aa r X i v : . [ m a t h . A T ] S e p Segal Group Actions
Matan PrasmaOctober 9, 2018
Abstract
We define a model category structure on a slice category of simplicialspaces, called the "Segal group action" structure, whose fibrant-cofibrantobjects may be viewed as representing spaces X with an action of a fixedSegal group (i.e. a group-like, reduced Segal space). We show that thismodel structure is Quillen equivalent to the projective model structure on G -spaces, S B G , where G is a simplicial group corresponding to the Segalgroup. One advantage of this model is that if we start with an ordinarygroup action X ∈ S B G and apply a weakly monoidal functor of spaces L : S −→ S (such as localization or completion) on each simplicial degreeof its associated Segal group action, we get a new Segal group action of LG on LX which can then be rigidified via the above-mentioned Quillenequivalence. Contents
The development and use of homotopy-coherent versions of classical notionsis by now widespread in several parts of mathematics. It is often beneficialto augment an "up-to-homotopy" notion with a "rigidification" procedure that1ompares it back to a classical (often enriched) notion. Such a comparison isuseful since one can use results that were proven for classical notions in or-der to establish properties of their homotopy-coherent counter-parts which areusually harder to manage. Let us demonstrate this by the following example.For a simplicial group G , the methods of higher category theory enable one tohave a flexible model of a group action by simply considering the ∞ -categoryof ∞ -functors Fun ( N ( B G ) , N ( S o )) from the homotopy coherent nerve of B G to the homotopy coherent nerve of the category of spaces. This ∞ -categorycan be thought of as spaces with a "group action" up to coherent homotopy,and moreover, comes with a rigidification functor [Lur09, Proposition 5.1.1.1]to (ordinary) G -spaces Fun ( N ( B G ) , N ( S o )) −→ S B G .We can use this rigidification to show that Example 1.1.
The Moore space construction M ( − , n ) cannot be lifted to an ∞ -functor M ( − , n ) : N ( A b ) −→ N ( S o ) .Proof. If there was a functor M ( − , n ) : A b −→ S , it would induce, for everygroup G , an "equivariant Moore space" functor M G ( − , n ) : A b B G −→ S B G but [Car] shows that there are (discrete) groups G (e.g. all non-cyclic groups)for which such a functor cannot exist. Similarly, if there was an ∞ -functor M ( − , n ) : N ( A b ) −→ N ( S o ) ,it would induce, for any discrete group G , an ∞ -functor Fun ( N ( B G ) , N ( A b )) −→ Fun ( N ( B G ) , N ( S o )) .But the latter may be rigidified to an ordinary functor A b G −→ S B G which cannot exist by [Car].The purpose of this work is to provide a point-set model for coherent groupactions and to establish a rigidification procedure for them. We will provide amodel-categorical framework for an existing notion, defined and studied in[Pre] under the name "homotopy action" and which will be referred to here as Segal group action . A Segal group action aims to encode a coherent action ofa loop space Ω Y (together with its coherent homotopies) on a space X . Moreprecisely, such an ’action’ is a map of simplicial spaces π : A • → B • in which2 ≃ X , the codomain B • is a Segal group representing Ω Y , i.e B • is a group-like reduced Segal space with Y ≃ | B • | , and certain ‘Segal-like’ maps A n −→ A × hB B n are weak equivalences.The definition we give here for a Segal group action is simpler than [Pre,Defintion 5.1] and our first concern is to show that these two definitions coin-cide. Then, given a Segal group B • , we shall construct a model category struc-ture on the slice category s S / B • whose fibrant-cofibrant objects are preciselythe Segal group actions. Using the diagonal functor d ∗ : s S → S we can alsoconsider the canonical model structure on S / d ∗ B • , induced by slicing under theKan-Quillen model structure. For B • as above, we further show that d ∗ : s S / B • / / S / d ∗ B • : d ∗⊥ o o constitutes a Quillen equivalence between these two model structures. By com-posing with a Quillen equivalence induced by a "rigidification map" and theQuillen equivalence of [DFK], the above-mentioned equivalence shows thatthe Segal group action model structure is Quillen equivalent to the projectivemodel structure on S B G (where G is a simplicial group satisfying BG ≃ d ∗ B • ),in which weak equivalences (resp. fibrations) are the maps whose underlyingmap of spaces is a weak equivalence (resp. fibration).One technical advantage of Segal group actions is their invariance undera weakly monoidal endofunctor of spaces, namely, functors L : S → S whichpreserve weak equivalences, contractible objects and finite products up to equiv-alence. Key examples of such functors are localization by a map, p -completiona la Bousfield-Kan, and the derived mapping space map h S ( C , − ) . Applying aweakly monoidal functor L : S → S on each simplicial degree of a Segal groupaction A • −→ B • yields a new Segal group action LA • → LB • , now thoughtof as a coherent action LB on the space LA . This invariance property can beapplied (see 5.1.1) to obtain a Postnikov tower for a G -space X , composed outof the P n X , but viewed as P n G -spaces. Related work
This work complements the treatment of the notion of "homotopy action" whichwas developed in [Pre]. In the meantime, two related works came out. The firstis the work of [NSS], which develops, in the context of an ∞ -topos, what theycalled "principal ∞ -bundles". The treatment of Segal group actions here showsthat they constitute a model-categorical presentation of principal ∞ -bundles(see Corollary 3.4). The second related work was published recently as [ ? ].There the authors develop the notion of an " A ∞ -action" in an operadic mannerand thus provide a way to model an action of an ∞ -monoid on a space. Al-though the work in loc. cit. does not give a model-categorical framework, itdoes provide a rigidification result which resembles the one in this paper.3 Preliminaries (a) Throughout, a space will always mean a simplicial set. Let S (resp. S )be the category of simplicial sets (resp. reduced simplicial sets) and s S the category of simplicial spaces ; we shall denote an object of s S withvalues [ n ] X n by X • . We let c ∗ : S → s S be the functor which sends asimplicial set to a degree-wise discrete simplicial space. On the other hand,a simplicial set K may also be viewed as a constant simplicial space whichhas K in each degree; we shall denote the latter by K again.(b) The category s S is a simplicial category; for X , Y ∈ s S we let map s S ( X , Y ) ∈ S denote the mapping space . It has the property that for a simplicial set K ,and simplicial spaces X , Y , map s S ( K × X , Y ) ∼ = map S ( K , map s S ( X , Y )) where map S ( − , − ) is the mapping space of S . If ∆ n ∈ S is the standard n -simplex, then, by the Yoneda lemma for bisimplicial sets, c ∗ ∆ n gives riseto the n -th space functor in that map s S ( c ∗ ∆ n , X ) ∼ = X n .(c) The category s S is also cartesian closed; for X , Y ∈ s S there is an internal-hom object Y X ∈ s S with the propertys S ( X × Y , Z ) ∼ = s S ( X , Z Y ) .A routine check shows that for a space K and a simplicial space X , the twopossible meanings for X K coincide.(d) By a model category structure we mean a bicomplete category satisfyingQuillen’s axioms [Qui] and having functorial factorizations .(e) We let s S Reedy denote the Reedy model structure on simplicial spaces (see[Ree]). This makes s S into a simplicial combinatorial model category, inwhich a map X → Y in s S is a Reedy fibration if for each n ≥ map s S ( c ∗ ∆ n , Y ) −→ map s S ( c ∗ ∆ n , X ) × map s S ( c ∗ ∂ ∆ n , X ) map s S ( c ∗ ∂ ∆ n , Y ) is a Kan fibration.(f) It is well-known that the Reedy and the injective model structures on s S coincide (see [GJ, IV.3, Theorem 3.8]) so that every object of s S Reedy is Reedycofibrant. On the other hand, every Reedy fibrant object in s S has a Kancomplex in each simplicial degree, with face maps being Kan fibrations.For a simplicial space B • ∈ s S , the Reedy model structure s S Reedy induces a4simplicial, combinatorial) model structure, denoted ( s S / B • ) Reedy , of whichall objects are cofibrant and the fibrant objects are precisely Reedy fibra-tions A • ։ B • . If furthermore B • was Reedy fibrant, then the domain A • of such a fibrant object is also Reedy fibrant.(g) Similarly, for a fixed space B , the Kan-Quillen model structure S KQ inducesa (simplicial, combinatorial) model structure on the slice category, denoted ( S / B ) KQ , of which all objects are cofibrant and the fibrant objects are pre-cisely Kan fibrations A ։ B . As before, if B was a Kan complex, it followsthat for every fibrant object A ։ B , the domain A is a Kan complex.(h) The diagonal functor d ∗ : s S → S (induced by d : ∆ → ∆ × ∆ ) is partof an adjoint triple d ! ⊣ d ∗ ⊣ d ∗ (left adjoints on the left). The functor d ∗ : S → s S is given by d ∗ ( A ) • = A ∆ • (i.e. d ∗ ( A ) n = A ∆ n ) and the functor d ! : S → s S is defined by extending the formula d ! ( ∆ n ) = ∆ n , n via colimits(here ∆ n , n is the representable presheaf on ([ n ] , [ n ]) ). These adjunctions arecompatible with the simplicial enrichments on s S and S mentioned above.(i) There is an adjunction (see [Kan]) G : S / / sGp : B ⊥ o o (2.0.1)where B is the classifying space functor (often denoted by W ) and G isthe Kan loop group. Furthermore, since the pair 2.0.1 is in fact a Quillenequivalence, all objects in S are cofibrant and all objects in sGp are fibrant,it follows that the unit map of this adjunction K −→ B G K is a weak equiva-lence. The category S is a reflective subcategory of S , with the left adjointof the pair d ( − ) : S / / S : ι ⊥ o o defined by identifying all the 0-simplicies. For a connected space K , theunit map K → b K is a weak equivalence, and we shall refer to the com-posite of these equivalences ρ : K → b K → B G b K as the rigidification map. The counterpart of the rigidification map relates the loop functor Ω : = map ∗ ( S , − ) : S −→ S to the Kan loop group.(j) For every Kan complex K ∈ S one has a weak equivalence Ω K ∼ → G K ofsimplicial sets. Thus, we define an ∞ -group to be a triple ( G , B G , η ) where G is a space, B G is a pointed connected space and η : G ≃ −→ Ω B G is a weakequivalence. We will often refer to G itself as an ∞ -group when B G and η are clear from the context.We say that the composite G ≃ −→ Ω B G ≃ −→ Ω d B G ≃ −→ G d B G rigidifies G into a simplicial group.(k) A Segal space is a Reedy fibrant simplicial space B • such that for each Notice the slight deviation from the original definition in [Seg] ≥ B n −→ holim ( B d −→ B d ←− · · · d −→ B d ←− B ) ≃ B × B B × B · · · × B B ( n times ) ,(induced by the maps p i : [ ] −→ [ n ] i −
1, 1 i ( ≤ i ≤ n ) )are weak equivalences.A Segal space B • is called a Segal groupoid (or: group-like) if the map ( d , d ) : B −→ holim ( B d −→ B d ←− B ) ≃ B × B B is a weak equivalence. If furthermore B ≃ ∗ we shall say that B • is a Segalgroup .(l) G. Segal essentially showed [Seg] that one can present any ∞ -group G asa Segal group. More precisely, he showed that if B • is a Segal group, thecanonical map B −→ Ω ( d ∗ B • ) is a weak equivalence. Given an ∞ -group G , a Segal group for G is a Segal group B • together with an equivalence G ∼ −→ B .(m) A homotopy fiber sequence is a sequence of spaces X −→ Y −→ Z hav-ing a null-homotopic composite and such that the associated map to thehomotopy fiber X → F h ( Y → Z ) is a weak equivalence.(n) For a simplicial group G , we denote by B G the simplicial groupoid withone object associated to G . We can then consider the category of simplicialfunctors S B G and we shall refer to an object X ∈ S B G as a G -space . Weshall refer to the projective model structure on the category of G -spaces asthe Borel model structure and denote it by (cid:16) S B G (cid:17) Borel . In other words,this model structure has as weak equivalences (resp. fibrations) the G -maps X → X ′ which are weak equivalences (resp. fibrations) in S KQ . Thecofibrant objects of ( S B G ) Borel are precisely the spaces with a free G -action.Thus, given X ∈ S B G , a model for its cofibrant replacement is X × EG (where EG : = WG is the free contractible G -space) and the homotopy quo-tient X // G : = X × G EG may be viewed as the right derived functor of the quotient ( − ) / G : ( S B G ) Borel −→ ( S / BG ) KQ .Every G -space X gives rise to the Borel (homotopy) fiber sequence X −→ X // G −→ BG and conversely, any (homotopy) fiber sequence of the form X −→ A −→ BG is equivalent to some Borel fibration. More concisely:6 heorem 2.1. [DFK] There is a Quillen equivalence ( − ) × BG ∗ : ( S / BG ) KQ / / ( S B G ) Borel : ( − ) / G . ⊥ o o Of course, in order to get hands-on calculations, it is useful to have a presen-tation of the ∞ -category Fun ( N ( B G ) , N ( S o )) as a model category. One suchmodel is the Borel model structure on S B G and another is the slice model struc-ture S / BG (see 2.1). The advantage of the first is that it gives a direct access tothe group and the space on which it acts but its disadvantage is that one cannotwork with a "flexible" model of the group, e.g., Ω BG nor of the space. In thesecond model the roles switch in that one may take any space of the homo-topy type of BG but there is no direct access to the group G nor to the space onwhich it acts (which can only be obtained after taking homotopy fiber). As weshall see below, Segal group actions, have, to certain extent, the advantages ofboth of the models above, since on the one hand a Segal group is a "flexible"model for a simplicial group, and on the other hand, Segal group actions havethe homotopy types of the group G and the space on which it (coherently) actsas part of their initial data. We will make use of this advantage to obtain aninvariance property of Segal group actions under weak monoidal functors (see5.3).We now come to the main notion of this work. Let α , α n : [ ] −→ [ n ] be the maps defined by 0 n respectively. Alternatively, α = d n d n − · · · d and α n = d · · · d . Definition 3.1. A Segal group action is a Reedy fibration of simplicial spaces π : A • −→ B • such that:1. B • is a Segal group;2. for every n, the map A n ( α ∗ , π n ) / / A × B B n is a weak equivalence.In this case, we say that the Segal group B • acts on A • , or that the ∞ -group ( B , | B • | , η : B ≃ −→ Ω | B • | ) acts on A . Remark 3.2.
One technical advantage of the Reedy fibrancy condition of 3.1 is that thestructure maps are fibrations. This means that the ordinary notions of fibers, sectionsetcetara for these maps are homotopy invariant.
The origin of Definition 3.1 is [Pre, Definition 5.1] where it was called ho-motopy action . However, the reader may wonder about a difference betweenDefinition 3.1 and [Pre, Definition 5.1]. Namely, the definition we give heredoes not include the condition that the map A n ( α ∗ n , π n ) / / A × B B n
7s a weak equivalence. We will now show that this additional condition is im-plied by the conditions of Definition 3.1 and is thus redundant. This was kindlypointed-out to us by Thomas Nikolaus. The proof we give here is independent.
Proposition 3.3.
Let π : A • −→ B • be a Segal group action. Then A • is a Segalgroupoid and the map A n ( α ∗ n , π n ) / / A × B B n is a weak equivalence.Proof. Note that our fibrancy assumption implies that the map A n ( α ∗ n , π n ) / / A × B B n is a weak equivalence if and only if the square A n / / α ∗ n (cid:15) (cid:15) B n (cid:15) (cid:15) A / / B is homotopy cartesian.Consider the following commutative cube A d / / d (cid:15) (cid:15) π ❆❆❆ ❆❆❆ A π ❇❇❇ ❇❇❇ (cid:15) (cid:15) B d / / d (cid:15) (cid:15) B d (cid:15) (cid:15) A / / π ❆❆❆ ❆❆❆ A π ❇❇❇ ❇❇❇ B d / / B . (3.0.2)Since B • is a Segal-group and in particular group-like, the outer face is homo-topy cartesian. Consider A π / / d (cid:15) (cid:15) B d (cid:15) (cid:15) A π / / d (cid:15) (cid:15) B d (cid:15) (cid:15) A / / B .8ince π : A • −→ B • is a Segal group action, the lower square is homotopycartesian, and since d d = d d the outer rectangle is homotopy cartesian. Itfollows that the upper square is homotopy cartesian; this square is the left-hand face of the cube 3.0.2. Consider the following commutative diagram ofsolid arrows. F / / d ∗ ✤✤✤ (cid:15) (cid:15) ✤✤✤ α ❅❅❅ ≃ ❅❅❅ A π / / ✤✤✤ d (cid:15) (cid:15) ✤✤ ( d , π ) ❍❍❍ ❍❍❍ B ❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅ d (cid:15) (cid:15) A / / A × B pr / / pr (cid:15) (cid:15) B (cid:15) (cid:15) F / / β ❆❆❆ ≃ ❆❆❆ s ∗ G G A π / / ❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍ s G G B ❅❅❅❅❅❅❅❅ A / / A / / ∗ Here, F and F are fibers of π and π (we assume a base-point in B waschosen), the maps s ∗ and d ∗ are the ones induces by s and d and β is themap induced between the fibers. Since B is contractible, β is an equivalenceand it follows from 2-out-of-3 that s ∗ is an equivalence. Since d ∗ s ∗ = id itfollows that d ∗ is an equivalence, which means that the lower face of 3.0.2 ishomotopy cartesian. We now deduce that all the faces of the cube 3.0.2 arehomotopy cartesian and in particular, cartesianess of the inner face means that A • satisfies the group-like condition.Consider now the following commutative cube. A d / / d (cid:15) (cid:15) π ❆❆❆ ❆❆❆ A π ❆❆❆ ❆❆❆ (cid:15) (cid:15) B d / / d (cid:15) (cid:15) B d (cid:15) (cid:15) A / / π ❆❆❆ ❆❆❆ A π ❆❆❆ ❆❆❆ B d / / B (3.0.3)The outer face is homotopy cartesian since B • is a Segal space and the right-hand face is homotopy cartesian since π : A • −→ B • is a Segal group action.We showed that the lower face is homotopy cartesian and it follows that all thefaces of 3.0.3 are homotopy cartesian. In particular, cartesianess of the innerface means that the Segal map for n = A • is a Segal groupoid. The homotopy cartesianess ofthe upper and right-hand faces of the cube 3.0.2 means that the map A n ( α ∗ n , π n ) / / A × B B n is a weak equivalence for n =
1, 2 and a similar argument shows this holds forany n ≥ ∞ -categorical notion of a "group action", as was de-fined in [NSS]. Corollary 3.4.
Let π : A • −→ B • be a Segal group action, viewed as a functor ∆ op −→ s S [ ] . Then the underlying ∞ -functor of π is a group action in the sense of[NSS, Definition 3.1]. Example 3.5.
Let G be a simplicial group and X a (right) G-space. The Bar con-struction [May, §7] provides, up to a Reedy fibrant replacement, a Segal group actionBar • ( X , G ) −→ Bar • ( G ) . The mapsX × G n ( α ∗ n , π n ) / / ( α ∗ , π n ) / / X × G n are given by the identity and ( x , g , ..., g n ) ( xg · · · g n , g , ..., g n ) (respectively). For an ∞ -group G together with a fixed choice of a Segal group B • for G ,we can thus consider the full subcategory of s S / B • spanned by the Segal groupactions. This category in meant to give a ’soft’ model for G -actions where G issome simplicial group with BG ≃ B G . Throughout, we fix a Segal group B • . Definition 4.1.
The
Segal group action model structure, ( s S / B • ) SegAc is the leftBousfield localization of ( s S / B • ) Reedy with respect to the mapsc ∗ ∆ " " ❋❋❋❋ α ∗ / / c ∗ ∆ n σ { { ①①①① B • defined for all pairs ( n , σ ) where n ≥ and σ : c ∗ ∆ n −→ B • . roposition 4.2. The fibrant-cofibrant objects of ( s S / B • ) SegAc are precisely the Segalgroup actions.Proof. In ( s S / B • ) Reedy all objects are cofibrant and since left Bousfield localiza-tions do not change the class of cofibrations, all objects in ( s S / B • ) SegAc are cofi-brant.An object π : A • −→ B • is fibrant if and only if it is local with respect to themaps of definition 4.1. Unwinding the definitions, we see thatmap / B • ( c ∗ ∆ n , A • ) ∼ −→ map / B • ( c ∗ ∆ , A • ) ⇔ Fib ( A n π n −→ B n ) ∼ −→ Fib ( A π −→ B ) . (4.0.4)This in turn is the map of associated fibers on vertical arrows in the square A n / / (cid:15) (cid:15) A (cid:15) (cid:15) B n / / B (4.0.5)with the horizontal maps being α ∗ . The equivalence 4.0.4 is precisely the ho-motopy cartesianess of 4.0.5, which in turn is just the condition that A n ( α ∗ , π n ) / / A × B B n are equivalences. Hence, A • −→ B • is a Segal group action.Recall that our goal is to compare the Segal group action model structureto the Borel model model structure. In light of 2.1, we would like to compare ( s S / B • ) SegAc to ( S / d ∗ B • ) KQ . Before that, it is worth verifying that the latter indeedmodels the Borel homotopy theory: Proposition 4.3.
The rigidification map ρ : d ∗ B • −→ BG (§2i) induces a Quillenequivalence ρ ∗ : ( S / d ∗ B • ) KQ / / ( S / BG ) KQ : ρ ! ⊥ o o Proof.
The space d ∗ B • is a 0-connected Kan complex so that ρ is a weak equiv-alence between fibrant-cofibrant objects.That settled, we recall a standard Observation 4.4.
Let C , D be categories andF : C / / D : U ⊥ o o an adjoint pair. Then for every object c ∈ C there is an induced adjunction on slicecategories F c : C / c / / D / Fc : U c ⊥ o o where U c is defined by applying U and then pulling back along the unit ⇒ UF. d ∗ : s S / B • / / S / d ∗ B • : d ∗ . ⊥ o o We are now at a state to formulate the main assertion of this paper:
Theorem 4.5.
Let B • be a Segal group. The adjoint paird ∗ : ( s S / B • ) SegAc / / ( S / d ∗ B • ) KQ : d ∗ . ⊥ o o (4.0.6) is a Quillen equivalence. We begin with a
Proposition 4.6.
If B • is a Segal group, the space d ∗ B • is a Kan complex.Proof. A simplicial space B • satisfying the extension condition with respect tothe maps of degree-wise discrete simplicial spaces c ∗ Λ ni −→ c ∗ ∆ n , for 0 ≤ i ≤ n , is fibrant in the diagonal model structure of [Jar, Corollary 1.6]. Moreover,the realization | B • | of such a simplicial space is a Kan complex by [Jar, Theorem2.14]. Since B • is a Segal space, it satisfies the extension condition with respectto c ∗ Λ ni −→ c ∗ ∆ n for 0 < i < n and since B • is group-like, it satisfies theextension condition with respect to c ∗ Λ ni −→ c ∗ ∆ n for i = n and i = n .The following is a well-known result, that can be deduced, for example,from [RSS, Theorem 5.2]. Proposition 4.7.
The adjunction d ∗ ⊣ d ∗ is a Quillen paird ∗ : s S Reedy / / S KQ : d ∗ . ⊥ o o Corollary 4.8.
The Quillen pair of proposition 4.7 induces a Quillen pair on slicemodel categories d ∗ : ( s S / B • ) Reedy / / ( S / d ∗ B • ) KQ : d ∗ . ⊥ o o We would like to use Corollary 4.8 as a stepping stone in order to provethat 4.0.6 is indeed a Quillen pair. For this, we use the simplicial structure asfollows.
Lemma 4.9.
Let F : M / / N : U ⊥ o o be an adjoint pair of simplicial model categories in which all objects of M are cofibrant.Then F ⊣ U is a Quillen pair if and only if F preserves cofibrations and U preservesfibrant objects.
Since left Bousfield localization does not change the class of cofibrations, itis clear that d ∗ : ( s S / B • ) SegAc −→ ( S / d ∗ B • ) KQ preserves cofibrations.12 roposition 4.10. For a Segal group B • the functord ∗ : ( S / d ∗ B • ) KQ −→ ( s S / B • ) SegAc preserves fibrant objects.
The proof of Proposition 4.10 relies on a folklore result which we addressfirst.
Lemma 4.11.
For a simplicial group G and a co-span of G-spaces X −→ Y ←− Z,the map ( X × hY Z ) // G −→ X // G × hY // G Z // Gis a weak equivalence.Proof.
We have a map of (homotopy) fiber sequences X × hY Z / / (cid:15) (cid:15) ( X × hY Z ) // G / / (cid:15) (cid:15) BGF h ( p ) / / X // G × hY // G Z // G p / / BG and it is thus enough to show that X × hY Z −→ F h ( p ) is a weak equivalence.Consider the 3 × X // G / / (cid:15) (cid:15) BG (cid:15) (cid:15) ∗ (cid:15) (cid:15) o o Y // G / / BG ∗ o o Z // G O O / / BG O O ∗ o o O O Taking homotopy limits of all rows and then of the resulting column, gives X × hY Z and taking homotopy limits of all columns and then of the resultingrow, gives F h ( p ) . The result now follows from commutation of homotopy lim-its. of 4.10. By Ken Brown’s lemma, d ∗ : ( S / d ∗ B • ) KQ −→ ( s S / B • ) Reedy preserves fibrant objects and it is thus left to verify that for a fibrant object A ։ d ∗ B • ∈ ( S / d ∗ B • ) KQ the map d ∗ ( A ։ d ∗ B • ) satisfies condition ( ) of definition 3.1. Let P • ∈ s S bethe domain of d ∗ ( A ։ d ∗ B • ) . The n -th level of P n is given by the pullback P n p / / π n (cid:15) (cid:15) A ∆ n (cid:15) (cid:15) (cid:15) (cid:15) B n / / ( d ∗ B • ) ∆ n . (4.0.7)13ince B • is a Segal group, d ∗ B • is a connected Kan complex. Thus, the rigidifi-cation map described in §2.j gives an equivalence d ∗ B • ≃ −→ BG for a simplicial group G and we let X be the homotopy fiber F h ( A −→ BG ) . ByTheorem 2.1, we have X // G ≃ A so that the square 4.0.7 is equivalent to X // G × BG G n / / (cid:15) (cid:15) X // G (cid:15) (cid:15) G n / / BG .This in turn may by rewritten as X // G × hEG // G G n + // G / / (cid:15) (cid:15) X // G (cid:15) (cid:15) G n / / BG where G acts on G n + via the inclusion to the last coordinate G −→ G n + . ByLemma 4.11 we can write P n ≃ X // G × EG // G G n + // G ≃ ( X × hEG G n + ) // G ≃ ( X × G n + ) // G ≃ X × G n (the last equivalence here is a straightforward identification) so that the equiv-alence is indeed induced by the projection maps ( π n , p ) . Since P ≃ X and theface maps d i of P n are defined via the above-mentioned pullbacks, it followsthat the maps of 3.1 (2) are weak equivalences. of 4.5. We shall show that the unit and counit maps, 1 ⇒ d ∗ d ∗ and d ∗ d ∗ ⇒ A ։ d ∗ B • be a fibrant object of S / d ∗ B • . As wesaw in 4.10, d ∗ ( A ։ d ∗ B • ) is a Segal group action P • −→ B • and thus has ineach simplicial degree F h ( P n −→ B n ) ≃ P ≃ F h ( A ։ d ∗ B • ) .Thus, by [Pup], F h ( d ∗ P • −→ d ∗ B • ) ≃ P and it follows by the five lemma that d ∗ d ∗ A −→ A is a weak equivalence over d ∗ B • . On the other hand, if we are given a Segal group action A • −→ B • , then F h ( d ∗ A • −→ d ∗ B • ) ≃ A and the proof of 4.10 shows that we have a weak equivalence of simplicialspaces A • ≃ d ∗ d ∗ A • ,which is compatible with the maps to B • and hence a weak equivalence ofSegal group actions. 14 orollary 4.12. There is a Quillen equivalence of simplicial combinatorial model cat-egories St : ( s S / B • ) SegAc / / ( S B G ) Borel : Un . ⊥ o o Proof.
One simply compose the Quillen equivalences of Theorem 2.1, Proposi-tion 4.3 and Theorem 4.5 as follows: ( s S / B • ) SegAc d ∗ / / ( S / d ∗ B • ) KQ ρ ∗ / / ⊥ d ∗ o o ( S / BG ) KQ ⊥ ρ ! o o ( − ) × BG ∗ / / ( S B G ) Borel . ⊥ ( − ) / G o o In algebraic topology one often applies constructions to spaces with a groupaction. Of course, for a G -space X , and an endofunctor of spaces L : S −→S there is no canonical group action on LX . However, many functors underconsideration admit additional properties such as the following. Definition 5.1.
An endofunctor L : S −→ S is said to be weakly monoidal if:1. L ( ∗ ) ∼ ∗ ;2. L preserves weak equivalences; and3. for any X , Y ∈ S the map L ( X × Y ) ∼ −→ LX × LY is a weak equivalence.
Example 5.2.
For convenience, we mention a few common weakly monoidal endo-functors of spaces:1. Any (homotopy) (co)localization functor in the sense of [Far]. These include then-th Postnikov piece and its dual, sometimes called the n-th Whitehead piece.2. The p-completion functor ( Z / p ) ∞ á la Bousfield-Kan.3. The (derived) mapping space functor from a fixed space map h S ( A , − ) . Any weak monoidal endofunctor L : S −→ S takes ∞ -groups to ∞ -groups.This is so since for any ∞ -group G we can construct a Segal group B • for G (i.e.with an equivalence B ∼ −→ G as ∞ -groups). Applying L on each simplicialdegree, we see that LB • becomes a Segal group for L G so that the latter isagain an ∞ -group. The same argument implies: Observation 5.3.
If A • −→ B • is a Segal group action and L : S −→ S is a weakmonoidal endofunctor of spaces, then (the Reedy fibrant replacement of) LA • −→ LB • is a Segal group action. X ∈ S B G and denote by A • ( X , G ) −→ B • ( G ) the Segal group action R Un ( X ) obtained from applying the total right derivedfunctor of 4.12 on X . The notation is meant to suggest the equivalent map ofsimplicial spaces Bar • ( X , G ) −→ Bar • ( G ) .For L : S −→ S weakly monoidal, denote by B L • ( G ) the Reedy fibrant replace-ment of LB • ( G ) which is a Segal group. Similarly, denote by A L • ( X , G ) −→ B L • ( G ) the Segal group action obtained from replacing LA • ( X , G ) −→ B L • ( G ) by a Reedy fibration. This is a fibrant-cofibrant object of s S / B L • ( G ) so that wecan apply St of Corollary 4.12 to obtain a space LX ∈ S B LG where LG is thesimplicial group obtained from applying the Kan loop group functor G on theconnected Kan complex d ∗ B L • ( G ) . Note that we have a weak equivalence LG ≃ −→ B L ( G ) ≃ −→ Ω d ∗ ( B L • ( G )) ≃ −→ G d ∗ ( B L • ( G )) = : LG (5.0.8)where the first map is obtained from the Reedy fibrant replacement, the secondis the map of § 2. l and the third is the "rigidification map" from §2. j. Moreover,the space LX is canonically equivalent to A L ( X , G ) which in turn is equivalentto LX . This should be viewed as endowing LX with a coherent action of the ∞ -group LG . Example 5.4.
Take L = P n , the Postnikov n-th piece functor, modeled bycosk n + ( Ex ∞ ( − )) . For X ∈ S B G we get an action of the simplicial group P n G onP n X. A natural question arising from our previous considerations is whether it ispossible to extend Example 5.4 to obtain an "equivariant Postnikov tower" forany group action. More specifically, denote Γ n : = P n G so that P n X becomes a Γ n -space. When we let n vary, the maps Γ n −→ Γ n − arising from P n G −→ P n − G are group maps, and we wish to obtain maps p n : P n X −→ P n − X and τ n : X −→ P n X ,16rising from P n X −→ P n − X and X −→ P n X which are Γ n − Γ n − -equivariant(here X arises from L = Ex ∞ ), thus giving rise to a tower... P n X p n (cid:15) (cid:15) ... (cid:15) (cid:15) P X p (cid:15) (cid:15) P X p (cid:15) (cid:15) X τ / / τ > > ⑥⑥⑥⑥⑥⑥⑥⑥ τ F F ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ τ n K K ✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗ P X . (5.1.1)In order to obtain such a tower, one needs to show that the straighteningconstructions done in this paper are functorial in an appropriate sense. How-ever the mere existence of the tower 5.1.1 is not satisfactory, and one would liketo know that it converges to the original G -space X . Moreover, if we insteadstart with a Segal group action, it is desirable to obtain a similar tower of Segalgroup actions and to show that these two towers are equivalent in the appro-priate sense. The difficulty in answering such a question is that this tower isnot a diagram in any category of G -spaces with a fixed group G . Rather, it is adiagram in the Grothendieck construction of the functor S B ( − ) : sGp −→ AdjCat(where AdjCat stands for categories and adjunctions) which associates to everysimplicial group G the category S B G of G -spaces and to a simplicial group mapthe extension-restriction adjunction. One is then lead to consider the homotopytheory of theGrothendieck construction Z G ∈ sGp S B G which takes into account the homotopy theory of the base sGp , and of eachof the fibers S B G . It is convenient to have a model structure that presents thehomotopy theory at hand but it is not clear a-priori that such a model structureexists.With these questions in mind, the author and Yonatan Harpaz developedin [HP1] general machinery that, in particular, enables one to obtain a model17tructure on R G ∈ sGp S B G from the model structure on the base and on each ofthe fibers. This was further developed in [HP2] where the authors showed thatan analogous "global" model structure can be constructed for Segal group ac-tions and that the two model structures are Quillen equivalent, thus extendingthe Quillen equivalence of 4.12 to this case. The main result is then that apply-ing P n to each simplicial degree as in Example 5.4 gives an n -truncation functorin the integral model structure for Segal group actions. It follows [HP2, §5.1]that the tower 5.1.1 of group actions converges to the the initial group action. Acknowledgements
I would like to thank Thomas Nikolaus for a usefuldiscussion (see Proposition 3.3) and to the referee for many valuable remarks.The author was supported by the Dutch Science Foundation (NWO), grantnumber 62001604. [Car] G. Carlsson,
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M. Prasma: Radboud Universiteit Nijmegen, Institute for Mathematics, Astrophysics, and Par-ticle Physics, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
E-mail address: [email protected]@gmail.com