It is important to note that since N : P ic ω −→ sAb is an equivalence of model categories, a presheaf A ∈
P re ω satisfies ( ˇCech) descent if and only if its nerve N A ∈
P re sSet satisfies ( ˇCech) descent.To prove Theorem 6.4, we will show that for each open cover U , a presheaf A of Picard ω -categories on S satisfies the unique glueing condition with respect to U on X if and only if it satisfies ˇCech descent withrespect to U . For the rest of the section, fix an object X ∈ Ob ( S ) and open cover U = { U i } i ∈ I of X . We define a pair of adjoint functors [ −
1] : Ch + ( Ab ) ⇆ Ch + ( Ab ) : Ω, which form a Quillen pair for themodel category Ch + ( Ab ). The functor [ −
1] is defined as follows. Let B ∈ Ch + ( Ab ). Then ([ − B ) i = B i − for i >
0, and ([ − B ) = 0. Thus, [ − B = ... −→ B −→ B −→
0, where B is in degree 1. On theother hand, Ω is defined by letting (Ω A ) i = A i +1 for i >
0, and (Ω A ) = Ker ( A d −→ A ). Thus,([1] , A ) = ... −→ A −→ A −→ Ker ( d ), with Ker ( d ) ⊂ A in degree 0. Clearly, Hom ch + ( Ab ) ([ − B, A ) ≃ Hom ch + ( Ab ) ( B, Ω A ), so [ −
1] is left adjoint to Ω. Since [ −
1] preserves weak equivalences and cofibrations in Ch + ( Ab ), ([ − , Ω) is a Quillen pair (Prop 8.5.3 in [19]).
Lemma 6.5.
Given the equivalences
P ic ω ≃ Ch + ( Ab ) ≃ sAb , the following pairs of endofunctors correspondto each other:1. ([ − , Ω) on Ch + ( Ab ) defined above,2. ([ − , [1] , ) on P ic ω , where [1] , is defined after Definition 13 and [ − is defined in Proposition 3.11,and3. ( M, L ) , where L is given by L ( X ) n = Ker ( d X : X n +1 −→ X n ) ∩ Ker ( d n ) and d L ( X ) n = − d Xn +1 , s L ( X ) n = − s Xn +1 . One can describe L ( X ) n as the ( n + 1) -simplices in X such that the 0-th face and 0-th vertexof x is . On the other hand, M ( X ) n = X n − for n > and M = 0 . The structure maps are d M ( X ) i = d Xi − for i > and d = 0 .Proof. Using the equivalences P : Ch + ( Ab ) −→ P ic ω and K : sAb −→ Ch + ( sAb ), the result follows easily.Lemma 6.5 suggests that we should think of Ω( A ) as loops in A based at 0. We can also consider pathfunctors. Lemma 6.6.
Given the equivalences,
P ic ω ≃ Ch + ( Ab ) ≃ sAb , the following functors correspond to eachother.1. In P ic ω the path functor [1] : P ic ω −→ P ic ω is the restriction of [1] : ωCat −→ ωCat defined immediatelyafter Definition 13. . Π : Ch + ( Ab ) −→ Ch + ( Ab ) defined by Π( A ) = ... −→ A −→ A −→ A ⊕ A , with A ⊕ A in degree anddifferential d ⊕ A −→ A ⊕ A .3. P ath : sAb −→ sAb defined by P ath ( X ) = ˆ S ( X ) ⊕ X , where X is a discrete simplicial set with X in degree and ˆ S : sAb −→ sAb is given by ˆ S ( X ) n := Ker ( d X : X n +1 −→ X n ) and d ˆ S ( X ) n = − d Xn +1 , s ˆ S ( X ) n = − s Xn +1 .Proof. To make the identification of Π with [1], let P denote the equivalence P : Ch + ( Ab ) −→ P ic ω . Identify-ing (( a + n +1 , a − n +1 ) , ..., ( a +2 , a − ) , ( a +1 + a +0 , a − + a − )) ∈ P (Π( A )) n with (( a + n +1 , a − n +1 ) , ..., ( a +2 , a − ) , ( a +1 , a − ) , ( a +0 , a +0 − da +1 )) ∈ ( P A ) n +1 = ( P A )[1] n establishes that [1] ◦ P = P ◦ Π.To make the identification of
P ath with Π, let S : Ch + ( Ab ) −→ Ch + ( Ab ) denote the functor A ( ... −→ A −→ A −→ A ). It is easily verified using the Dold-Kan correspondence that ˆ S corresponds with S .Observe that Π( A ) = S ( A ) ⊕ ( ... −→ A ) so that K Π( A ) = K ( S ( A )) ⊕ K ( A ), where K ( A ) is the discretesimplicial abelian group with A in degree 0.For A ∈ P ic ω and 0-objects a, b ∈ A , the sub- ω -category A [1] a,b of A [1] is not a Picard ω -category.However, we can still describe its nerve as a sub-object N A [1] a,b of N A [1] in sSet . Denote the path functor sAb −→ sSet corresponding to [1] a,b : P ic ω −→ ωCat by P ath a,b . First we argue that the n-simplices of
N A [1]can be viewed as the ( n + 1)-simplices of N A for which the 0th face is s n +10 b for some vertex b ∈ N A . ByLemma 6.6, N A [1] n = { y ∈ ( N A ) n +1 | d y = 0 } ⊕ A , and there is an inclusion N A [1] n ֒ → N A n +1 given by( y, b ) y + s n +10 b . Lemma 6.7.
Let A be a Picard ω -category. Then N A [1] a,bn consists of simplices x ∈ N A n +1 for which the0-th vertex v x = ( d ) n +1 x = a and d x = b .Proof. We prove by induction that for an n-simplex in x ∈ N A [1] n ⊂ N A n +1 such that v x = a and d x = b , x ∈ N A [1] a,b . For n = 0, let x be a 0-simplex in N A [1] such that v x = a and d x = b . Since ( N A ) = A ,we can think of the 1-simplex x in N A as a 1-morphism in A , where the source of x is s x = d x = v x = a and the target of x is t x = d x = b . Hence, x ∈ N [1] a,b .Now let x be an n-simplex in N A [1] n ⊂ N A n +1 such that v x = a and d x = b . Observe that v x = v ( d i x ) = a and d x = d d i x = b for every 0 < i ≤ n + 1, and in fact, v w = a and d w = b forevery r-dimensional face w of x . We make use of the notation defined in § N A [1] is a morphism x ∈ Hom ωCat ( O ( ˜∆ n ) , A [1]), where O ( ˜∆ n )is Street’s n-th oriental, defined in § ω -category B = ( Ob ( B ) , ∗ i , s i , t i ) i ∈ N , and m ≥ | B | m = ( B m , ∗ i , s i , t i ) ≤ i ≤ m − denote the m -category formed by taking all k -morphisms for k ≤ m .In [43], Street shows that a choice of morphism x ∈ Hom ωCat ( O ( ˜∆ n ) , A [1]) is equivalent to a morphism g ∈ Hom ωCat ( |O ( ˜∆ n ) | n − , A [1]) and α ∈ A [1] n such that s [1] n − α = g ( s n − h (01 ...n ) i ) and t [1] n − α = g ( t n − h (01 ...n ) i ), where h (01 ...n ) i is the unique non-trivial n-morphism in O ( ˜∆ n ). Hence, the 0-source and0-target of α in A are determined by g because s α = s s [1] n − α = s g ( s n − h (0 ...n ) i ) = g ( s s n − h (0 ...n ) i ) = g ( s h (0 ...n ) i ), and similarly, t α = g ( t h (0 ...n ) i ). Street shows that g ( s n − h (0 ...n i ) and g ( t n − h (0 ...n i ) arecompositions of of g applied to h β i for some β : ∆ r ֒ → ∆ n with r < n . Using properties (1d) and (2b)of Definition 1, taking s or t of a composition simply applies s or t to the last morphism in the chainof compositions. Thus, s α = s g ( h β i ) and t α = t 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 x = v ( xβ ∗ ) and b = d ( xβ ∗ ). By induction hypothesis, s ( g h β i ) = a and t ( g h β i ) = b , so s α = a and t α = b . Also, for r < n , every r-dimensional face xγ ∗ for γ : ∆ r ֒ → ∆ n , v ( xγ ∗ ) = v x = a and d ( xγ ∗ ) = d x = b , whence s ( g h γ i ) = a and t ( g h γ i ) = b by induction hypothesis. Thus, for r ≤ n , everyr-face w of x , the r-morphism g ( h γ w i )in A [1] representing w has a as its 0-source and b as its 0-target.Therefore, x ∈ N A [1] a,b . 21 emark 6.8.
For X ∈ sAb , P ath a,b X (the paths in P ath ( X ) from a to b ) is isomorphic to the left mappingspace Hom LX ( a, b ) of Lurie [31]. Lemma 6.9.
Let A be a presheaf of ω -categories. The unique 0-glueing condition is equivalent to thecondition that π ( N A ( X )) ≃ π ( holim ˇ N A ) .Proof. In [5] ch.10-11, it was proved that for G ∈ sAb ∆ , hom ( Y , G ) −→ holimG is a weak equivalence. There-fore, π ( hom ( Y , ˇ N A )) ≃ π holim ˇ N A . We conclude the proof by arguing that π ( N A ( X )) ≃ π ( hom ( Y , ˇ N A ))if and only if A satisfies the unique 0-glueing condition.The 0-glueing condition states that for all f ∈ Hom sSet ∆ ( Y , ˇ N A ), there exists H ∈ Hom sSet ∆ ( Y × ∆ , ˇ N A ) such that H |Y×{ } = f and H |Y×{ } = ρg for some g ∈ A ( X ) . Since Hom sSet ∆ ( Y × ∆ , ˇ N A ) = hom ( Y , ˇ N A ) , this is equivalent to asking that for every vertex f ∈ hom ( Y , ˇ N A ) , there exists H ∈ hom ( Y , ˇ N A ) such that d H = f and d H = ρg for some g ∈ N A ( X ) . In other words, The 0-glueingcondition states that ρ ∗ : π ( N A ( X )) −→ π ( hom ( Y , ˇ N A )) is a surjection. The uniqueness part of theunique 0-glueing condition states that ρ ∗ is also injective. Lemma 6.10.
Let A be a presheaf of Picard ω -categories. Then A satisfies the unique glueing conditionfor all loops if and only if it satisfies the unique glueing condition for loops at . Furthermore, for any a ∈ A ( X ) , A [1] , ≃ A [1] a,a in P re ω .Proof. First we show that for A ∈ P ic ω and a ∈ A , addition by a , p a : A−→A (mapping x x + a ) definesan isomorphism of ω -categories, though not of Picard ω -categories. To see that p a is a morphism of ω -categories, 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-object. Composition 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 . Therefore, p a is a morphism of ω -categories. Since p a has inverse p − a , it is an isomorphism. If A is a presheaf of ω -categories and a ∈ A ( X ) , then p a : A−→A is an isomorphism of presheaves of ω -categories sending basepoint 0 to basepoint a . Lemma 6.11.
Let G ∈ sAb and L : sAb −→ sAb be the functor described above, corresponding to [1] , : P ic ω −→ P ic ω . Then π n +1 L k ( G ) ≃ π n L k +1 G for all k, n ≥ .Proof. It follows from the definition of L that Hom sSet (∆ n , LG ) ≃ { f ∈ Hom sSet (∆ n +1 , G ) | (0) and f d (012 ...n ) = 0 } . Therefore, { f ∈ Hom sSet (∆ n , LG ) | ∂ ∆ n −→ } ≃ { f ∈ Hom sSet (∆ n +1 , G ) | ∂ ∆ n +1 −→ } .Not only are they isomorophic as sets, but homotopies in { f ∈ Hom sSet (∆ n , LG ) | ∂ ∆ n −→ } coincide withthose in { f ∈ Hom sSet (∆ n +1 , G ) | ∂ ∆ n +1 −→ } . To see this, we will show that f and g are homotopic in { f ∈ Hom sSet (∆ n , LG ) | ∂ ∆ n −→ } if and only if the corresponding simplices in { f ∈ Hom sSet (∆ n +1 , G ) | ∂ ∆ n +1 −→ } are homotopic. However, since G and LG are simplicial abelian groups, it suffices to show that f being ho-motopic to 0 is the same in both sets. It is a standard fact, found in [22], that for f ∈ G n with d i f = 0 forall 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 = 0for all i ≤ n . Clearly then, f : ∆ n −→ LG is homotopic to 0 if and only if the corresponding map ∆ n +1 −→ G is homotopic to 0. Proposition 6.12.
Let A be a presheaf of Picard ω -categories. Then A is a satisfies ˇCech descent if andonly if A satisfies the unique glueing condition for loops.Proof. Let G = N A be the corresponding presheaf of simplicial abelian groups. Then G satisfies descentfor all hypercovers if and only if G ( X ) −→ holim ˇ G is a weak equivalence, i.e. π n G ( X ) −→ π n holim ˇ G is anisomorphism for each n ≥
0. By lemma 6.11, π n G ( X ) −→ π n holim ˇ G is an isomorphism for each n ≥ π ( L n G ( X )) ≃ π ( L n ( holim G )) for all n ≥
0. Since L is right Quillen, it preserves homotopylimits. Therefore, π ( L n ( holim G )) ≃ π ( holim ( L n G )). In summary, G satisfies ˇCech descent if and only if π ( L n G ( X )) ≃ π ( holim ( L n ˇ G )) for all n ≥
0. By Lemma 6.9, we conclude that G satisfies ˇCech descent if22nd only if L n G satisfies the unique 0-glueing condition for each n. To say that L n G satisfies the unique0-glueing condition is just to say that G satisfies the unique n-glueing condition. Therefore, G satisfies ˇCechdescent if and only if G satisfies the unique glueing condition for loops based at 0. However, Lemma 6.10implies that this is equivalent to the unique glueing condition for all loops. Corollary 6.13.
Let be a A is a presheaf of Picard ω -categories. If A satisfies ω -descent, then it satisfiesˇCech descent. To complete the proof of Theorem 6.4, we now show that if a presheaf of simplicial abelian groups satisfiesˇCech descent for all hypercovers, it satisfies the unique glueing condition, not just for loops.
Lemma 6.14.
Let A be a presheaf of Picard ω -categories, X ∈ S , and a, b ∈ A ( X ) . If there exists f ∈ A ( X ) such that s f = a , t f = b , then A [1] a,b ≃ A [1] , in P re ω , whence A [1] a,b is a presheaf ofPicard ω -categories.Proof. There is an isomorphism A [1] a,b −→A [1] a,a sending a section y to x − ∗ y = y +( x − − b ). First, let ussee that for y ∈ A [1] a,b , x − ∗ y ∈ A [1] a,a . We easily see that s ( y + x − − b ) = s y + s x − − s b = a + b − b = a and t o ( y + x − − b ) = t y + t x − − t b = b + a − b = a so that x − ∗ y ∈ A [1] a,a . Since x − − b ∈ A [1]( X ) ,addition by x − − b is an isomorphism (The proof is identical to that of Lemma 6.10). Recall from Lemma6.10 that A [1] a,a ≃ A [1] , . Lemma 6.15.
Let A be a sheaf of ω -categories and a, b ∈ A ( X ) . Then A [1] a,b ( X ) −→ holim ˇ A [1] a,b is aweak equivalence if and only if A [1] a,b ( X ) −→ hom sSet ∆ ( Y , ˇ A [1] a,b ) is a weak equivalence.Proof. Let B = N A [1] a,b ∈ P re sAb . If some B ( U i ...i n ) = ∅ , then ˇ B = ∅ , and B ( X ) = ∅ , so both A [1] a,b ( X ) −→ holim ˇ A [1] a,b and A [1] a,b ( X ) −→ hom sSet ∆ ( Y , A [1] a,b ) are weak equivalences. Now supposethat each B ( U i ...i n ) = ∅ . By the two out of three axiom for weak equivalences, it suffices to show that hom sSet ∆ ( Y , A [1] a,b ) −→ holim ˇ B is a weak equivalence. By Ch. 11, § B is a fibrant object in sSet ∆ with Bousfield and Kan’s model structure.For X ∈ sSet ∆ , let the n-th matching space M n X = { ( x , ...x n ) ∈ X n × X n × ... × X n | s i x j = s j − x i if ≤ i < j ≤ n } , where s i denotes the i-th coface map X ( σ i ). There is a map X n +1 −→ M n X in sSet given by x ( s x, ..., s n x ). By definition (Ch. 10 § X is fibrant if and only if each X n +1 −→ M n X , n ≥ X −→∗ are fibrations in sSet . We now proceed to show that ˇ B is fibrant.First we show that we can endow each each ˇ B n with the structure of a simplicial abelian group such thatall s i : ˇ B n +1 −→ ˇ B n is a morphism in sAb . For the open cover U = { U i } i ∈ I , we are assuming that for each n ≥ α ∈ I [ n ] , B ( U α ) = ∅ , so by Lemma 6.14, we can choose a group structure on ˇ B n by choosinga 1-simplex f = { f α } α ∈ I [ n ] with d f = a , d f = b . The goal is to choose a group structure on each ˇ B so thatthe coface maps are all morphisms of simplicial abelian groups. First declare an equivalence relationship on I [ n ] by setting α ∼ β if U α = U β , and let I [ n ] denote the set of equivalence classes. Observe that the cofacemaps 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 monotonicmap which repeats only m .To choose the group stuctures on ˇ B n , we start with n = 0. Choosing any group structure f = { f α } α ∈ I [0] such that f α = f β for α ∼ β . Now, having chosen group structures for all ˇ B k for k ≤ n such that all cofacemaps are morphisms of simplicial abelian groups, choose any group 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 . To see that such a choice exists, we simplyobserve that if α ∼ γ such that α = σ ∗ m β and 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 morphisms of simplicial abelian groupsbecause for β ∈ I [ n ] , π β s m is the composition of the identity map B ( U σ ∗ m β ) −→B ( U β ) with the projection π σ ∗ m β : ˇ B n +1 −→B ( U σ ∗ m β ), both of which are group maps since the group stuctures f σ ∗ m β and f β were chosento coincide. 23e now demonstrate that each ˇ B n +1 −→ M n ˇ B is surjective. It will follow that these maps are levelwiseepimorphisms of abelian groups, hence fibrations in sSet . Choose any n ≥
0. The proof that ˇ B n +1 −→ M n ˇ B issurjective is very much the same as the proof in the previous paragraph. For x ... x ∈ ˇ B n ( x i = { x αi } α ∈ I [ n ] ),( x , ..., x m ) ∈ M n ˇ B if and only if for all 0 ≤ l < m ≤ n and α ∈ I [ n − , x σ ∗ l αm = x σ ∗ m − αl . Choose any y = { y α } α ∈ I [ n +1] such that for all 0 ≤ m ≤ n , β ∈ I [ n ] , y σ ∗ β = x βm . Hence the map ˇ B n +1 −→ M n ˇ B sends y to ( x , ...x n ). However, we need to check that such a choice exists. We need to show that if σ ∗ m β = σ ∗ l γ , then x βm = x γl . Suppose that σ ∗ m β = σ ∗ l γ . Clearly α = σ ∗ m σ ∗ l δ for some δ , so α = ( σ l σ m ) ∗ δ =( σ m − σ l ) ∗ δ = σ ∗ l σ ∗ m − γ . In general, if σ ∗ m τ = σ ∗ m µ , then τ = µ . Hence, β = σ ∗ l δ and γ = σ ∗ m − γ , so x βm = x σ ∗ l δm = x σ ∗ m − δl = x γl . Therefore, y α is well-defined. Lemma 6.16.
Let A be a presheaf of Picard ω -categories that satisfies ˇCech descent, and let a, b ∈ A ( X ) .If Hom sSet ∆ ( Y , ˇ N A [1] a,b ) is non-empty, then there exists a path x ∈ A ( X ) from a to b , and A [1] a,b ≃A [1] a,a ≃ A [1] , as sheaves of ω -categories. It follows that A [1] a,b ∈ P re ωAb .Proof.
Choose any F ∈ Hom sSet ∆ ( Y , ˇ N A [1] a,b ). Let Y (1) denote the object in sSet ∆ 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 − n +1 σ i +1 −→ ∆ n . First we observethat Hom sSet ∆ ( Y , ˇ N A [1] a,b ) is isomorphic to Hom sSet ∆ ( Y (1) , ˇ N A ) a,b := { f ∈ Hom sSet ∆ ( Y (1) , ˇ N A ) | f n :∆ n +1 −→ ˇ N A n satisfies f ((0)) = a, d f (01 ...n + 1) = b } .We will define a morphism p : ∆ n × ∆ in sSet which sends ∆ n ×{ } to a and ∆ n ×{ } to b . First observethat ∆ n × ∆ has non-degenerate ( n + 1)-simplices z k := (012 .., k − , k, k, k + 1 , ...n, s k (01) ∈ ∆ nn +1 × ∆ n +1 ,0 ≤ k ≤ n , which satisfy relations d k z k = d k z k − , for k ≥
1. For any X ∈ sSet , to give a morphism f ∈ Hom sSet (∆ n × ∆ , X ), it is necessary and sufficient to give ( n + 1)-simplices y k = f ( z k ) ∈ X n +1 suchthat d k y k = d k y k − . Let p : ∆ n × ∆ −→ ∆ n +1 be the morphism given by y k = s k d k (012 ...n + 1). One mayverify that ∆ n × { }−→ s n (0) and ∆ n × { }−→ d (01 ...n + 1). An easy but tedious calculation shows that p extends to a morphism p : Y × ∆ −→Y (1) in sSet ∆ .Now, F ∈ Hom sSet ∆ ( Y , ˇ N A [1] a,b ) corresponds to some f ∈ Hom sSet ∆ ( Y (1) , ˇ N A ) a,b . We can form thecomposition f p ∈ Hom sSet ∆ ( Y × ∆ , ˇ N A [1]) = hom ( Y , ˇ N A ) , which sends Y × { } to a and Y × { } to b . Thus, f p ∈ hom ( Y , ˇ N A ) is a 1-simplex in hom ( Y , ˇ N A ) from a to b . Since A satisfies ˇCech descent, ρ : N A ( X ) −→ hom ( Y , ˇ N A ) is a weak equivalence of fibrant simplicial sets. Therefore, there exists a morphism G : hom ( Y , ˇ N A ) −→ N A ( X ) such that Gρ is homotopic to the identity. Since N A ( X ) is fibrant, one canalso find a 1-simplex in N A ( X ) from a to b , i.e. a path in A ( X ) from a to b . The result now follows fromLemma 6.14. Remark 6.17.
For a, b ∈ A ( X ) as in Lemma 6.16, the (presheaf of) abelian group structure of A [1] a,b isnot the one endowed from being a sub- ω -category. Lemma 6.18.
Let A be a presheaf of Picard ω -categories which satisfies ˇCech descent. Suppose a, b ∈ A ( X ) k are such that s k − a = s k − b and t k − a = t k − b . Then A [ k + 1] a,b satisfies ˇCech descent. Additionally, ifthere exists a ( k + 1) -morphism x from a to b , then A [ k + 1] a,b is a presheaf of Picard ω -categories.Proof. We prove the statement by induction on k . First let k = 0. If Hom sSet ∆ ( Y , ˇ N A [1] a,b ) is non-empty,then there exists a path x ∈ A ( X ) from a to b and A [1] a,b ≃ A [1] a,a ≃ A [1] , ∈ P re ωAb . Since A satisfiesˇCech descent, it satisfies the unique k-glueing condition for loops at 0, and since ( A [1] , )[ k ] , = A [ k + 1] , , A [1] , satisfies the unique k-glueing condition for loops at 0. Therefore, A [1] , is a presheaf of Picard ω -categories which satisfies ˇCech descent. It follows that since A [1] , ≃ A [1] a,b , A [1] a,b is a presheaf ofPicard ω -categories satisfying ˇCech descent.If, on the other hand, Hom sSet ∆ ( Y , ˇ N A [1] a,b ) = ∅ , then the simplicial set hom sSet ∆ ( Y , ˇ N A [1] a,b ) = ∅ .But Hom sSet ∆ ( Y , ˇ N A [1] a,b ) = ∅ also implies that N A [1] a,b ( X ) = ∅ , whence N A [1] a,b ( X ) = ∅ , so A [1] a,b trivially satisfies ˇCech descent. This proves the base case ( k = 0).24ow suppose that a, b ∈ A ( X ) k are such that s k − a = s k − b and t k − a = t k − b . Then for any open U ⊂ X , as a set, A [ k + 1] a,b ( U ) = { x ∈ Ob ( A ( U )) | s k x = a, t k x = b } = { x ∈ Ob ( A ( U )) | s k x = a, t k x = b, s k − x = s k − a, t k − x = t k − b } = { x ∈ Ob ( A [ k ] s k − a,t k − b ( U )) | s k x = a, t k x = b } = { x ∈ Ob ( A [ k ] s k − a,t k − b ( U )) | s [ k ] x = a, t [ k ] x = b } = ( A [ k ] s k − a,t k − b )[1] a,b ( U ) . Hence, A [ k + 1] a,b = ( A [ k ] s k − a,t k − b )[1] a,b . But since a is a k -morphism from s k − a to t k − b , A [ k ] s k − a,t k − b is a prehseaf of Picard ω -categories satisfying ˇCech descent, by induction hypothesis. By the base case, A [ k +1] a,b = ( A [ k ] s k − a,t k − b )[1] a,b satisfies ˇCech descent, and if there exists a ( k +1)-morphism x ∈ A ( X ) k +1 from a to b , then A [ k + 1] a,b ∈ P re ωAb .Theorem 6.4 now follows directly from Lemma 6.18.
Proof.
Suppose that
A ∈
P re ωAb and satisfies ˇCech descent. Then by Lemma 6.9, A satisfies the unique0-glueing condition. Lemma 6.18 ensures that each A [ k ] a,b satisfies ˇCech descent hence the unique 0-glueingproperty by Lemma 6.9. Therefore A satisfies the unique k -glueing property for each k , i.e. satisfies ω -descent. ∞ -torsors We have added this section for completeness, as using ω -descent as a way to make ∞ -torsors accessible wasa primary motivation for establishing the equivalence of the two descent conditions. The central ideas inthis section (Definition 16 and Propositions 7.1 and 7.4) are due to Fiorenza, Sati, Schreiber, and Stasheff[15, 37, 41]. We simply formulate them in a slightly different way, prefering to define objects up to homotopy. Definition 16.
Let G be a presheaf of simplicial abelian groups on a site C . Let BG denote any deloopingobject of G in the homotopy category of P re proj,locsSet ( C ). This means that BG is an object with a point ∗−→ BG , and G is the homotopy pullback of the diagram ∗ y ∗ −−−−→ BG We define
T ors G = hom ( − , g BG ), where hom denotes simplicial enrichment in P re sSet ( C ), and g BG denotesa sheafification (i.e. fibrant replacement) of BG [15]. Remark 7.1.
Note that
T ors G is well-definied up to weak equivalence due to the uniqueness up to homotopyof looping and delooping functors [18, 21]. Furthermore, Lemma 7.2 shows that for two different choices T , T for T ors G , T ( X ) is weakly equivalent to T ( X ) for any X ∈ C . Lemma 7.2.
Let
F, G ∈ P re proj,locsSet be fibrant objects which are weakly equivalent. Then For any X ∈ C , F ( X ) and G ( X ) are weakly equivalent.Proof. Let V −→ X be a hypercover of X , and let V ′ be a cofibrant replacement of V . Then By [14] Lemma4.4, F ( X ) ≃ hom ( X, F ) −→ hom ( V ′ , F ) and G ( X ) −→ hom ( V ′ , G ) are weak equivalences. If there is a weakequivalence F −→ G , then it is a general fact [19] Corollary 9.3.3 that in a simplicial model category withcofibrant V ′ and a weak equivalence of fibrant objects F −→ G then hom ( V ′ , F ) −→ hom ( V ′ , G ) is a weakequivalence of simplicial sets. Therefore, F ( X ) is weakly equivalent to G ( X ). If there is a zigzag of weakequivalences from F to G , then the result is the same: F ( X ) and G ( X ) are weakly equivalent.25he central observations of this section are Propositions 7.1 and 7.4. The proof Proposition 7.1 is amodification of the proof of Proposition 3.2.17 in [15], which is for complexes concentrated in one degreeonly. First we make use of the following fact about homotopy limits for presheaves.A homotopy pullback is simply the homotopy limit in the model category P re proj,locsSet . However, sinceevery sectionwise weak equivalence is a local weak equivalence, the identity map i : P re projsSet −→ P re proj,locsSet preserves weak equivalence and is adjoint to itself. It is a general fact that if a functor U between modelcategories is a right adjoint and preserves weak equivalences, then U preserves homotopy limits. Hence, tocompute the homotopy limit in the local model structure, it is enough to compute it in the global projectivemodel structure.Given a complex A ∈ Ch + ( A b ) of presheaves of abelian groups concentrated in non-negative degrees,recall that A [1] is A shifted up one degree so that A [1] n = A n − . For a presheaf of simplicial groups G , let G [1] be the presheaf of simplicial groups corresponding to the shift functor in Ch + ( A b ) by the Dold-Kancorrespondence. Proposition 7.3.
Let G be a presheaf of simplicial groups. Then G [1] is a delooping object of G .Proof. We use the Dold-Kan correspondence between
P re sAb ( C ) and Ch + ( A b ). Given a complex A ∈ Ch + ( A b ) of presheaves of abelian groups concentrated in non-negative degrees, recall that A [1] is A shifted upone degree so that A [1] n = A n − . Define B = B ( A ) as follows. Let B n = A n × A n − , and let d : B n −→ B n − be d ( x, y ) = ( da + ( − n b, db ). Clearly, d = 0 so that B ∈ Ch + ( A b ). We define f : B −→ A [1] as the obviousmap: f n : A n −→ A n − −→ A n − is just π . It is obvious that this is a chain map. Furthermore, B is acyclicin the sense that each homology class H n B = 0. Using the preceding facts about homotopy pullbacks, since B −→ A [1] is a fibration, the pullback of the diagram B y −−−−→ A [1]is in fact the homotopy pullback. The pullback P has the property that any g : C −→ B such that f g = 0factors through P . First we see that there is a map h : A −→ B given by in degree n by x ( x, f h = 0. It is easy to see that a map g such that f g = 0 is precisely a map C −→ A −→ B . Hence, A is thepullback. Since there is a weak equivalence from the diagram B y −−−−→ A [1]to 0 y −−−−→ A [1] ,A is the homotopy pullback of the latter diagram, whence A [1] is a delooping of A .Since G [1] is a delooping of G , it follows that T ors G ( X ) = hom( X, g G [1]) ≃ g G [1]( X ), whence T ors G = g G [1]. Having defined T ors G , we define T ors nG , which is well defined up to local weak equivalence. Let T ors nG := g G [ n ], where the fibrant replacement g G [ n ] is chosen to be a sheaf of simplicial abelian groups. Itwas shown in Lemma 5.5 that it is possible to make such a choice.26 roposition 7.4. Let G be a presheaf of simplicial abelian groups. Then T ors nG = g G [ n ] = T ors
T ors n − G .Proof. This follows by induction. By definition,
T ors
T ors n − G = T ors ^ G [ n − = (( G [ n − ∼ [1]) ∼ . Since theDold-Kan correspondence commutes with taking stalks, which is to say that the diagram Ch + ( p A b ) −−−−→ P re sAb y y Ch + ( Ab ) −−−−→ sAb commutes if the vertical arrows represent taking stalks at a point x . Therefore, for a local weak equivalence A −→ B in P re sAb , the induced map A [1] −→ B [1] is a local weak equivalence. Therefore, given A ∈ P re sAb with sheafification ˜ A , the local weak equivalence A −→ ˜ A induces a local weak equivalence A [1] −→ ˜ A [1].Hence, g A [1] is weakly equivalent to ˜ A [1]. From here it is easy to see that g G [ n ] is weakly equivalent to(( G [ n − ∼ [1]) ∼ . The result follows. Jardine and Luo have taken a different approach to principal bundles [26, 27], and we now compare theirformulations with those of [15]. Here we consider only the case where C is the site of open sets on a space X so that X is the terminal object in C . Throughout this section, fix a space X and a sheaf G of simplicialgroups on X . Let G − sP re ( C ) denote the simplicial presheaves on X with G -action. Lemma 7.5.
There is a cofibrantly generated closed model structure on the category G − sP re ( C ) of sim-plicial G-presheaves, where a map is a fibration (resp. weak equivalence) if the underlying map of simplicialpresheaves is a global fibration (resp. local weak equivalence). Definition 17.
1. Let G − T ors be the category of cofibrant fibrant simplicial G-presheaves P such that P/G −→∗ is a hypercover (i.e. local trivial fibration). We call the objects in G − T ors
G-principalbundles . A G-principal bundle P is called a principal bundle over X = P/G .2. Choose a factorization ∅−→ EG −→ X in G − sP re ( C ) where the first map is a cofibration and thesecond is a trivial fibration. Let B G = EG/G .Note that EG and B G are defined up to weak homotopy equivalence in G − sP re ( C ). Lemma 7.6.
Any B G is a delooping object of G .Proof. We would like to see that G is a homotopy pullback of ∗ y ∗ −−−−→ B G in (some/any) model structure where the weak equivalences are the local weak equivalences.In Remark 4 of [27], the following observation is made. Let W G and
W G be defined sectionwise. That isto say, (
W G )( U ) = W ( G ( U )) and ( W G )( U ) = W ( G ( U )), where W and W are the standard constructions[18, 11, 28]. By taking cofibrant replacements and factorizing maps, we can find objects EG and ˜ W G in G − sP re ( C ) together with maps W G p ←− ˜ W G j −→ EG such that p is a trivial fibration in G − sP re ( C ), j is atrivial cofibration in G − sP re ( C ), and ˜ W G is cofibrant. Since p is a weak equivalence of cofibrant simplicialG-spaces, it induces a weak equivalence ˜ W G/G −→ W G/G = W G . Similarly, j induces a weak equivalence˜ W G/G −→ EG/G = B G . We therefore have a sequence of weak equivalences in the diagram category: from˜ W G y ∗ −−−−→ ˜ W G/G
W G y ∗ −−−−→
W G and also to EG y ∗ −−−−→ B G, which maps to ∗ y ∗ −−−−→ B G. by a weak equivalence. Replacing an object in the diagram category by a weakly equivalent one does notchange the homotopy pullback. Hence, we need only show that G is the homotopy pullback of W G y ∗ −−−−→
W G.
First we show that the pullback of ∗−→
W G ←− W G is in fact a homotopy pullback. The homotopypullback is simply the homotopy limit in the model category
P re proj,locsSet . From the paragraph precedingProposition , we know that to compute the homotopy limit in the local model structure, it is enough tocompute it in the global projective model structure. We know that
P re projsSet is proper [14]. It is well knownthat in a right proper model category, the pullback of a diagram X −→ Z ←− Y is the homotopy pullbackprovided that one of the morphisms X −→ Z or Y −→ Z is a fibration. In the global projective model struc-ture, W G −→ W G is a fibration, so the homotopy pullback of the original diagram is simply the pullback of ∗−→
W G ←− W G .The pullback can be taken sectionwise, and the reader can consult [18] for an exposition of the fact thatsectionwise, G is isomorphic to the pullback of W G y ∗ −−−−→
W G.
Lemma 7.7.
Any delooping object BG is weakly equivalent to a classifying space.Proof. Take a classifying space B G . By Lemma 7.6, B G is a delooping of G . However, delooping is well-defined up to weak equivalence, so since BG and B G are both deloopings, they are isomorphic in thehomotopy category.Alternately, a straightforward calculation shows that G [1] is weakly equivalent to W G and hence B G .28 Appendix
Let
P ic ω = { A ∈ P ic ω | A = A } , and let P ic denote the category of Picard categories in the sense ofDeligne [12]. That is, a Picard category C is a quadruple ( C , + , σ, τ ), where + : C × C−→C is a functor suchthat for all objects x ∈ C , x + : C−→C is an equivalence, and addition is commutative and associative up toisomorphisms τ and σ . In this section we explain in detail the relationship between P ic ω and P ic . Lemma 8.1.
Let
C ∈
P ic . For any objects x , y ∈ C , id x + id y = id x + y whenever x is isomorphic to y .Proof. Take any g ∈ Hom C ( x, y ). Then( g ◦ id x ) + ( g − ◦ id y ) = ( g + g − ) ◦ ( id x + id y ) g + g − = ( g + g − )( id x + id y ) id x + y = id x + id y Lemma 8.2.
Assume
C ∈
P ic such that + is strictly commutative and associative.1. For any f ∈ Hom C ( x, y ) in C , f + f − = id x + y .2. For any f ∈ H C ( a, b ) , g ∈ Hom C ( b, c ) , id b + g ◦ f = g + f .3. Assume that for every x ∈ Ob ( C ) , x + : C−→C is actually an isomorphism. Then Ob ( C ) is an abeliangroup. For all f ∈ Hom C ( x, y ) , f + id = f , and there exists a unique h ∈ Hom C ( − x, − y ) such that f + h = id . Hence, Hom C = ∪ x,y ∈ ob ( C ) ( x, y ) is also an abelian group.Proof. id x + y = id x + id y = id x + f ◦ f − = ( id x ◦ id x ) + ( g ◦ g − ) = ( id x + g ) ◦ ( id x + g − ) =( id x + g ) ◦ ( g − + id x ) = ( id x ◦ g − ) + ( g ◦ id x ) = g − + g g ◦ f + id b = ( g ◦ f ) + ( g − ◦ g ) = ( g + g − ) ◦ ( f + g ) = id x + y ◦ ( f + g ) = f + g
3. The first claim is obvious. Since 0 + 0 = 0, id + id = id . Therefore, for f ∈ Hom C ( x, y ), id +( id + f ) = id + f . However, since 0+ : C−→C is an equivalence of categories, it gives a bijection
Hom C ( x, y ) −→ Hom C ( x, y ) sending f id + f . Since this is injective, id + f = f . We showed that f + f − = id x + y , so f + ( f − + id − x − y ) = id x + y + id − x − y = id . We have showed the existenceof an additive inverse h = f − + id − x − y to f . To show uniqueness. If f + g = id = f + h , g + id = g + ( f + h ) = ( g + f ) + h = id + h . Again, since 0+ is an equivalence, h = g . Proposition 8.3.
P ic strict consists of all small Picard categories in P ic such that + is strictly associativeand commutative (i.e. τ and σ are identities) and for each x ∈ ob ( C ) , x + : C−→C is an isomorphism, notjust an equivalence.Proof.
Clearly any 1-category in
P ic strict ⊂ P ic is a small Picard category satisfying these properties. On theother hand, if a small Picard category
C ∈
P ic satisfies these properties, lemma 8.2 shows that C = Hom C isactually an abelian group, and sending the inclusion ob ( C ) ֒ → C ( x id x ) is an inclusion of abelian groups.Furthermore, lemma 8.2 demonstrates that the second property f ◦ g = f + g − s f is satisfied.29 .2 Nerve and Path Functors for ω -categories We wish to see that ω -descent and ˇCech descent are equivalent for ω -groupoids generally, not just Picard ω -categories. Integral to our proof for Picard ω -categories was a description of the nerve of A [1] a,b . As astep towards extending the proof to ω -groupoids, we give a characterization of the nerve of A [1] a,b for any A ∈ ωCat and a, b ∈ A . Proposition 8.4.
Let A be an ω -category. Then Hom ωCat ( O ( ˜∆ n )) , A [1] a,b ) ≃ { f ∈ Hom ωCat ( O ( ˜∆ n × ˜∆ ) , A ) | f ( h u, i ) = a, f ( h u, i ) = b for all u ∈ ˜∆ n × ˜∆ } . Proof.
Let C = ˜∆ n × ˜∆ . For ( M, P ) ∈ N ( C ), let Θ(( M, P )) = ( M ∩ ˜∆ n × ˜∆ , P ∩ ˜∆ n × ˜∆ ) ∈ N ( C ).Notice that ˜∆ n × ˜∆ = ˜∆ n × { (01) } . Recall from § O ( C ) is generated by atoms h c i for c ∈ C . Thisimplies that every element in α ∈ O ( C ) is a gotten by a composition of atoms α = h c i ∗ k c ∗ k ... ∗ k t − h c t i for c , ...c t ∈ C . We omit parentheses in such compositions for generality and ease of exposition. ViewingΘ as a map of sets Θ : O ( C ) −→N ( C ), for α ∈ O ( C ), let α denote the set Θ − (Θ α ). We will prove thefollowing statements:1. Θ( s n ( M, P )) = s n (Θ( M, P )), and Θ( t n ( M, P )) = t n (Θ( M, P )).2. If N , Q ⊂ ˜∆ n × ˜∆ , then Θ neglects composition 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 compositions are defined. More generally, for any(
N, Q ), (
M, P ), Θ((
N, Q ) ∗ k ( M, P )) = Θ(
N, Q ) ∗ k Θ( M, P )3. For u ∈ ˜∆ n , Θ h u, (01) i = h u i × { (01) } , and s n ( h u i × { (01) } ) = s n − h u i × { (01) } (similarly for t n ).4. If h u, (01) i ∈ ( h x, (01) i ), h v, (01) i ∈ ( h y, (01) i ) and we are able to compose h x, (01) i ∗ k h y, (01) i , thenone can form the composition h u i ∗ k − h v i in O ( ˜∆ n ). In this case, Θ( h x, (01) i ∗ k h y, (01) i ) = h u i ∗ k − h v i × { (01) } .The proofs of (1)-(3), are essentially based on the observation that the operations involved in taking thesource and target and composition respect the decomposition of a subset S n ⊂ C n = F p + q = n ˜∆ np × ˜∆ q into S n = F p + q = n S n ∩ ( ˜∆ np × ˜∆ q ). To be more precise, the operations consist of replacing a set M by M n , | M | n , or ( M \ M n ) as well as taking unions. Let C ∗ , = ˜∆ n × ˜∆ . Clearly, ( M ∩ C ∗ , ) n = M n ∩ C ∗ , . Also,( P ∪ M ) ∩ C ∗ , = ( P ∩ C ∗ , ) ∪ ( M ∩ C ∗ , ). That | M ∩ C ∗ , | n = | M | n ∩ C ∗ , follows from the previous twoproperties together with the fact that | M | n = S nk =0 M k . Finally, the first property shows that intersectingwith C ∗ , respects grading so that ( M \ M n ) ∩ C ∗ , = ( M ∩ C ∗ , ) \ ( M n ∩ C ∗ , ) = ( M ∩ C ∗ , ) \ ( M ∩ C ∗ , ) n .Statements (1) - (3) now follow easily.1. Θ( s n ( M, P )) = Θ( | M | n , M n ∪ | P | n − ) = ( | M | n ∩ ˜∆ n × ˜∆ , ( M n ∪ | P | n − ) ∩ ˜∆ n × ˜∆ ) = ( | M ∩ ˜∆ n × ˜∆ | n , ( M n ∩ ˜∆ n × ˜∆ ) ∪ | P ∩ ˜∆ n × ˜∆ | n − ) = s n (Θ( M, P )). The proof for t n is similar.2. First suppose that we can form the composition ( N, Q ) ∗ k ( M, P ) = ( M ∪ ( N \ N n ) , ( P \ P n ) ∪ Q ). Itfollows from the observations in the previous paragraph together with the fact that Θ( N, Q ) = ( ∅ , ∅ )that Θ(( N, Q ) ∗ k ( M, P )) = ( M ∩ C ∗ , , ( P ∩ C ∗ , ) \ ( P n ∩ C ∗ , ). In order for ( N, Q ) and (
M, P ) to becomposable, we require that s n ( N, Q ) = t n ( M, P ), which implies that P n = N n , whence P n ∩ C ∗ , = ∅ .Therefore, Θ(( N, Q ) ∗ k ( M, P ) = ( M ∩ C ∗ , , P ∩ C ∗ , ) = Θ( M, P ). Showing that Θ(
M, P ) ∗ n ( N, Q ) =Θ(
M, P ) follows similarly. More generally, for any (
M, P ), (
N, Q ),Θ((
N, Q ) ∗ k ( M, P )) = ( M ∩ C ∗ , , ( P ∩ C ∗ , ) \ ( P n ∩ C ∗ , )= (( M ∪ ( N \ N n )) ∩ C ∗ , , ( Q ∪ ( P \ P n )) ∩ C ∗ , = (( M ∩ C ∗ , ) ∪ ( N \ N n ) ∩ C ∗ , , ( Q ∩ C ∗ , ) ∪ ( P \ P n ) ∩ C ∗ , )= (( M ∩ C ∗ , ) ∪ ( N ∩ C ∗ , \ N n ∩ C ∗ , ) , ( Q ∩ C ∗ , ) ∪ ( P ∩ C ∗ , \ P n ∩ C ∗ , ))= ( N ∩ C ∗ , , Q ∩ C ∗ , ) ∗ k ( M ∩ C ∗ , , P ∩ C ∗ , )= Θ( N, Q ) ∗ k Θ( M, P )30. First we verify the claim that for u ∈ ˜∆ nr and ( M, P ) = Θ( h u, (01) i ) ∈ N ( C ), M n − k = µ ( u ) n − k − ×{ (01) } and P n − k = π ( u ) n − k − × { (01) } for 0 ≤ k ≤ n , where µ ( z ), and π ( z ) are as in § k . Let z = ( u, (01)) so that < z > = ( µ ( z ) , π ( z )). For k = 0, µ ( z ) n = { ( u, (01) } , so Θ( µ ( z )) n = { u, (01) } . Now suppose that the claim is true up for all i ≤ k . Then µ ( z ) n − k = µ ( u ) n − k − × { (01) } ∪ S k × { (0) } ∪ T k × { (1) } for some subsets S k , T k ⊂ ˜∆ nn − 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 − × { (01) } ∪ S k +1 × { (0) } ∪ T k +1 × { (1) } for some sets T k +1 , S k +1 and where ± means, that it can be + or − depending on the parity of n and k . Therefore, Θ( µ ( z )) n − ( k +1) = µ ( u ) n − k − × { (01) } . It follows by induction that Θ( µ ( z )) n − k = µ ( u ) n − k − ×{ (01) } for all 0 ≤ k ≤ n . A similar shows that Θ( π ( z )) n − k = π ( u ) n − k − ×{ (01) } for all 0 ≤ k ≤ n . We conclude that Θ( µ ( z ) , π ( z )) = h u i×{ (01) } . Since ( h u i×{ (01) } ) n = h u i n − ×{ (01) } , an easycalculation shows that s n ( h u i × { (01) } ) = ( s n − h u i ) × { (01) } and t n ( h u i × { (01) } ) = ( t n − h u i ) × { (01) }
4. Observe that Θ( h u, (01) 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 composable, 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 canform the composition Θ( h u, (01) i ) ∗ k Θ( h v, (01) i ). But we just showed 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 composable since s k − h u i × { (01) } = s k ( h u i × { (01) } ) = s k Θ h u, (01) i = t k Θ h v, (01) i = t k ( h v i × { (01) } ) = t k − 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 − ) × { (01) } , π ( u ) × { (01) } ∪ ( π ( v ) \ π ( v ) k − ) × { (01) } )= (( µ ( v ) ∪ ( µ ( u ) \ µ ( u ) k − )) × { (01) } , ( π ( u ) ∪ ( π ( v ) \ π ( v ) k − )) × { (01) } )= ( h u i ∗ k − h v i ) × { (01) } . We are now able to prove the main proposition. We will use induction to give a bijection
Hom ωCat ( |O ( ˜∆ n ) | k , A [1] a,b ) ˜ −→H nk := { f ∈ Hom ( |O ( ˜∆ n × ˜∆ ) | 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 nk 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 all u ∈ ˜∆ n , and f ∈ H nk corresponds to g such that g ( h u i ) = f ( h u, (01) i ). We now proceed by induction.For k = 0, a functor g ∈ Hom ωCat ( |O ( ˜∆ n ) | , A [1] a,b ) consists of a choice of n + 1 objects g (0),..., g ( n ) ∈ A [1] a,b . Equivalently, this is a choice of n + 1 1-morphisms in A from a to b . Since O ( ˜∆ n × ˜∆ ) is freelygenerated by its atoms h c i for c ∈ ˜∆ n × ˜∆ , an f in H n ⊂ Hom ωCat ( |O ( ˜∆ n × ˜∆ ) | , A ) is freely determinedby f ( h c i ) for c ∈ ( ˜∆ n × ˜∆ ) i , i = 0 , f is compatible with source and target maps s , t . Sincewe require that f ( h u, (0) i ) = a and f ( h u, (1) i ) = b for all u , we can freely choose f ( h u, (01) i ) ∈ A as longas s f ( h u, (01) i ) = a and t f ( h u, (01) i ) = b for u ∈ ˜∆ n . Therefore, a choice of f ∈ H n is a choice of ( n + 1)1-morphisms in A from a to b . It is evident that under this identification, for u ∈ ˜∆ n , g ( h u i ) = f ( h u, (01) i ).Now assume that Hom ωCat ( |O ( ˜∆ n ) | i , A [1] a,b ) is identified with H ni as above for all i ≤ k . Since O ( ˜∆ n × ˜∆ ) is freely generated by its atoms, a functor ˆ f ∈ H nk +1 is equivalent to a functor f ∈ H nk togetherwith any choice of ( k +2)-morphisms ˆ f ( h u, (01) i ) for u ∈ ˜∆ nk +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 because all f ( h u, ( i ) i ) for u ∈ ˜∆ nk +2 are forced to be a or b . Also, O ( ˜∆ n ) is freely generated by its atoms, so a choice of ˆ g ∈ Hom ωCat ( |O ( ˜∆ n ) | k +1 , A [1] a,b ) is equiv-alent to a choice of g ∈ Hom ωCat ( |O ( ˜∆ n ) | k , A [1] a,b ) together with a choice of ˆ g h u i for u ∈ ˜∆ nk +1 with the31ppropriate k -source and target.Let us then begin with f ∈ H nk , which corresponds to g ∈ Hom ωCat ( |O ( ˜∆ n | k , A [1] a,b ). First we showthat if x ∈ O ( ˜∆ n × ˜∆ ) is not in the sub- ω -category generated by elements of the form h u, ( i ) i , then f ( x ) = gp Θ x , where p denotes projection onto the first factor. Choose such an x ∈ O ( ˜∆ n × ˜∆ ). Since O ( ˜∆ n × ˜∆ ) is freely generated by its atoms, it is also generated by its atoms, so is a composition of atoms: x = h x i ∗ l h x i ∗ l ... ∗ l e − h x e i for some x , ..., x e ∈ ˜∆ n × ˜∆ (omitting parentheses). Since f ( h u, ( i ) i )is a 0-object in A for i = 0 ,
1, the value of f is not affected by composition with elements of the form h u, ( i ) i . Similarly, Θ also neglects composition with these elements. Let x i ,... x i r be the ones of the form x i k = h u i k , (01) i , so f ( x ) = f h x i ∗ l f h x i ∗ l ... ∗ l e − h f x e i = f h x i i ∗ l i f h x i i ∗ l i ... ∗ l ir f h x i r i = f h u i , (01) i ∗ l i f h u i , (01) i ∗ l i ... ∗ l ir f h u i r , (01) i , whereas Θ( x ) = Θ h x i ∗ l θ h x i ∗ l ... ∗ l e − h Θ x e i = Θ h x i i ∗ l i Θ h x i i ∗ l i ... ∗ l ir Θ h x i r i = Θ h u i , (01) i ∗ l i Θ h u i , (01) i ∗ l i ... ∗ l ir Θ h u i r , (01) i = ( h u i i ∗ l i − h u i i ∗ l i − ... ∗ l ir − h u i r i ) × { (01) } . Therefore, gp Θ x = g h u i i ∗ l i − h gu i i ∗ l i − ... ∗ l ir − h gu i r i ), and f ( x ) = f h u i , (01) i ∗ l i f h u i , (01) i ∗ l i ... ∗ l ir f h u i r , (01) i . By induction hypothesis, f h u i j , (01) i = g h u i j i for each j , and since the compositionsappearing in gp Θ x are in A [1] a,b , ∗ [1] n = ∗ n +1 for any n so that the compositions coincide too.Now, for u ∈ ˜∆ nk +1 , f ( t k +1 ( h u, (01) i )) = gp θt k +1 h u, (01) i = gp t k +1 θ h u, (01) i = gp t k +1 ( h u i × { (01) } = gp 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 ). Similarly, f ( s k +1 ( h u, (01) i )) = s k +1 g ( h u i ). In sum-mary, an extension ˆ f ∈ H nk +1 of f is equivalent to a choice of ( k + 2)-morphisms ˆ f ( h u, (01) i ) for u ∈ ˜∆ nk +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 comparison, an extension ˆ g of g consists of any choice of ( k + 1)-morphisms g h u i ∈ A [1] a,b for u ∈ ˜∆ nk +1 such that s [1] k ˆ g h u i = gs k h u i and t [1] k ˆ g h u i = gt 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 and g ( t k h u i ) = t k +1 g h u i , an extension ˆ g of g is the same as a choice of ( k + 2)-morphisms 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 + 1ˆ g h u i = t k +1 g h u i . Note that this implies that s ˆ g h u i = a and t ˆ g h u i = b . It isnow evident that the choice for an extension ˆ f of f is equivalent to a choice of an extension ˆ g of g . It followsthat H nk +2 is in bijection with Hom ωCat ( |O ( ˜∆ n )) | k +1 , A [1] a,b ), thus completing our proof by induction. Corollary 8.5.
For A ∈ ωCat ,1. N A [1] a,bn ≃ { f ∈ Hom Cs (∆ n ⊗ ∆ , N A ) | f | ∆ n × = a, f | ∆ n × = b } .2. If A is an ω -groupoid, there is a weak homotopy equivalence N ( A [1] a,b ) −→ Hom
LNA ( a, b ) .Proof.
1. Let ∆ n ⊗ ∆ denote the complicial set with underlying simplicial set ∆ n × ∆ and for whichthe thin r-simplices are ( x, y ) ∈ ∆ nr × ∆ r such that for some i ≤ j in [ r ], x is degenerate at i and y is degenerate at j . In [47], Verity proves that there is an isomorphism of ω -categories c n, : F ω (∆ n ⊗ ∆ ) −→O ( ˜∆ n × ˜∆ ). Since F ω is left adjoint to the nerve functor N : ωCat −→ Cs , Hom ωCat ( O ( ˜∆ n × ˜∆ ) , A ) ≃ Hom ωCat ( F ω (∆ n ⊗ ∆ ) , A ) ≃ Hom Cs (∆ n ⊗ ∆ , N A ).By Proposition 8.4, the only thing left to prove is that { f ∈ Hom ωCat ( O ( ˜∆ n × ∆ ) , A ) | f ( h u, i ) = a, f ( h u, i ) = b for all u ∈ ˜∆ n × ˜∆ } ⊂ Hom ωCat ( O ( ˜∆ n × ∆ ) , A ) corresponds to Hom Cs (∆ n ⊗ ∆ , N A ) | f | ∆ n ×{ } = a, f | ∆ n ×{ } = b } under the isomorphism Hom ωCat ( O ( ˜∆ n × ˜∆ ) , A ) ≃ Hom Cs (∆ n ⊗ , N A ). To describe this isomorphism, we will first need the following two facts, which can be foundin [47]. One can take products of maps of parity complexes, so given morphisms [ r ] φ −→ [ n ], [ s ] ψ −→ [ m ],there is a morphism ˜∆( φ ) × ˜∆( ψ ) : ˜∆ r × ˜∆ l −→ ˜∆ n × ˜∆ m . In fact, O ( ˜∆( φ ) × ˜∆( ψ ))( h u, v i ) = h φu, ψv i .Secondly, there is a morphism ∇ r : ˜∆ r −→ ˜∆ r of parity complexes, which sends v = ( v v ..v r ) to { ( v ...v s , v s ...v r ) ∈ ˜∆ rs × ˜∆ rr − s | s = 0 , ..., r } . Now, given f ∈ Hom ωCat ( O ( ˜∆ n × ˜∆ ) , A ), we de-scribe the corresponding F ∈ Hom Cs (∆ n ⊗ ∆ , N A ). Given an r-simplex ( α, β ) ∈ ∆ nr × ∆ r , whichwe think of as a pair ([ r ] α −→ [ n ] , [ r ] β −→ [1]), F ( α, β ) is the composition O ( ˜∆ r ) O ( ∇ r ) −→ O ( ˜∆ r × ˜∆ r ) O ( ˜∆( α ) × ˜∆( β )) −→ O ( ˜∆ n × ˜∆ ) f −→ A (an r-simplex in N A ). By the universal property of F ω , f ◦ O ( ˜∆( α ) × ˜∆( β )) ◦ O ( ∇ r ) = f ◦ c n, ◦ ι ( α,β ) . Additionally, F ω (∆ n ⊗ ∆ ) is, by definition [47],a quotient q : F ω (∆ n × ∆ ) −→F ω (∆ n ⊗ ∆ ) of F ω (∆ n × ∆ ), and c n, ◦ q = c n, .Now we will show that f satisfies 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 and F ( x, (1)) = b for all r and all r-simplices x ∈ ∆ n . For i ∈ { , } , we write ( i ) asshorthand for the r-simplex ( ii...i ) with i listed r times. Suppose f satisfies this condition. To show that F has the desired property, it is enough to verify the statement for non-degenerate simplices ( x, ( i ))since if it is true for non-degenerate simplices, then F ( σ k ( x, ( i )) = σ k F ( x, ( i )) = σ k a = a , where σ k isthe k-th degeneracy map. Since it is enough to verify the statement for non-degenerate simplices, wemay also assume that x is non-degenerate, since x is degenerate if and only if ( x, ( i )) is degenerate.We know that F ( α, β ) = f ◦ c n, ◦ ι ( α,β ) . Let α = ( u u ...u r ) non-degenerate and β = ( i ). Bydefinition, ι ( α,β ) ( h ...r i ) = [[ α, β ]], so F ( α, β ) h ...r i = f ◦ c n, ([[ α, β ]]), which equals f ( h u, ( i ) i ) byTheorem 255 of [47]. Hence, F ( α, β ) h ...r i = a if i = 0 or equals b if i = 1. For the general case, weshow that F ( α, β ) h v i = a or b . A k-dimensional v ∈ ˜∆ rk corresponds to a strictly increasing morphism[ k ] φ −→ [ r ]. We know that ι ( α,β ) ◦ O ( ˜∆( φ )) = ι φ ∗ ( α,β ) , whence ι ( α,β ) h v i = ι ( α,β ) ◦ O ( ˜∆( φ )) h ...k i = ι φ ∗ ( α,β ) h ...k i = [[ φ ∗ a, φ ∗ β ]] = [[ φ ∗ α, ( i )]]. Then, F ( α, β ) h v i = f ◦ c n, ι ( α,β ) h v i = f ◦ c n, ([[ φ ∗ α, ( i )]]) = f ( h u, ( i ) i ) (again by Theorem 255 of [47]), which is equal to a if i = 0 or b if i = 1. Thus, F is asdesired. This argument can be run backwards to show that if F ( α, (0)) = a , F ( α, (1)) = b for all α ∈ ∆ n , then f ( h u, (0) i ) = a and f ( h u, (1) i ) = b for all u ∈ ˜∆ n .2. ∆ n +1 is a retract of ∆ n × ∆ , with the inclusion ∆ n +1 ֒ → ∆ n × ∆ given by (0 , , ..., n + 1) (012 ...nn, ....
01) and p : ∆ n × ∆ −→ ∆ n +1 chosen to send thin simplices to degenerate ones. In this section we make some basic observations about homotopies between morphisms of chain complexesfrom the perspective of ω -categories. Using the equivalence of Ch + ( Ab ) with P ic ω , we see that homotopies of maps of complexes correspond tothe following notion of homotopy between maps F , G : A −→ B in P ic ω . A homotopy consists of maps H n : A n −→ B n +1 such that H n ( A n − ) ⊂ B n and ( t n − s n ) H n + H n − ( t n − − s n − ) agrees with G − F whenwe pass to the quotient A n /A n − −→ B n /B n − . Lemma 8.6.
For complexes A , B ∈ Ch + ( Ab ) , there is a C ∈
P ic ω such that C = { maps of complexes A −→ B } and Hom ( F, G ) = { homotopies from F to G } .Proof. For i >
0, let H i consist of all maps from A −→ B [ − i ]. By this we mean h ∈ H i consists of a sequenceof maps h k : A k −→ B k − i but not necessarily commuting with d . Define H = Hom Ch + ( Ab ) ( A, B ). Nowdefine a differential ∂ : H i −→ H i − by ∂f = df + ( − i − f d . This makes H ∗ into a complex of abeliangroups. ˜ H is the desired ω -category. 33emma 8.6 gives the following corollary, an observation also made by Street [46]. Corollary 8.7.
There is an ω category, A such that A = Ch + ( Ab ) , A = { maps of complexes } , and A = { homotopies of maps of complexes } . For any ω -category in abelian groups A , there is a group homomorphism D : A −→ A given by D = P n ≥ t n − s n . Since we assume that A is a union of the A n , this sum makes sense because it is a finitesum on A n . Now we can efficiently define a homtopy between two functors F, G ∈ Hom
P ic ω ( A, B ) byreformulating the description at the beginning of this section.
Definition 18.
Let
A, B ∈ P ic ω and F, G ∈ Hom
P ic ω ( A, B ). We say that a group homomorphism H : A −→ B is a homotopy if1. H ( A n ) ⊂ B n +1 ,2. DH + HD = G − F , and3. s n H ( s n − s n − ) = 0 for all n ≥ Lemma 8.8.
Let f, g : A −→ B be maps in Ch + ( Ab ) . A homotopy h : f −→ g is equivalent to a homotopy H from P f to P g in P ic ω .Proof. Since
P A = ⊕ K i ≃ A i K i = σ ( A i ) as in § 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 − ) for all n if and only if H i ( K i ) ⊂ L i +1 . Thenfor x = ((0 , , ... (0 , dx n ) , ( x n , x n ) , (0 , , ... ) ∈ K n , Hx = ((0 , , ... (0 , dy n +1 ) , ( y n +1 , y n +1 , (0 , , ... ) ∈ L n +1 .Defining h : A n −→ B n +1 by h = π n +1 Hσ gives a bijection between maps h : A −→ B [1] (not necessarilycommuting with the differential d ) and homomorphisms H : P A −→ P B such that s n H ( s n − s n − ) and H ( P A n ) ⊂ P B n +1 . A direct computation shows that with x = ((0 , , ... (0 , dx n ) , ( x n , x n ) , (0 , , ... ) ∈ K n ,( DH + HD ) x = ((0 , , ... (0 , ( hd + dh ) dx n ) , (( dh + hd ) x n , ( dh + hd ) x n ) , (0 , , ... ) ∈ L n . Hence HD + DH = P g − P f if and only if hd + dh = g − f . Remark 8.9.
P A is a projective abelian group if and only if each A i is projective. Proposition 8.10.
Let P denote the full subcategory of P ic ω consisting of objects which are projectiveabelian groups, and let P denote the category the objects of which are ob ( P ) and the morphisms of which aremorphisms in P modulo homotopy. There is an equivalence of categories D ≤ ( Ab ) −→P .Proof. Let K ≤ ( P roj ) denote the homotopy category of complexes of projective abelian groups in degrees ≤
0. Lemma 8.8 shows that H : K ≤ ( P roj ) −→P is an equivalence of categories. The theorem now followssince K ≤ ( P roj ) −→ D ≤ ( Ab ) is an equivalence. ω -Categories The homotopies described above are algebraic in nature, but we can still define homotopies for ordinary ω -categories. The following definition extends the concept of homotopy for from P ic ω to ω -cat. Definition 19.
Let f, g : A −→ B be two functors of ω -categories. A homotopy H : F −→ G is a map H : A −→ B with H ( A n ) ⊂ B n +1 , and for x ∈ A n , Hx is an n + 1 isomorphism Ht n − x ∗ n − ( Ht n − x ∗ n − ( ... ( Ht x ∗ ( Ht x ∗ f x )) .. ) Hx −→ n +1 ( ... (( gx ∗ Hs x ) ∗ Hs x ) ∗ ... ) ∗ n − Hs n − x Intuitively, a homotopy H : f −→ g looks like a “natural transformation” from f to g , except any diagram of n -morphisms which should commute only commutes up to an ( n + 1)-isomorphism specified by H . Proposition 8.11.
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