aa r X i v : . [ m a t h . A T ] J a n COMMUTATIVE SIMPLICIAL BUNDLES
C˙IHAN OKAY AND P ´AL ZS ´AMBOKI
Abstract.
In this paper we introduce a simplicial analogue of principal bundles with com-mutativity structure and their classifying spaces defined for topological groups. Our con-struction is a variation of the W -construction for simplicial groups. We show that thecommutative W -construction is a classifying space for simplicial principal bundles with acommutativity structure and geometric realization relates our constructions to the topolog-ical version. Introduction
Let G be a topological group. Principal G -bundles with a commutativity structure on theirtransition functions are introduced in [AG15]. The classifying space B com G for such bundles,first introduced in [ACTG12], is a variant of the ordinary classifying space BG , which isconstructed from pairwise commuting group elements. This construction is a particular caseof a class of constructions, denoted by B ( τ, G ), that depend on a cosimplicial group τ • .There is a cosimplicial group free in each degree so that this construction gives BG , and forthe level-wise abelianization of the cosimplicial group we obtain B com G .In this paper we carry over this construction to the simplicial category. Let K be asimplicial group. The classifying space of K is given by the bar construction W K [May67, § sGrp of simplicial groups to the category sSet ofsimplicial sets whose left adjoint is given by Kan’s loop group functor G . We introduce apair of adjoint functors G ( τ, − ) : sSet ⊣ sGrp : W ( τ, − )generalizing the well-known adjunction G ⊣ W .We show that W ( τ, − ) is a classifying space for certain types of simplicial principal bun-dles. This is the simplicial analogue of a result due to Adem–G´omez [AG15], which statesthat B ( τ, G ) classifies principal G -bundles whose transition functions have a commutativitystructure. We introduce the notion of a τ -structure for a principal K -bundle and define theset H τ ( X, K ) of equivalence ( τ -concordance) classes of such bundles. Theorem 4.14.
There is a bijection of sets
Θ : [
X, W ( τ, K )] → H τ ( X, K ) . Let | − | denote the geometric realization functor. We compare our construction to thetopological version.
Theorem 5.4.
There is a natural homotopy equivalence B ( τ, | K | ) → | W ( τ, K ) | . Thus our construction has the correct homotopy type. We also show that
Date : January 14, 2020. orollary 5.7. For a compact Lie group G , there is a natural homotopy equivalence B ( τ, G ) → | W ( τ, SG ) | where S ( − ) denotes the singular functor, the right adjoint of the geometric realization. As a consequence we can import the results regarding all the interesting examples studiedpreviously, such as [AG15, ACGV17, AGLT17, Oka18, GH19, OW19], to the simplicialcategory.The structure of the paper is as follows. In § G = π Dec is proved in Proposition 2.3. In § W ( τ, K ). Under some assumptions on τ we describe its set of n -simplices(Proposition 3.6). We also prove that after looping once the natural map W ( τ, K ) → W K splits up to homotopy (Proposition 3.7). Simplicial bundles with τ -structure are introducedin §
4. The classification result that identifies H τ ( X, K ) with the set of homotopy classes ofmaps X → W ( τ, K ) is proved in Theorem 4.14. We compare our simplicial construction tothe topological version B ( τ, G ) in §
5. The homotopy equivalence B ( τ, | K | ) ≃ | W ( τ, K ) | isproved in Theorem 5.4 and B ( τ, G ) ≃ | W ( τ, SG ) | is proved in Corollary 5.7. Acknowledgement:
We thank Danny Stevenson for sharing a proof of Corollary 2.5.The first author is partially supported by NSERC, and would like to thank Alfr´ed R´enyiInstitute of Mathematics for their hospitality during a visit in the summer of 2019. Thesecond author is partially supported by the project NKFIH K 119934.2.
Kan’s loop group and d´ecalage
The standard n -simplices ∆[ n ] as n varies can be assembled into a cosimplicial simplicialset. Applying Kan’s loop group functor G level-wise gives a cosimplicial simplicial group. Inthis section we describe this object using the factorization G = π Dec. This description willbe essential later on when we introduce variations of the W -construction. For the propertiesof the loop group functor and the d´ecalage functor we refer to [Ste12].For a category C we write sC for the category of simplicial objects in that category.2.1. D´ecalage.
Let ∆ denote the simplex category. Let ∆ + denote the category obtainedfrom ∆ by adjoining the empty ordinal denoted by [ − ∆ + is a monoidalcategory with unit [ − ∆ + × ∆ + → ∆ + which sends a pair ([ m ] , [ n ]) to the ordinal [ m + n + 1], and a pair of morphisms ϕ :[ m ] → [ m ′ ] and θ : [ n ] → [ n ′ ] to the morphism( ϕ + θ )( i ) = (cid:26) ϕ ( i ) 0 ≤ i ≤ mθ ( i − m −
1) + m ′ + 1 m + 1 ≤ i ≤ m + n + 1Let sSet ( sSet + ) denote the category of (augmented) simplicial sets. Given a simplicialset X the augmented simplicial set Dec X is defined by pre-composing X with the functor∆ + → ∆ defined by [ n ] [ n ] + [0]. Note that (Dec X ) n = X n +1 and the simplical structuremaps d i : (Dec X ) n → (Dec X ) n − and s j : (Dec X ) n → (Dec X ) n +1 are given by d i : X n +1 → X n , s j : X n +1 → X n +2 where 0 ≤ i, j ≤ n . Dec X is an augmented simplicial set where the augmentation map d : Dec X → X is induced in degree n by the n + 1-fold composition d n +10 = d ◦ · · · ◦ d :[0] → [ n ] + [0]. et us describe Dec (∆[ k ]) in more detail. Lemma 2.1.
There is a natural isomorphism of augmented simplicial sets ` ≤ l ≤ k ∆[ l ] Dec (∆[ k ]) { , , · · · , k } ∆[ k ] d (2.1.1) which maps ∆[ l ] isomorphically onto the pre-image ( d ) − ( l ) .Proof. Dec (∆[ k ]) is the coproduct of ( d ) − ( l ) over l since the augmentation map d is adeformation retraction [Ste12]. An m -simplex of ( d ) − ( l ) is a functor ϕ : [ m + 1] → [ k ]such that ϕd m +10 (0) = l , and such functors are in one-to-one correspondence with functors[ m ] → [ l ]. (cid:3) Total d´ecalage.
The d´ecalage functor Dec is a comonad whose structure maps aredescribed as follows: Dec → (Dec ) is induced by [ n ] + [0] + [0] → [ n ] + [0] given by the sumof the identity map on [ n ] and s : [0] + [0] → [0], and the other structure map Dec → id isinduced by d n : [ n ] → [ n ] + [0]. The total d´ecalage of the simplicial set X is defined to bethe comonadic resolution Dec n X = (Dec ) n +1 X. If we think of the bisimplicial set Dec X as a vertical bisimplicial set with horizontal simplicialsets then the set of ( m, n )-simplices is X m + n +1 . The horizontal face and degeneracy maps aregiven by d hi = d i : (Dec n X ) m → (Dec n X ) m − and s hi = s i : (Dec n X ) m → (Dec n X ) m +1 where0 ≤ i ≤ m , the vertical face and degeneracy maps are given by d vi = d m +1+ j : (Dec n X ) m → (Dec n − X ) m and s vi = s m +1+ j : (Dec n X ) m → (Dec n +1 X ) m where 0 ≤ j ≤ n .We will give a description of Dec ∆[ k ]. For this we introduce a bisimplicial set D [ k ] definedby D n [ k ] = a ϕ :[ n ] → [ k ] ∆[ ϕ (0)]together with the simplicial structure d i ( ϕ, θ ) = (cid:26) ( ϕd , ιθ ) i = 0( ϕd i , θ ) 0 < i ≤ n where ι : ∆[ ϕ (0)] → ∆[ ϕ (1)] is induced by the inclusion [ ϕ (0)] ⊂ [ ϕ (1)] and s j ( ϕ, θ ) = ( ϕs j , θ )for all 0 ≤ j ≤ n . Proposition 2.2.
There is an isomorphism of bisimplicial sets g : D [ k ] → Dec ∆[ k ] . Proof.
Applying Lemma 2.1 gives an isomorphism g n : a ϕ :[ n ] → [ k ] ∆[ ϕ (0)] → Dec n ∆[ k ] (2.2.1) hich in degree m sends ( ϕ, θ ) to the functor [ m + n + 1] → [ k ] defined using θ on the subset[ m ] ⊂ [ m + n + 1] and on the rest using ϕ . More precisely it is the unique functor which fillsthe diagram [ m ][ m + n + 1] [ k ][ n ] θ ( d ) m +1 ϕ On the other hand the inverse to this map is given by g − n : Dec n ∆[ k ] → a ϕ :[ n ] → [ k ] ∆[ ϕ (0)] (2.2.2)which in degree m sends a functor α : [ m + 1 + n ] → [ k ] to the pair of the functors given by α ( d ) m +1 : [ n ] → [ k ] and α ′ : [ m ] → [ α ( m + 1)] that fits into the diagram[ m + 1 + n ] [ k ][ m ] [ α ( m + 1)] αα ′ It remains to check that { g n } n ≥ gives a morphism of simplicial sets. It is straightforwardto check this for s j and d i with i >
0. When i = 0 the face map d changes ϕ (0) to ϕ (1),and this is accounted for by adding the inclusion ι . (cid:3) Nerve functor and its adjoint.
We can take the nerve of a group to obtain a sim-plicial set. Let
Grp denote the category of groups. This construction gives a functor N : Grp → sSet . In fact the image of the functor lives in the category sSet of reducedsimplicial sets i.e. those simplicial sets whose set of 0-simplices is a singleton. Let us define afunctor π : sSet → Grp by assigning a simplicial set X the quotient of the free group F ( X )on the set of 1-simplices by the relations [ s x ] = 1 for all x ∈ X , and [ d σ ][ d σ ] = [ d σ ] forall σ ∈ X . The functor π is the left adjoint of N .Let us compute π of a k -simplex. Note that the set of 1-simplices is given by functors[1] → [ k ]. The relation coming from s will kill those which are not injective. Each 2-simplexwill introduce a relation among the edges in its boundary. As a result there is an isomorphismof groups F k → π ∆[ k ]which sends a generator e i of the free group F k to the 1-simplex [1] → [ k ] specified by theimage { i − , i } . We will identify these groups and given an ordinal map α : [ k ] → [ l ] we willwrite α ∗ : F k → F l for the induced map π ∆[ k ] → π ∆[ l ]. Considering all the simplices at once defines acosimplicial group F • : ∆ → Grp where F • [ k ] is the free group F k = π ∆[ k ]. At this point e also observe that the nerve functor is represented by the cosimplicial group F • in thesense that ( N G ) • = Grp ( F • , G ) . This approach allows us to describe the adjunction π ⊣ N from the general theory of Kanextensions.2.4. Loop group.
Given a (reduced) simplicial set X the loop group GX is a simplicialgroup which homotopically behaves as the loop space. The group of n -simplices is the freegroup F ( X n +1 − s X n ). The face maps are defined by d i [ x ] = (cid:26) [ d x ][ d x ] − i = 0[ d i +1 x ] 0 < i ≤ n and the degeneracy maps are defined by s j [ x ] = [ s j +1 x ] for all 0 ≤ j ≤ n . In fact thisconstruction is related to the d´ecalage construction.Recall that the bisimplicial set Dec ∆[ k ] has a nice description whose simplicial structureis given in Proposition 2.2. Let us consider the simplicial group π Dec∆[ k ] obtained byapplying π to each vertical degree. From the isomorphism 2.2.1 we see that the resultingsimplicial group consists of the free groups( π Dec∆[ k ]) n ∼ = ( π D [ k ]) n = a ϕ :[ n ] → [ k ] F ϕ (0) (2.4.1)whose simplicial structure maps are induced by the ones of D [ k ]. Let us describe thisstructure explicitly. We write [ ϕ, e j ] to denote the generator of F ϕ (0) corresponding to the ϕ term in the coproduct. The simplicial structure maps are given by d i [ ϕ, e j ] = (cid:26) [ ϕd , ι ∗ ( e j )] i = 0[ ϕd i , e j ] 0 < i ≤ n (2.4.2)and s i [ ϕ, e j ] = [ ϕs i , e j ] for the degeneracy maps. Recall that ι ∗ is the map π ∆[ ϕ (0)] → π ∆[ ϕ (1)] induced by the inclusion [ ϕ (0)] ⊂ [ ϕ (1)].2.5. Relation to d´ecalage.
Let ∆ • : ∆ → sSet denote the functor defined by sending[ k ] to the k -simplex ∆[ k ] = ∆ ( − , [ k ]). We can think of ∆ • as a cosimplicial object in thecategory of simplicial sets. Applying the functor Dec to the cosimplicial object ∆ • in eachdegree and taking π of the resulting simplicial set gives a cosimplicial object π Dec ∆ • : ∆ → sGrp . The object [ n ] is sent to the cosimplicial group π Dec ∆[ n ]. Another cosimplicial object canbe obtained by composing with Kan’s loop group functor G ∆ • : ∆ → sGrp . Proposition 2.3.
There is a natural isomorphism of functors ǫ • : G ∆ • → π Dec ∆ • . Proof.
In the cosimplicial degree k we have( ǫ k ) n : ∆[ k ] n +1 → a ϕ :[ n ] → [ k ] ∆[ ϕ (0)] efined by sending β : [ n +1] → [ k ] to the pair given by βd : [ n ] → [ k ] and β | [1] , the restrictionof β along the natural inclusion [1] ⊂ [ n + 1]. For simplicity of notion we will suppress thecosimplicial degree and write ǫ n = ( ǫ k ) n . The map ǫ n induces a group homomorphism F ǫ n : G (∆[ k ]) n −→ π Dec n ∆[ k ]since a generator [ αs ] in the image of the degeneracy map s is mapped to the generatorgiven by the pair of the map αs d = α and the map obtained by restricting αs to [1].The latter generator corresponds to [( d ) α (0) s ] in the free group F α (0) and is equivalent tothe identity element under the identification on the target. Therefore the homomorphism F (∆[ k ] n +1 ) → ` ϕ :[ n ] → [ k ] F (∆[ ϕ (0)] ) induced by ǫ n on the free groups factors through F ǫ n since the image of the degeneracy map s is sent to the identity element. Moreover, ǫ n isan isomorphism since both groups are free groups and the map is a bijection between thegenerators. Next we check that the map is compatible with the simplicial structure. For0 < i ≤ n we have ǫ n d i ( α ) = ǫ n ( αd i +1 ) = αd i +1 d = αd d i , similarly for degeneracy maps wehave ǫ n s j ( α ) = αd s j for all 0 ≤ j ≤ n . Thus for the face map we have ǫ n ( d i [ α ]) = [ αd d i , αd i +1 | [1] ] = d i [ αd , α | [1] ] = d i ǫ n [ α ]and similarly ǫ n commutes with the degeneracy maps. Finally d on the loop group is definedby d [ α ] = [ αd ][ αd ] − . Therefore we have ǫ n ( d [ α ]) = ǫ n ([ αd ]) ǫ n ([ αd ]) − = [ α ( d ) , αd | [1] ] [ α ( d ) , αd | [1] ]where we used the simplicial identity d d = ( d ) . Note that the first coordinates are thesame that is the generators under ǫ n both [ αd ] and [ αd ] map to the same free group F ( α (2))indexed by α ( d ) . Let σ denote the restriction of α to [2] (note that α lives in degree ≥ π ∆[ α (2)] ∼ = F ( α (2)) we can write[ αd | [1] ][ αd | [1] ] − = [ d σ ][ d σ ] − = [ d σ ] = [ α | [1] ]which is the same, after adding the coproduct index, as d of the generator ǫ n [ α ].Thus we showed that we have an isomorphism of simplicial groups ǫ k : G ∆[ k ] → π Dec∆[ k ]where we remembered the cosimplicial index, and suppressed the simplicial index. The re-sulting map is compatible with the cosimplicial structure since all the constructions involvedare functorial in ∆[ k ]. (cid:3) Remark 2.4.
One can check that the inverse of ǫ k is induced by the map a ϕ :[ n ] → [ k ] ∆[ ϕ (0)] → ∆[ k ] n +1 (2.5.1)which sends ( ϕ, γ ) to the functor α : [ n + 1] → [ k ] defined by α (0) = γ (0) and α ( i ) = ϕ ( i − < i ≤ n + 1. An argument similar to the proof of Proposition 2.3 can be used tosee that after taking the appropriate quotients of the free groups the resulting map is anisomorphism. We will denote this map simply as the inverse( ǫ • ) − : π Dec∆ • → G ∆ • . Almost immediately we obtain the following result which is also proved in [Ste12, Propo-sition 16], however, without an explicit isomorphism. orollary 2.5. For any simplicial set X there is a natural isomorphism of simplicial groups G ( X ) → π Dec ( X ) . Proof.
An arbitrary simplicialset can be written as a colimit of ∆[ n ] over the simplex cate-gory. All the functors in sight G , π , Dec are left adjoints, thus they preserve colimits. Thenthe result follows from Proposition 2.3. (cid:3) Remark 2.6.
In Proposition 16 of [Ste12] Kan’s loop group G is compared to π R Dec where R is the left adjoint of the inclusion ssSet → ssSet . We avoided this functor by defining π for an arbitrary simplicial set not necessarily a reduced one.2.6. Simplicial group homomorphisms.
Let K be a simplicial group. The set of simpli-cial group homomorphisms G ∆[ k ] → K is well-known [GJ99]. This set is precisely the setof k -simplices of the W -construction of K . For variations of this construction we need anexplicit description of such simplicial group homomorphisms.By Proposition 2.3 we can instead consider a morphism f : π Dec∆[ k ] → K of simplicialgroups. The degree n -part f n belongs to the set Grp ( π Dec∆[ k ] n , K n ) = Y ϕ :[ n ] → [ k ] Grp ( F ϕ (0) , K n ) = Y ϕ :[ n ] → [ k ] K × ϕ (0) n where K × ϕ (0) n is understood to be the trivial group if ϕ (0) = 0. Therefore f n is determinedby the tuple of elements f n [ ϕ, e j ] in K n where e j ∈ F ϕ (0) and 1 ≤ j ≤ ϕ (0) . The map ϕ factors as follows [ n ] [ k ][ k − ϕ (0)] ϕφ ( d ) ϕ (0) where φ (0) = 0 i.e. the canonical decomposition of φ does not involve d . The simplicialstructure in 2.4.2 implies that φ ∗ [( d ) ϕ (0) , e j ] = [( d ) ϕ (0) φ, e j ] = [ ϕ, e j ] . and thus f n [ ϕ, e j ] = φ ∗ f k − ϕ (0) [( d ) ϕ (0) , e j ]. As a result the elements f k − l [( d ) l , e j ] where e j ∈ F l for l = 1 , · · · , k completely determine f n . It remains to consider the effect of d . Asa consequence of 2.4.2 we have d [( d ) l , e j ] = [( d ) l +1 , d l +1 ∗ e j ] . Note that in this case ι is induced by the inclusion d l +1 : [ l ] → [ l + 1]. By abuse of notationwe will identify d l +1 ∗ e j with the generator e j in F l +1 . After this identification we see that allthe other generators are determined once we fix f k − l [( d ) l , e l ] for 1 ≤ l ≤ k , where e l is thetop generator of F l . We package this information as a diagram F F · · · F k K k − K k − · · · K d ∗ f k − d ∗ f k − d k ∗ f d d d (2.6.1)where d l ∗ is the map induced by d l : [ l − → [ l ]. valuating the homomorphisms f k − l at the generator e l of each free group gives a k -tuples( x k − , x k − , · · · , x ) where x k − l belongs to K k − l . Then the images of the generators of F l are given by f k − l ( e j ) = ( d ) l − j x k − j (2.6.2)where 1 ≤ j ≤ l . Proposition 2.7.
There is a bijection of sets sGrp ( π Dec∆[ k ] , K ) −→ K k − × K k − × · · · × K . defined by sending f ( x k − , x k − , · · · , x ) where x k − l = f k − l [( d ) l , e l ] . Remark 2.8.
In particular, for K = G ∆[ k ] we can find the k -tuple corresponding to themap ( ǫ • ) − defined in Remark 2.4. Consider the restriction ∆[ l ] → ∆[ k ] n +1 of the map in2.5.1 to the factor ϕ = ( d ) l : [ k − l ] → [ k ] in the coproduct. The generator e l in F l isrepresented by the map ( d ) l − : [1] → [ l ]. Under 2.5.1 the pair (( d ) l , ( d ) l − ) is mapped to( d ) l − : [ k − l + 1] → [ k ]. As a result we obtain f k − l ( e l ) = [( d ) l − ] ∈ G ∆[ k ] k − l Thus, under the bijection in Proposition 2.7 we have( ǫ • ) − ([id] , [ d ] , · · · , [( d ) k − ]) . Variations of the W -construction We introduce the simplicial set W ( τ, K ) that depends on an endofunctor τ on the categoryof groups. Under some niceness conditions on τ we describe the set of n -simplices. Thisobject comes with a map W ( τ, K ) → W K , which we show splits up to homotopy afterlooping once.When τ is the identity functor we recover W K . Other examples come mainly from the de-scending central series, and in particular, the abelianization functor, which gives a simplicialversion of B ( Z , G ).3.1. W -construction. The loop functor G has a well-known right adjoint W : sGrp → sSet . For a simplicial group K the set of k -simplices of W K consists of simplicial grouphomomorphisms G ∆[ k ] → K , and the simplicial structure is determined by the cosimplicialstructure of G ∆ • . More explicitly, W ( K ) k can be identified with the product K k − × K k − ×· · · × K . Under this identification the simplicial structure is described as follows: the facemaps are given by d i ( x k − , · · · , x ) = ( x k − , x k − , · · · , x ) k = 0( d i − x k − , d i − x k − , · · · , d x k − i x k − i − , x k − i − , · · · , x ) 0 < i < k ( d k − x k − , d k − x k − , · · · , d x ) i = k and the degeneracy maps are given by s i ( x k − , · · · , x ) = (cid:26) (1 , x k − , · · · , x ) i = 0( s i − x k − , s i − x k − , · · · , s x k − i , , x k − i − , x k − i − , · · · , x ) 0 < i ≤ k. s a consequence of the natural isomorphism G ∆ • → π Dec∆ • proved in Proposition 2.3the functor π Dec has also a right adjoint isomorphic to the functor W . The right adjointof π Dec∆ • is defined by K sGrp ( π Dec∆ • , K )where the simplicial group homomorphisms are taken level-wise. Let us describe the cosim-plicial structure of π Dec∆ • . Given an ordinal map α : [ l ] → [ k ] the induced map ond´ecalage Dec∆[ l ] → Dec∆[ k ] sends a pair ( ϕ, θ ) to composition by α , namely to the pair( αϕ, αθ ). The map between the fundamental groups is determined when θ is a 1-simplex of∆[ ϕ (0)], that is, we have α ∗ [ ϕ, e j ] = [ αϕ, α ∗ ( e j )] where e j belongs to the free group F ϕ (0) .Given this general description for arbitrary ordinal maps let us figure out the effect of thecoface maps d i : [ k − → [ k ] first. It suffices to consider the generators [( d ) m , e m ], where1 ≤ m ≤ k −
1, since the rest is determined by the simplicial structure. Using the cosimplicialidentity d i d = d d i − for i > d i [( d ) m , e m ] = [( d ) m d i − m , e m ] 1 ≤ m < i ≤ k [( d ) m +1 , e m ] [( d ) m +1 , e m +1 ] m = i [( d ) m +1 , e m +1 ] i < m ≤ k − s i : [ k + 1] → [ k ], using s i d = d s i − for i > s d = 1,we have s i [( d ) m , e m ] = [( d ) m s i − m , e m ] 1 ≤ m ≤ i ≤ k [( d ) m − , m = i + 1[( d ) m − , e m − ] i + 1 < m ≤ k + 1 . (3.1.2) Proposition 3.1.
There is a natural identification of simplicial sets W ( K ) = sGrp ( π Dec ∆ • , K ) . Proof.
Using the cosimplicial structure of π Dec∆ • described in 3.1.1 and 3.1.1 we checkthat the right adjoint is equal to the W -construction. In Proposition 2.7 we have seen thatsimplicial group homomorphisms f : π Dec∆[ k ] → K are in one-to-one correspondence with k -tuples ( x k − , x k − , · · · , x ) where x i ∈ K i . The correspondence is obtained by letting x k − m denote the image of [( d ) m , e m ] under f . Using the cosimplicial structure of π Dec∆ • we firstcompute the coface maps d i : [ k − → [ k ]: f ( d i [( d ) m , e m ]) = d i − m x k − m ≤ m < i ≤ kd ( x k − i ) x k − i − m = ix k − m − i < m ≤ k − f ( d i [( d ) m , e m ]) = s i − m x k − m ≤ m ≤ i ≤ k m = i + 1 x k − m +1 i + 1 < m ≤ k − . This shows that the simplicial structure on the k -tuples ( x k − , x k − , · · · , x ) is exactly theone of the W -construction. (cid:3) .2. Endofunctors.
Recall that we used the cosimplicial group F • , where F k = π ∆[ k ], inthe definition of the nerve functor, namely N = Grp ( F • , − ). We will introduce a variant ofthis construction with respect to an endofunctor τ : Grp → Grp .Given an endofunctor τ we consider the cosimplicial group τ • defined by τ k = τ F k . Definition 3.2.
We say that τ is of quotient type , if there exists a natural transformationid η −→ τ such that the map of groups F n η F n −−→ τ n is surjective for all n ≥
0. In this case wealso say that η is surjective on finitely generated free groups .Left Kan extension [Rie14, Chapter 1] of τ • : ∆ → Grp along the natural inclusion∆ • : ∆ → sSet gives a functor π ( τ, − ) : sSet → Grp . By the general theory of left Kan extensions there is a corresponding right adjoint N ( τ, − ) : Grp → sSet defined by N ( τ, G ) n = Grp ( τ n , G ). The group π ( τ, ∆[ k ]) is naturally isomorphic to τ F k = τ π ∆[ k ]. Note that we recover the adjunction π ⊣ N when τ is the identity functor.Let us give a list of endofunctors that are of interest to us. • Descending central series endofunctor Γ q : The descending central series of a group H is defined by Γ ( H ) = H, Γ q ( H ) = [Γ q − ( H ) , H ] . We will denote the q -th stage H/ Γ q H of the descending central series by Γ q H . • Γ will have a special importance. We introduce the notation Z • = Γ F • . • Mod- p version Γ p,q : For a group H mod- p descending central series is defined byΓ p ( H ) = H, Γ qp ( H ) = [Γ q − p ( H ) , H ](Γ q − p ( H )) p . The q -th stage H/ Γ qp H is denoted by Γ p,q H . • Γ p, is used to define ( Z /p ) • = Γ p, F • . • Let Γ p k , denote the composition of Γ with the mod- p k reduction functor that sendsan abelian group to the largest p k -torsion quotient. We write( Z /p k ) • = Γ p k , F • . • Let H H ∧ p denote the p -adic completion functor. Let ˆΓ p,q denote the functor H Γ q ( H ∧ p ). • The p -adic cosimplicial group is defined by( Z p ) • = ˆΓ p,q F • . Remark 3.3.
The endofunctors Γ q and Γ p k ,q are of quotient type. The completed versionˆΓ p,q fails to satisfy the surjectivity assumption. xample 3.4. Let X ( n ) denote the quotient of ∆[ n ] by the set of vertices ∆[ n ] . Then onecan check that π ( Z , − ) of the inclusion ∨ n X (1) → X ( n )is the abelianization map F n → Z n . Higher simplices has the effect of abelianizing theordinary fundamental group.3.3. Variants of W -construction. Similarly we can generalize the adjunction between the W -construction and Kan’s loop group functor G with respect to a given endofunctor τ onthe category of groups.We define the functor G ( τ, − ) : sSet → sGrp X π ( τ, Dec X )which, up to natural isomorphism, is the left Kan extension of π ( τ, Dec∆ • ) along the inclu-sion of the simplex category into the category of simplicial sets. On a k -simplex this functoris given by G ( τ, ∆[ k ]) n = a ϕ :[ n ] → [ k ] τ F ϕ (0) and the simplicial structure is induced from the one of G ∆[ k ].There exists a right adjoint of the functor G ( τ, − ) which we denote by W ( τ, − ) : sGrp → sSet . Observe that these constructions recover the usual adjunction G ⊣ W when τ is the identityfunctor. Example 3.5.
It is instructive to look at W ( Z , K ). The set of k -simplices consists of( x k − , x k − , · · · , x )such that the elements ( d ) l − x k − , ( d ) l − x k − , · · · , d x k − l − , x k − l pairwise commute for all1 ≤ l ≤ k .This easily generalizes to W (Γ q , K ). When K is discrete, i.e. K n = G for some discretegroup G and the simplicial maps are all identity, the geometric realization of W (Γ q , G ) isprecisely the space B ( q, G ) introduced in [ACTG12]. Proposition 3.6.
Let
Grp τ −→ Grp be an endofunctor, and id η −→ τ a natural transformationthat is surjective on finitely generated free groups. Then the set of k -simplices of W ( τ, K ) isgiven by tuples ( x k − , x k − , · · · , x ) ∈ K k − × K k − × · · · × K that satisfy the following property: For ≤ l ≤ k let S l denote the subgroup generated by ( d ) l − x k − , ( d ) l − x k − , · · · , d x k − l − , x k − l then the map η S l : S l → τ S l is an isomorphism.Proof. The set W ( K ) k , equivalently the set of simplicial group homomorphisms π Dec∆[ k ] → K , is described in Proposition 2.7. Since η F n is surjective by assumption the induced map η ∗ : Grp ( τ F n , H ) → Grp ( F n , H ) is injective for any group H . We see that there is an nclusion of simplicial sets W ( τ, K ) ⊂ W K and the elements of W ( τ, K ) k corresponds todiagrams τ F τ F · · · τ F k K k − K k − · · · K d ∗ f k − d ∗ f k − d k ∗ f d d d Let R n denote the kernel of F n → τ F n . It is generated by certain relations among thegenerators e , · · · , e n . These relations are also satisfied among f ( e ) , · · · , f ( e n ) where f is ahomomorphism τ F n → H . As a consequence we observe that in the diagram1 R n F n τ F n K Im( f ) τ Im( f ) 1 f K = 1, or equivalently τ Im( f ) = Im( f ). This applies to each map f k − l and the resultfollows. (cid:3) Descending central series filtration.
We introduce a simplicial version of the filtra-tion introduced in [ACTG12] for the classifying space of a topological group. This filtrationis obtained from the sequence of endofunctorsid = Γ ∞ → · · · → Γ q → Γ q − → · · · Γ associated to the stages of the descending central series. For each endofunctor we have acosimplicial group Γ • q = Γ q F • . We write W ( q, − ) = W (Γ q , − ) , G ( q, − ) = G (Γ q , − ) . The resulting sequence G ( X ) → · · · → G ( q, X ) → G ( q − , X ) → · · · → G (2 , X ) = G ( Z , X )consists of surjective of simplicial group homomorphisms since Γ q H → Γ q − H is surjectivefor any group H .On the other hand, we have a sequence of inclusions of simplicial sets W ( Z , K ) = W (2 , K ) → · · · → W ( q − , K ) → W ( q, K ) → · · · → W ( K )that yields a filtration of the W -construction.3.5. Kan suspension.
Let X be a pointed simplicial set. Kan suspension of X is thesimplicial set Σ X whose set of n -simplices is given by the wedge X n − ∨ X n − ∨ · · · ∨ X ([GJ99, page 189]). An ordinal map θ : [ m ] → [ n ] induces θ ∗ : (Σ X ) n → (Σ X ) m which maps wedge summand X n − i to the base point if θ − ( i ) is empty, otherwise it is determined by θ ∗ i : X n − θ ( i ) → X m − i where θ i is defined by the diagram[ m − i ] [ m ][ n − θ ( i )] [ n ] ( d ) i θ i θ ( d ) θ ( i ) Let K be a simplicial group pointed by the identity element of each K n . There is a canonicalmap of simplicial sets κ : Σ K → W K induced by the inclusion K n − ∨ · · · ∨ K → K n − × · · · × K at the n -th level. Proposition 3.7.
Assume that τ in Proposition 3.6 satisfies the property that η F : F → τ F is an isomorphism. Then the natural map GW ( τ, K ) → G ( W K ) splits (naturally) up to homotopy. This is a simplicial analogue of [ACTG12, Theorem 6.3] that applies to the topological B com construction. Proof.
First observe that under the assumption on τ we have that η C : C → τ C is anisomorphism for any cyclic group C . Then by the description of the simplices of W ( τ, K )given in Proposition 3.6 the map κ factors through κ τ : Σ K → W ( τ, K ) . The splitting is given by the following diagram GW ( τ, K ) G (Σ K ) G ( W K ) K F K G ( κ τ ) ∼ ∼ = Let us explain the maps. The weak equivalence is the counit of the adjunction betweenthe loop group and bar construction. In degree n the simplicial group F K , known as theMilnor’s construction [GJ99, page 285], is the free group generated on K n − {∗} . The map K → F K sends an n -simplex to the corresponding generator of the free group. The setof n -simplices of G (Σ K ) is given by F (Σ K ) n +1 /F ( s (Σ K ) n ) which can be identified withthe n -simplices of F K since s (Σ K ) n maps onto the wedge summands of (Σ K ) n +1 otherthan X n . This isomorphism is compatible with the simplicial structure. The last two mapsare already described above. Starting from K the composition of the five maps gives theidentity. The splitting is natural with respect to K since each construction is functorial in K . (cid:3) Corollary 3.8.
Under the assumption of Proposition 3.7 the natural map W ( τ, K ) → W K induces a split surjection on homotopy groups. . Simplicial bundles with τ -structure In this section K denotes a simplicial group and τ • denotes a cosimplicial group obtainedfrom an endofunctor τ , i.e. τ k = τ F k , which is equipped by a natural transformation id η −→ τ that is surjective on finitely generated free groups.We introduce simplicial principal bundles with τ -structure. This is the simplicial analogueof a topological definition that generalizes the transitional commutative structure definedfor topological bundles. We prove a classification theorem for such simplicial bundles where W ( τ, K ) is the classifying object.4.1. Simplicial fiber bundles.
First, following [May67], we introduce simplicial fiber bun-dles and simplicial principal bundles.
Definition 4.1.
Let F be a nonempty simplicial set. Then a map of simplicial sets E p −→ B is an F -fiber bundle , if for each n -simplex ∆[ n ] b −→ B , there exists a pullback square of theform F × ∆[ n ] E ∆[ n ] B. y β ( b ) π b p Such a projection map F × ∆[ n ] → E is called a local trivialization . A collection of a localtrivialization F × ∆[ n ] β ( b ) −−→ E for each simplex b ∈ B n for each n ≥ atlas of p . Notation 4.2.
The automorphism simplicial group Aut F ⊆ Map(
F, F ) has a natural rightaction on the mapping simplicial set Map(
F, E ), which is defined as follows. For m -simplices F × ∆[ m ] α −→ F of Aut F and F × ∆[ m ] β −→ E of Map( F, E ), the product β · α is defined asthe composite F × ∆[ m ] ( α, pr) −−−→ F × ∆[ m ] β −→ E. Proposition 4.3.
Let F be a nonempty simplicial set, E p −→ B an F -fiber bundle, b ∈ B m an m -simplex of B , and β ( b ) ∈ Map ( F, E ) m a local trivialization of p over b . Then the setof local trivializations of p over b is a torsor over the group Aut ( F ) m . That is, the followingassertions hold.(1) Let β ′ ( b ) ∈ Map ( F, E ) m be another local trivialization of p over b . Then there exists aunique m -simplex α ∈ Aut ( F ) m such that we have β ′ ( b ) = β ( b ) · α .(2) Let α ∈ Aut ( F ) m be an m -simplex. Then the m -simplex β ( b ) · α ∈ Map ( F, E ) m is alsoa local trivialization of p over b .Proof. By the Pasting Lemma, in the commutative diagram below, the square below β ′ ( b )is Cartesian if and only if the square below ( α, pr) is Cartesian. × ∆[ n ] F × ∆[ n ] E ∆[ n ] ∆[ n ] B. yy ( α, pr ) β ′ ( b ) π β ( b ) π id ∆[ n ] b p (cid:3) Definition 4.4.
Let F be a nonempty simplicial set, E p −→ B an F -fiber bundle, b ∈ B m an m -simplex, β ( b ) ∈ Map(
F, E ) m a local trivialization of p over b , and [ m ] θ −→ [ n ] a map ofordinals. Then the pasting diagram F × ∆[ m ] F × ∆[ n ] E ∆[ m ] ∆[ n ] B. yy id × ∆[ θ ] θ ∗ β ( b ) π β ( b ) π ∆[ θ ] b p shows that F × ∆[ n ] θ ∗ β ( b ) −−−→ E is a local trivialization. Therefore, it corresponds to a transitionmap α ∗ ( θ ) ∈ Aut( F ) m . In particular, we let r i ( b ) = α ∗ ( d i ) for 0 ≤ i ≤ n . Remark 4.5.
Every fiber bundle has an atlas β such that r i ( b ) is constant identity for all n ≥ , b ∈ B n and ≤ i ≤ n [May67, Lemma 19.2] Remark 4.6.
Suppose that r i ( b ) is constant identity for all n ≥ , b ∈ B n and ≤ i ≤ n .Then the atlas β defines an isomorphism F × B = colim b ∈ ∆ ↓ B ( F × ∆[ n ]) → E [May67, Theorem19.4] . We will organize the assignment θ α ∗ ( θ ) into a functor. For this we construct twocategories from the ordinal category ∆ . First one is the simplex category ∆ ↓ X associatedto a simplicial set X : • objects are simplicial set maps x : ∆[ n ] → X for n ≥ • morphisms are commuting triangles∆[ m ] ∆[ n ] X θθ ∗ x x Next category involves a simplicial group K . Let ∆ ⋉ K denote the category • objects are the ordinals [ n ] where n ≥ • morphisms are pairs ( θ, k ) : [ m ] → [ n ]where θ is an ordinal map and k ∈ K m . The composition is defined by( θ, k )( ϕ, k ′ ) = ( θϕ, k ′ ϕ ∗ ( k )) . oth categories come with a forgetful functor U with target ∆ . Then we can construct thefunctor ∆ ↓ B ∆ ⋉ Aut( F ) ∆ α ∗ U U (4.1.1)by sending an n -simplex b to the object [ n ] and a morphism θ to the pair ( θ, α ∗ ( θ )). Lemma 4.7.
The assignment specified by α ∗ defines a functor.Proof. Let [ k ] φ −→ [ m ] θ −→ [ n ] be a diagram in ∆, and b ∈ B n . We need to show that( θ ◦ φ, α ∗ ( θ ◦ φ )) = ( θ, α ∗ θ ) ◦ ( φ, α ∗ φ ) = ( θ ◦ φ, α ∗ φ · φ ∗ α ∗ θ ) . By definition, the transition map α ∗ θ ∈ Aut( F ) m is the unique element such that θ ∗ β ( b ) = β ( θ ∗ b ) · α ∗ θ in Map( F, E ) m . Therefore, restricting along φ we get φ ∗ θ ∗ β ( b ) = φ ∗ β ( θ ∗ b ) · φ ∗ α ∗ θ, substituting into which the definition of α ∗ φ , we get φ ∗ θ ∗ β ( b ) = β ( φ ∗ θ ∗ b ) · α ∗ φ · φ ∗ α ∗ θ, which by definition shows that α ∗ ( θ ◦ φ ) = α ∗ φ · φ ∗ α ∗ θ . (cid:3) τ -structure. Let N ( τ, ∆ ⋉ K ) denote the simplicial subset of the nerve N (∆ ⋉ K )whose k -simplices are given by[ n ] ( θ ,x ) −→ [ n ] ( θ ,x ) −→ · · · ( θ k ,x k − ) −→ [ n k ]such that ( x , θ ∗ x , · · · , θ ∗ θ ∗ · · · θ ∗ k − x k − )belongs to N ( τ, K n ) k = Grp ( τ k , K n ). The simplicial structure is induced from the nerve. Definition 4.8.
Let p : E → B be a fiber bundle. An atlas α for p is said to have a τ -structure if the nerve N α ∗ of the associated transition functor α ∗ (4.1.1) factors through N ( τ, ∆ ⋉ Aut( F )). Example 4.9.
We will construct examples of simplicial fiber bundles with τ -structure thatare obtained from the topological fiber bundles. In the topological context the definitionof a τ -structure is given in Definition 5.10. However, we will use an alternative definition.Suppose G is a topological group. Let p : E → B denote a principal G -bundle over aHausdorff paracompact space B together with a trivialization { U i } i ∈ I . Let U denote thepartially ordered set (poset) of finite intersections of U i for i ∈ I . We can assume that eachfinite intersection is contractible by refining the open cover if necessary. The collection oftransition functions can be seen as a functor ρ : U → G as observed in [Seg68]. Then having a τ -structure can be equivalently defined by saying thatthe map of simplicial spaces N ( U ) → BG • factors through the simplicial subspace B ( τ, G ) • . ince all the nonempty finite intersections of the U i are contractible, there exists a homotopyequivalence N U ≃ X [Hat02, Corollary 4G.3].Let Top denote any convenient category of topological spaces such as the category ofcompactly generated Hausdorff spaces. Using the singular functor S : Top → sSet , theright adjoint to the geometric realization functor, we can produce a simplicial bundle withfiber SG . Let us define p U as a pull-back E U SEGN ( U ) S | N ( U ) | SBG p U ǫ Sρ where ǫ is the counit of the adjunction | | ⊣ S . The resulting simplicial bundle is a principal SG -bundle. Over a simplex σ : ∆[ n ] → N ( U ) the preimage p U ( σ ) − consists of SG × ( ρσ , · · · , ρσ n ) where σ is regarded as a sequence of inclusions U i σ −→ · · · σ n −→ U i n . We can define an atlas α by sending the simplex σ to the simplicial map α ( σ ) : SG × ∆[ n ] → E U defined, in degree m , by sending ( g m , θ ) to ( g m , θ ∗ ( ρσ , · · · , ρσ n )), where g m ∈ SG m and θ : [ m ] → [ n ]. The resulting transition functor α ∗ : ∆ ↓ N ( U ) → ∆ ⋉ SG is determined bythe original topological transition functions. Therefore N α ∗ factors through N ( τ, ∆ ⋉ SG )and gives a τ -structure for α .4.3. Simplicial principal bundles.
A principal K -bundle is a map of simplicial sets π : P → X where K acts freely on P and X is isomorphic to the quotient space. Free actioncan be formulated by saying that we have a pull-back diagram K × P PP X π m ππ where π is the projection and m is the action map. Every principal K -bundle is a Kanfibration and, in particular, a fiber bundle [May67, § d are non-trivial. Equivalently, we can express this by saying that the principal bundleadmits a pseudo-section. A pseudo-section s is a simplicial set map that is a section ofDec π Dec P PDec X X πs A principal K -bundle together with a choice of a pseudo-section is called a principal twistedcartesian product (PTCP).The pseudo-section can be used to define a map s ∗ : X → W K as described in [May67, Theorem 21.7], also see [Ste12, § π is isomorphic tothe pull-back of the universal bundle W K → W K [May67, Lemma 21.9]. For different seudo-sections the resulting maps are homotopic to each other [May67, Theorem 21.12]so we can talk about a classifying map X → W K defined up to homotopy. Conversely, amap X → W K induces a pseudo-section for π by pulling-back the canonical pseudo-section s can of the universal bundle W K → W K which, in degree n , is given by ( g n − , · · · , g ) (1 n , g n − , · · · , g ) where 1 n is the identity element of K n . Definition 4.10.
We say that a pseudo-section has a τ -structure if the corresponding atlashas a τ -structure.For principal K -bundles the transition functor has the form α ∗ : ∆ ↓ X → ∆ ⋉ K since all the transition functions specify simplices in K .4.4. Classification of principal bundles.
The category of simplicial sets comes with theQuillen model structure [Qui06]. As it is well-known
W K is fibrant in this model structurei.e. a Kan complex. However, W ( τ, K ) is not necessarily fibrant. This is even the casefor a discrete non-abelian group if we take τ • = Z • . Therefore the homotopy classes ofmaps [ X, W ( τ, K )], the set of morphisms in the homotopy category Ho( sSet ), cannot berepresented as simplicial homotopy classes of maps. To fix this we choose a fibrant replace-ment functor R , for instance Kan’s Ex ∞ functor would do. Then [ X, W ( τ, K )] is in bijectivecorrespondence with the set of simplicial homotopy classes of maps X → RW ( τ, K ).The inclusion map ι : W ( τ, K ) → W K induces a map ι ∗ : [ X, W ( τ, K )] → [ X, W K ] (4.4.1)which is, in general, neither injective nor surjective. Under the assumption of Corollary 3.8 itis surjective when | X | ≃ S n . However, it fails to be injective most of the time, see Example5.9.We prove an analogue of [AG15, Theorem 2.2] in the simplicial category. Lemma 4.11.
A principal K -bundle π : P → X has a τ -structure if and only if the classify-ing map X → W K factors through the inclusion W ( τ, K ) ⊂ W K in the homotopy categoryHo ( sSet ) .Proof. Suppose that π has a pseudo-section s with a τ -structure. By definition the cor-responding atlas α has a τ -structure. The nerve of the functor α ∗ : ∆ ↓ X → ∆ ⋉ K factors through the space N ( τ, ∆ ⋉ K ). This implies that the image of the induced map s ∗ : X → W K is contained in W ( τ, K ) by the description given in Proposition 3.6.Conversely, suppose that the classifying map f : X → W K , say defined with respectto a pseudo-section s , factors through W ( τ, K ) in the homotopy category. Let f τ : X → RW ( τ, K ) represent this map, where R is a fibrant replacement functor. The canonicalmap W ( τ, K ) ∼ → RW ( τ, K ) has a homotopy inverse, which we denote by r . The diagramcommutes up to homotopy X W KW ( τ, K ) frf τ ι hen the pseudo-section obtained by the pull-back of the canonical one along ιrf τ has a τ -structure. (cid:3) A τ -structure for a principal K -bundle with a classifying map f : X → W K is a choice ofa homotopy class in ι ∗ ([ f ]) − where ι ∗ is as defined in (4.4.1). Definition 4.12.
Consider two principal K -bundles π i : P i → X , i = 0 ,
1, together with τ -structures [ f iτ ]. We say π is τ -concordant to π , written as π ∼ τ π , if there exists aprincipal K -bundle π : P → X × [0 ,
1] with a τ -structure [ f τ ] such that π | X ×{ i } = π i , [ f τ | X ×{ i } ] = [ f iτ ] . Let H τ ( X, K ) denote the set of τ -concordance classes of principal K -bundles with a chosen τ -structure. Remark 4.13.
When τ • = F • the set H τ ( X, K ) coincides with the set H ( X, K ) of iso-morphism classes of principal K -bundles. Theorem 4.14.
There is a bijection of sets
Θ : [
X, W ( τ, K )] → H τ ( X, K ) . Proof.
Let f : X → RW ( τ, K ) be a map representing a homotopy class. The image ι ∗ [ f ] isrepresented by a map g : X → W K which is unique up to homotopy. The map Θ sends [ f ]to the class of the principal bundle g ∗ W K . This bundle has a τ -structure by Lemma 4.11.Conversely, we can define a map Ψ : H τ ( X, K ) → [ X, W ( τ, K )] by forgetting the principalbundle P but retaining the τ -structure [ f τ ]. The composite ΨΘ is clearly identity. On theother hand, ΨΘ is also the identity by the uniqueness of g up to homotopy. (cid:3) Comparison to the topological version
In this section G denotes a topological group.We introduced W ( τ, K ) and claimed that this is a simplicial analogue of B ( τ, G ). In thissection we turn this claim into a theorem. We prove two homotopy equivalences B ( τ, | K | ) ≃ | W ( τ, K ) | , B ( τ, G ) ≃ | W ( τ, SG ) | where in the latter one G is required to be a compact Lie group.5.1. Classifying spaces.
The nerve construction of a discrete group can be extended to thecategory of topological groups [Seg68]. Given a topological group G the set of n -simplicesof N G is the n -fold direct product G × n = G × · · · × G , hence a topological space when G has topology. The simplicial object N G • is a simplicial space, and its geometric realizationis denoted by BG = | N G • | which is called the classifying space of G . We can think of the n -fold product as the spaceof group homomorphisms N G n = Hom( F n , G ) . We use Hom( − , − ), rather than Grp ( − , − ), to emphasize the topology.Given a simplicial group K , the geometric realization | K | is a topological group, and wecan consider B | K | . We would like to compare this space to the geometric realization of W ( K ). t is well-known that W ( K ) is naturally isomorphic to T N ( K ), see [Dus75, Ehl91], where T is the Artin–Mazur totalization functor [AM66] from bisimplicial sets to simplicial sets.The factorization W = T N follows immediately from Proposition 3.1 since T is the rightadjoint of the total d´ecalage functor Dec. Here N K is the bisimplicial set with ( p, q )-simplicesgiven by N ( K q ) p .Another way of obtaining a simplicial set from a bisimplicial set Y is to take the diagonal dY whose n -simplices are Y n,n . There is a natural weak equivalence dY → T Y for all bisimplicial sets Y [CR05]. Applying this to the bisimplicial set N K we obtain thefollowing result.
Proposition 5.1.
There is a natural weak equivalence dN K → W ( K ) . Next we consider the classifying space B | K | of the geometric realization of a simplicialgroup, and compare it to | W K | . The following result is well-known. Proposition 5.2.
There is a natural homotopy equivalence B | K | → | W ( K ) | . Proof.
Taking the geometric realization of Proposition 5.1 we obtain a homotopy equivalence | dN K | → | W K | . We can realize the bisimplicial set N K in different ways. All of these arehomeomorphic to each other [Qui73]. There is a sequence of natural homeomorphisms | dN K | ∼ = | [ p ]
7→ | [ q ] N ( K q ) p || = | [ p ]
7→ | K p ||∼ = | [ p ]
7→ | K | p | = B | K | where we also used the fact that geometric realization preserves finite products [Mil57]. (cid:3) τ -version. Given an endofunctor τ on the category of groups we can define B ( τ, G ) = | [ n ] Hom( τ n , G ) | where τ • = τ F • as usual. Note that this definition works for an arbitrary cosimplicial groupnot necessarily coming from an endofunctor. For the rest of the section we assume that τ is of quotient type. In this case the homomorphism space Hom( τ n , G ) is topologized as asubspace of G × n . The natural transformation η induces an inclusion B ( τ, G ) ⊂ BG . Wedenote by E ( τ, G ) → B ( τ, G ) the pull-back of the universal bundle EG → BG .We would like to prove an analogue of Proposition 5.2. To prepare we need a preliminaryresult. Observe that the set of group homomorphisms Hom( τ F n , K m ) can be assembled intoa simplicial set by using the simplicial structure of K . Lemma 5.3.
There is a natural homeomorphism | Hom ( τ F n , K • ) | → Hom ( τ F n , | K | ) . roof. Recall [May67, §
14] that the geometrical realization | K | is the quotient space of thedisjoint union ⊔ n ≥ K n × ∆ n by the equivalence relations( d i k n , u n − ) ∼ ( k n , d i u n − ) k n ∈ K n , u n − ∈ ∆ n − , and ( s i k n , u n +1 ) ∼ ( k n , s i u n +1 ) k n ∈ K n , u n +1 ∈ ∆ n +1 . We denote by [ k m , u m ] ∈ | K | the equivalence class given by ( k m , u m ) ∈ K m × ∆ m .Let H be a discrete group. For m ≥
0, by construction, we have Hom(
H, K ) m = Grp ( H, K m ). The construction | Hom(
H, K ) | φ H −→ Hom( H, | K | )mapping a pair of a homomorphism H f −→ K m and a point u m ∈ ∆ m to the homomorphism H h [ f ( h ) ,u m ] −−−−−−−→ | K | gives a morphism of pointed spaces, which is natural in H . In particular,the composite | K × n | ∼ = −→ | Hom( F n , K ) | φ F n −−→
Hom( F n , | K | ) ∼ = −→ | K | × n is the map induced by the projection maps, which is a homeomorphism as geometricalrealization preserves finite products. Therefore, the map φ F n is a homeomorphism too.Let κ n denote the kernel of F n η −→ τ n . Then we get an exact sequence of pointed simplicialsets 1 → Hom( τ n , K ) → Hom( F n , K ) → Hom( κ n , K ) . Since geometrical realization preserves equalizers, we get an exact sequence of pointed spaces1 → |
Hom( τ n , K ) | → | Hom( F n , K ) | → | Hom( κ n , K ) | . We also have another exact sequence of pointed spaces1 → Hom( τ n , | K | ) → Hom( F n , | K | ) → Hom( κ n , | K | ) . By naturality of φ , we get a commutative diagram of exact sequences1 | Hom( τ n , K ) | | Hom( F n , K ) | | Hom( κ n , K ) | τ n , | K | ) Hom( F n , | K | ) Hom( κ n , | K | ) . φ τ n φ F n φ κ n Since φ F n is an isomorphism, so is φ τ n , which is what we wanted to prove. (cid:3) Theorem 5.4.
There is a natural homotopy equivalence B ( τ, | K | ) → | W ( τ, K ) | . Proof.
Recall that we have a weak equivalence dY → T Y for any bisimplicial set Y . Weapply this to the bisimplicial set N ( τ, K ), and use the identification W ( τ, K ) ∼ = T N ( τ, K ).This gives a homotopy equivalence | dN ( τ, K ) | → | W ( τ, K ) | fter realization. It remains to identify | dN ( τ, K ) | with B ( τ, | K | ). Using Lemma 5.3 andthe homeomorphism between different ways of realizing a bisimplicial space we obtain | dN ( τ, K ) | ∼ = | [ p ]
7→ | [ q ] Hom( τ F p , K q ) ||∼ = | [ p ] Hom( τ F p , | K | ) | = B ( τ, | K | ) . (cid:3) Compact Lie groups.
In Theorem 5.4 we started from a simplicial group K and com-pared the simplicial and topological constructions. Conversely we can do such a comparisonfor a topological group G . However, for such a comparison to work we need to restrict ourattention to a nice class of groups such as compact Lie groups. Remark 5.5.
The main reason for this restriction is that the geometric realization functordoes not respect homotopy equivalences in general. A better behaving realization functor isthe fat realization which is obtained by forgetting the degeneracies when gluing the simplices.For a simplicial space X its fat realization is denoted by || X || . If the simplicial space is good ,i.e. all degeneracy maps s i : X n − → X n are closed cofibrations in the sense of Hurewicz,then the natural map || X || → | X | is a homotopy equivalence. Any simplicial set is a goodsimplicial space, or more generally the simplicial space [ n ]
7→ | S ( X n ) | is good where X isan arbitrary simplicial space. If G is a compact Lie group then [ n ] Hom( τ n , G ) is good.Thus up to homotopy we can replace geometric realization by the fat realization in theconstruction of B ( τ, G ), see [OW19]. This property will be crucial in our consideration.We need a version of Lemma 5.3 for the singular functor. Lemma 5.6.
Let H be a discrete group, and G a topological group. Then there is a naturalisomorphism of simplicial sets S ( Hom ( H, G )) → Hom ( H, S ( G ) • ) . Proof.
I) Since geometrical realization of simplicial sets preserves direct products, the ad-junction ( | | , S ) gives a natural isomorphism
Top (∆ m × H, G ) ∼ = −→ sSet (∆[ m ] × H, SG ) . II) By construction, a continuous map ∆ m × H f −→ G gives a continuous map ∆ m → Hom(
H, G ) if and only if for all points x ∈ ∆ m , the restriction H f |{ x }× H −−−−−→ G is a ho-momorphism. This is equivalent to the commutativity of the diagram∆ m × H × H ∆ m × H × ∆ m × H G × G ∆ m × H G, g id ∆ m × m H f × f m G f where g is constructed using the diagonal map ∆ m → ∆ m × ∆ m and the shuffle map ∆ m × H → H × ∆ m , and m H , m G are multiplication maps.III) By construction, a map of simplicial sets ∆[ m ] × H φ −→ SG gives a map of simplicialsets ∆[ m ] → Hom(
H, SG ) if and only if the restriction H φ |{ id ∆[ m ] }× H −−−−−−−−→ SG to the unique ondegenerate m -simplex id ∆[ m ] ∈ ∆[ m ] m is a homomorphism. This is equivalent to thecommutativity of the diagram∆[ m ] × H × H ∆[ m ] × H × ∆[ m ] × H SG × SG ∆[ m ] × H SG, ψ id ∆[ m ] × m H φ × φ m SG φ where ψ is constructed using the diagonal map ∆[ m ] → ∆[ m ] × ∆[ m ] and the shuffle map∆[ m ] × H → H × ∆[ m ], and m H , m SG are multiplication maps.IV) Let ∆ m × H f −→ G be a continuous map which gives an m -simplex of the simplicial set S Hom(
H, G ). Then the adjoint ∆[ m ] × H φ −→ SG can be given as the composite∆[ m ] × H u ∆[ m ] × id H −−−−−−→ S ∆ m × H Sf −→ SG, where ∆[ m ] u ∆[ m ] −−−→ S | ∆[ m ] | = S ∆ m is the unit map. Applying the limit-preserving functor S to the commutative diagram in II) and precomposing with unit maps we get the commutativediagram∆[ m ] × H × H ∆[ m ] × H × ∆[ m ] × HS ∆ m × H × H S ∆ m × H × S ∆ m × H SG × SG ∆[ m ] × H S ∆ m × H SG, ψu ∆[ m ] × id H × H id ∆[ m ] × m h φ × φu ∆[ m ] × id H × u ∆[ m ] × id H Sg id ∆ m × m H Sf × Sfu ∆[ m ] × id H m SG Sf showing that the adjoint φ gives an m -simplex of the simplicial set Hom( H, SG ).V) Let ∆[ m ] × H φ −→ SG be a map of simplicial sets which gives an m -simplex of thesimplicial set Hom( H, SG ). Then the adjoint ∆ m × H f −→ G can be given as the composite∆ m × H = | ∆[ m ] | × H | φ | −→ | SG | v G −→ G, where | SG | v G −→ G is the counit map. Applying the direct product-preserving functor | | to thecommutative diagram in III) and postcomposing with counit maps we get the commutativediagram∆ m × H × H ∆ m × H × ∆ m × H | SG | × | SG | G × G ∆ m × H | SG | G, | ψ | = g id ∆ m × m H | φ | × | φ | f × f v G × G m SG m G φ v G where the commutativity of the right square is implied by the naturality of the counit maps,thus showing that the adjoint f gives an m -simplex of the simplicial set S Hom(
H, G ). Corollary 5.7.
Let G be a compact Lie group. Then there is a natural zigzag of homotopyequivalences B ( τ, G ) ← B ( τ, | SG | ) → | W ( τ, SG ) | . Proof.
By Remark 5.5 we can use fat realization at any time. Lemma 5.3, Lemma 5.6,and the canonical homotopy equivalence | SX | → X for a topological space X gives us thediagram | S Hom( τ F n , G ) | Hom( τ F n , G ) | Hom( τ F n , ( SG ) • ) | Hom( τ F n , | SG | ) ∼∼ = ∼ = Thus we obtain a level-wise homotopy equivalence, which after realization, gives a homotopyequivalence B ( τ, | SG | ) → B ( τ, G ) . (5.3.1)The other homotopy equivalence is provided by Theorem 5.4. (cid:3) Remark 5.8.
In general, there is no reason to expect that a homomorphism G → G ′ which is a homotopy equivalence induces a homotopy equivalence B ( τ, G ) → B ( τ, G ′ ). Animportant exception is the inclusion K ⊂ G of a maximal compact subgroup K in a complex(or real) reductive algebraic group G . In this case the map Hom( Z n , K ) → Hom( Z n , G )is a homotopy equivalence [PS13]. This implies that B ( Z , K ) → B ( Z , G ) is a homotopyequivalence [AG15]. However, the homotopy equivalence between the homomorphism spacesfails when Z n is replaced by an arbitrary finitely generated group, see [PS13] for an example.Another example where B ( τ, − ) respects a homotopy equivalence is given in the proof ofCorollary 5.7. The canonical map | SG | → G induces the homotopy equivalence 5.3.1. Example 5.9.
We list some of the known results on the homotopy type of B ( τ, G ) where G is a topological group. We can use them to deduce results about W ( τ, K ). We focus onthe commutative case τ • = Z • .It is better to consider discrete and continuous cases separately. Let G be discrete fornow. Then | W ( τ, G ) | = B ( τ, G ) by definition. Most of the known results come from thefinite case.(1) Suppose that G is a finite transitively commutative (TC) group, i.e. the commutatordefines a transitive relation for elements outside the center of the group. For suchgroups B ( Z , G ) is homotopy equivalent to the classifying space of π B ( Z , G ), asshown in [ACTG12]. Thus it is an Eilenberg–Maclane space of type K ( π, on 3 letters is a TC group.(2) Let G be an extraspecial p -group of order p n +1 where p is a prime. If n ≥ B ( Z , G ) is a wedge of equidimensional spheres S n [Oka18].Thus there are non-zero higher homotopy groups. A similar phenomenon occurs forthe general linear groups GL n ( F q ), n ≤
4, over a field of characteristic p , and thesymmetric groups Σ k on k letters where k ≥ [Oka15].Next we consider Lie groups. Let G be a compact Lie group. Using Corollary 5.7, we cantranslate the results to W ( Z , SG ).
1) The rational cohomology of B ( Z , G ) is computed in [AG15]. These calculationsapply to the classical groups U ( n ), SU ( n ) and Sp ( n ) to describe the ring structureexplicitly. Not much is known integrally. The integral cohomology rings of SU (2), U (2) and O (2) are calculated in [ACGV17]. However, a lot more is known stably. Forthe stable groups U , O , Sp the space B ( τ, G ) gives rise to generalized cohomologytheories [AGLT17, Vil17]. In the commutative case its homotopy type is describedin [GH19].(2) At a prime ℓ the homotopy type of the ℓ -completion of B ( Z , G ) in the sense ofBousfield–Kan depends only on the mod- ℓ homology of BG when π G is a finite ℓ -group [OW19]. When ℓ is odd this gives a mod- ℓ homology isomorphism B ( Z , Sp ( n )) → B ( Z , SO (2 n + 1)) . Topological bundles with τ -structure. We will generalize the definition of a transi-tionally commutative bundle introduced in [AG15]. Let π : P → X be a principal G -bundleof topological spaces. Assume that { U i } i ∈ I is a trivialization of π . Let U = ` i U i andconsider q : U → X induced by the inclusions. The ˇCech nerve of q is defined to be thesimplicial space ˇ C ( U ) n = Y ≤ i ≤···≤ i n U i ∩ · · · ∩ U i n with the obvious simplicial structure maps. The (topological) transition functions ρ ij : U i ∩ U j → G induce a map of simplicial spaces ρ U : ˇ C ( U ) • → BG • sending x ∈ U i ∩ · · · ∩ U i n to the tuple ( ρ i ,i ( x ) , · · · , ρ i n − ,i n ( x )) in simplicial degree n . Definition 5.10.
A trivializing cover { U i } i ∈ I of a principal G -bundle π : P → X is saidto have a τ -structure if ρ U factors through the simplicial space B ( τ, G ) • . We say that theprincipal bundle π has a τ -structure if there exists a trivializing cover with a τ -structure.A τ -structure for π is a choice of a lift of [ f ], where f : X → BG is a classifying mapfor the principal bundle, under the map [ X, B ( τ, G )] → [ X, BG ] induced by the inclusion B ( τ, G ) ⊂ BG . Remark 5.11.
Note that when τ • = Z • having a Z -structure is the same as being a tran-sitionally commutative cover in the sense of [AG15]. A Z -structure is the same as a transi-tionally commutative structure defined in [Gri17, Definition 1.1.6].The definition of a τ -structure for a principal bundle is motivated by the following theorem.Originally the statement involves transitionally commutative bundles. We state it moregenerally for principal bundles with a τ -structure, its proof is completely analogous. Theorem 5.12. [AG15, Theorem 2.2]
Let G be a Lie group and X be a CW complex. Aprincipal G -bundle π : P → X has a τ -structure if and only if the classifying map f : X → BG factors, up to homotopy, through B ( τ, G ) . Remark 5.13.
In the original statement of Theorem 5.12 X is required to be a finite CWcomplex. However, with a little more work the techniques of the proof can be improved toremove the finiteness condition [G´om]. he set H τ ( X, G ) also makes sense in
Top and defined using the τ -concordance relationwhose definition is completely analogous to the simplicial version given in Definition 4.12.Note that a τ -structure for a principal G -bundle in the topological context is a choice of alift of the classifying map under the natural map [ X, B ( τ, G )] → [ X, BG ]. Theorem 5.12says that the classifying map factors through B ( τ, G ) if and only if there exist a τ -structure.Therefore we can identify H τ ( X, G ) with the set [
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Department of Physics and Astronomy, University of British Columbia, Vancouver BCV6T 1Z4, Canada
E-mail address : [email protected]/[email protected] Alfr´ed R´enyi Institute of Mathematics, Re´altanoda street 13-15, H-1053, Budapest, Hun-gary
E-mail address : [email protected]@renyi.hu