REPRESENTATIONS OF Spaces

WOJCIECH CHACHÓLSKI AND JÉRÔME SCHERER

Abstract. We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Spaces.

1. Introduction

Let C be a category with a chosen class of weak equivalences W (W is a collection of morphisms in C that satisfies the “2 out of 3” property and contains all isomorphisms). What does it mean that one can do homotopy theory on (C, W)? For us it means three things. First, we should be able to perform basic operations in C such as push-outs/pull-backs and more generally colimits/limits. Second, the derived versions of these functors (hocolims and holims) should exist. Third, these derived functors should yield a certain action of the category of simplicial sets Spaces on (C, W) and lead to mapping spaces. Our attitude is that colimits/limits and their derived functors are the basic fundamental ingredients of homotopy theory and the rest should follow from their properties. This is an alternative to the approach of M. Hovey presented in [9] and more in the spirit of what we believe we have learned from Bousfield, Dwyer, Kan and others, [1] and [4].

In this paper we study the following functor:

⊗_{l} : C × Spaces → Ho(C) X ⊗_{l} K := hocolim_{K}cX

where Ho(C) = C[W^{−1}] and cX : K → C is the constant functor indexed by the simplex category of K (see 2.6) whose value is X. We would like ⊗_{l} to be the composition of the localization functor C × Spaces → Ho(C) × Ho(Spaces) and a functor denoted by the same symbol ⊗_{l} : Ho(C) × Ho(Spaces) → Ho(C) (homotopy invariance). The later should be a homotopy left action (see 2.11 and 2.15), which implies in particular that, for any X ∈ Ho(C), the functor X ⊗_{l} − : Ho(Spaces) → Ho(C) has a right adjoint map(X, −). This is in our view the universal characterization of what mapping spaces should be.

2000 Mathematics Subject Classification. Primary 55U35,18G55 Secondary 18G10.

The first author is supported in part by NSF grant DMS-0296117, Vetenskapsrådet grant 2001-4296, and Göran Gustafssons Stiftelse, the second author by the program Ramón y Cajal, MCyT, Spain.

How can this be realized? In order to insure the existence of the localized category C[W^{−1}] and the derived functors hocolims and holims satisfying the above conditions, we need to be able to do some homotopical algebra on (C, W). One way of formalizing this is to assume that (C, W) can be given a Quillen (simplicial) model structure [11]. Dwyer and Kan explained in [5] how to enrich such a category with an action of Spaces using the so called “hammocks”. They also introduce the technique of taking (co)simplicial resolution of the source (target), further developed by M. Hovey in [9], as well as P. Hirschhorn in [8] to obtain mapping spaces. Alternatively one could try to find conditions under which a model category is Quillen equivalent to a simplicial model category, as defined by Quillen in [11, Section II.1]. This has been done by D. Dugger in [3] and C. Rezk, B. Shipley, and S. Schwede in [12]. These approaches however do not seem to emphasis the fundamental role of hocolims and holims.

Since in our view hocolims and holims play such an important role we need a suitable set up in which such constructions can be effectively studied. It turns out that for that purpose, instead of putting additional assumption on a model category (cellularity [8], realization axiom [12] etc...), it is more advantageous to relax some of the requirements. The aim of this paper is to explain the less restrictive approach of model approximations as introduced in [2]. Recall:

Definition 1.1. A left model approximation of (C, W) is a pair of adjoint functors l : M ⇄ C : r satisfying the following conditions:

(1) M is a model category;

(2) l is left adjoint to r;

(3) if f is a weak equivalence in C, then rf is a weak equivalence in M;

(4) if f is a weak equivalence between cofibrant objects in M, then lf is a weak equivalence in C;

(5) for any A ∈ C and cofibrant X ∈ M, if X → rA is a weak equivalence in M, then its left adjoint lX → A is a weak equivalence in C.

A pair of adjoint functors l : C ⇄ M : r is a right model approximation of (C, W) if the duals r^{∨} : M^{op} ⇄ C^{op} : l^{∨} form a left model approximation of (C^{op}, W).

Model approximations have several crucial properties. A category admitting a model approximation (left or right) can be localized with respect to weak equivalences to form the homotopy category Ho(C) [2, Proposition 5.5]. Weak equivalences in such categories are saturated, i.e., a morphism in C belongs to W if and only if the induced morphism in Ho(C) is an isomorphism. The category Ho(C) can be identified with a full subcategory of Ho(M) via the functor r or l, depending on whether the approximation is left or right ([2, Proposition 5.5]).

A left model approximation l : M ⇄ C : r leads to a “cofibrant replacement” in C given by the adjoint lQrA → A of a cofibrant replacement QrA → rA in M [2, Remark 5.10]. In a dual way a right model approximation leads to a fibrant replacement. These replacements can be used to construct derived functors. If F : C → D is a functor for which the composition F l sends weak equivalences between cofibrant objects in M to isomorphisms in D, then the left derived functor of F is given by A ↦ F (lQrA) [2, Proposition 5.9, Remark 5.10]. In the same way right model approximations can be used to construct right derived functors. A left model approximation l : M ⇄ C : r induces a natural left model approximation on the category of functors Fun(I, C), for any small category I (see Theorem 6.5). A cofibrant replacement of this model approximation is good for constructing the total left derived functor of colim_{I} : Fun(I, C) → C. Same applies to right model approximations.

To make the approach of model approximations viable we need one more thing. We need to show that a model approximation leads in a natural way to an action of Spaces. The purpose of [2] was to introduce simple geometric techniques to study hocolims and holims. In this paper we are going to illustrate how to use these techniques further, which is what motivated us originally. We did not want to just state and prove some properties of mapping spaces, but more importantly explain an approach and provide coherent tools to study them. This will be illustrated by proving the following theorem (see Proposition 9.3 and 10.7):

Theorem Assume that (C, W) admits a left model approximation. The functor ⊗_{l} : C × Spaces → Ho(C) is homotopy invariant and for any X ∈ C the functor X⊗_{l}− : Ho(Spaces) → Ho(C) has a right adjoint map(X, −).

Assume that (C, W) admits both a left and a right model approximations. Then, for any space K, the functor − ⊗_{l} K : Ho(C) → Ho(C) is left adjoint to X ↦ holim_{K}cX.

As an application we prove that the homotopy category of any monoidal model category (as defined in [9, Definition 4.2.6]) is always a central algebra over the homotopy category of Spaces, as conjectured in [9].

Proposition 11.1 Let M be a monoidal model category. Then the functor ⊗_{l} : Ho(M) × Ho(Spaces) → Ho(M) describes a central algebra over Ho(Spaces).

2. Notation and set up

2.1. Cat denotes the category of small categories and Spaces the category of simplicial sets. A simplicial set is also called a space.

2.2. From Section 3 on categories denoted by the symbols C and D are assumed to contain all colimits and limits, compare with Quillen’s axiom (MC0) [11].

2.3. A category with weak equivalences is a category C with a chosen collection of morphisms W, called weak equivalences, which satisfies the “2 out of 3” property and contains all isomorphisms. Elements in W are denoted by the symbol ⟶^{∼}. The category C[W^{−1}], if it exists, is called the homotopy category and is denoted by Ho(C). By π_{C} : C → Ho(C) we denote the localization functor.

2.4. The category of all C-valued functors indexed by a small category I, with natural transformations as morphisms, is denoted by Fun(I, C). The symbol cX : I → C denotes the constant functor indexed by I with value X ∈ C. If (C, W) is a category with weak equivalences, then Fun(I, C) stands for a category with weak equivalences given by the object-wise weak equivalences. A functor F : I → C is called homotopically constant if there is an object X ∈ C and a weak equivalence F → cX in Fun(I, C).

2.5. Let I and J be small categories and f : I → J be a functor. The left and right adjoints to f^{∗} : Fun(J, D) → Fun(I, D), F ↦ F ∘f, if they exist, are called Kan extensions. They are denoted respectively by f^{k} : Fun(I, D) → Fun(J, D) and f_{k} : Fun(I, D) → Fun(J, D). These functors can be described explicitly in terms of the over-categories f ↓ j and under-categories j ↓ f [10, II.6]: (f^{k}F)(j) = colim_{f↓j}F and (f_{k}F)(j) = lim_{j↓f}F.

2.6. The simplex category of a simplicial set K, denoted by the same symbol K, is a category whose objects are simplices of K, or equivalently maps of the form σ : Δ[n] → K. The set of morphisms in K, between σ : Δ[n] → K and τ : Δ[m] → K, consists of those maps f : Δ[n] → Δ[m] for which σ = τf. Subject to standard cosimplicial identities, the morphisms in K are generated by the face and degeneracy morphisms:

\xymatrix{Δ[n] \ar[rr]^{d_{i}} \ar[dr] &  & Δ[n + 1] \ar[dl] & Δ[n + 1] \ar[rr]^{s_{i}} \ar[dr] &  & Δ[n] \ar[dl] \\  & K &  &  & K}

A map of spaces f : A → B induces a functor f : A → B. For any simplex σ ∈ B, the over category f ↓ σ → A is the simplex category given by the map of spaces lim(Δ[n] →^{σ} B ←^{f} A) → A. In particular the over category B ↓ σ → B can be identified with the map σ : Δ[n] → B

2.7. The functor Spaces → Cat, K ↦ K, has a right adjoint and therefore converts colimits in Spaces into colimits in Cat. This functor also takes pullbacks of spaces into pull-backs of categories. It does not however preserve products since the simplex category of a point is the category of finite ordered sets Δ. For that reason we use the symbol A ⊠ B to denote the product of the simplex categories of A and B. The natural inclusion A × B → A ⊠ B is never an equivalence of categories.

2.8. N : Cat → Spaces denotes the nerve construction [2, Definition 6.3]. It has a left adjoint and thus it converts limits in Cat into limits in Spaces.

2.9. The composition Spaces → Cat ⟶^{N} Spaces is called the barycentric subdivision. It has a right adjoint and so it commutes with colimits.

2.10. A functor f : I → J between small categories is called a weak equivalence if N(f) is a weak equivalence in Spaces. A map of spaces f : A → B is a weak equivalence if and only if the induced functor of simplex categories is a weak equivalence. For any spaces A and B, the inclusion A × B → A ⊠ B is a weak equivalence of categories.

2.11. A functor ⊗ : C × Spaces → C is called a left action (of Spaces on C) if: (1) there are isomorphisms X ⊗ Δ[0] ≅ X natural in X;

(2) there are coherent isomorphisms X ⊗ (K × L) ≅ (X ⊗ K) ⊗ L natural in X ∈ C and K, L ∈ Spaces (the explicit coherency diagrams can be found in [9, Definition 4.1.6]);

(3) for any X ∈ C, X ⊗ − : Spaces → C has a right adjoint map(X, −). Note that if C is closed under taking colimits, then condition (3) can be

replaced by the requirement:

(3’) for any X ∈ C, X ⊗ − : Spaces → C commutes with arbitrary colimits.

2.12. A simplicial functor between left actions on C and D is, by definition, a functor ϕ : C → D together with a natural transformation μ : ϕ(X) ⊗ K → ϕ(X ⊗ K) for which the composition ϕ(X) ⊗ Δ[0] ⟶^{μ} ϕ(X ⊗ Δ[0]) ≅ ϕ(X) is the natural isomorphism in 2.11(1).

A simplicial functor can also be described as a functor ϕ : C → D together with a natural transformation map(X, Y) → map(ϕ(X), ϕ(Y)) which on 0dimensional simplices coincides with ϕ (by adjunction the 0-simplices identify with morphisms out of X ⊗ Δ[0]).

2.13. A functor hom : Spaces^{op}×C → C is called a right action (of Spaces on C) if hom^{op} is a left action. Explicitly it has to satisfy the following conditions:

(1d) there are isomorphisms X ≅ hom(Δ[0], X) natural in X;

(2d) there are coherent isomorphism hom(K × L, X) ≅ hom(K, hom(L, X))

natural in X ∈ C and K, L ∈ Spaces; (3d) for any X ∈ C, hom(−, X) : Spaces^{op} → C has a left adjoint map(−, X). A simplicial functor between right actions on C and D is a functor ϕ : C →

D together with a natural transformation ϕ(hom_{C}(K, Y)) → hom_{D}(K, ϕ(X)) satisfying an identity analogous to the one in 2.12.

2.14. A left action ⊗ : C × spaces → C which satisfies the extra condition:

(4) The functor −⊗K : C → C has a right adjoint hom(K, −) for any space

K. is called a simplicial structure on C or simply an action. If such an action is fixed C is called a simplicial category.

A simplicial structure can also be described in dual terms as a right action hom(−, −) : Spaces^{op} × C → C for which:

(4d) hom(K, −) : C → C has a left adjoint − ⊗ K, for any space K.

2.15. If instead of Spaces we take the homotopy category Ho(Spaces) in the above definitions, then we get the notions of homotopy (left/right) actions of Spaces on C and simplicial functors. A weak simplicial structure on C is, by definition, a homotopy action. If such an action is fixed, C is called a weak simplicial category.

2.16. If ⊗ : C × Ho(Spaces) → C is a homotopy left action, then the set of morphisms mor_{C}(X, Y) can be identified with the set of connected components π_{0}map(X, Y). It follows that a simplicial functor between homotopy left actions on C and D can be described either as in 2.12 or as a pair consisting of a functor ϕ : C → D and a natural transformation map_{C}(X, Y) → map_{D}(ϕ(X), ϕ(Y)) which on the set of connected components coincides with ϕ. The same applies to simplicial functors of homotopy right actions.

2.17. Let (C, W) and (D, V) be categories with weak equivalences whose homotopy categories exist. The total left derived functor of Φ : C → D is the left Kan extension [10] of π_{C} : C → Ho(C) applied to the composition of Φ and π_{D} : D → Ho(D). Explicitly, it is a functor LΦ : C → Ho(D) with a natural transformation LΦ → π_{D}Φ which is terminal among natural transformations with the range π_{D}Φ and whose domains are functors factoring through π_{C} : C → Ho(C).

A left replacement for Φ : C → D is a functor Q : C → C and a natural morphism QX → X in C which is a weak equivalence such that π_{D}ΦQ → π_{D}Φ is the total left derived functor of Φ.

The total right derived functor of Φ : C → D is the right Kan extension [10] of π_{C} : C → Ho(C) applied to the composition of π_{D} : D → Ho(D) and Φ. A right replacement for Φ is a functor R : C → C and a natural morphism X → RX in C which is a weak equivalence such that π_{D}Φ → π_{D}ΦR is the total right derived functor of Φ.

2.18. We say that (C, W) can be left localized if, for any small category I, the homotopy category of Fun(I, C) exists and the functor colim_{I} : Fun(I, C) → C has a left replacement. In particular the total left derived functor of colim_{I} exists and is denoted by hocolim_{I} : Fun(I, C) → Ho(C). It is given by the formula hocolim_{I}F = π_{C}(colim_{I}QF).

We say that (C, W) can be right localized if, for any small category I, the homotopy category of Fun(I, C) exists and the functor lim_{I} : Fun(I, C) → C has a right replacement. In particular the total right derived functor of lim_{I} exists and is denoted by holim_{I} : Fun(I, C) → Ho(C).

2.19. Assume that (C, W) can be left localized. By the universal property, for any F : J → C and f : I → J, there is a morphism hocolim_{I}f^{∗}F → hocolim_{J}F in Ho(C) natural with respect to f and F. This morphism can be described in terms of left replacements Q_{I} and Q_{J} of colim_{I} and colim_{J}. It is represented by the following sequence of morphisms in C:

colim_{I}(Q_{I}f^{∗}F) ⟵^{∼} colim_{I}(Q_{I}f^{∗}Q_{J}F) → colim_{I}(f^{∗}Q_{J}F) → colim_{J}(Q_{J}F)

In the case J is the category with one object and one morphism, this leads to a functor:

⊗_{l} : C × Cat → Ho(C) X ⊗_{l} I := hocolim_{I}cX

Dually, when (C, W) can be right localized, by the universal property, for any F : J → C and f : I → J, there is a morphism holim_{J}F → holim_{I}f^{∗}F in Ho(C) natural with respect to f and F. In the case J is the category with one object and one morphism, this leads to a functor:

rhom : Cat^{op} × C → Ho(C) rhom(I, X) := holim_{I}cX

2.20. Assume that (C, W) and (D, V) are categories that can be left localized. Let I be a small category, F : I → C be a functor, and Q_{C} and Q_{D} be left replacements of colim_{I} : Fun(I, C) → C and colim_{I} : Fun(I, D) → D respectively. If Φ : C → D is a functor that sends weak equivalences in C to weak equivalences in D, then the following sequence of morphisms in D:

colim_{I}Q_{D}ΦF ⟵^{∼} colim_{I}Q_{D}ΦQ_{C}F → colim_{I}ΦQ_{C}F → Φ(colim_{I}Q_{C}F)

leads to a morphism hocolim_{I}ΦF → Φ(hocolim_{I}F) in Ho(D). This morphism is natural in F and does not depend on the choice of the left replacements Q_{C} and Q_{D}.

Dually, let (C, W) and (D, V) be categories that can be right localized. Assume that Φ : C → D sends weak equivalences in C to weak equivalences in D. As in the case of hocolims, for any small category I and any functor F : I → C, there is a natural morphism Φ(holim_{I}F) → holim_{I}ΦF in Ho(D).

3. Simplex Categories

To understand actions of Spaces one can try to look at the problem locally and study functors indexed by simplex categories (2.6). Such functors are called representations of the indexing space. What is so special about simplex categories that enables us to say something about their representations?

Simplex categories have one important advantage over arbitrary small categories of which we give now an elementary illustration. Let g : I → X be a functor of small categories and G : I → C be a functor. Let us look for conditions under which the operation G ↦ g^{k}G commutes with the base change. Explicitly, consider a pull-back of small categories:

\xymatrix{P \ar[r]^{f_{1}} \ar[d]_{g_{1}} & I \ar[d]^{g} \\ J \ar[r]^{f} & X}

For any j ∈ J, there is a functor of over-categories g_{1} ↓ j → g ↓ f(j). These functors induce a natural transformation:

(g_{1}^{k}f_{1}^{∗}G)(j) = colim_{g_{1}↓j}G → colim_{g↓f(j)}G = (f^{∗}g^{k}G)(j)

We ask: when is this natural transformation an isomorphism? If G = g^{∗}F, for some F : X → C, this natural transformation is of the form:

g_{1}^{k}g_{1}^{∗}f^{∗}F = g_{1}^{k}f_{1}^{∗}g^{∗}F → f^{∗}g^{k}g^{∗}F

We could then apply f^{k} to it and ask: when is (fg_{1})^{k}(fg_{1})^{∗}F → f^{k}f^{∗}g^{k}g^{∗}F an isomorphism? In particular, when is the operation F ↦ f^{k}f^{∗}g^{k}g^{∗}F symmetric with respect to f and g?

Let us look at two examples:

Example 3.1. First a negative case. Consider a pull-back of categories:

\xymatrix{∅ \ar[r]^{f_{1}} \ar[d]_{g_{1}} & 1 \ar[d]^{g} \\ 0 \ar[r]^{f} & \{0 → 1\}}

and a functor {A → B} : {0 → 1} → Spaces. It can be verified that:

g^{k}g^{∗}{A → B} = {∅ → B} and f^{k}f^{∗}{A → B} = {A →^{id} A}

In particular:

f^{k}f^{∗}g^{k}g^{∗}{A → B} = {∅ → ∅} ≅ g̸^{k}g^{∗}f^{k}f^{∗}{A → B} = {∅ → A}

Here is a positive case. Consider a pull-back of categories:

\xymatrix{∅ \ar[rr]^{f_{1}} \ar[d]_{g_{1}} &  & 1 \ar[dl]^{g} \\ 0 \ar[r]^{f} & \{0 → 01 ← 1\}}

and a functor {A → B ← C} : {0 → 01 ← 1} → Spaces. In this case:

g^{k}g^{∗}{A → B ← C} = {∅ → C ←^{id} C}

f^{k}f^{∗}{A → B ← C} = {A →^{id} A ← ∅}

In particular:

f^{k}f^{∗}g^{k}g^{∗}{A → B ← C} = {∅ → ∅ ← ∅}
g^{k}g^{∗}f^{k}f^{∗}{A → B ← C} = {∅ → ∅ ← ∅}

which agrees with (gf_{1})^{k}(gf_{1})^{∗}{A → B ← C}.

What is so special about the second example? It turns out that it is an example of functors indexed by simplex categories. The reason this does not work in general is the fact that arbitrary categories can have vastly different over-categories. Simplex categories on the other hand look the same locally. For any simplex σ : Δ[n] → K, the over-category K ↓ σ → K is isomorphic to the functor induced by the map σ : Δ[n] → K. Thus locally, all simplex categories look like standard simplices.

Proposition 3.2. Let the following be a pull-back square of spaces:

\xymatrix{P \ar[r]^{f_{1}} \ar[d]_{g_{1}} & A \ar[d]^{g} \\ B \ar[r]^{f} & D}

For any functor G : A → C, the natural transformation g_{1}^{k}f_{1}^{∗}G → f^{∗}g^{k}G is an isomorphism. In particular, for any F : D → C, the natural transformations g_{1}^{k}g_{1}^{∗}f^{∗}F → f^{∗}g^{k}g^{∗}F and (fg_{1})^{k}(fg_{1})^{∗}F → f^{k}f^{∗}g^{k}g^{∗}F are isomorphisms and the operation F ↦ f^{k}f^{∗}g^{k}g^{∗}F is symmetric with respect to f and g.

Proof. Note that for any simplex σ : Δ[n] → B, the over-categories g_{1} ↓ σ and g ↓ f(σ) can be identified with the simplex category of the pull-back:

lim(Δ[n] →^{σ} B ←^{g_{1}} P)

Moreover the functor g_{1} ↓ σ → g ↓ f(σ) is an isomorphism. □

4. Bounded functors

Simplex categories have another advantage. There are a lot of geometric constructions that can be performed on spaces. These constructions often translate well to operations on representations. For example, fix a functor H : I → Spaces. The simplex category of colim_{I}H can be identified with the colimit of H in the category of small categories. A covariant functor F indexed by colim_{I}H consists of the following data:

(1) for every object i in I, a functor F_{i} : H(i) → C;

(2) for every morphism α : i → j in I, F_{i} : H(i) → C should coincide with

the composition H(i) ⟶^{H(α)} H(j) ⟶^{F_{j}} C.

This decomposition yields an isomorphism colim_{colim_{I}H}F ≅ colim_{I}colim_{H(i)}F_{i}, see [2, Proposition 8.2].

By applying the above procedure inductively, we can calculate colimits using the cell decomposition of the indexing space. Diagrams indexed by the standard simplices Δ[n] play the role of fundamental building blocks in this process. Unfortunately representations of Δ[n] are complicated. For example, the simplex category of Δ[0] is equivalent to the category of finite ordered sets and so its representations are cosimplicial and simplicial objects, depending on variance. For that reason we need to make additional assumptions on considered functors. We restrict our attention to diagrams that reflect the geometry of the indexing space more directly. A functor F, indexed by A, is called bounded [2, Definition 10.1] if it assigns an isomorphism to any morphism of the form s_{i} : Δ[n + 1] → Δ[n] in A. If S denotes the set of all these degeneracy morphisms in A, then a bounded functor is nothing else but a functor indexed by the localized category A[S^{−1}]. Depending on the variance, the categories of bounded diagrams are denoted by Fun^{b}(A, C) and Fun^{b}(A^{op}, C).

An important observation is that the degeneracy morphisms in A do not form any essential loop. Any bounded functor is naturally isomorphic to a functor that assigns identities to morphisms in S [2, Proposition 10.3]. This can be therefore assumed about any considered bounded functor.

Example 4.1. The “triviality” of S does not imply that the localization functor A → A[S^{−1}] is a weak equivalence (see 2.10). Let:

A = colim(Δ[3]/Δ[2] ⟵^{π} Δ[3, 1] ↪ Δ[3])

where π : Δ[3, 1] → Δ[3]/Δ[2] is the composition of Δ[3, 1] ↪ Δ[3] and the quotient map Δ[2] ⟶^{d_{1}} Δ[3] → Δ[3]/Δ[2]. Although A is contractible, the category A[S^{−1}] is not. The nerve N(A[S^{−1}]) is weakly equivalent to S^{2}.

Let A = Δ[n]/∂Δ[n]. For n > 1, the category A[S^{−1}] is equivalent to ∗ → ∗ (the category with two objects and one non-identity morphism).

It is clear that, for any map of spaces f : A → B, f^{∗} : Fun(B, C) → Fun(A, C) takes bounded functors to bounded functors. The same is also true for the left Kan extension f^{k} : Fun(A, C) → Fun(B, C) [2, Theorem 33.1]. If follows that the functors f^{k} : Fun^{b}(A, C) ⇄ Fun^{b}(B, C) : f^{∗} form an adjoint pair.

5. Subdivision

In the previous section we briefly described what effect a cell attachment has on representations of spaces. In this section we look at another geometric operation: barycentric subdivision. Recall that the subdivision of A is given by the nerve N(A) of the simplex category of A.

A map of spaces f : A → B is called reduced if it sends non-degenerate simplices in A to non-degenerate simplices in B [2, Definition 12.9]. Reduced maps can be characterized using a lifting test: the map f : A → B is reduced if and only if in any commutative diagram of the form:

\xymatrix{Δ[n + 1] \ar[d]_{s_{i}} \ar[r] & A \ar[d]^{f} \\ Δ[n] \ar[r] & B}

there is a lift, i.e. a morphism Δ[n] → A for which the resulting diagram with five arrows commutes [2, Proposition 18.5]. It follows that a pull-back of a reduced map is also reduced.

The subdivision operation has two crucial properties. First:

Proposition 5.1 ([2, Example 12.10]). For any map of spaces f : A → B, the induced map N(f) : N(A) → N(B) is reduced.

Second, the subdivision N(A) contains enough combinatorial information to induce a non-trivial simplicial structure on Fun^{b}(N(A), C) (the non-triviality follows from Corollary 8.6). Here is its description. Let K be a space and p_{K} : A×K → A be the projection onto the first factor. Denote the subdivision of p_{K} by π : N(A × K) → N(A). Define:

F ⊗ K := π^{k}π^{∗}F

This construction is natural in F and K and hence defines a functor:

⊗ : Fun^{b}(N(A), C) × Spaces → Fun^{b}(N(A), C)

Example 5.2. We do not know any explicit formula describing F ⊗K in terms of the values of F ∈ Fun^{b}(N(A), C). However if F : N(A) → C is a composition of G : A^{op} → C and ϵ : N(A) → A^{op}, ϵ(σ_{n} → σ_{n−1} → ⋯ → σ_{0}) := σ_{n}, then F ⊗ K is given by:

(σ_{n} → σ_{n−1} → ⋯ → σ_{0}) ↦ ∐_{K_{|σn|}} G(σ_{n})

This formula should be compared with the simplicial structure introduced by Quillen on simplicial objects in [11, II.1 Proposition 2]

Proposition 5.3. For any space A, the functor ⊗ defines a simplicial structure on Fun^{b}(N(A), C) (see 2.11 and 2.14).

Proof. We need to check 4 conditions (see 2.14). It is clear that the natural morphism F ⊗ Δ[0] = id^{k}id^{∗}F → F is an isomorphism and so the first requirement is satisfied.

Consider two spaces K and L and a pull-back square:

\xymatrix{N(A × K × L) \ar[r]^{π̄_{L}} \ar[d]_{π̄_{K}} & N(A × K) \ar[d]^{π_{K}} \\ N(A × L) \ar[r]^{π_{L}} & N(A)}

According to Proposition 3.2, for any F : N(A) → C, the transformation:

F ⊗ (K × L) = (π_{L}π̄_{K})^{k}(π_{L}π̄_{K})^{∗}F → (π_{L})^{k}(π_{L})^{∗}(π_{K})^{k}(π_{K})^{∗}F = (F ⊗ K) ⊗ L

is an isomorphism. Coherency of these isomorphisms follows directly from coherency of products in Spaces. That proves the second requirement.

To prove condition (3’), we need to show that, for a fixed F ∈ Fun^{b}(N(A), C), the functor F ⊗ − commutes with arbitrary colimits. Let H : I → Spaces be a functor. Observe that the map colim_{I}(A × H) → A × colim_{I}H is an isomorphism of spaces and so is the map colim_{I}N(A × H) → N(A × colim_{I}H) as the subdivision commutes with arbitrary colimits (see 2.9). Next, choose a simplex σ : Δ[n] → N(A) and define:

P (i) := lim(Δ[n] →^{σ} N(A) ←^{π_{i}} N(A × H(i)))

where π_{i} is the subdivision of the projection pr_{1} : A × H(i) → A. The colimit of P can be identified with the pull-back:

colim_{I}P ≅ lim(Δ[n] →^{σ} N(A) ←^{π} N(A × colim_{I}H))

These isomorphisms yield the following identifications:

(F ⊗ colim_{I}H)(σ) = (π^{k}π^{∗}F)(σ) ≅ colim_{colim_{I}P} F ≅

≅ colim_{i∈I}colim_{P (i)}F = colim_{i∈I}((π_{i})^{k}(π_{i})^{∗}F )(σ) = colim_{I}(F ⊗ H)(σ)

Note that for a fixed K, the functor − ⊗ K has a right adjoint as it is a composition of two functors that admit right adjoints (here we use our general assumption that C has all limits, see 2.2). This proves the fourth requirement.

□

Taking the subdivision in Proposition 5.3 is crucial. In a similar way one can define a simplicial structure on Fun^{b}(A, C) by F ⊗ K := (p_{K})^{k}(p_{K})^{∗}F, where p_{K} : A × K → A is the projection onto the first factor. This however does not give anything interesting as the obtained structure is trivial.

5.4. Naturality. How natural is this simplicial structure on Fun^{b}(N(A), C)? Consider the following pull-back of spaces associated with a map f : A → B and a space K:

\xymatrix{N(A × K) \ar[d]_{N(f×id)} \ar[r]^{π} & N(A) \ar[d]^{N(f)} \\ N(B × K) \ar[r]^{π_{1}} & N(B)}

where the horizontal maps are the subdivisions of the projections. For any F ∈ Fun^{b}(N(B), C), there is a natural transformation:

(N(f)^{∗}F ) ⊗ K = π^{k}π^{∗}N(f)^{∗}F → N(f)^{∗}(π_{1})^{k}(π_{1})^{∗}F = N(f)^{∗}(F ⊗ K)

which is an isomorphism by Proposition 3.2. Note that this isomorphism is functorial with respect to K and F. Together with this isomorphism the functor N(f)^{∗} : Fun^{b}(N(B), C) → Fun^{b}(N(A), C) becomes a simplicial functor (see 2.12).

For any G ∈ Fun(N(A), C), there is a natural transformation:

N(f × id)^{k}π^{∗}G → (π_{1})^{∗}N(f)^{k}G

which is an isomorphism again by Proposition 3.2. We can apply to it (π_{1})^{k} and get a natural isomorphism:

N(f)^{k}(G ⊗ K) = N(f)^{k}π^{k}π^{∗}G = (π_{1})^{k}N(f × id)^{k}π^{∗}G →
→ (π_{1})^{k}(π_{1})^{∗}N(f)^{k}G = (N(f)^{k}G) ⊗ K

It is not difficult to see that this isomorphism is functorial with respect to K and G. The functor N(f)^{k} : Fun^{b}(N(A), C) → Fun^{b}(N(B), C), together with the inverse of this isomorphism, becomes a simplicial functor.

Let ϕ : C → D be a functor and K be a simplicial set. Consider the following pull-back of spaces:

\xymatrix{P_{σ} \ar[r] \ar[d] & N(A × K) \ar[d]^{π} \\ Δ[n] \ar[r]^{σ} & N(A)}

By the universal property of the colimit construction, for any F : N(A) → C, there is a morphism in D:

((ϕF) ⊗ K)(σ) = colim_{P_{σ}}(ϕF π) → ϕ(colim_{P_{σ}}F π) = ϕ(F ⊗ K)(σ)

These morphisms form a natural transformation (ϕF) ⊗ K → ϕ(F ⊗ K) functorial with respect to K and F. With this natural transformation, the functor ϕ_{∗} : Fun^{b}(N(A), C) → Fun^{b}(N(A), D) becomes a simplicial functor.

14 WOJCIECH CHACHÓLSKI AND JÉRÔME SCHERER

6. Homotopy theory of bounded functors

Simplex categories are also rather special from the homotopy point of view. Theorem 6.1 ([2, Theorem 21.1]). Let M be a model category. The following describes a model structure on Fun^{b}(A, M):

• ϕ : F → G is a weak equivalence (fibration) if, for any simplex σ ∈ A, ϕ_{σ} : F (σ) → G(σ) is a weak equivalence (fibration) in M;

• ϕ : F → G is a (acyclic) cofibration if, for any non-degenerate simplex

σ : Δ[n] → A, the morphism:

colim(F (σ) ← colim_{∂Δ[n]}F → colim_{∂Δ[n]}G) → G(σ) is a (acyclic) cofibration in M.

The description of (acyclic) cofibrations in Theorem 6.1 can be used by induction on the cell attachment to show: Proposition 6.2 ([2, Proposition 20.1]). If ϕ : F → G is a (acyclic) cofibration in Fun^{b}(A, M), then, for any subspace X ⊂ A, the morphism:

colim(colim_{X}G ←^{ϕ} colim_{X}F → colim_{A}F ) ↪ colim_{A}G

is an (acyclic) cofibration in M.

How functorial is this model structure? Proposition 6.3. Let f : A → B be a map of Spaces.

(1) If ϕ is an (acyclic) cofibration in Fun^{b}(A, M), then f^{k}ϕ is an (acyclic) cofibration in Fun^{b}(B, M) [2, Theorem 11.2.]. In particular colim_{A}ϕ is a (acyclic) cofibration in M.

(2) If f is reduced and ϕ is an (acyclic) cofibration in Fun^{b}(B, M), then so is f^{∗}ϕ in Fun^{b}(A, M).

(3) If ϕ is an (acyclic) cofibration in Fun^{b}(N(B), M), then so is N(f)^{∗}ϕ in Fun^{b}(N(A), M).

The model structure given in Theorem 6.1 was introduced to study homotopy invariance of the colimit operation. The motivation was to explain: Theorem 6.4 ([2, Theorem 20.3, Corollary 21.4]). Let ϕ : F → G be a natural transformation in Fun^{b}(A, M). If ϕ is a weak equivalence between cofibrant objects, then so is colim_{A}ϕ.

Theorem 6.4 implies that the functor colim_{A} : Fun^{b}(A, M) → M has a left replacement (see 2.17) given by a cofibrant replacement Q in Fun^{b}(A, M). In particular its total left derived functor denoted by ocolim_{A} : Fun^{b}(A, M) → Ho(M) assigns to a bounded functor F the object colim_{A}(QF) in M [2, Proposition 14.2]. The functor ocolim_{A} coincide with hocolim_{A[S^{−1}]} (the total left derived functor of colim_{A[S^{−1}]} : Fun(A[S^{−1}], M) → M). In general ocolim_{A} is not the restriction of the total left derived functor of colim_{A} : Fun(A, M) → M. To construct hocolim_{A} an additional step is necessary: A needs to be subdivided first. This works in general for any small category I and any category that admits a left model approximation.

Theorem 6.5 ([2, Theorem 11.3]). Let I be a small category, ϵ : N(I) → I the forgetful functor (i_{n} → ⋯ → i_{0}) ↦ i_{0} and l : M ⇆ C : r a left model approximation. Then the following functors form a left model approximation:

lϵ^{k} : Fun^{b}(N(I), M) ⇄ Fun(I, C) : rϵ^{∗}

Moreover, F ↦ lϵ^{k}Q(rϵ^{∗}F) is a left replacement for colim_{I} : Fun(I, C) → C.

Corollary 6.6. If (C, W) admits a left model approximation, then it can be left localized (see 2.18). In particular the assignment (X, I) ↦ hocolim_{I}cX defines a functor ⊗_{l} : C × Cat → Ho(C).

Proof. According to Theorem 6.5, any functor F : I → C can be functorially modified to Q_{I}F := lϵ^{k}Q(rϵ^{∗}F) so that colim_{I}Q_{I}F represents hocolim_{I}F in Ho(C). □

For categories admitting left model approximations, homotopy colimits can be characterized in an alternative way:

Corollary 6.7. If (C, W) admits a left model approximation, then the functor hocolim_{I} : Ho(Fun(I, C)) → Ho(C) is left adjoint to the constant diagram functor c : Ho(C) → Ho(Fun(I, C)).

Proof. If M is a model category, then colim_{K} : Fun^{b}(K, M) → M preserves (acyclic) cofibrations (Proposition 6.3). Thus by [6, Theorem 9.7], the total derived functors of the adjoint functors colim_{K} : Fun^{b}(K, M) ⇄ M : c also form an adjoint pair ocolim_{K} : Ho(Fun^{b}(K, M)) ⇄ Ho(M) : c.

Choose a left model approximation l : M ⇆ C : r. Let F : I → C be a functor and Y an object in C. We have the following bijections between sets of morphisms respectively in Ho(Fun(I, C)), Ho(Fun^{b}(N(I), M)), Ho(M), and Ho(C):

[F, cY] ≅^{(a)} [rϵ^{∗}F, rϵ^{∗}cY] = [rϵ^{∗}F, crY] ≅^{(b)} [ocolim_{N(I)}rϵ^{∗}F, rY] ≅^{(c)}

≅^{(c)} [L(l)ocolim_{N(I)}rϵ^{∗}F, Y] = [hocolim_{I}F, Y]

where (a) follows from the fact that lϵ^{k} : Fun^{b}(N(I), M) ⇄ Fun(I, C) : rϵ^{∗} is a left model approximation (Theorem 6.5) and [2, Proposition 5.5], (b) is a consequence of adjointness of ocolim_{N(I)} and c, and the last bijection (c) follows from the following Lemma 6.8. □

Lemma 6.8. If l : M ⇄ C : r is a left model approximation, then the total derived functors induce an adjoint pair L(l) : Ho(M) ⇄ Ho(C) : r.

Proof. The lemma is a consequence of two facts. First r : Ho(C) → Ho(M) is a fully-faithful embedding [2, Proposition 5.5]. Second L(l)(rX) → X is an isomorphism in Ho(C), for any X. □

7. Constant functors

Assume that (C, W) admits a left model approximation. To understand how this structure leads to an action of Spaces on C we are going to study properties of homotopy colimits of constant functors. The fundamental observation is:

Proposition 7.1. If f : I → J is a weak equivalence of small categories, then, for any X ∈ C, X ⊗_{l} f : X ⊗_{l} I → X ⊗_{l} J is an isomorphism in Ho(C).

Corollary 7.2. If f : K → L is a weak equivalence of spaces, then, for any X ∈ C, X ⊗_{l} f : X ⊗_{l} K → X ⊗_{l} L is an isomorphism in Ho(C).

Corollary 7.2 implies that the functor (X, K) ↦ hocolim_{K}cX = X ⊗_{l} K is a composition of the localization C × Spaces → Ho(C) × Ho(Spaces) and a functor which we denote by the same symbol:

⊗_{l} : Ho(C) × Ho(Spaces) → Ho(C)

The aim of this paper is to understand to what extent ⊗_{l} is a homotopy action. We start by proving that it satisfies the first two requirements of 2.16:

Corollary 7.3.

(1) There are isomorphisms X ⊗_{l}Δ[0] ≅ X in Ho(C) natural in X ∈ Ho(C).

(2) There are coherent isomorphisms X ⊗_{l} (K × L) ≅ (X ⊗_{l} K) ⊗_{l} L in Ho(C) natural in X ∈ Ho(C) and K, L ∈ Ho(Spaces).

Proof. Part (1) follows from Proposition 7.1 and the fact that the functor between the simplex category of Δ[0] and the category with only one object and one morphism is a weak equivalence of categories.

Recall that K × L ↪ K ⊠ L is a weak equivalence of categories. Thus by Proposition 7.1, X ⊗_{l} (K × L) → X ⊗_{l} (K ⊠ L) is an isomorphism in Ho(C). According to [2, Theorem 24.9] there is a further isomorphism in Ho(C):

X ⊗_{l} (K ⊠ L) = hocolim_{K⊠L}cX ≅ hocolim_{L}hocolim_{K}cX = (X ⊗_{l} K) ⊗_{l} L

Coherency of these isomorphisms is straightforward. This concludes the proof. □ Proof of 7.1. Since ϵ : N(I)^{op} → I, (i_{n} → ⋯ → i_{0}) ↦ i_{n} is terminal [2, Theorem 29.1], the map hocolim_{N(I)^{op}}cX → hocolim_{I}cX is an isomorphism in Ho(C) for any X ∈ C. To prove the proposition it remains to show that a weak equivalence of spaces f : K → L induces an isomorphism X ⊗_{l}f^{op} : X ⊗_{l}K^{op} → X ⊗_{l} L^{op} in Ho(C).

Let S be the collection of maps of spaces f : K → L such that, for any category (C, W) admitting a left model approximation and for any object X ∈ C, X ⊗_{l} f^{op} is an isomorphism in Ho(C). We claim that:

(1) If K and L are contractible, then any f : K → L belongs to S.

(2) Maps of the form Δ[n, k] ↪ Δ[n] belong to S.

(3) S is closed under retracts.

(4) S is closed under arbitrary sums.

(5) If an inclusion f : K ↪ L belongs to S, then, for any map K → A, so

does the inclusion A ↪ colim(A ← K ↪^{f} L).

(6) If a sequence of inclusion K_{i} ↪ K_{i+1} (i > 0) belong to S, then so does K_{1} ↪ colim_{i>0}K_{i}.

Once the claim is proved, the proposition clearly follows as properties (2)-(6) imply that any weak equivalence of spaces belongs to S.

Property (1) is a consequence of [2, Corollary 29.2] and (2) is a particular case of it. Property (3) is obvious. Since the arguments for (4), (5), and (6) are basically the same, we present the details of how to prove (5) only.

Let D := colim(A ← K ↪^{f} L) and fix a left model approximation l : M ⇄: C : r. For an object X ∈ C, choose a cofibrant replacement QX in Fun^{b}(N(D^{op}), M) of the constant diagram crX. Consider next the following push-out square of spaces:

\xymatrix{N(K^{op}) \ar@{^{(}->}[r]^{N(f^{op})} \ar[d] & N(L^{op}) \ar[d]^{h} \\ N(A^{op}) \ar@{^{(}->}[r] & N(D^{op})}

Note that all the maps in this diagram are reduced and the horizontal maps are cofibrations. It follows that h^{∗}QX is cofibrant in Fun^{b}(N(L^{op}), M) and consequently the morphism colim_{N(K^{op})}QX → colim_{N(L^{op})}QX is a cofibration (Proposition 6.2). Moreover it is an acyclic cofibration as f is assumed to be in S. This shows that colim_{N(A^{op})}QX → colim_{N(D^{op})}QX is also an acyclic cofibration between cofibrant objects. The morphism l(colim_{N(A^{op})}QX) → l(colim_{N(D^{op})}QX) is therefore a weak equivalence in C. Since it can be identified with X ⊗_{l} A^{op} → X ⊗_{l} D^{op}, the claim is proven.

8. An almost simplicial model structure

Let M be a model category and A be a space. The category Fun^{b}(N(A), M) has both a model structure (Theorem 6.1) and a simplicial structure (Proposition 5.3). How compatible are they? To what extent is it a simplicial model category [11]? How related are the functors:

⊗_{l} : Fun^{b}(N(A), M) × Spaces → Ho(Fun^{b}(N(A), M))
⊗ : Fun^{b}(N(A), M) × Spaces → Fun^{b}(N(A), M)?

These are the questions we are going to answer in this section.

Example 8.1. Let us choose a simplex σ : Δ[0] → N(A). By definition it is given by an object in A, i.e., a map Δ[n] → A. Let F ∈ Fun^{b}(N(A), Spaces) be a cofibrant diagram such that F (σ) ≠ ∅. Note that:

lim(Δ[0] ⟶^{σ} N(A) ⟵^{π} N(A × Δ[1])) = ∐_{mor(Δ[n],Δ[1])} Δ[0]

Therefore (F ⊗ Δ[1])(σ) = π^{k}π^{∗}F = ∐_{mor(Δ[n],Δ[1])} F (σ) which is not weakly equivalent to F (σ). It follows that the map Δ[1] → Δ[0] does not induce a weak equivalence of functors F ⊗ Δ[1] → F ⊗ Δ[0] = F. In particular the objects F ⊗ Δ[1] and F ⊗_{l} Δ[1] are not isomorphic in Ho(Fun^{b}(N(A), Spaces)). Hence Fun^{b}(N(A), Spaces) is not a simplicial model category with the structures given by Theorem 6.1 and Proposition 5.3

Despite the above negative example, the model and the simplicial structures on Fun^{b}(N(A), M) are closely related and almost compatible. Let ϕ : F → G be a morphism in Fun^{b}(N(A), M) and f : K → L be a map of spaces. Define:

H := colim(F ⊗ L ⟵^{F ⊗f} F ⊗ K ⟶^{ϕ⊗K} G ⊗ K)

Let ψ : H → G ⊗ L in Fun^{b}(N(A), M) be induced by f, ϕ, and the universal property of the colimit. The following proposition shows that half of Quillen’s axiom (SM7) [11] holds for the simplicial structure introduced in Section 5.

Proposition 8.2. If f : K ↪ L is an inclusion of spaces and ϕ : F → G an (acyclic) cofibration in Fun^{b}(N(A), M), then ψ : H → G ⊗ L is an (acyclic) cofibration in Fun^{b}(N(A), M).

Proof. We need to show that, for a non-degenerate σ : Δ[n] → N(A), the morphism:

i : colim(H(σ) ← colim_{∂Δ[n]}H → colim_{∂Δ[n]}(G ⊗ L)) ⟶ (G ⊗ L)(σ)

is an (acyclic) cofibration in M (Theorem 6.1). To identify both sides of this morphism consider the following commutative diagram, where all the squares are pull-backs:

\xymatrix{∂P \ar@{^{(}->}[r] \ar@{^{(}->}[d] & P \ar[r] \ar@{^{(}->}[d] & N(A × K) \ar@{^{(}->}[d] \\ ∂Q \ar@{^{(}->}[r] \ar[d] & Q \ar[r] \ar[d]^{p} & N(A × L) \ar[d] \\ ∂Δ[n] \ar@{^{(}->}[r] & Δ[n] \ar[r]^{σ} & N(A)}

All the vertical maps in this diagram are reduced, as they are pull-backs of reduced maps (Proposition 5.1). Moreover, since ∂Q ↪ Q and P ↪ Q are inclusions, the map R := colim(∂Q ← ∂P → P) → Q is a monomorphism. The fact that p : Q → Δ[n] is reduced, implies that the only non-degenerate simplices in Q \ ∂Q are of dimension n and p sends them to the only nondegenerate simplex of dimension n in Δ[n]. As σ is non degenerate in N(A) and (Q\R) ⊂ (Q\∂Q), it follows that the composition σp takes non-degenerate simplices in Q \ R to non-degenerate simplices in N(A).

The spaces in this diagram can be used to make the following identifications:

colim_{∂Δ[n]}(F ⊗ L) = colim_{∂Q}F  (F ⊗ L)(σ) = colim_{Q}F
colim_{∂Δ[n]}(G ⊗ L) = colim_{∂Q}G  (G ⊗ L)(σ) = colim_{Q}G
colim_{∂Δ[n]}(F ⊗ K) = colim_{∂P} F  (F ⊗ K)(σ) = colim_{P} F
colim_{∂Δ[n]}(G ⊗ K) = colim_{∂P} G  (G ⊗ K)(σ) = colim_{P} G

Thus the left hand side of the morphism i coincide with the colimit of the cube:

[figure: colim_{∂P} F; colim_{P} F; colim_{∂P} G; colim_{P} G; colim_{∂Q}F; colim_{Q}F; colim_{∂Q}G]

Note that:

colim(colim_{∂Q}F ← colim_{∂P} F → colim_{P} F ) = colim_{R}F
colim(colim_{∂Q}G ← colim_{∂P} G → colim_{P} G) = colim_{R}G

Thomason’s Theorem [2, Theorem 26.8] permits in general to decompose the (ho)colimit over a Grothendieck construction into simpler (ho)colimits. In the particular case of a cube as above, [2, Example 38.2] shows that the colimit can be also expressed as a pushout (see also [7, Claim 2.8]):

colim(colim_{Q}F ← colim_{R}F ↪ colim_{R}G)

Moreover i can be identified with the morphism induced by the inclusion R ⊂ Q and the natural transformation ϕ : F → G:

colim(colim_{Q}F ← colim_{R}F ↪ colim_{R}G) → colim_{Q}G

We can now apply Proposition 6.2 to conclude that this morphism is an (acyclic) cofibration in M. □

Although in general the operation K ↦ F ⊗ K is not homotopy invariant (Example 8.1), we still have:

Proposition 8.3. Let F ∈ Fun^{b}(N(A), M) be homotopically constant (2.4) and cofibrant. If f : K → L a weak equivalence of spaces, then the morphism:

colim_{N(A)}(F ⊗ f) : colim_{N(A)}(F ⊗ K) → colim_{N(A)}(F ⊗ L)

is a weak equivalence in M.

Proof. Choose a weak equivalence F → cX in Fun^{b}(N(A), M). Let π be the subdivision of the projection A × K → A. Note that there is an isomorphism colim_{N(A)}(F ⊗ K) ≅ colim_{N(A×K)}π^{∗}F. Since π is reduced, the functor π^{∗}F is cofibrant in Fun^{b}(N(A × K), M). It follows that there is an isomorphism colim_{N(A)}(F ⊗ K) ≅ X ⊗_{l} (A × K) in Ho(M). By the same argument, the morphism X ⊗_{l} (A × f) : X ⊗_{l} (A × K) → X ⊗_{l} (A × L) can be identified with colim_{N(A)}(F ⊗f). Since A×f is a weak equivalences, according to Corollary 7.2, X ⊗_{l} (A × f) is an isomorphism in Ho(M) and so colim_{N(A)}(F ⊗ f) is a weak equivalence in M. □

How do these properties of the operation ⊗ in Fun^{b}(N(A), M) translate to the mapping spaces?

Corollary 8.4. If ϕ is a weak equivalence between cofibrant objects and ψ is a weak equivalence between fibrant objects in Fun^{b}(N(A), M), then map(ϕ, ψ) is a weak equivalence of spaces.

Proof. By standard K. Brown lemma type of arguments [6, Lemma 9.9], without loss of generality, we can assume in addition that ϕ is an acyclic cofibration and ψ is an acyclic fibration. In this case the corollary follows from Proposition 8.2 and the usual adjointness argument. □

The mapping space functor satisfies also the following duality property:

Proposition 8.5. If H ∈ Fun^{b}(K, Fun^{b}(N(A), M)) is a cofibrant functor whose values are homotopically constant functors in Fun^{b}(N(A), M) and Z is fibrant in M, then map(H, cZ) is fibrant in Fun^{b}(K^{op}, Spaces) [2, Section 31.3].

Proof. We need to show that, for a non-degenerate simplex σ : Δ[n] → K, map(H(σ), cZ) → lim_{(∂Δ[n])^{op}}map(H, cZ) = map(colim_{∂Δ[n]}H, cZ) is a fibration of spaces. To do that one needs to construct certain lifts. By adjointness this problem reduces to showing that, for a push-out square of the form:

\xymatrix{(colim_{∂Δ[n]}H) ⊗ Δ[k, l] \ar[r] \ar[d]_{α} & H(σ) ⊗ Δ[k, l] \ar[d] \\ (colim_{∂Δ[n]}H) ⊗ Δ[k] \ar[r] & P (σ) \ar[r] & H(σ) ⊗ Δ[k]}

any natural transformation P → cZ factors through P → H(−) ⊗ Δ[k]. Since cZ is a constant functor and Z is fibrant, it is enough to prove that the morphism colim_{N(A)}P → colim_{N(A)}(H(σ) ⊗ Δ[k]) is an acyclic cofibration in M. Cofibrancy is a consequence of Propositions 8.2 and 6.3.(1). Using Proposition 8.3, acyclicity would follow if colim_{N(A)}α is an acyclic cofibration in M. Since H is cofibrant, same is true for colim_{∂Δ[n]}H and so α is a cofibration by Proposition 8.2. This implies that colim_{N(A)}α is also a cofibration. It remains to show that colim_{N(A)}α is a weak equivalence.

Note that, for any L, H ⊗L is cofibrant in Fun^{b}(K, Fun^{b}(N(A), M)) (another consequence of Proposition 8.2). The values of H are cofibrant and homotopically constant, we can therefore apply Proposition 8.3 to conclude that the natural transformation colim_{N(A)}(H ⊗Δ[k, l]) → colim_{N(A)}(H ⊗Δ[k]) is a weak equivalence between cofibrant objects in Fun^{b}(K, M) [2, Proposition 24.2]. Consequently colim_{K}colim_{N(A)}(H ⊗ Δ[k, l]) → colim_{K}colim_{N(A)}(H ⊗ Δ[k]) is a weak equivalence in M. As this last morphism can be identified with colim_{N(A)}α, the proposition is proven. □

This implies that the connected components of certain mapping spaces coincide as expected with homotopy classes of morphisms.

Corollary 8.6. If Z is fibrant in M and F is homotopically constant and cofibrant in Fun^{b}(N(A), M), then map(F, cZ) is Kan. Moreover there is a bijection, natural in F and Z, between the set of components π_{0}map(F, cZ) and the set of morphism [F, cZ] in Ho(Fun^{b}(N(A), M)).

Proof. The first part is a particular case of Proposition 8.5 when K = Δ[0]. To prove the second part, we will show that the map which assigns to the component of a morphism ϕ : F → cZ in map(F, cZ) the homotopy class of ϕ in [F, cZ], is a bijection.

Since F is cofibrant, the natural transformation F ⊗ ∂Δ[1] → F ⊗ Δ[1], induced by the inclusion ∂Δ[1] ↪ Δ[1], is a cofibration. Consider two natural transformations ϕ_{0} : F → cZ and ϕ_{1} : F → cZ, choose a cylinder object H for F, and consider the following commutative diagram:

\xymatrix{F ⊗ ∂Δ[1] \ar[d]_{_{id ∐ id}} \ar@{^{(}->}[dr] \ar[rr]^{ϕ_{0} ∐ ϕ_{1}} &  & cZ \\ F & F ⊗ Δ[1] \ar@{^{(}->}[d] \\  & H \ar[ul]_{∼}}

We claim that there is a natural transformation F ⊗ Δ[1] → cZ which makes the above diagram commutative if and only if there is a natural transformation H → cZ with the same property. Since the target cZ is a constant functor any such F ⊗ Δ[1] → cZ is induced by a morphism colim_{N(A)}(F ⊗ Δ[1]) → Z in M. Note that, by Proposition 8.3 and Proposition 6.3(1), the morphisms colim_{N(A)}(F ⊗ Δ[1]) → colim_{N(A)}F and colim_{N(A)}H → colim_{N(A)}F are weak equivalences. It follows that colim_{N(A)}(F ⊗ Δ[1]) → colim_{N(A)}H is an acyclic cofibration in M and our claim is proven, as Z is assumed to be fibrant. This argument shows that ϕ_{0} and ϕ_{1} are left homotopic if and only if they are in the same component of map(F, cZ). □

9. Resolutions and mapping spaces

A standard way of studying derived or homotopical properties of various operations is by considering resolutions. This resolution principle is playing a significant role in our approach to mapping spaces too. In our setting it is given by:

Proposition 9.1. Let M be a model category. If A is a contractible space, then the pair of adjoint functors colim_{N(A)} : Fun^{b}(N(A), M) ⇄ M : c is a left model approximation of M.

Proof. Conditions (1), (2), and (3) of Definition 1.1 are obviously satisfied. Condition (4) is a consequence of Theorem 6.4. To prove (5) consider a weak equivalence F → cX in Fun^{b}(N(A), M) with a cofibrant domain. According to Theorem 6.5, colim_{N(A)}F is isomorphic to X ⊗_{l} A in Ho(M). Since A is contractible, the functor from the simplex category of A to the category with only one object and one morphism is a weak equivalence. Thus, by Proposition 7.1, X ⊗_{l} A → X is also an isomorphism in Ho(M) and so colim_{N(A)}F → X is a weak equivalence in M. □

Taking the subdivision of A is a crucial assumption in the statement of Proposition 9.1. It is not true in general that if A is contractible, then the pair of adjoint functors colim_{A} : Fun^{b}(A, M) ⇄ M : c forms a left model approximation. Take M to be Spaces and A to be the space defined in Example 4.1. Consider a weak equivalence F → cΔ[0] in Fun^{b}(A, Spaces) with a cofibrant domain. The space colim_{A}F is weakly equivalent to hocolim_{A[S^{−1}]}Δ[0] ≃ S^{2}, which is not weakly contractible. Thus the requirement (5) of Definition 1.1 is not satisfied.

How to construct mapping spaces in a category C that admits a left model approximation? Here is the recipe:

• choose a left model approximation l : M ⇄ C : r;

• choose a contractible space A;

• choose a cofibrant replacement Q in Fun^{b}(N(A), M) and a fibrant replacement R in M;

• define a functor map : C^{op} × C → Spaces by the formula:

map(X, Y) := map(Qcr(X), cRr(Y))

where the mapping space on the right comes from the simplicial structure on Fun^{b}(N(A), M) introduced in Proposition 5.3.

These mapping spaces have the following properties:

Proposition 9.2.

(1) For any X and Y in C, map(X, Y) is Kan.

(2) There is a bijection, natural in X and Y in C, between the set of connected components π_{0}map(X, Y) and the set of morphism [X, Y] in Ho(C).

(3) If α : X_{0} → X_{1} and β : Y_{0} → Y_{1} are weak equivalences in C, then map(α, β) : map(X_{1}, Y_{0}) → map(X_{0}, Y_{1}) is a weak equivalence of spaces.

Proof. Statement (1) is a direct consequence of the first part of Corollary 8.6.

According to the second part of this corollary there is a natural bijection between the set of connected components π_{0}map(X, Y) and the set of morphism [cr(X), cr(Y)] in Ho(Fun^{b}(N(A), M)). Since A is contractible, Proposition 9.1 implies that the pair lcolim_{N(A)} : Fun^{b}(N(A), M) ⇄ C : cr forms a left model approximation of C. It follows that there is a bijection between the set of morphisms [X, Y] in Ho(C) and [cr(X), cr(Y)] in Ho(Fun^{b}(N(A), M)) [2, Proposition 5.5], which shows (2).

Statement (3) is a direct consequence of Corollary 8.4. □

Part (3) of Proposition 9.2 implies that the mapping space functor induces a functor on the level of homotopy categories, denoted by the same symbol:

map : Ho(C)^{op} × Ho(C) → Ho(Spaces)

How dependent is this functor on the choices we have made: the model approximation l : M ⇄: C : r, the contractible space A, and the cofibrant and fibrant replacements? It can be characterized by the following universal property:

Proposition 9.3. Let X ∈ C. The functor map(X, −) : Ho(C) → Ho(Spaces) is right adjoint to X ⊗_{l} − : Ho(Spaces) → Ho(C). In particular it does not depend on the choices of the model approximation l : M ⇄: C : r, the contractible space A, and the cofibrant and fibrant replacements.

Proof. Consider the following sequence of adjoint functors:

\xymatrix{Spaces \ar[r]^{QcrX⊗−} & Fun^{b}(N(A), M) \ar[l]^{map(QcrX,−)} \ar[r]^{^{colim^{N(A)}}} & M \ar[l]^{c} \ar[r]^{l} & C \ar[l]^{r}}

According to Propositions 6.3, 8.2, and 8.3, the functor colim_{N(A)}(QcrX ⊗ −) preserves cofibrations and acyclic cofibrations. Thus, by [6, Theorem 9.7], the total derived functors also form adjoint pairs (see also Lemma 6.8):

\xymatrix{Ho(Spaces) \ar[r]^{L(colim_{N(A)}(QcrX⊗−))} & Ho(M) \ar[l]^{R(map(QcrX,c−))} \ar[r]^{L(l)} & Ho(C) \ar[l]^{r}}

Note that the composition of L(colim_{N(A)}(QcrX⊗−)) and L(l) can be identified with K ↦ X ⊗_{l} (A×K) (see the proof of Proposition 8.3), and the composition of r and R(map(QcrX, c−)) with map(X, −) : Ho(C) → Ho(Spaces). Since A is assumed to be contractible, the morphism X⊗_{l}(A×K) → X⊗_{l}K, induced by the projection π : A×K → K, is an isomorphism in Ho(C) (Corollary 7.2). □

Corollary 9.4. If (C, W) admits a left model approximation, then the functor ⊗_{l} : Ho(C) × Ho(Spaces) → Ho(C) is a homotopy left action (2.15).

How natural is this homotopy left action? Let (C, W) and (D, V) admit left model approximations and Φ : C → D be a functor that sends weak equivalences in C to weak equivalences in D. The universal property of the total left derived functor hocolim_{I} yields a morphism hocolim_{I}ΦF → Φ(hocolim_{I}F) in Ho(D) natural in F : I → C (see 2.20). For X ∈ C and K ∈ Spaces, define Φ(X) ⊗_{l} K → Φ(X ⊗_{l} K) to be the morphism hocolim_{K}cΦ(X) → Φ(hocolim_{K}cX) in Ho(D).

Corollary 9.5. Assume that (C, W) and (D, V) admit left model approximations and Φ : C → D is a functor that sends weak equivalences in C to weak equivalences in D. Then the induced functor Φ : Ho(C) → Ho(D) together with the morphisms Φ(X) ⊗_{l} K → Φ(X ⊗_{l} K) is a simplicial functor (2.16).

We prove finally the classical property relating mapping spaces out of a homotopy colimit with the corresponding homotopy limit of mapping spaces, see [1, XII Proposition 4.1] and [8, Theorem 19.4.4].

Proposition 9.6. Assume that (C, W) admits a left model approximation. Let Y ∈ C and F : I → C be a functor. Then there is an isomorphism map(hocolim_{I}F, Y) ≅ holim_{I^{op}}map(F, Z) in Ho(Spaces) natural in Y and F.

Proof. Let us choose a left model approximation l : M ⇄ C : r and a contractible space A. Apply then Proposition 8.5 to a fibrant replacement Z ∈ M of rY and a cofibrant replacement H ∈ Fun^{b}(N(I), Fun^{b}(N(A), M)) of the composition N(I) ⟶^{ϵ} I ⟶^{F} C ⟶^{r} M ⟶^{c} Fun^{b}(N(A), M). □

10. Right model approximations and duality

In this section we study categories admitting both a left and a right model approximations. These structures lead to homotopy left and right actions. How related are these actions? Are the induced mapping spaces isomorphic? We answer these questions in Theorem 10.7 below, but need first some dual statements to those we have proven so far for left model approximations. They can be restated by simply taking the opposite categories.

10.1. One can define a simplicial structure on Fun^{b}(N(A)^{op}, C) by identifying it with Fun^{b}(N(A), C^{op}) and apply Proposition 5.3. Explicitly, this simplicial structure is given by a functor:

hom : Spaces^{op} × Fun^{b}(N(A)^{op}, C) → Fun^{b}(N(A)^{op}, C)

described as follows. Denote by π_{k} : Fun^{b}(N(A × K)^{op}, C) → Fun^{b}(N(A)^{op}, C) the right adjoint of π^{∗} : Fun^{b}(N(A)^{op}, C) → Fun^{b}(N(A × K)^{op}, C), where as before π : N(A × K) → N(A) stands for the subdivision of the projection onto the first factor. Define then hom(K, F) := π_{k}π^{∗}F.

10.2. Assume that (C, W) admits a right model approximation. Then it can be right localized. For any small category I, holim_{I} : Ho(Fun(I, C)) → Ho(C) is right adjoint to c : Ho(C) → Ho(Fun(I, C)). The assignment (I, X) ↦ holim_{I}cX defines a functor rhom : Cat^{op} × C → Ho(C). This functor is homotopy invariant: if f : I → J is a weak equivalence of small categories, then rhom(f, X) is an isomorphism in Ho(C) for any X ∈ C, compare with Proposition 7.1.

10.3. If f : K → L is a weak equivalence of spaces, then the morphism rhom(f, X) : rhom(L, X) → rhom(K, X) is an isomorphism in Ho(C), for any X ∈ C. The functor rhom : Spaces^{op} ×C → Ho(C) is therefore a composition of the localization Spaces^{op} × C → Ho(Spaces)^{op} × Ho(C) and a functor denoted by the same symbolrhom : Ho(Spaces)^{op} × Ho(C) → Ho(C). This functor is a homotopy right action (2.16), i.e., it has the following properties, dual to those in Corollary 7.3 and Proposition 9.3:

(1d) there are isomorphisms X → rhom(Δ[0], X) in Ho(C) natural in X; (2d) there are coherent isomorphisms rhom(K×L, X) ≅ rhom(K, rhom(L, X))

in Ho(C) natural in X ∈ Ho(C) and K, L ∈ Ho(Spaces)^{op}; (3d) for any X ∈ Ho(C), rhom(−, X) : Ho(Spaces)^{op} → Ho(C) has a left

adjoint map(−, X) : Ho(C) → Ho(Spaces)^{op}.

10.4. Let (C, W) and (D, V) be categories that admit right model approximations and Φ : C → D be a functor that sends weak equivalences in C to weak equivalences in D. This and the universal property of holim, as a total right derived functor, implies that, for any F : I → C, there is a morphism Φ(holim_{I}F) → holim_{I}ΦF in Ho(D) natural in F (see 2.20). For X ∈ C and K ∈ Spaces, define Φ(rhom(K, X)) → rhom(K, ΦF) to be the morphism Φ(holim_{K}cX) → holim_{K}cΦX in Ho(D). In this way Φ : Ho(C) → Ho(D) together with the morphism Φ(rhom(K, X)) → rhom(K, ΦF) is a simplicial functor.

10.5. The functor map(−, X) : Ho(C) → Ho(Spaces)^{op}, defined in 10.3, can be rigidified as follows:

• choose a right model approximation l : C ⇄ M : r;

• choose a contractible space A;

• choose a fibrant replacement R in Fun^{b}(N(A)^{op}, M) and a cofibrant replacement Q in M;

• define a functor map : C^{op} × C → Spaces by the formula:

map(X, Y) := map(cQl(X), Rcr(Y))

where the mapping space on the right comes from the simplicial structure on Fun^{b}(N(A)^{op}, M).

The functor map : C^{op} × C → Spaces, defined by the above procedure, is homotopy invariant and its values are Kan spaces. The induced functor on the level of homotopy categories is naturally isomorphic to the functor described in 10.3 and therefore it does not depend on the choice of the right model approximation l : C ⇄ M : r, the contractible space A, and the fibrant and cofibrant replacements.

10.6. The mapping space into a homotopy limit satisfies a property analogous to Proposition 9.6. Let Y ∈ C and F : I → C be a functor. Then there is an isomorphism map(Y, holim_{I}F) ≅ holim_{I}map(Z, F) in Ho(Spaces) natural in Y and F.

We are now ready to compare the homotopy left and right actions arising from left and right model approximations.

Theorem 10.7. Assume that (C, W) admits both a left and a right model approximations. Then, for any space K, the functor − ⊗_{l} K : Ho(C) → Ho(C) is left adjoint to rhom(K, −) : Ho(C) → Ho(C). The weak simplicial structures on Ho(C) (2.15), induced by the homotopy actions ⊗_{l} and rhom, are isomorphic and so are the induced mapping spaces.

Proof. According to Corollary 6.7, we have the following adjoint pairs:

\xymatrix{Ho(C) \ar[r]^{c} & Ho(Fun(K, C)) \ar[l]^{_{holim_{K}}} \ar[r]^{^{hocolim^{K}}} & Ho(C) \ar[l]^{c}}

Since, by definition, hocolim_{K}cX = X ⊗_{l} K and holim_{K}cX = rhom(K, X) the theorem is proven. □

11. Monoidal model categories

Recall that a monoidal model category is a model category M together with a “compatible” monoidal structure M ⊗ M → M, see [9, Definition 4.2.6]. M. Hovey asks whether the homotopy category of such a monoidal model category always forms a central algebra over Ho(Spaces), i.e. if there is a monoidal functor i : Ho(Spaces) → Ho(M) and natural isomorphims t : i(K) ⊗ X → X ⊗ i(K) satisfying certain coherence conditions introduced in [9, Definition 4.1.10]. We show this is indeed so.

Proposition 11.1. Let M be a monoidal model category. Then the functor ⊗_{l} : Ho(M) × Ho(Spaces) → Ho(M) describes a central algebra over Ho(Spaces).

Proof. Let us denote by ⊗_{M} : M×M → M the monoidal structure in M and by S ∈ M the unit. By definition, for any cofibrant X ∈ M, the operations X ⊗_{M} − and − ⊗_{M} X preserve cofibrations. Consequently, for any such X, if F ∈ Fun^{b}(A, M) is cofibrant, then so are the functors X ⊗_{M} F and F ⊗_{M} X.

Let F : I → M be a functor and QF ∈ Fun^{b}(N(I), M) be a cofibrant replacement of the composition of ϵ : N(I) → I and F. The above remark implies that, for a cofibrant X, QF ⊗_{M} X and X ⊗_{M} QF are cofibrant replacements of F ϵ ⊗_{M} X and X ⊗_{M} F ϵ respectively. Since ⊗_{M} commutes with colimits we have isomorphisms in M:

colim_{N(I)}(QF ⊗_{M} X) ≅ (colim_{N(I)}QF) ⊗_{M} X
colim_{N(I)}(X ⊗_{M} QF) ≅ X ⊗_{M} (colim_{N(I)}QF)

The corresponding isomorphisms in Ho(M) are easily seen to be natural in F : I → M and X ∈ Ho(M):

hocolim_{I}(F ⊗_{M} X) ≅ (hocolim_{I}F) ⊗_{M} X
hocolim_{I}(X ⊗_{M} F) ≅ X ⊗_{M} (hocolim_{I}F)

It is now straightforward to check that the following isomorphism t satisfy the coherence conditions for Ho(M) to be central:

\xymatrix{(S ⊗_{l} K) ⊗_{M} X = (hocolim_{K}cS) ⊗_{M} X & hocolim_{K}c(S ⊗_{M} X) \ar[l] \ar[d] \\  & hocolim_{K}cX \\ X ⊗_{M} (S ⊗_{l} K) = X ⊗_{M} (hocolim_{K}cS) & hocolim_{K}c(X ⊗_{M} S) \ar[l] \ar[u]}

□

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Wojciech Chachólski K.T.H., Matematik,

S-10044 Stockholm, Sweden

 E-mail: wojtek@math.kth.se,

Jérôme Scherer Universitat Autònoma de Barcelona, Departament de Matemàtiques, E-08193 Bellaterra, Spain

                                   E-mail: jscherer@mat.uab.es