aa r X i v : . [ m a t h . AG ] J u l Higher Descent Data as a Homotopy Limit
Matan Prasma
Abstract
We define the 2-groupoid of descent data assigned to a cosimplicial2-groupoid and present it as the homotopy limit of the cosimplicial spacegotten after applying the 2-nerve in each cosimplicial degree. This canbe applied also to the case of n -groupoids thus providing an analogouspresentation of “descent data” in higher dimensions. In this note we reinterpret algebro-geometric information, namely descent data,in a homotopically-invariant way. Given a cosimplicial 2-groupoid G ● , its descentdata is (the 2-nerve of) a 2-groupoid Desc (G ● ) ∶= T ot r ( N G ● ) (where T ot r means“totalization without degeneracies”) whose path components coincide with theset of descent data modulo the gauge equivalence relation (see [BGNT] andalso [Ye1, Definitions 1.4, 1.5]). We show that this 2-groupoid is (canonicallyequivalent to) the homotopy limit holim ∆ N G ● where N is the 2-nerve applied oneach level. Thus, given a weak equivalence of cosimplicial 2-groupoids G ● → H ● ,the map Desc (G ● ) → Desc (H ● ) is a weak equivalence of 2-groupoids; thisgeneralizes [Ye1, Theorem 0.1]. We know of two situations in which this setupcan arise.The first concerns Maurer-Cartan equations. Consider a cosimplicial DGLA,which shows up for instance as the ˇCech construction for a sheaf of nilpotentparameter DGLAs. Taking the Deligne 2-groupoid (which encodes solutions toMaurer-Cartan equations) of each cosimplicial degree gives rise to a cosimplicial2-groupoid. As follows from [Ye2, Theorem 0.4], a quasi-isomorphism of cosim-plicial pronilpotent DGLAs of quantum type (i.e. concentrated in degrees ≥ − G -gerbes for a sheaf of groups G (see[Br1], [Br2]). There, the cosimplicial 2-groupoid arises via the ˇCech construc-tion (with respect to a cover) from the sheaf of 2-groups (or crossed mod-ules) G → Aut (G) and the descent data approximates isomorphism classes of G− gerbes. In some cases, for example when the cover totally trivializes all G -gerbes, π Desc (G) will classify all G− gerbes and a refinement will yield a weakequivalence of cosimplicial 2-groups.Descent data is intimately related to non-abelian cohomology. For this rea-son, the role of codegeneracies is degenerate and we can consider the restricted1otalization (see §
2) which simplifies the homotopical framework. This elim-inates the difficulty arising from the fact that the cosimplicial simplicial setgotten by taking the 2-nerve of each level of a cosimplicial 2-groupoid need notbe Reedy fibrant (see [Ja, Example 9]) and gives an argument which is also validfor the case of n -groupoids; this is discussed in § Let ∆ be the category whose objects are non-empty finite ordinals [ ] , [ ] , ..., [ n ] where [ n ] = { , , ..., n } and whose morphisms are weakly order preserving func-tions. Every morphism in ∆ is a composition of face maps d i ∶ [ n − ] → [ n ] and degeneracies s i ∶ [ n + ] → [ n ] , i = , ..., n . A simplicial set is a functor X ∶ ∆ op → Set and we write X n ∶= X ([ n ]) , d i ∶= X ( d i ) , s i ∶= X ( s i ) . Write s S et for the category whose objects are simplicial sets and whose morphismsare natural transformations.A cosimplicial object in a category C is a functor ∆ → C . In particular, a cosimplicial simplicial set is a cosimplicial object in s S et . We write s S et ∆ forthe category whose objects are cosimplicial simplicial sets and whose morphismsare natural transformations. If X is a cosimplicial object we will denote theobject assigned to [ n ] by X n . The maps d i ∶= X ( d i ) and s i ∶= X ( s i ) are calledcofaces and codegeneracies respectively. The cosimplicial standard simplex ∆has ∆ n in its n -th cosimplicial degree and cofaces and codegeneracies inducedby precomposition. For X, Y ∈ s S et ∆ , the product X × Y is the cosimplicialsimplicial set with ( X × Y ) n ∶= X n × Y n and for A ∈ s S et , we write, by abuseof notation, A for the constant cosimplicial simplicial set with A n ∶= A for all n and cofaces and codegeneracies being identities.The category s S et ∆ is enriched over simplicial sets. Given X, Y ∈ s S et ∆ , the‘internal hom’ s S et ∆ ( X, Y ) is the simplicial set whose n -simplices are s S et ∆ ( X, Y ) n = s S et ∆ ( X × ∆ n , Y ) Here, X × ∆ n is the product of X with the constant cosimplicial simplicial set∆ n .With this enrichment, s S et ∆ is a simplicial category in the sense of [GJ,II,2.1] or in our terminology, tensored and cotensored over s S et (see [GJ, II,2.5]). For A, B ∈ s S et ∆ we denote the tensor and cotensor functors by A × ( − ) ∶ s S et → s S et ∆ and B (−) ∶ ( s S et ) op → s S et ∆ respectively; these are the left adjoints of s S et ∆ ( A, − ) and s S et ∆ ( − , B ) .2 efinition 2.1. The totalization
T ot ∶ s S et ∆ → s S et is the simplicial set T ot ( X ● ) = s S et ∆ ( ∆ ● , X ● ) .We let ∆ r denote the subcategory of ∆ with the same objects but only in-jective maps i.e. compositions of face maps d i . A restricted cosimplicial object in a category C is a functor ∆ r → C ; it is also called a semi-cosimplicial object by some authors. In particular, a restricted cosimplicial object in s S et is calleda restricted cosimplicial simplicial set . There is an obvious ‘restriction’ func-tor r ∶ s S et ∆ → s S et ∆ r and in particular we have r ∆ ∈ s S et ∆ r . The category s S et ∆ r is again enriched over simplicial sets so that if X, Y ∈ s S et ∆ r we denote s S et ∆ r ( X, Y ) ∈ s S et . Its n -simplices are s S et ∆ r ( X, Y ) n ∶= s S et ∆ r ( X × r ∆ n , Y ) and given θ ∶ [ m ] → [ n ] in ∆ , the map θ ∗ ∶ s S et ∆ r ( X, Y ) n → s S et ∆ r ( X, Y ) m is induced by composing with the map θ ∗ ∶ r ∆ m → r ∆ n . Simplicial identitieshold since their opposites hold in ∆. The arguments in [GJ, II,2.5] may be usedverbatim to show that s S et ∆ r is tensored and cotensored over s S et . Definition 2.2.
The restricted totalization is the functor
T ot r ∶ s S et ∆ r → s S et defined by T ot r ( X ● ) = s S et ∆ r ( r ∆ ● , X ● ) .More generally we can use ends (see [Ma, IX.5]) to get: Definition 2.3.
Let C be a category cotensored over simplicial sets.1. The totalization of G ● ∈ C ∆ is the object of C is given by the end T ot ( G ● ) ∶= ∫ [ n ]∈ ∆ ( G n ) ∆ n .
2. The restricted totalization of G ● ∈ C ∆ r is the object of C is given by theend T ot r ( G ● ) ∶= ∫ [ n ]∈ ∆ r ( G n ) ∆ n . We assume the reader is familiar with the definition of a model category. Letus shortly spell out the definition of a simplicial model category.
Definition 3.1.
A model category M is called simplicial if it is enriched withtensor and cotensor over s S et and satisfies the following axiom [Qu, II.2 SM7]:If f ∶ A → B is a cofibration in M and i ∶ K → L is a cofibration in s S et thenthe map q ∶ A ⊗ L ∐ A ⊗ K B ⊗ K → B ⊗ L
1. is a cofibration;2. is a weak equivalence if either(a) f is a weak equivalence in M or3b) i is a weak equivalence in s S et . Definition 3.2.
A category R is called a Reedy category if it has two subcat-egories R + , R − ⊆ R and a degree function d ∶ ob ( R ) → α where α is an ordinalnumber such that: • Every non-identity morphism in R + raises degree; • Every non-identity morphism in R − lowers degree; • Every morphism in R factors uniquely as a map in R − followed by a mapin R + .The category ∆ is a Reedy category with ∆ + = ∆ inj ( = ∆ r ) , ∆ − = ∆ surj andthe obvious degree function.Let R be a Reedy category and C any category. Given a functor X ∶ R → C and an object n ∈ R we set X n ∶= X ( n ) (to relate to the case R = ∆), and definethe n -th latching object to be L n X = colim L (R) X s where L ( R ) is the full subcategory of the over category R + / n containing allobjects except the identity id n .Dually, define the n -th matching object to be M n X = lim M (R) X s where M ( R ) is the full subcategory of the under category n / R − containing allobjects except id n . We have natural morphisms L n X → X n → M n X. The importance of a Reedy structure on R is due to the following: Theorem 3.3. [Re] Let R be a Reedy category and M a model category. Thefunctor category M R admits a structure of a model category, called Reedy modelstructure in which a map X → Y is a • Weak equivalence iff X n → Y n is a weak equivalence in M for every n . • Cofibration iff the map L n Y ∐ L n X X n → Y n is a cofibration in M for every n . • Fibration iff the map X n → M n X × M n Y Y n is a fibration in M for every n .In particular, an object X is • Fibrant iff X n → M n X is a fibration in M for every n . • Cofibrant iff L n X → X n is a cofibration in M for every n . oreover [An, Theorem 4.7], if the model structure on M is simplicial, so isthe Reedy model structure on M R . Corollary 3.4.
The Kan-Quillen model structure s S et K − Q and the Reedy struc-ture on ∆ (respectively ∆ r ) induce a simplicial model structure on s S et ∆ (respec-tively s S et ∆ r ).Example . The object X = ∆ ● ∈ s S et ∆ is Reedy cofibrant. The map L n X → X n is the inclusion ∂ ∆ n ↪ ∆ n which is a cofibration of simplicial sets.Next, we recall another model structure on s S et ∆ r . Theorem 3.5.
The simplicial enrichment of s S et ∆ r can be extended to a sim-plicial model structure, called the projective model structure , in which a map X → Y is a • weak equivalence if for each n , X n → Y n is a weak equivalence. • fibration if for each n , X n → Y n is a Kan fibration. • cofibration if it has the left lifting property with respect to trivial fibrations.In particular, X is a fibrant object iff X n is a Kan complex for every n . Suppose R is a Reedy category and M is a model category. In general, ifthe projective model structure on M R exists (e.g. when M is sufficiently nice)it will be very different than the Reedy model structure. However, in specialcases the two may coincide. Proposition 3.6. If R = R + the projective and Reedy model structures on M R coincide.Proof. In this case, for every X ∈ M the n -th matching object M n X equal theterminal object, being the limit over the empty diagram, so that a map X → Y is a Reedy fibration iff X n → Y n is a fibration in M . This means that the twomodel structures have the same classes of weak equivalences and fibrations, andhence coincide.For R = ∆ r we obtain: Corollary 3.7.
The Reedy and projective model structures on s S et ∆ r coincide.Thus, an object X ∈ s S et ∆ r is Reedy fibrant iff X n is a Kan complex for each n .Remark . By example 1, ∆ ● is Reedy cofibrant in s S et ∆ and since the index-ing category defining L n ∆ ● depends only on ∆ + = ∆ r , we have L n ∆ ● = L n r ∆ ● .Thus, the map L n r ∆ n → r ∆ n is again the inclusion ∂ ∆ n ↪ ∆ n so that r ∆ ● isReedy cofibrant in s S et ∆ r . 5 Definition 4.1.
A (strict) is a groupoid-enriched (small) categoryin which all morphisms are invertible.Explicitly, a 2-groupoid consists of: • a set of objects ; • for every pair of objects x, y , a set of , written as f ∶ x → y ;and, for every object x , a distinguished 1-morphism 1 x ∶ x → x ; • for every pair of 1-morphisms f, g ∶ x → y a set of , writtenas a ∶ f ⇒ g ; and, for every 1-morphism f , a distinguished 2-morphism1 f ∶ f ⇒ f together with a composition law for 1-morphisms and vertical and horizontalcomposition laws for 2-morphisms (denoted by ∗ and ○ respectively) subjectto three axioms, expressing associativity of composition and left and right unitlaws and in addition satisfy the ‘interchange law’: ( b ∗ a ) ○ ( b ′ ∗ a ′ ) = ( b ′ ○ b ) ∗ ( a ′ ○ a ) . All morphisms are invertible with respect to these composition laws.There are 2-categorical analogues for the notions of a functor and naturaltransformation. However, since 2-categories have 2-morphisms, an additional‘level of arrows’ reveals itself, namely, the one of modifications. There is someambiguity regarding these notions, since one can consider also their weak ver-sions. For the sake of clarity, we spell out the definitions we use, which aretaken from [Gr, I,2.2;I,2.3].
Definition 4.2.
Let G , H be a pair of 2-groupoids.(I) A (strict) Φ ∶ G → H is a groupoid-enriched functor betweenthe underlying groupoids of G and H . Explicitly, Φ assigns: • to each object x ∈ G , an object Φ x ∈ H , • to each 1-morphism f ∶ x → y ∈ G , a 1-morphism Φ f ∶ Φ x → Φ y ∈ H , • to each 2-morphism a ∶ f ⇒ g ∈ G a 2-morphism Φ a ∶ Φ f ⇒ Φ g ∈ H and this assignment respects all compositions and units.(II) Given a pair of 2-functors Φ , Ψ ∶ G → H between 2-groupoids, a (strict) Θ ∶ Φ ⇒ Ψ consists of a 1-morphism η x ∶ Φ x ⇒ Ψ x x ∈ G which is natural in the sense that for every 2-morphism a ∶ f ⇒ g in G , the diagramΦ ( x ) η x (cid:15) (cid:15) Φ g * * Φ f ✤✤ ✤✤ (cid:11) (cid:19) Φ a Φ ( y ) η y (cid:15) (cid:15) Ψ ( x ) Ψ f * * Ψ g ✤✤ ✤✤ (cid:11) (cid:19) Ψ a Ψ ( y ) is commutative in that 1 η y ○ Φ a = Ψ a ○ η x as 2-morphisms in H .(III) Given a pair of 2-natural transformations η, θ ∶ Φ ⇒ Ψ, a (strict) modi-fication µ ∶ η ⇛ θ consists of a 2-morphism µ x ∶ η x ⇒ θ x in H for everyobject x ∈ G such that for every 1-morphism f ∶ x → y in G , the diagramΦ ( x ) Φ f (cid:15) (cid:15) θ x * * η x ✤✤ ✤✤ (cid:11) (cid:19) µ x Ψ ( x ) Ψ f (cid:15) (cid:15) Φ ( y ) η y * * θ y ✤✤ ✤✤ (cid:11) (cid:19) µy Ψ ( y ) is commutative in the sense of (II).We denote by G pd the category of 2-groupoids and strict 2-functors betweenthem. The collection of 2-functors from G to H , their 2-natural transformationsand their modifications is naturally a 2-category (see [Gr, 2.3]) which is in facta 2-groupoid because of invertibility of 1-and 2-morphisms in the codomain H .We denote this 2-groupoid by G pd ( G , H ) . Theorem 4.3. (cf. [Gr, 2.3]) The category G pd is cartesian closed with respectto G pd ( G , H ) . Let ∆ ≤ n be the full subcategory of ∆ with objects [ ] , ..., [ n ] and let s S et ≤ n be the category of functors ( ∆ ≤ n ) op → S et . Objects of s S et ≤ n are called n -truncated simplicial sets . The inclusion ∆ ≤ n → ∆ induces a ‘truncation functor’ tr n ∶ s S et → s S et ≤ n which admits right and left adjoints cosk n ∶ s S et ≤ n → s S et and sk n ∶ s S et ≤ n → s S et respectively. We denote by Cosk n ∶ s S et → s S et thecomposition cosk n ○ tr n and by Sk n the composition sk n ○ tr n . The functor Sk n takes a simplicial set and creates a new simplicial set from its n -truncation byadding degenerate simplices in all levels above n ; it is the simplicial analogueof the n -skeleton of a CW complex. The functor Cosk n has a more involvedsimplicial description; it is the simplicial analogue of the ( n − ) th Postnikovpiece P n − .By abstract considerations, one can show that Cosk n is right adjoint to Sk n .Thus, a map X → Cosk n Y correspond precisely to a map Sk n X → Y .7 efinition 4.4. A simplicial set X is called n -coskeletal if the canonical map X → Cosk n X is an isomorphism.In particular, given an n -truncated simplicial set X , cosk n X is an n -coskeletal simplicial set. Thus, in order to define an n -coskeletal simplicial setit is enough to define its n -truncation.In § §
2] with the notations relevant for our formulae.
Definition 4.5.
The is the functor N ∶ G pd → s S et which takes a2-groupoid G to the 3-coskeletal simplicial set N G whose • G ; • G ; • x g ! ! ❈❈❈❈❈❈❈ x ⇑ a g = = ④④④④④④④ g / / x where g ij ∶ x i → x j and α ∶ g ⇒ g ○ g are 1-and 2-morphisms (respec-tively) in G ; • x x ⇒ a ⇒ a g O O g ❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ x ⇒ a g G G ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ g ? ? g / / ⇑ a x g W W ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ a ijk ∶ g ik ⇒ g jk ○ g ij . Commutativity of this tetrahedron means that the diagram of 2-morphisms g a / / a (cid:15) (cid:15) g ○ g g ○ a (cid:15) (cid:15) g ○ g a ○ g / / g ○ g ○ g (1)commutes.We will need four well-known properties of the 2-nerve:8 roposition 4.6. [MS]1. N preserves products.2. For every 2-groupoid G , N G is a Kan complex.3. A map of 2-groupoids G → H is a weak equivalence iff N G → N H is a weakequivalence of simplicial sets.4. N admits a left adjoint W ∶ s S et → G pd , called the Whitehead 2-groupoid . The category G pd admits a natural simplicial enrichment via N G pd ( − , − ) .This enrichment is nicely behaved in the following sense: Proposition 4.7.
The simplicially-enriched category G pd is tensored and coten-sored over s S et .Proof. We need to verify the conditions of [GJ, II,2.1]. The functor (( − ) × G ) ○ W is a left adjoint to N G pd ( G , − ) and the functor G pd ( W ( − ) , H ) is a left adjointto N G pd ( − , H ) . Remark . It is worth notice that [No, 5.1] shows the inner hom described in4.2 does not induce a simplicial model category structure on G pd via settingthe simplicial mapping space to be N G pd ( G , H ) . However in the current note,the main homotopical part is done in the category of (cosimplicial) simplicialsets so that we do not need a full-fledged homotopy theory of G pd .We shall need a slight generalization of proposition 4.7: Proposition 4.9. If C is tensored and cotensored over s S et and I is any smallcategory, then the functor category C I is again tensored and cotensored over s S et .Proof. Denote the inner homs and their adjoints by C C( X, −) / / s S et X ⊗(−) o o and C op C(− ,Y ) / / s S et Y (−) o o .One defines for X ̃ ∈ C I and K ∈ s S et , ( X ̃ ⊗ K ) α ∶= X ̃ α ⊗ K and ( X ̃ K ) α ∶= ( X ̃ α ) K for every α ∈ I . Then, C I ( X ̃ , Y ̃) n = C I ( X ̃ ⊗ ∆ n , Y ̃) with the obviousface and degeneracy maps provides the desired inner hom. Corollary 4.10.
The categories G pd ∆ and G pd ∆ r are tensored and cotensoredover simplicial sets. The last corollary enables us to express the totalization as an end via Defi-nition 2.3.By abuse of notations, we denote by N , W the prolongation of the 2-nerve andWhitehead 2-groupoid functors to the categories G pd ∆ , s S et ∆ (respectively).Since (level-wise) coproducts define the tensoring (over s S et ) in G pd and s S et ∆ and W commutes with coproducts, the premisses of [GJ, Lemma 2.9(1)] aresatisfied and we have: 9 roposition 4.11. There is an enriched adjunction G pd ∆ ( W G ● , H ● ) ≅ s S et ∆ ( G ● , N H ● ) Following [BGNT], descent data of a cosimplicial crossed groupoid is defined in[Ye1]. Since crossed groupoids can be viewed precisely as 2-groupoids (e.g. asa special case of [BH1]), a translation leads to the following:
Definition 5.1. (cf. [Ye1, Definition 1.4]) Given a cosimplicial 2-groupoid G ● = { G n } , a descent datum is a triple ( x, g, a ) in which:1. x is an object of G ;2. g ∶ d x → d x is a 1-morphism in G and3. a ∶ d g ⇒ d g ○ d g is a 2-morphism in G .such that ( d d g ○ d a ) ∗ d a = ( d a ○ d d g ) ∗ d a. (twisted 2-cocycle)Let ( x, g, a ) be a descent datum of G ● .Write x i ≡ x ( ) i ( i = , ) for the object of G corresponding to the vertex ( i ) of ∆ , i.e. x i = d j x where { j } = { , } ∖ { i } ; thus g ∶ x → x .Similarly, write x i ≡ x ( ) i ( i = , , ) and g ij ≡ g ( ) ij ( ≤ i < j ≤ ) for (respec-tively) the object and 1-morphism of G corresponding to the vertex ( i ) andedge ( ij ) of ∆ . In other words, x i = d k d j x where { j < k } = { , , } ∖ { i } and g ij = d k g where { k } = { , , } ∖ { i, j } ; thus g ij ∶ x i → x j and a ∶ g ⇒ g ○ g .Finally, write x i ≡ x ( ) i ( i = , ..., ) , g ij ≡ g ( ) ij ( i < j ) and a ijk ≡ a ( ) ijk ( i < j < k ) for (respectively) the object, 1-morphism and 2-morphism of G correspond-ing to the vertex ( i ) , edge ( ij ) and face ( ijk ) of ∆ ; thus g ij ∶ x i → x j and a ijk ∶ g ik ⇒ g jk ○ g ij .With these notations in mind, one can immediately see that the twisted co-cycle condition corresponds precisely to the commutativity of a tetrahedron t in G as in 4.5. Thus, such triples are in 1-1 correspondence with diagrams ofsimplicial sets of the form∆ x (cid:15) (cid:15) / / / / ∆ g (cid:15) (cid:15) / / / / / / ∆ a (cid:15) (cid:15) / / / / / / / / ∆ t (cid:15) (cid:15) N G / / / / N G / / / / / / N G / / / / / / / / N G . (2)10ince N G n is 3-coskeletal, diagrams as above are in turn the 0-simplices T ot r ( r N G ● ) = s S et ∆ r ( r ∆ ● , r N G ● ) = s S et ∆ r ( r ∆ ● , r N G ● ) . Definition 5.2. (cf. [Ye1, definition 1.5]) Let d = ( x, g, a ) , d ′ = ( x ′ , g ′ , a ′ ) bea pair of descent data of G ● . A gauge transformation d ❀ d ′ is a pair ( f, c ) inwhich:1. f ∶ x → x ′ is a 1-morphism in G and2. c ∶ d f ○ g ⇒ g ′ ○ d f is a 2-morphism in G (see diagram 3) x g / / ⇙ cf (cid:15) (cid:15) x f ( f ∶ = d f, f ∶ = d f ) (cid:15) (cid:15) x ′ g ′ / / x ′ (3)such that the prism in G x f (cid:15) (cid:15) g ❇❇❇❇ ! ! ❇❇❇❇❇ x f (cid:15) (cid:15) g ⑤⑤⑤⑤⑤ = = ⑤⑤⑤⑤ g / / c i q ⇑ a x f ( c ij ∶ f j ○ g ij ⇒ g ′ ij ○ f i ) (cid:15) (cid:15) ⇙ c ⇙ c x ′ g ′ (cid:31) (cid:31) x ′ g ′ ? ? g ′ / / ⇑ a ′ x ′ (4)is commutative in the sense of 1.Let Desc ( G ● ) denote the set of descent data of G ● . The relation d R d ′ ⇔ ∃ d ❀ d ′ is an equivalence relation on Desc ( G ● ) and we denote by Desc ( G ● ) itsquotient (cf. [Ye1], definition 1.8). We now claim that: Theorem 5.3.
For any cosimplicial 2-groupoid G ● , there is a (natural) isomor-phism Desc ( G ● ) ≅ π T ot r ( r N G ● ) In order to prove Theorem 5.3 we would like to view a gauge transformation d ❀ d ′ as a path between two vertices of T ot r ( r N G ● ) but there is a slight11roblem. Given a pair of descent data, thought of as 4-tuples d = ( x, g, a, t ) and d ′ = ( x ′ , g ′ , a ′ , t ′ ) of the form 2, a path between them is an element of T ot r ( r N G ● ) = s S et ∆ r ( r ∆ ● , r N G ● ) = s S et ∆ r ( r ∆ ● × r ∆ , r N G ● ) that restricts to d and d ′ via the maps d , d ∶ ∆ / / / / ∆ . Since N G n is3-coskeletal, such elements correspond to diagrams of the form∆ × ∆ f (cid:15) (cid:15) / / / / ∆ × ∆ c ′ (cid:15) (cid:15) / / / / / / ∆ × ∆ p ′ (cid:15) (cid:15) N G / / / / N G / / / / / / N G (5)that restrict to that restrict to ( x, g, a ) and ( x, g, a ) .The last diagram carries an automatic ‘triangulation’. The map c ′ is adiagram in G of the form x g / / h ❆❆❆ ⇙⇙ ❆❆❆ f (cid:15) (cid:15) x f (cid:15) (cid:15) x ′ g ′ / / x ′ (6)which is a triangulation of 3; and similarly, the map p ′ is a diagram in G whichis a triangulation of 4.There are two possible solutions for that. The first (which was suggestedby the referee) is to change the framework into crossed complexes, relying on[BH2, Theorem 2.4] and obtain a description of gauge transformations as mapsof crossed complexes. The second, which we will adopt for the sake of simplicity,is to notice the following: Lemma 5.4.
Every gauge transformation d ❀ d ′ gives rise to a canonical pathin T ot r ( N G ● ) between d and d ′ and every such path gives rise to a canonicalgauge transformation.Proof. Given a path between d and d ′ , represented by a triple ( f, c ′ , p ′ ) as in5, one can compose the 2-morphisms appearing in c ′ and in the squares of p ′ toobtain a triple ( f, c, p ) and hence a gauge transformation d ❀ d ′ . Conversely,given a gauge transformation ( f, c ) ∶ d ❀ d ′ , one obtains, from condition 4 ofdefinition 5.2 a prism p in G . Then, by inserting the 1-morphism f ○ g asthe diagonal in 3 and 1 f ○ g in the upper triangle, one obtains a diagram of theform 6 and a similar procedure on 4 yields a prism p ′ . The triple ( f, c ′ , p ′ ) isthe resulting path.Expressing the totalization as an end allow us to reveal its higher structure: Proposition 5.5.
For a cosimplicial 2-groupoid G ● , there are natural isomor-phisms1. T ot ( N G ● ) ≅ N T ot ( G ● ) ; . T ot r ( r N G ● ) ≅ N T ot r ( r G ● ) ;(see definition 2.3).Proof. We only prove (1) as the proof of (2) is identical. Since N is a rightadjoint, it commutes with limits. Relying on [Ma, IX.5], N T ot ( G ● ) = N (∫ [ n ] ∈ ∆ ( G n ) ∆ n ) ≅ ∫ [ n ] ∈ ∆ N (( G n ) ∆ n ) ≅ ∫ [ n ] ∈ ∆ ( N G n ) ∆ n = T ot ( N G ● ) (7)where the last isomorphism comes from [GJ, II, Lemma 2.9(2)] relying on thefact that W commutes with arbitrary coproducts.Thus, we define: Definition 5.6.
Given a cosimplicial 2-groupoid G ● , its descent 2-groupoid is Desc ( G ● ) ∶= T ot r ( N G ● ) . Proof of Theorem 5.3.
Since
T ot r ( r N G ● ) is a Kan complex (being the 2-nerveof a 2-groupoid), π T ot r ( N G ● ) = T ot r ( N G ● ) / ∼ where d ∼ d ′ iff there is a pathbetween them. By lemma 5.4 this equivalence relation is equal to the gaugeequivalence relation. Theorem 5.3 enables us to use homotopy-theoretic tools to prove invariance ofdescent data under weak equivalence. We need one more simple theorem:
Theorem 6.1.
For any cosimplicial 2-groupoid G ● , there is a (natural) weakequivalence T ot r ( N G ● ) ≃ holim ∆ N G ● ,Proof. In the simplicial model category s S et ∆ r proj , the homotopy limit (over ∆ r ) of a fibrant object can be described as the internal mapping space froma weakly contractible cofibrant object [Hi, Theorem 19.4.6(2)]. In our case, N G ● is fibrant and r ∆ ● is (weakly contractible and) cofibrant (see remark 3.8).Thus, T ot r ( r N G ● ) = s S et ∆ r ( r ∆ ● , N G ● ) ≃ holim ∆ r r N G ● . By ([DF, Lemma 3.8]),holim ∆ r r N G ● ∼ holim ∆ N G ● .In light of definition 5.6 and the previous theorem it now follows that: Corollary 6.2.
A weak equivalence of cosimplicial 2-groupoids G ● → H ● inducesa weak equivalence of 2-groupoids Desc ( G ● ) → Desc ( H ● ) . In particular, we have:
Corollary 6.3. (cf. [Ye1, Theorem 2.4]) If G ● → H ● is a weak equivalence ofcosimplicial 2-groupoids, the induced map Desc ( G ● ) → Desc ( H ● ) is an isomor-phism of setsProof. By Theorems 5.3 and 6.1, the map
Desc ( G ● ) → Desc ( H ● ) coincides with π ( holim ∆ N G ● ) → π ( holim ∆ N H ● ) and N and holim ∆ preserve weak equiva-lences. 13 Descent data of cosimplicial n -groupoids The techniques of § § n -groupoids and the corresponding n -nerve N ( n ) in the sense of [St] but the same arguments work for weaker notions of n -groupoids, e.g. Tamsamani n -groupoids. We only need two ingredients. Thefirst is that N ( n ) admits a left adjoint (and hence commutes with limits); thisis true since the inclusion n G pd ↪ nCat admits a left adjoint Π n ∶ nCat → n G pd and thus the composite Π n ○ τ n (where τ n is the fundamental n -category) is thedesired left adjoint. The second ingredient is that N ( n ) G is a Kan complex forevery n -groupoid G ; this goes back to [Da].In light of theorem 5.3, it makes sense to define: Definition 7.1.
Let G ● be a cosimplicial n -groupoid. Its n -descent data is thesimplicial set Desc n ( G ● ) ∶= T ot r ( N ( n ) G ) .Definition 7.1 makes sense formally, but its geometric meaning is unknownto us. Nevertheless, the formal reasoning of proposition 5.5 implies: Proposition 7.2.
The simplicial set
Desc n ( G ● ) is the n -nerve of an n -groupoid. Moreover, theorem 6.1 generalizes immediately.
Theorem 7.3.
Let G ● be a cosimplicial (strict) n -groupoid. There is a naturalweak equivalence T ot r ( r N ( n ) G ● ) ≃ holim ∆ N ( n ) G ● . References [An] V. Angeltveit,
Enriched Reedy categories , Proceedings of the AmericanMathematical Society, Vol. 136, No. 7, pp. 2323–2332 (2008). [BGNT] P. Bressler, A. Gorokhovsky, R. Nest and B. Tsygan,
Deformationquantization of gerbes , Adv. Math. 214, Issue 1, pp. 230–266 (2007). [BH1] R. Brown and P. J. Higgins,
The equivalence of ∞ -groupoids andcrossed complexes, Cah. Top. G´eom. Diff., 22 (1981). The classifying space of a crossed complex , Mathematical Proceedings of the Cambridge Philosophical Society, 110,pp. 95–120 (1991). [Br1] L. Breen,
Notes on 1- and 2-Gerbes , available at arXiv 0611317 (2006). [Br2] L. Breen, On the classification of 2-gerbes and 2-stacks , Soc. Math. deFrance, 225 (1994). [BK] A. K. Bousfield and D. Kan
Homotopy limits, completions and lo-calizations , Lecture Notes in Mathematics, Vol. 304. Springer-Verlag,Berlin-New York (1972).
Kan complexes and multiple groupoid structures , Math-ematical sketches, 32, Paper No. 2, xi+92 pp., Esquisses Math., 32,Univ. Amiens, Amiens, (1983). [DF] B. Dwyer and E. D. Farjoun,
A long homology localization tower , Com-ment. Math. Helvet. (52), 186-210 (1977). [Gr] J. W. Gray
Formal category theory: adjointness for 2-categories , Lec-ture Notes in Mathematics 391, Springer-Verlag (1974). [GJ] P. Goerss and R. Jardine,
Simplicial homotopy theory , Progress inMathematics 174, Birkhauser, (1999). [Hi] P. S. Hirschhorn,
Model categories and their localizations , Mathemati-cal Surveys and Monographs, 99. AMS, xvi+457 pp. (2003). [Ja] R. Jardine,
Cosimplicial spaces and cocycles , preprint, available atHomepage (2010). [Ma] S. MacLane, Categories for the working mathematician , GraduateTexts in Mathematics, Springer-Verlag (1971). [MS] I. Moerdijk and J. A. Svensson,
Algebraic classification of equivarianthomotopy 2-types, I , Journal of Pure and Applied Algebra Vol. 89. pp.187-216 (1993). [No] B. Noohi,
Notes on 2-groupoids, 2-groups and crossed modules , Ho-mology, Homotopy Appl. 9 (1) pp. 75–106 (electronic) (2007). [Qu] D. G. Quillen,
Homotopical algebra , Lecture Notes in Math., Vol. 43,Springer-Verlag, New York (1967). [Re] C. L. Reedy,
Homotopy theory of model categories , [St] R. Street, The algebra of oriented simplexes , J. Pure Appl. Algebra 49,no. 3, pp. 283–335 (1987). [Ye1] A. Yekutieli,