Equivariant homotopy equivalence of homotopy colimits of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}-functors

Equivariant homotopy equivalence of homotopy colimits of G\documentclass[12pt]{minimal}... Arab. J. Math. https://doi.org/10.1007/s40065-023-00424-1 Arabian Journal of Mathematics Rafael Villarroel-Flores Equivariant homotopy equivalence of homotopy colimits of G-functors Received: 17 October 2021 / Accepted: 10 February 2023 © The Author(s) 2023 Abstract Given a group G and a G-category C, we give a condition on a diagram of simplicial sets indexed by C that allows us to deﬁne a natural action of G on its homotopy colimit, and some other simplicial sets deﬁned in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to equivariant versions. Mathematics Subject Classiﬁcation 55U10 1 Introduction Let G be a group and C a small category. Consider a C-diagram of simplicial sets, where the values of the diagram have a G-action. Then several structures deﬁned in terms of the diagram, like the colimit and the homotopy colimit, have an induced structure of G-object. However, it is often the case that one has a diagram F : C → D where C is a small G-category, D is an arbitrary category and the values of F do not necessarily have a G-action, but nevertheless the homotopy colimit of F does have one. This situation was considered in [6], and independently, by this author in his Ph. D. thesis [11], where the concept of an action of a group G on a functor F by natural transformations is introduced. Here we deﬁne this concept formally in Sect. 3,after the basic deﬁnitions in Sect. 2. We show that there are induced G-actions on colimits, coends, and bar constructions of G-functors. In Sect. 4 we consider more closely the homotopy colimit, and show some basic identities involving the constructions deﬁned so far. One important feature of the homotopy colimit (and of the bar construction, which is its generalization) that the usual colimit lacks, is the fact that it induces a homotopy equivalence from a “pointwise” homotopy equivalence, that is, if F, F are two diagrams of simplicial sets, and η is a natural transformation from F to F such that η : FX → F X is a homotopy equivalence for all C-objects X, then hocolim F and hocolim F are homotopy equivalent. In Sect. 5 we prove an equivariant version of the homotopy invariance of the bar construction, and this is the main result in this paper. Finally, in Sect. 6 we prove the equivariant versions of the four theorems listed in [2, p. 154] about the homotopy colimit. Some of them were noted in [6], however a mild additional hypothesis lets us obtain a more precise result. 2 Preliminaries In this section, C denotes a small category, and D an arbitrary category. R. Villarroel-Flores ( ) Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, Carr. Pachuca-Tulancingo km. 4.5, Pachuca Hgo. 42184, Mexico E-mail: rafaelv@uaeh.edu.mx 123 Arab. J. Math. Let G be a group. We will denote by G the category with a single object ∗, in which hom (∗, ∗) = G and the composition corresponds to group multiplication. For n ≥ 0, let [n] be the small category with object set given by {0, 1,..., n}, and with hom (i, j ) a singleton whenever i ≤ j, otherwise empty. [n] An object D ∈ obj D is called a G-object if there is a functor J : G → D such that J (∗) = D. Clearly this is equivalent to the existence of a collection of D-morphisms {g : D → D} , such that the morphism g∈G corresponding to the identity element in G is the identity 1 , and composition of morphisms corresponds to group multiplication. Let SCat be the category of small categories and  be the subcategory of SCat with objects obj  = { [n]| n ≥ 0 } and morphisms given by the non-decreasing maps. We denote by D the category of functors op C → D ([7, page 40]). The category of simplicial sets (see [8]), denoted sSet, is equal to Set .The category of small G-categories is deﬁned as SCat . We consider the nerve functor N : SCat → sSet,given op by C → (hom (−, C) :  → Set). The nerve functor sends G-categories to G-simplicial sets. There is SCat also a geometric realization functor |·|: sSet → Top, that sends G-simplicial sets to G-topological spaces. We denote |N (C)| simply as |C|. In the case that C and D are small G-categories, and F : C → D is a functor such that F (gC ) = gF (C ), F (gφ) = gF (φ) for all g ∈ G, C ∈ obj C and all C-morphisms φ,wewill saythat F is an equivariant functor. If X and Y are G-topological spaces, a G-homotopy from X to Y is a continuous map H : X ×[0, 1]→ Y such that H (gx , t ) = gH (x , t ) for all g ∈ G, x ∈ X and t ∈[0, 1].Two G-maps f , f : X → Y are 1 2 G-homotopic if there is a G-homotopy H from X to Y such that H (x , 0) = f (x ) and H (x , 1) = f (x ). 1 2 In this case we write f f .The G-topological spaces X and Y are G-homotopy equivalent if there are 1 G 2 G-maps f : X → Y and f : Y → X such that f f 1 and ff 1 . G X G Y We say that two small G-categories C , C are G-homotopy equivalent if the spaces |C |, |C | are. It is not 1 2 1 2 required that the map |C |→|C | deﬁning the homotopy equivalence is induced from a functor C → C . 1 2 1 2 Similarly, we say that two G-simplicial sets are G-homotopic if their realizations are G-homotopy equivalent. op If S : C ×C → D is a functor and D is an object in D,a wedge from S to D is a dinatural transformation ([7, §IX.4]) ζ ={ζ : S(A, B) → D} from S to the constant functor D.A coend of S is a pair (coend S,ζ ), A,B where coend S is an object of D and ζ is a wedge from S to coend S, which is universal among all wedges op from S to a constant functor. If for any small category C any functor S : C × C → D has a coend, we say op that D has small coends. The category sSet has small coends. If F : C → sSet, T : C → sSet are functors, then coend(T × F ) is denoted by T ⊗ F (as in [5], for the analogous case of diagrams of topological spaces). Proposition 2.1 Let D be a category with small coends. Then op (1) Let C , C be small categories, and S : C × C → D be functors for i = 1, 2. Suppose we also 1 2 i i op have a functor U : C → C and η a natural transformation η : S → S ◦ (U × U ). Then there is a 1 2 1 2 unique morphism coend(U,η) : coend S → coend S that makes the following diagram commute for all 1 2 C ∈ obj C: (ζ ) 1 (C,C ) S (C, C ) coend S 1 1 (C,C ) coend(U,η) (ζ ) 2 (UC,UC ) S (UC, UC ) coend S 2 2 (2) There is a category Bif (D), with class of objects: op (C, S) | C is a small category, S : C × C → D , and a morphism (C , S ) → (C , S ) is a pair (U,η),asin(1). 1 1 2 2 (3) There is a functor coend : Bif (D) → D, deﬁned on objects by (C, S) → coend S, and on a morphism (U,η) by the morphism coend(U,η) of (1). Proof Statement (1) is a straightforward generalization of the dual of [7, Proposition IX.7.1]. Statements (2) and (3) are analogous to [7, Exercise V.2.5]. The composition in (2) is given by (U ,η ) ◦ (U,η) = (U ◦ op U,η (U × U ) ◦ η). op Following [7], we say that a functor S : C × C → D is dummy on its ﬁrst variable if its equal to a op composition of the functor projection on the second factor Q : C × C → C with a functor F : C → D.In this case, we will identify the functor S with the functor F, since the latter determines the former. Note that 123 Arab. J. Math. Bif (D) has a subcategory isomorphic to the “super comma” category of [7], namely the full subcategory on the functors which are dummy in their ﬁrst variable. The coend of such a functor, say the one determined by F : C → D, can be identiﬁed with its colimit colim F. op Let S : C × C → sSet be a functor. There is then a simplicial set B(C, S), called the (simplicial) bar construction (see [9]), such that B(C, S) = S(X , X ) (1) n n 0 n φ φ 1 2 n X − →X − →···− →X ∈N (C) 0 1 n n = { (φ ,...,φ ; z) | z ∈ S(X , X ) } (2) 1 n n 0 n with boundaries and degeneracies given by: φ ,...,φ ; d (S(1 ,φ )(z)) i = 0, ⎨ 2 n 0 X 1 d (φ ,...,φ ; z) = (3) (φ ,...,φ φ ,...,φ ; d z) 1 ≤ i ≤ n − 1, i 1 n 1 i +1 i n i φ ,...,φ ; d (S(φ , 1 )(z)) i = n 1 n−1 n n X s (φ ,...,φ ; z) = (φ ,...,φ , 1 ,φ ,...,φ ; s z), 0 ≤ i ≤ n. (4) i 1 n 1 i X i +1 n i op If the functor S is of the form T × F, with T : C → sSet, F : C → sSet, then we denote B(C, S) as B(T , C, F ). Proposition 2.2 The bar construction can be interpreted as a functor B : Bif (sSet) → sSet. (5) Proof Given a morphism (U,η) : (C , S ) → (C , S ) in Bif (sSet), the morphism of simplicial sets 1 1 2 2 B(U,η) : B(C , S ) → B(C , S ) is deﬁned in n-simplices: 1 1 2 2 S (X , X ) → S (UX ,UX ) (6) 1 n 0 n 2 n 0 n X →X →···→X ∈N (C ) UX →UX →···→UX ∈N (C ) 0 1 n 1 n 0 1 n 2 n by means of the coproduct of morphisms η : S (X , X ) → S (UX ,UX ). (X ,X ) 1 n 0 2 n 0 n 0 3 G-functors We deﬁne a G-functor analogously to Deﬁnition 2.2 in [6]. Deﬁnition 3.1 Let G be a group, and C beasmall G-category, and D be an arbitrary category. We say that F : C → D is a G-functor if for each g ∈ G we have a natural transformation η : F → F ◦ g in such a way that the components of η are identities, and η = (η ◦ g ) ◦ η for all g , g ∈ G. 1 g g g 1 g 1 2 1 2 2 1 Deﬁnition 3.2 Let F , F : C → D be two G-functors, with collections of natural transformations indexed by 1 2 1 2 G given by η ,η respectively. A morphism of G-functors is a natural transformation  : F → F such that 1 2 1 2 ( ◦ g) ◦ η = η ◦  for all g ∈ G. g g We have an equivalent deﬁnition of G-functor. Proposition 3.3 Let C be a small category, D an arbitrary category and F : C → D be a functor, such that (C, F ) is a G-object in the category Bif (D). Then F has a structure of a G-functor. Proof If (C, F ) is a G-object in the category Bif (D), there is a functor J : G → Bif (D) with J (∗) = (C, F ). Then, for each g ∈ G we obtain a morphism in the category Bif (D) corresponding to g, from (C, F ) to itself. Such a morphism has two components: a functor C → C which may be just denoted as g,and anatural transformation from F to F ◦ g, which we can denote as η . The fact that J is a functor imply the properties required in the deﬁnition of G-functor. Unraveling the deﬁnition, we see that if F : C → D is a G-functor, then C has a structure of small G-category, and there is a family of D-morphisms η ={η : F (X ) → F (gX )} indexed by g ∈ G and g,X X ∈ obj C such that 123 Arab. J. Math. (1) η = 1 for all X ∈ obj C, 1,X F (X ) (2) η ◦ η = η for any X ∈ obj C, g , g ∈ G, g ,g X g ,X g g ,X 1 2 1 2 2 1 2 (3) η ◦ F ( f ) = F (gf ) ◦ η for any g ∈ G and f : X → Y a morphism in C. g,Y g,X Conversely, these conditions determine the G-object (C, F ) in Bif (D). Note that, by Propositions 2.1.(3), 2.2 and 3.3, it follows that in the case where D = sSet,if S is an object in Bif (D), then both coend S and B(C, S) have deﬁned a G-action on them. It is often the case that we can derive a G-functor a G-object in some other category. We show some examples that will be of use later. Example 3.4 Let C and D be G-categories, and F : C → D an equivariant functor. Then, for D, D ∈ obj D, we have a category D\F/D with objects u v obj(D\F/D ) = (u, C,v) | C ∈ obj C, D − → FC − → D and a morphism p : (u, C,v) → (u , C ,v ) givenbya C-morphism p : C → C such that F ( p) ◦ u = u and v ◦ F (p) = v. op There is a functor D × D → SCat deﬁned on objects by (D, D ) → D\F/D .If (φ, ψ ) : (D, D ) → op (E , E ) is a morphism in D ×D, the associated functor D\F/D → E \F/E sends (u, C,v) to (uφ, C,ψv). op op Then D and D ×D have an obvious structure of G-categories, and there is an action of G on the functor op D × D → SCat we just deﬁned: for g ∈ G,set η  as the functor D\F/D → gD\F/gD given by g,(D,D ) (u, C,v) → (gu, gC, gv). op In this context, we can also consider the comma categories D\F and F/D (see [7]) as functors D → SCat, D → SCat with a G-action. If ν : F → F is an equivariant natural transformation (that is, a natural 1 2 transformation such that gν = ν ), then there is an induced morphism of G-functors ν ¯ :−\F →−\F , C gC 1 2 given by ν ¯ : D\F → D\F , (u, C ) → (ν u, C ). D 1 2 C Example 3.5 Again, let C and D be G-categories, and F : C → D an equivariant functor. There is a functor op C × C → Set given on objects by (X, Y ) → hom (FX, FY ) and on morphisms by (φ, ψ ) → ( f → F ψ ◦ f ◦ Fφ).Ithas a G-action deﬁned by η : hom (FX, FY ) → hom (gF X, gFY ), f → gf . g,(X,Y ) D D Since any set X can be considered as a simplicial set Y such that Y = X for all n and all faces and degeneracies equal to the identity, we can as well consider the last functor as taking values in the category of simplicial sets. Example 3.6 Consider G a group, H ≤ G a subgroup, and let C be the discrete small G-category with object set G //H ={a H, a H,..., a H }, that is, the set of left cosets of H in G with the usual action by left 1 2 n translation, with a = 1. Let Z be an H-simplicial set, and consider the constant functor F : C → sSet with value Z.Wedeﬁne a G-action η on F as follows: Let η : F (H ) → F (gH ) be deﬁned as z → hz,where g,H g = a h, h ∈ H;and then η (z) = η (z). It is straightforward to check that this deﬁnes an action of G i g,aH ga,H on F, and so colim F is a G-simplicial set. This construction is usually known as the induced action from H to G. We will denote colim F in this case as Z ↑ . We ﬁnish this section by stating some basic and easily provable properties of G-functors. Proposition 3.7 If F : C → D is a functor with a G-action given by η and X is an object in C,then FX is aG -object, where G is the stabilizer of X under the action of G on obj C. The action is deﬁned by the X X morphisms η : FX → FX. g,X Proposition 3.8 ((2.3) from [6]) Let F : C → D be a G-functor, U : C → C an equivariant functor, and T : D → E any functor. Then both F ◦ U and T ◦ F have induced structures of G-functors. For example, for any G-category C,wehavea G-functor N (−\C) : C → sSet. (7) 123 Arab. J. Math. 4 The homotopy colimit Deﬁnition 4.1 If F : C → sSet is a functor, its homotopy colimit is deﬁned as B(∗, C, F ),where ∗ is the op constant functor C → sSet with value the simplicial set with exactly one simplex in each dimension. op Let C be a G-category and S : C × C → sSet a G-functor. Proposition 4.2 We have an equivariant isomorphism: op N (−\C/−) ⊗ S B(C, S), (8) C ×C G Proof This can be proven by showing that B(C, S) satisﬁes the deﬁnition of coend of the functor N (−\C/−)× op op op S : (C × C) × (C × C) → sSet. In a much greater generality, this was shown in [9] in a non-equivariant setting, however, all the maps involved are equivariant. For example, a wedge from N (−\C/−) × S to B(C, S) is determined by α : N (B\C/A) × S(A, B) → B(C, S) that is given on an n-simplex deﬁned A,B by f : B → A, φ : C → C for i = 0, 1,..., n − 1, g : C → A and z ∈ S(A, B) , sending it to 0 i i i +1 n n n (φ ,φ ,...,φ , S(g , f )(z) 1 2 n n 0 op If it is the case that S = T × F where F : C → sSet, T : C → sSet are G-functors, then using Fubini’s theorem for coends [7, p. 230], this leads to T ⊗ N (−\C/−) ⊗ F B(T , C, F ). (9) C C G op op Using that, we can prove that for the G-functors F : C → sSet, T : C × D → sSet and U : D → sSet, we have B(B(F, C, T ), D, U ) = B(F, C, T ) ⊗ N (−\D) ⊗ U G D D = F ⊗ N (−\C/−) ⊗ T ⊗ N (−\D/−) ⊗ U G C C D D B(F, C, B(T , D, U )). (10) whose non-equivariant version, and in the context of diagrams of topological spaces, is 3.1.3 from [5]. The constant functor ∗ clearly has a unique structure of G-functor. One has the isomorphism of G-functors: ∗⊗ N (−\C/−) = N (−\C) (11) which by (9), leads to the equivariant isomorphism, for a functor F : C → sSet: hocolim F = B(∗, C, F ) = N (−C) ⊗ F, (12) G C and so this last expression could also be taken as the deﬁnition of the homotopy colimit. Note that the morphism of G-functors N (−\C) →∗ induces an equivariant map ∼ ∼ hocolim F N (−\C) ⊗ F →∗ ⊗ F colim F, (13) = = C C and the morphism of G-functors F →∗ induces an equivariant map ∼ ∼ hocolim F N (−\C) ⊗ F → N (−\C) ⊗ ∗ N (C) (14) = = C C Finally, we note that by similar categorical arguments, one can obtain: Proposition 4.3 Let U : D → C be an equivariant functor between G-categories, and let F : C → sSet a G-functor. Then, with the induced G-functor structure on F ◦ U, we have: ∼ ∼ (1) N (−\D) ⊗ hom (C, U −) B(∗, D, hom (C, U −)) N (C \U ) as G-functors on C. = = D C C ∼ ∼ (2) (F ◦ U )(D) = hom (−,UD) ⊗ F = B(hom (−,UD), C, F ), as G-functors on D. C C C Here we consider a functor with values in the category of sets, such as hom (−,UD),asafunctorwith op values in sSet, by identifying a set X with the simplicial set given by the constant functor  → Set with value X. As a consequence of Proposition 4.3,ifwetake U = 1 to be the identity functor, we obtain that for all C ∈ obj C, B(∗, C, hom (C, −)) = N (C \C ) ∗, (15) C G G C C given that [\FC]C has an initial object 1 : C → ﬁxed by G ([12, (4.3)]). C C 123 Arab. J. Math. 5 The homotopy invariance theorem The proofs of the theorems of the next section are based on Theorem 5.2. The reader may refer to [10, Chapter I, Section 4.] for the basic properties of induced topological spaces. For completeness, we state here results needed from [12]. A morphism between G-simplicial sets is a weak G-homotopy equivalence if it induces a G-homotopy equivalence of topological spaces when passing to geometric realizations. Theorem 5.1 (Theorem 3.5 from [12]) Let φ : X → Y a map of bisimplicial G-sets. Suppose that for all n we have that φ : X → Y is a weak G-homotopy equivalence. Then diag φ is a weak G-homotopy equivalence. n n n Sketch of proof The non-equivariant version of this statement is [4, Chapter IV, Proposition 1.9.]. From there, one obtains the desired result using a result from Bredon ([1]), namely, that a G-equivariant map is a G- homotopy equivalence whenever the induced map between ﬁxed point sets is a homotopy equivalence for every subgroup H of G. op Theorem 5.2 Let S, S : C × C → sSet be G-functors. Let  : S → S be a morphism of G-functors such op that  : S(X, Y ) → S (X, Y ) is a G -homotopy equivalence for all X ∈ obj C, Y ∈ obj C . Then the (X,Y ) (X,Y ) map  ¯ induced by : ¯ : B(C, S) → B(C, S ) (16) is a G-homotopy equivalence. Proof From [9], we know that B(C, S) is the diagonal of a bisimplicial set B(C, S) with (m, n)-simplices the set S(X , X ) . (17) m 0 n φ φ 1 2 m X − →X − →···−→X 0 1 m This coproduct has an action of G deﬁned by: g(φ ,...,φ ; z) = gφ ,..., gφ ; η (z) , (18) 1 m 1 m g,(X ,X ) m 0 ˜ ˜ ˜ which makes B(C, S) a bisimplicial G-set. It follows that  induces a map  ˜ : B(C, S) → B(C, S ), sending (φ ,...,φ ; z) → φ ,...,φ ;  (z) , (19) 1 m 1 m (X ,X ) m 0 The map  ˜ is equivariant, and so if we deﬁne  ¯ as diag  ˜,then  ¯ is equivariant as well. φ φ φ 1 2 m Let us denote X − → X −→· · · −→ X ∈ N (C) by X. According to Theorem 5.1, to prove that  ¯ is a 0 1 m m G-homotopy equivalence, it is sufﬁcient to prove that ˜ : S(X , X ) → S (X , X ) (20) m,− m 0 m 0 ¯ ¯ X ∈N (C) X ∈N (C) m m is a G-homotopy equivalence for all m. Taking geometric realization on both sides of (20), since geometric realization commutes with coproducts, we obtain: |˜  |: |S(X , X )|→ |S (X , X )| (21) m,− m 0 m 0 ¯ ¯ X ∈N (C) X ∈N (C) m m Let E be a set of representatives for the orbits of the action of G on N (C) . Then the map in (21) can be m m written as: G G  G |˜  |↑ : |S(X , X )|↑ → |S (X , X )|↑ (22) m,− m 0 m 0 G G G ¯ ¯ ¯ Y Y Y ¯ ¯ Y ∈E Y ∈E m m Since by hypothesis, each  is a G -homotopy equivalence, given that G ≤ G ,theyare (X ,X ) (X ,X ) ¯ (X ,X ) m 0 m 0 Y m 0 also G -homotopy equivalences, and so each map G  G |S(X , X )|↑ →|S (X , X )|↑ (23) m 0 m 0 G G ¯ ¯ Y Y is a G-homotopy equivalence. Therefore the map in (22) is a coproduct of G-homotopy equivalences, hence a G-homotopy equivalence. 123 Arab. J. Math. 6 Equivariant coﬁnality and push-down theorems for G-functors Some of the proofs in these section are as those in [5], adapted for the case of the group action, and in the context of diagrams of simplicial sets. We include more details than in the cited paper, given that homotopy colimit methods have been used by combinatorialists, see for example [13]. Theorem 6.1 (Equivariant Homotopy Invariance Of The Homotopy Colimit).Let F, F : C → sSet G- functors, and  : F → F a morphism of G-functors such that each  : FX → F Xis a G -homotopy X X equivalence. Then the induced morphism  ¯ : hocolim F → hocolim F is a G-homotopy equivalence. Proof Straightforward from Theorem 5.2, since the homotopy colimit is a special case of a bar construction. Theorem 6.2 (Reduction Theorem) Let U : D → C be an equivariant functor between G-categories, and let F : C → sSet a G-functor. Then we have the equivariant isomorphism. hocolim F ◦ U N (−\U ) ⊗ F (24) G C Proof hocolim F ◦ U N (−\D) ⊗ (F ◦ U ) Equation (12) = N (−\D) ⊗ (hom (−,UD) ⊗ F ) Proposition 4.3.(2) G D C C (N −\D) ⊗ hom (−,UD)) ⊗ F Fubini’s theorem G D C C N (−\U ) ⊗ F Proposition 4.3.(1) G C In [6, (2.6)], this result is given as a homotopy equivalence. However, as noted in [5, 4.4] in the context of diagrams of topological spaces, this is even an isomorphism, which in this case is equivariant. Theorem 6.3 (Coﬁnality Theorem) Let U : D → C be an equivariant functor between G-categories, and let F : C → sSet a G-functor. Consider the induced G-functor structure on F ◦U. If N (C \U ) is G -contractible for all objects C in C,then hocolim F ◦ U hocolim F. Proof hocolim F ◦ U = B(∗, D, F ◦ U ) = B(∗, D, B(hom (−,UD), C, F )) Proposition 4.3.(2) = B(B(∗, D, hom (C, U −)), C, F ) Equation 10 G C B(N (−\U ), C, F ) Proposition 4.3..(1) B(∗, C, F ) = hocolim F Hypothesis Here we also used the equivariant homotopy invariance (Theorem 5.2) of the bar construction in the last step, together with the G -contractibility of the ﬁber N (C \S) to obtain the equivariant homotopy. In [6, (2.7)], without assuming the equivariant contractibility, this result is given as a homotopy equivalence not necessarily equivariant. Theorem 6.4 (Homotopy Pushdown Theorem) Let U : D → C be an equivariant functor and F : D → sSet a G-functor. Let U (F ) : C → sSet the functor given by C → B(hom (U −, C ), D, F ).Then U (F ) is a h C h ∗ ∗ G-functor and hocolim U (F ) hocolim F. h G Proof hocolim U (F ) = B(∗, C, B(hom (U −, C ), D, F )) h C B(B(∗, C, hom (UD, −)), D, F ) Equation 10 G C B(∗, D, F ) = hocolim F Equation 15 123 Arab. J. Math. We again used Theorem 5.2 in the last step. Hence in [6, (2.5)] we do have a G-homotopy equivalence. As an example of the application of Theorem 6.1, consider a simplicial set X that has two isomorphic proper subsimplicial sets X , X that have union X and nonempty intersection X ∩ X . The isomorphism 1 2 1 2 φ : X → X must be the identity when restricted to X ∩ X .Let P be the partially ordered set with three 1 2 1 2 points x , y, z,and let x < y, x < z be the only nontrivial order relations. The poset P can be seen as a category P ([7, Section 2 of Chapter I]), and in fact is a G-category when G = C =a, the cyclic group of order 2, interchanges b with c and ﬁxes a.Wehaveadiagram D : P → sSet that sends x to X ∩ X , y to X and z 1 2 1 to X . The morphism corresponding to x < y is sent by D to the inclusion of X ∩ X into X , and similarly 2 1 2 1 x < z. In this case, hocolim D is equal to X ∪ X = X (see [3, Section 4]). To make D a G-functor, we deﬁne 1 2 −1 η as the identity, η : X → X as φ,and η as φ . Suppose there is another simplicial set Y that can a,x a,y 1 2 a,z be written as union of Y , Y , analogously to X, with an isomorphism ψ : Y → Y , and there are homotopy 1 2 1 2 equivalence f : X → Y ,for i = 1, 2, such that they coincide on X ∩ X and the common restriction gives i i i 1 2 a homotopy equivalence X ∩ X → Y ∩ Y . We may then form a G-diagram D with hocolim D = Y,and 1 2 1 2 have a morphism of G-functors  : D → D with  = f ,  = f ,and  the restriction of f to X ∩ X . y 1 z 2 x 1 1 2 Since both stabilizers G and G are trivial, by Theorem 6.1 we conclude that X is homotopy equivalent (in y z fact, G-homotopy equivalent) to Y . Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funding There was no outside funding or grants received that assisted in this project. Declarations Conﬂict of interest The author declares that he has no conﬂict of interest. References 1. Bredon, G.E.: Equivariant Cohomology Theories. Lecture Notes in Mathematics, No. 34 (Springer, Berlin, 1967). 2. Dwyer, W.G.; Kan, D.M.: A classiﬁcation theorem for diagrams of simplicial sets. Topology 23(2), 139–155 (1984) 3. Dwyer, W.G.: Classifying Spaces and Homology Decompositions. Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser, Basel, 2001), x+98. 4. Goerss, P.G.; Jardine, J.F.: Simplicial Homotopy Theory. Birkhäuser Verlag, Basel (1999) 5. Hollender, J.; Vogt, R.M.: Modules of topological spaces, applications to homotopy limits and E structures. Arch. Math. (Basel) 59(2), 115–129 (1992) 6. Jackowski, S.; Słominska, ´ J.: G-functors, G-posets and homotopy decompositions of G-spaces. Fund. Math. 169(3), 249–287 (2001) 7. Saunders, M.L.: Categories for the working mathematician, Graduate Texts in Mathematics, Volume 5 (Springer, New York, 1998), 2nd edn. 8. May, J.P.: Simplicial Objects in Algebraic Topology. Van Nostrand, Princeton (1967) 9. Meyer, J.-P.: Bar and cobar constructions. I. J. Pure Appl. Algebra 33(2), 163–207 (1984) 10. Dieck, T.: Transformation Groups. Walter de Gruyter, Berlin (1987) 11. Villarroel-Flores, R.: Equivariant Homotopy Type of Categories and Preordered Sets. Ph.D. thesis, University of Minnesota. (1999) 12. Villarroel-Flores, R.: Homotopy equivalence of simplicial sets with a group action. Bol. Soc. Mat. Mexicana (3) 6(2), 247–262 (2000) 13. Welker, V.; Ziegler, G.M.; Živaljevic, ´ R.T.: Homotopy colimits–comparison lemmas for combinatorial applications. J. Reine Angew. Math. 509, 117–149 (1999) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arabian Journal of Mathematics Springer Journals

Equivariant homotopy equivalence of homotopy colimits of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}-functors

, Volume OnlineFirst – Mar 25, 2023
8 pages

Loading next page...

/lp/springer-journals/equivariant-homotopy-equivalence-of-homotopy-colimits-of-g-xQM0hfchzB
Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2023
ISSN
2193-5343
eISSN
2193-5351
DOI
10.1007/s40065-023-00424-1
Publisher site
See Article on Publisher Site

Abstract

Arab. J. Math. https://doi.org/10.1007/s40065-023-00424-1 Arabian Journal of Mathematics Rafael Villarroel-Flores Equivariant homotopy equivalence of homotopy colimits of G-functors Received: 17 October 2021 / Accepted: 10 February 2023 © The Author(s) 2023 Abstract Given a group G and a G-category C, we give a condition on a diagram of simplicial sets indexed by C that allows us to deﬁne a natural action of G on its homotopy colimit, and some other simplicial sets deﬁned in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to equivariant versions. Mathematics Subject Classiﬁcation 55U10 1 Introduction Let G be a group and C a small category. Consider a C-diagram of simplicial sets, where the values of the diagram have a G-action. Then several structures deﬁned in terms of the diagram, like the colimit and the homotopy colimit, have an induced structure of G-object. However, it is often the case that one has a diagram F : C → D where C is a small G-category, D is an arbitrary category and the values of F do not necessarily have a G-action, but nevertheless the homotopy colimit of F does have one. This situation was considered in [6], and independently, by this author in his Ph. D. thesis [11], where the concept of an action of a group G on a functor F by natural transformations is introduced. Here we deﬁne this concept formally in Sect. 3,after the basic deﬁnitions in Sect. 2. We show that there are induced G-actions on colimits, coends, and bar constructions of G-functors. In Sect. 4 we consider more closely the homotopy colimit, and show some basic identities involving the constructions deﬁned so far. One important feature of the homotopy colimit (and of the bar construction, which is its generalization) that the usual colimit lacks, is the fact that it induces a homotopy equivalence from a “pointwise” homotopy equivalence, that is, if F, F are two diagrams of simplicial sets, and η is a natural transformation from F to F such that η : FX → F X is a homotopy equivalence for all C-objects X, then hocolim F and hocolim F are homotopy equivalent. In Sect. 5 we prove an equivariant version of the homotopy invariance of the bar construction, and this is the main result in this paper. Finally, in Sect. 6 we prove the equivariant versions of the four theorems listed in [2, p. 154] about the homotopy colimit. Some of them were noted in [6], however a mild additional hypothesis lets us obtain a more precise result. 2 Preliminaries In this section, C denotes a small category, and D an arbitrary category. R. Villarroel-Flores ( ) Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, Carr. Pachuca-Tulancingo km. 4.5, Pachuca Hgo. 42184, Mexico E-mail: rafaelv@uaeh.edu.mx 123 Arab. J. Math. Let G be a group. We will denote by G the category with a single object ∗, in which hom (∗, ∗) = G and the composition corresponds to group multiplication. For n ≥ 0, let [n] be the small category with object set given by {0, 1,..., n}, and with hom (i, j ) a singleton whenever i ≤ j, otherwise empty. [n] An object D ∈ obj D is called a G-object if there is a functor J : G → D such that J (∗) = D. Clearly this is equivalent to the existence of a collection of D-morphisms {g : D → D} , such that the morphism g∈G corresponding to the identity element in G is the identity 1 , and composition of morphisms corresponds to group multiplication. Let SCat be the category of small categories and  be the subcategory of SCat with objects obj  = { [n]| n ≥ 0 } and morphisms given by the non-decreasing maps. We denote by D the category of functors op C → D ([7, page 40]). The category of simplicial sets (see [8]), denoted sSet, is equal to Set .The category of small G-categories is deﬁned as SCat . We consider the nerve functor N : SCat → sSet,given op by C → (hom (−, C) :  → Set). The nerve functor sends G-categories to G-simplicial sets. There is SCat also a geometric realization functor |·|: sSet → Top, that sends G-simplicial sets to G-topological spaces. We denote |N (C)| simply as |C|. In the case that C and D are small G-categories, and F : C → D is a functor such that F (gC ) = gF (C ), F (gφ) = gF (φ) for all g ∈ G, C ∈ obj C and all C-morphisms φ,wewill saythat F is an equivariant functor. If X and Y are G-topological spaces, a G-homotopy from X to Y is a continuous map H : X ×[0, 1]→ Y such that H (gx , t ) = gH (x , t ) for all g ∈ G, x ∈ X and t ∈[0, 1].Two G-maps f , f : X → Y are 1 2 G-homotopic if there is a G-homotopy H from X to Y such that H (x , 0) = f (x ) and H (x , 1) = f (x ). 1 2 In this case we write f f .The G-topological spaces X and Y are G-homotopy equivalent if there are 1 G 2 G-maps f : X → Y and f : Y → X such that f f 1 and ff 1 . G X G Y We say that two small G-categories C , C are G-homotopy equivalent if the spaces |C |, |C | are. It is not 1 2 1 2 required that the map |C |→|C | deﬁning the homotopy equivalence is induced from a functor C → C . 1 2 1 2 Similarly, we say that two G-simplicial sets are G-homotopic if their realizations are G-homotopy equivalent. op If S : C ×C → D is a functor and D is an object in D,a wedge from S to D is a dinatural transformation ([7, §IX.4]) ζ ={ζ : S(A, B) → D} from S to the constant functor D.A coend of S is a pair (coend S,ζ ), A,B where coend S is an object of D and ζ is a wedge from S to coend S, which is universal among all wedges op from S to a constant functor. If for any small category C any functor S : C × C → D has a coend, we say op that D has small coends. The category sSet has small coends. If F : C → sSet, T : C → sSet are functors, then coend(T × F ) is denoted by T ⊗ F (as in [5], for the analogous case of diagrams of topological spaces). Proposition 2.1 Let D be a category with small coends. Then op (1) Let C , C be small categories, and S : C × C → D be functors for i = 1, 2. Suppose we also 1 2 i i op have a functor U : C → C and η a natural transformation η : S → S ◦ (U × U ). Then there is a 1 2 1 2 unique morphism coend(U,η) : coend S → coend S that makes the following diagram commute for all 1 2 C ∈ obj C: (ζ ) 1 (C,C ) S (C, C ) coend S 1 1 (C,C ) coend(U,η) (ζ ) 2 (UC,UC ) S (UC, UC ) coend S 2 2 (2) There is a category Bif (D), with class of objects: op (C, S) | C is a small category, S : C × C → D , and a morphism (C , S ) → (C , S ) is a pair (U,η),asin(1). 1 1 2 2 (3) There is a functor coend : Bif (D) → D, deﬁned on objects by (C, S) → coend S, and on a morphism (U,η) by the morphism coend(U,η) of (1). Proof Statement (1) is a straightforward generalization of the dual of [7, Proposition IX.7.1]. Statements (2) and (3) are analogous to [7, Exercise V.2.5]. The composition in (2) is given by (U ,η ) ◦ (U,η) = (U ◦ op U,η (U × U ) ◦ η). op Following [7], we say that a functor S : C × C → D is dummy on its ﬁrst variable if its equal to a op composition of the functor projection on the second factor Q : C × C → C with a functor F : C → D.In this case, we will identify the functor S with the functor F, since the latter determines the former. Note that 123 Arab. J. Math. Bif (D) has a subcategory isomorphic to the “super comma” category of [7], namely the full subcategory on the functors which are dummy in their ﬁrst variable. The coend of such a functor, say the one determined by F : C → D, can be identiﬁed with its colimit colim F. op Let S : C × C → sSet be a functor. There is then a simplicial set B(C, S), called the (simplicial) bar construction (see [9]), such that B(C, S) = S(X , X ) (1) n n 0 n φ φ 1 2 n X − →X − →···− →X ∈N (C) 0 1 n n = { (φ ,...,φ ; z) | z ∈ S(X , X ) } (2) 1 n n 0 n with boundaries and degeneracies given by: φ ,...,φ ; d (S(1 ,φ )(z)) i = 0, ⎨ 2 n 0 X 1 d (φ ,...,φ ; z) = (3) (φ ,...,φ φ ,...,φ ; d z) 1 ≤ i ≤ n − 1, i 1 n 1 i +1 i n i φ ,...,φ ; d (S(φ , 1 )(z)) i = n 1 n−1 n n X s (φ ,...,φ ; z) = (φ ,...,φ , 1 ,φ ,...,φ ; s z), 0 ≤ i ≤ n. (4) i 1 n 1 i X i +1 n i op If the functor S is of the form T × F, with T : C → sSet, F : C → sSet, then we denote B(C, S) as B(T , C, F ). Proposition 2.2 The bar construction can be interpreted as a functor B : Bif (sSet) → sSet. (5) Proof Given a morphism (U,η) : (C , S ) → (C , S ) in Bif (sSet), the morphism of simplicial sets 1 1 2 2 B(U,η) : B(C , S ) → B(C , S ) is deﬁned in n-simplices: 1 1 2 2 S (X , X ) → S (UX ,UX ) (6) 1 n 0 n 2 n 0 n X →X →···→X ∈N (C ) UX →UX →···→UX ∈N (C ) 0 1 n 1 n 0 1 n 2 n by means of the coproduct of morphisms η : S (X , X ) → S (UX ,UX ). (X ,X ) 1 n 0 2 n 0 n 0 3 G-functors We deﬁne a G-functor analogously to Deﬁnition 2.2 in [6]. Deﬁnition 3.1 Let G be a group, and C beasmall G-category, and D be an arbitrary category. We say that F : C → D is a G-functor if for each g ∈ G we have a natural transformation η : F → F ◦ g in such a way that the components of η are identities, and η = (η ◦ g ) ◦ η for all g , g ∈ G. 1 g g g 1 g 1 2 1 2 2 1 Deﬁnition 3.2 Let F , F : C → D be two G-functors, with collections of natural transformations indexed by 1 2 1 2 G given by η ,η respectively. A morphism of G-functors is a natural transformation  : F → F such that 1 2 1 2 ( ◦ g) ◦ η = η ◦  for all g ∈ G. g g We have an equivalent deﬁnition of G-functor. Proposition 3.3 Let C be a small category, D an arbitrary category and F : C → D be a functor, such that (C, F ) is a G-object in the category Bif (D). Then F has a structure of a G-functor. Proof If (C, F ) is a G-object in the category Bif (D), there is a functor J : G → Bif (D) with J (∗) = (C, F ). Then, for each g ∈ G we obtain a morphism in the category Bif (D) corresponding to g, from (C, F ) to itself. Such a morphism has two components: a functor C → C which may be just denoted as g,and anatural transformation from F to F ◦ g, which we can denote as η . The fact that J is a functor imply the properties required in the deﬁnition of G-functor. Unraveling the deﬁnition, we see that if F : C → D is a G-functor, then C has a structure of small G-category, and there is a family of D-morphisms η ={η : F (X ) → F (gX )} indexed by g ∈ G and g,X X ∈ obj C such that 123 Arab. J. Math. (1) η = 1 for all X ∈ obj C, 1,X F (X ) (2) η ◦ η = η for any X ∈ obj C, g , g ∈ G, g ,g X g ,X g g ,X 1 2 1 2 2 1 2 (3) η ◦ F ( f ) = F (gf ) ◦ η for any g ∈ G and f : X → Y a morphism in C. g,Y g,X Conversely, these conditions determine the G-object (C, F ) in Bif (D). Note that, by Propositions 2.1.(3), 2.2 and 3.3, it follows that in the case where D = sSet,if S is an object in Bif (D), then both coend S and B(C, S) have deﬁned a G-action on them. It is often the case that we can derive a G-functor a G-object in some other category. We show some examples that will be of use later. Example 3.4 Let C and D be G-categories, and F : C → D an equivariant functor. Then, for D, D ∈ obj D, we have a category D\F/D with objects u v obj(D\F/D ) = (u, C,v) | C ∈ obj C, D − → FC − → D and a morphism p : (u, C,v) → (u , C ,v ) givenbya C-morphism p : C → C such that F ( p) ◦ u = u and v ◦ F (p) = v. op There is a functor D × D → SCat deﬁned on objects by (D, D ) → D\F/D .If (φ, ψ ) : (D, D ) → op (E , E ) is a morphism in D ×D, the associated functor D\F/D → E \F/E sends (u, C,v) to (uφ, C,ψv). op op Then D and D ×D have an obvious structure of G-categories, and there is an action of G on the functor op D × D → SCat we just deﬁned: for g ∈ G,set η  as the functor D\F/D → gD\F/gD given by g,(D,D ) (u, C,v) → (gu, gC, gv). op In this context, we can also consider the comma categories D\F and F/D (see [7]) as functors D → SCat, D → SCat with a G-action. If ν : F → F is an equivariant natural transformation (that is, a natural 1 2 transformation such that gν = ν ), then there is an induced morphism of G-functors ν ¯ :−\F →−\F , C gC 1 2 given by ν ¯ : D\F → D\F , (u, C ) → (ν u, C ). D 1 2 C Example 3.5 Again, let C and D be G-categories, and F : C → D an equivariant functor. There is a functor op C × C → Set given on objects by (X, Y ) → hom (FX, FY ) and on morphisms by (φ, ψ ) → ( f → F ψ ◦ f ◦ Fφ).Ithas a G-action deﬁned by η : hom (FX, FY ) → hom (gF X, gFY ), f → gf . g,(X,Y ) D D Since any set X can be considered as a simplicial set Y such that Y = X for all n and all faces and degeneracies equal to the identity, we can as well consider the last functor as taking values in the category of simplicial sets. Example 3.6 Consider G a group, H ≤ G a subgroup, and let C be the discrete small G-category with object set G //H ={a H, a H,..., a H }, that is, the set of left cosets of H in G with the usual action by left 1 2 n translation, with a = 1. Let Z be an H-simplicial set, and consider the constant functor F : C → sSet with value Z.Wedeﬁne a G-action η on F as follows: Let η : F (H ) → F (gH ) be deﬁned as z → hz,where g,H g = a h, h ∈ H;and then η (z) = η (z). It is straightforward to check that this deﬁnes an action of G i g,aH ga,H on F, and so colim F is a G-simplicial set. This construction is usually known as the induced action from H to G. We will denote colim F in this case as Z ↑ . We ﬁnish this section by stating some basic and easily provable properties of G-functors. Proposition 3.7 If F : C → D is a functor with a G-action given by η and X is an object in C,then FX is aG -object, where G is the stabilizer of X under the action of G on obj C. The action is deﬁned by the X X morphisms η : FX → FX. g,X Proposition 3.8 ((2.3) from [6]) Let F : C → D be a G-functor, U : C → C an equivariant functor, and T : D → E any functor. Then both F ◦ U and T ◦ F have induced structures of G-functors. For example, for any G-category C,wehavea G-functor N (−\C) : C → sSet. (7) 123 Arab. J. Math. 4 The homotopy colimit Deﬁnition 4.1 If F : C → sSet is a functor, its homotopy colimit is deﬁned as B(∗, C, F ),where ∗ is the op constant functor C → sSet with value the simplicial set with exactly one simplex in each dimension. op Let C be a G-category and S : C × C → sSet a G-functor. Proposition 4.2 We have an equivariant isomorphism: op N (−\C/−) ⊗ S B(C, S), (8) C ×C G Proof This can be proven by showing that B(C, S) satisﬁes the deﬁnition of coend of the functor N (−\C/−)× op op op S : (C × C) × (C × C) → sSet. In a much greater generality, this was shown in [9] in a non-equivariant setting, however, all the maps involved are equivariant. For example, a wedge from N (−\C/−) × S to B(C, S) is determined by α : N (B\C/A) × S(A, B) → B(C, S) that is given on an n-simplex deﬁned A,B by f : B → A, φ : C → C for i = 0, 1,..., n − 1, g : C → A and z ∈ S(A, B) , sending it to 0 i i i +1 n n n (φ ,φ ,...,φ , S(g , f )(z) 1 2 n n 0 op If it is the case that S = T × F where F : C → sSet, T : C → sSet are G-functors, then using Fubini’s theorem for coends [7, p. 230], this leads to T ⊗ N (−\C/−) ⊗ F B(T , C, F ). (9) C C G op op Using that, we can prove that for the G-functors F : C → sSet, T : C × D → sSet and U : D → sSet, we have B(B(F, C, T ), D, U ) = B(F, C, T ) ⊗ N (−\D) ⊗ U G D D = F ⊗ N (−\C/−) ⊗ T ⊗ N (−\D/−) ⊗ U G C C D D B(F, C, B(T , D, U )). (10) whose non-equivariant version, and in the context of diagrams of topological spaces, is 3.1.3 from [5]. The constant functor ∗ clearly has a unique structure of G-functor. One has the isomorphism of G-functors: ∗⊗ N (−\C/−) = N (−\C) (11) which by (9), leads to the equivariant isomorphism, for a functor F : C → sSet: hocolim F = B(∗, C, F ) = N (−C) ⊗ F, (12) G C and so this last expression could also be taken as the deﬁnition of the homotopy colimit. Note that the morphism of G-functors N (−\C) →∗ induces an equivariant map ∼ ∼ hocolim F N (−\C) ⊗ F →∗ ⊗ F colim F, (13) = = C C and the morphism of G-functors F →∗ induces an equivariant map ∼ ∼ hocolim F N (−\C) ⊗ F → N (−\C) ⊗ ∗ N (C) (14) = = C C Finally, we note that by similar categorical arguments, one can obtain: Proposition 4.3 Let U : D → C be an equivariant functor between G-categories, and let F : C → sSet a G-functor. Then, with the induced G-functor structure on F ◦ U, we have: ∼ ∼ (1) N (−\D) ⊗ hom (C, U −) B(∗, D, hom (C, U −)) N (C \U ) as G-functors on C. = = D C C ∼ ∼ (2) (F ◦ U )(D) = hom (−,UD) ⊗ F = B(hom (−,UD), C, F ), as G-functors on D. C C C Here we consider a functor with values in the category of sets, such as hom (−,UD),asafunctorwith op values in sSet, by identifying a set X with the simplicial set given by the constant functor  → Set with value X. As a consequence of Proposition 4.3,ifwetake U = 1 to be the identity functor, we obtain that for all C ∈ obj C, B(∗, C, hom (C, −)) = N (C \C ) ∗, (15) C G G C C given that [\FC]C has an initial object 1 : C → ﬁxed by G ([12, (4.3)]). C C 123 Arab. J. Math. 5 The homotopy invariance theorem The proofs of the theorems of the next section are based on Theorem 5.2. The reader may refer to [10, Chapter I, Section 4.] for the basic properties of induced topological spaces. For completeness, we state here results needed from [12]. A morphism between G-simplicial sets is a weak G-homotopy equivalence if it induces a G-homotopy equivalence of topological spaces when passing to geometric realizations. Theorem 5.1 (Theorem 3.5 from [12]) Let φ : X → Y a map of bisimplicial G-sets. Suppose that for all n we have that φ : X → Y is a weak G-homotopy equivalence. Then diag φ is a weak G-homotopy equivalence. n n n Sketch of proof The non-equivariant version of this statement is [4, Chapter IV, Proposition 1.9.]. From there, one obtains the desired result using a result from Bredon ([1]), namely, that a G-equivariant map is a G- homotopy equivalence whenever the induced map between ﬁxed point sets is a homotopy equivalence for every subgroup H of G. op Theorem 5.2 Let S, S : C × C → sSet be G-functors. Let  : S → S be a morphism of G-functors such op that  : S(X, Y ) → S (X, Y ) is a G -homotopy equivalence for all X ∈ obj C, Y ∈ obj C . Then the (X,Y ) (X,Y ) map  ¯ induced by : ¯ : B(C, S) → B(C, S ) (16) is a G-homotopy equivalence. Proof From [9], we know that B(C, S) is the diagonal of a bisimplicial set B(C, S) with (m, n)-simplices the set S(X , X ) . (17) m 0 n φ φ 1 2 m X − →X − →···−→X 0 1 m This coproduct has an action of G deﬁned by: g(φ ,...,φ ; z) = gφ ,..., gφ ; η (z) , (18) 1 m 1 m g,(X ,X ) m 0 ˜ ˜ ˜ which makes B(C, S) a bisimplicial G-set. It follows that  induces a map  ˜ : B(C, S) → B(C, S ), sending (φ ,...,φ ; z) → φ ,...,φ ;  (z) , (19) 1 m 1 m (X ,X ) m 0 The map  ˜ is equivariant, and so if we deﬁne  ¯ as diag  ˜,then  ¯ is equivariant as well. φ φ φ 1 2 m Let us denote X − → X −→· · · −→ X ∈ N (C) by X. According to Theorem 5.1, to prove that  ¯ is a 0 1 m m G-homotopy equivalence, it is sufﬁcient to prove that ˜ : S(X , X ) → S (X , X ) (20) m,− m 0 m 0 ¯ ¯ X ∈N (C) X ∈N (C) m m is a G-homotopy equivalence for all m. Taking geometric realization on both sides of (20), since geometric realization commutes with coproducts, we obtain: |˜  |: |S(X , X )|→ |S (X , X )| (21) m,− m 0 m 0 ¯ ¯ X ∈N (C) X ∈N (C) m m Let E be a set of representatives for the orbits of the action of G on N (C) . Then the map in (21) can be m m written as: G G  G |˜  |↑ : |S(X , X )|↑ → |S (X , X )|↑ (22) m,− m 0 m 0 G G G ¯ ¯ ¯ Y Y Y ¯ ¯ Y ∈E Y ∈E m m Since by hypothesis, each  is a G -homotopy equivalence, given that G ≤ G ,theyare (X ,X ) (X ,X ) ¯ (X ,X ) m 0 m 0 Y m 0 also G -homotopy equivalences, and so each map G  G |S(X , X )|↑ →|S (X , X )|↑ (23) m 0 m 0 G G ¯ ¯ Y Y is a G-homotopy equivalence. Therefore the map in (22) is a coproduct of G-homotopy equivalences, hence a G-homotopy equivalence. 123 Arab. J. Math. 6 Equivariant coﬁnality and push-down theorems for G-functors Some of the proofs in these section are as those in [5], adapted for the case of the group action, and in the context of diagrams of simplicial sets. We include more details than in the cited paper, given that homotopy colimit methods have been used by combinatorialists, see for example [13]. Theorem 6.1 (Equivariant Homotopy Invariance Of The Homotopy Colimit).Let F, F : C → sSet G- functors, and  : F → F a morphism of G-functors such that each  : FX → F Xis a G -homotopy X X equivalence. Then the induced morphism  ¯ : hocolim F → hocolim F is a G-homotopy equivalence. Proof Straightforward from Theorem 5.2, since the homotopy colimit is a special case of a bar construction. Theorem 6.2 (Reduction Theorem) Let U : D → C be an equivariant functor between G-categories, and let F : C → sSet a G-functor. Then we have the equivariant isomorphism. hocolim F ◦ U N (−\U ) ⊗ F (24) G C Proof hocolim F ◦ U N (−\D) ⊗ (F ◦ U ) Equation (12) = N (−\D) ⊗ (hom (−,UD) ⊗ F ) Proposition 4.3.(2) G D C C (N −\D) ⊗ hom (−,UD)) ⊗ F Fubini’s theorem G D C C N (−\U ) ⊗ F Proposition 4.3.(1) G C In [6, (2.6)], this result is given as a homotopy equivalence. However, as noted in [5, 4.4] in the context of diagrams of topological spaces, this is even an isomorphism, which in this case is equivariant. Theorem 6.3 (Coﬁnality Theorem) Let U : D → C be an equivariant functor between G-categories, and let F : C → sSet a G-functor. Consider the induced G-functor structure on F ◦U. If N (C \U ) is G -contractible for all objects C in C,then hocolim F ◦ U hocolim F. Proof hocolim F ◦ U = B(∗, D, F ◦ U ) = B(∗, D, B(hom (−,UD), C, F )) Proposition 4.3.(2) = B(B(∗, D, hom (C, U −)), C, F ) Equation 10 G C B(N (−\U ), C, F ) Proposition 4.3..(1) B(∗, C, F ) = hocolim F Hypothesis Here we also used the equivariant homotopy invariance (Theorem 5.2) of the bar construction in the last step, together with the G -contractibility of the ﬁber N (C \S) to obtain the equivariant homotopy. In [6, (2.7)], without assuming the equivariant contractibility, this result is given as a homotopy equivalence not necessarily equivariant. Theorem 6.4 (Homotopy Pushdown Theorem) Let U : D → C be an equivariant functor and F : D → sSet a G-functor. Let U (F ) : C → sSet the functor given by C → B(hom (U −, C ), D, F ).Then U (F ) is a h C h ∗ ∗ G-functor and hocolim U (F ) hocolim F. h G Proof hocolim U (F ) = B(∗, C, B(hom (U −, C ), D, F )) h C B(B(∗, C, hom (UD, −)), D, F ) Equation 10 G C B(∗, D, F ) = hocolim F Equation 15 123 Arab. J. Math. We again used Theorem 5.2 in the last step. Hence in [6, (2.5)] we do have a G-homotopy equivalence. As an example of the application of Theorem 6.1, consider a simplicial set X that has two isomorphic proper subsimplicial sets X , X that have union X and nonempty intersection X ∩ X . The isomorphism 1 2 1 2 φ : X → X must be the identity when restricted to X ∩ X .Let P be the partially ordered set with three 1 2 1 2 points x , y, z,and let x < y, x < z be the only nontrivial order relations. The poset P can be seen as a category P ([7, Section 2 of Chapter I]), and in fact is a G-category when G = C =a, the cyclic group of order 2, interchanges b with c and ﬁxes a.Wehaveadiagram D : P → sSet that sends x to X ∩ X , y to X and z 1 2 1 to X . The morphism corresponding to x < y is sent by D to the inclusion of X ∩ X into X , and similarly 2 1 2 1 x < z. In this case, hocolim D is equal to X ∪ X = X (see [3, Section 4]). To make D a G-functor, we deﬁne 1 2 −1 η as the identity, η : X → X as φ,and η as φ . Suppose there is another simplicial set Y that can a,x a,y 1 2 a,z be written as union of Y , Y , analogously to X, with an isomorphism ψ : Y → Y , and there are homotopy 1 2 1 2 equivalence f : X → Y ,for i = 1, 2, such that they coincide on X ∩ X and the common restriction gives i i i 1 2 a homotopy equivalence X ∩ X → Y ∩ Y . We may then form a G-diagram D with hocolim D = Y,and 1 2 1 2 have a morphism of G-functors  : D → D with  = f ,  = f ,and  the restriction of f to X ∩ X . y 1 z 2 x 1 1 2 Since both stabilizers G and G are trivial, by Theorem 6.1 we conclude that X is homotopy equivalent (in y z fact, G-homotopy equivalent) to Y . Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funding There was no outside funding or grants received that assisted in this project. Declarations Conﬂict of interest The author declares that he has no conﬂict of interest. References 1. Bredon, G.E.: Equivariant Cohomology Theories. Lecture Notes in Mathematics, No. 34 (Springer, Berlin, 1967). 2. Dwyer, W.G.; Kan, D.M.: A classiﬁcation theorem for diagrams of simplicial sets. Topology 23(2), 139–155 (1984) 3. Dwyer, W.G.: Classifying Spaces and Homology Decompositions. Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser, Basel, 2001), x+98. 4. Goerss, P.G.; Jardine, J.F.: Simplicial Homotopy Theory. Birkhäuser Verlag, Basel (1999) 5. Hollender, J.; Vogt, R.M.: Modules of topological spaces, applications to homotopy limits and E structures. Arch. Math. (Basel) 59(2), 115–129 (1992) 6. Jackowski, S.; Słominska, ´ J.: G-functors, G-posets and homotopy decompositions of G-spaces. Fund. Math. 169(3), 249–287 (2001) 7. Saunders, M.L.: Categories for the working mathematician, Graduate Texts in Mathematics, Volume 5 (Springer, New York, 1998), 2nd edn. 8. May, J.P.: Simplicial Objects in Algebraic Topology. Van Nostrand, Princeton (1967) 9. Meyer, J.-P.: Bar and cobar constructions. I. J. Pure Appl. Algebra 33(2), 163–207 (1984) 10. Dieck, T.: Transformation Groups. Walter de Gruyter, Berlin (1987) 11. Villarroel-Flores, R.: Equivariant Homotopy Type of Categories and Preordered Sets. Ph.D. thesis, University of Minnesota. (1999) 12. Villarroel-Flores, R.: Homotopy equivalence of simplicial sets with a group action. Bol. Soc. Mat. Mexicana (3) 6(2), 247–262 (2000) 13. Welker, V.; Ziegler, G.M.; Živaljevic, ´ R.T.: Homotopy colimits–comparison lemmas for combinatorial applications. J. Reine Angew. Math. 509, 117–149 (1999) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

Journal

Arabian Journal of MathematicsSpringer Journals

Published: Mar 25, 2023

Keywords: 55U10

References

Access the full text.

Sign up today, get DeepDyve free for 14 days.