Chapter VIII Simplicial functors and homotopy coherence

Suppose that A is a simplicial category. The main objects of study in this chapter are the functors X : A → S taking values in simplicial sets, and which respect the simplicial structure of A. In applications, the simplicial category A is typically a resolution of a category I, and the simplicial functor X describes a homotopy coherent diagram. The main result of this chapter (due to Dwyer and Kan, Theorem 2.13 below) is a generalization of the assertion that simplicial functors of the form X : A → S are equivalent to diagrams of the form I → S in the case where A is a resolution of the category I. The proof of this theorem uses simplicial model structures for categories of simplicial functors, given in Section 1, and then the result itself is proved in Section 2.

The Dwyer-Kan theorem immediately leads to realization theorems for homotopy coherent diagrams in cases where the simplicial categories A model homotopy coherence phenomena. A realization of a homotopy coherent diagram X : A → S is a functor Y : I → S which is weakly equivalent to X in a strong sense. Insofar as the information arising from X typically consists of simplicial set maps X(α) : X(i) → X(j), one for each morphism α of I which only respect the composition laws of I up to some system of higher homotopies, a realization Y is a replacement of X, up to weak equivalence, by a collection of maps Y (α) : Y (i) → Y (j) which define a functor on the nose. Diagrams of spaces which are not quite functorial are really very common: the machines which produce the algebraic K-theory spaces, for example, are not functors (on scheme categories in particular), but they have homotopy coherent output for categorical reasons [49].

Approaches to homotopy coherence for simplicial set diagrams arising from some specific resolution constructions are discussed in Section 3; traditional homotopy coherence (in the sense of [90]) is one of the examples.

But more generally, we take the point of view that a homotopy coherent diagram on a fixed index category I is a simplicial functor X : A → S defined on any simplicial resolution A of I. We can further ask for realization results concerning homotopy coherent diagrams A → M taking values in more general simplicial model categories M. This is the subject of Section 4. We derive, in particular, realization theorems for homotopy coherent diagrams taking values in pointed simplicial sets, spectra and simplicial abelian groups (aka. chain complexes).

1. Simplicial functors.

We shall take the point of view throughout this chapter that a simplicial category A is a simplicial object in the category of categories having a discrete simplicial class of objects and satisfying the conditions of Definition 2.1 of Chapter II. This means, in particular that the objects A⊗K and hom(K, A) exist for every (finite) simplicial set K and every object A of A, and that these objects satisfy the exponential laws. The full simplicial set category S with the function complexes Hom(X, Y) is the canonical example of a simplicial category. In all that follows, we shall assume that the simplicial category A has a small set of objects.

The simplicial set of morphisms from A to B in a simplicial category A is denoted by Hom(A, B); from this point of view, the corresponding set of nsimplices Hom(A, B)_{n} is the set of morphisms from A to B in the category at level n. Any morphism α : B → C in Hom(B, C)_{0} induces a simplicial set map

α_{∗} : Hom(A, B) → Hom(A, C),

which one understands to be composition with α, and which can be defined to be the composite simplicial set map

Hom(A,B)≅Hom(A,B)×Δ^{0}⟶^{1×ια}Hom(A,B)×Hom(B,C)⟶^{∘}Hom(A,C).

Similarly, any morphism β : A → B induces a simplicial set map

β^{∗} : Hom(B, C) → Hom(A, C),

that one thinks of as precomposition. Composition and precomposition respect composition in α and β in the traditional sense.

A simplicial functor f : A → B is a morphism of simplicial categories. This means that f consists of a function f : Ob(A) → Ob(B) and simplicial set maps f : Hom(A, B) → Hom(f(A), f(B)) which respect identities and composition at all levels.

A natural transformation η : f → g of simplicial functors f, g : A → B consists of morphisms

η_{A} : f(A) → g(A)

in hom(f(A), g(A)) = Hom(f(A), g(A))_{0}, one for each object A of A, such that the following diagram of simplicial set maps commutes

Hom(A, B) w f u g

Hom(f(A), f(B)) uη_{B∗}

Hom(g(A), g(B)) w η_{A}^{∗} Hom(f(A), g(B))

for each pair of objects A, B of A. Notice that this is just another way of saying that the various degeneracies of the morphisms η_{A} are natural transformations between the functors induced by f and g at all levels. The collection of all simplicial functors from A to B and all natural transformations between them form a category, which we shall denote by B^{A}. Write Nat(f, g) for the set of all natural transformations from f to g.

The category S^{A} of simplicial functors taking values in the simplicial set category S are of particular interest, and can be given a much more explicit description. Suppose that A_{n} denotes the small category at level n within the simplicial category A. Then a simplicial functor X : A → S consists of a function X : Ob(A) → Ob(S) as before, together with a collection of simplicial set maps X(α) : X(A) × Δ^{n} → X(B), one for each morphism α : A → B in A_{n}, n ≥ 0, such that

(1) the simplicial set map X(βα) is the composite

X(A) × Δ^{n} ⟶^{1×Δ^{n}} X(A) × Δ^{n} × Δ^{n} ⟶^{X(α)×1} X(B) × Δ^{n} ⟶^{X(β)} X(C)

for each composable string of morphisms

A ⟶^{α} B ⟶^{β} C

in A_{n},

(2) the simplicial set map X(1_{A}) associated to the identity on A ∈ A_{n} is the projection X(A) × Δ^{n} → X(A), and

(3) for each ordinal number map θ : m → n and each morphism α : A → B of A_{n}, the following diagram commutes:

X(A) × Δ^{m}_{A A AA C} X(θ^{∗}(α)) u 1 × θ_{∗}

X(A) × Δ^{n} w X(α) X(B).

Then, from this point of view, a natural transformation η : X → Y of simplicial functors taking values in simplicial sets consists of simplicial set maps η_{A} : X(A) → Y (A), one for each A ∈ Ob(A), such that the diagram

\begin{matrix}X(A) × Δ^{n}  w X(α) u η_{A} × 1 \\ X(B) uη_{B} \end{matrix} ; \begin{matrix}Y (A) × Δ^{n}  w Y (α) \\ Y (B) \end{matrix}

commutes for each morphism α : A → B of A_{n}, and for all n ≥ 0.

Now it’s easy to see that the category S^{A} is the category at level 0 of a simplicial category. Given a simplicial functor X : A → S and a simplicial set K, there is a simplicial functor K × X which assigns to each A ∈ Ob(A) the simplicial set K × X(A) and to each morphism α : A → B in A_{n} the simplicial set map

K × X(A) × Δ^{n} ⟶^{1×X(α)} K × X(B).

Dually, there is a simplicial functor hom(K, X) which associates to each A ∈ Ob(A) the simplicial set hom(K, X(A)), and to each morphism α : A → B of A_{n} the simplicial set map hom(K, X(A)) × Δ^{n} → hom(K, X(B)) which is defined to be the adjoint of the composite

K × hom(K, X(A)) × Δ^{n} ⟶^{ev×1} X(A) × Δ^{n} ⟶^{X(α)} X(B),

where ev : K × hom(K, X(A)) → X(A) is the standard evaluation map. It follows immediately that the collection of all such evaluation maps induces a natural bijection

Nat(K × Y, X) ≅ Nat(Y, hom(K, X)).

There is also, plainly, a simplicial set hom(Y, X) whose set of n-simplices is the set Nat(Δ^{n} × Y, X). Finally, since the simplicial set K is a colimit of its simplices, there is a natural bijection (exponential law)

(1.1)  hom(K, Hom(Y, X)) ≅ Nat(K × Y, X)

relating morphisms in the simplicial set category to natural transformations of simplicial functors.

The representable simplicial functor Hom(A, ) is the simplicial functor which associates to each B ∈ Ob(A) the simplicial set Hom(A, B), and associates to each morphism β : B → C of A_{n} the composite simplicial set map

Hom(A, B) × Δ^{n} ⟶^{1×ι_{β}} Hom(A, B) × Hom(B, C) ⟶^{∘} Hom(A, C).

It is central to observe that the representable simplicial functors in S^{A} satisfy the Yoneda lemma:

Lemma 1.2. Suppose that A is an object of a simplicial category A, and that X : A → S is a simplicial functor. Then there is a natural isomorphism of simplicial sets

Hom(Hom(A, ), X) ≅ X(A).

Proof: The natural transformations η : Hom(A, ) → X are in one to one correspondence with the vertices of X(A). In effect, given a morphism α : A → B of A_{m}, there is a commutative diagram

Hom(A, A) × Δ^{m} w η_{A} × 1 u α_{∗} X(A) × Δ^{m} uX(α)

Hom(A, B)  w η_{B}  X(B),

so that

η_{B}(α) = X(α)(η_{A}(1_{A}) × ι_{m}),

where ι_{m} = 1_{m} ∈ Δ_{m}^{m} is the classifying simplex. But 1_{A} = s(1_{A,0}) where 1_{A,0} denotes the identity on A in A_{0}. Thus, η is completely determined by the vertex η_{A}(1_{A,0}) of X(A).

It follows that there are natural bijections

Nat(Δ^{n} × Hom(A, ), X) ≅ Nat(Hom(A, ), hom(Δ^{n}, X))
≅ hom(Δ^{n}, X(A))_{0}
≅ X(A)_{n}

and that these bijections respect the simplicial structure. ■

A natural transformation η : Δ^{n} × Hom(A, ) → X is completely determined by the n-simplex η(ι_{n}, 1_{A}) of the simplicial set X(A): this is a corollary of the proof of Lemma 1.2.

The functor S → S defined by K ↦ K × Δ^{n} preserves all small colimits of simplicial sets and takes limits to limits fibred over Δ^{n}. It follows immediately that the category S^{A} of simplicial functors from A to S is complete and cocomplete, and that all limits and colimits are formed pointwise.

There is an analogue of the simplex category for each simplicial functor X ∈ S^{A}. Write A ↓ X for the category whose objects are the transformations

Δ^{n} × Hom(A, ) → X,

and whose objects are all commutative triangles of transformations of the form

Δ^{n} × Hom(A, )_{‚ ‚ ‚‚ ›} u

X
Δ^{m} × Hom(B, )^{4 4 44 6}

The objects of this category are simplices of sections of X, and the morphisms form its structural data. It follows that X is a colimit of its simplices in S^{A} in the sense that there is a natural isomorphism

(1.3)  X ≅ \underrightarrow{lim}_{Δ^{n}×Hom(A, )→X} Δ^{n} × Hom(A, ).

Say that a map f : X → Y of S^{A} is a fibration (respectively weak equivalence) if the component maps f : X(A) → Y (A) are fibrations (respectively weak equivalences) of simplicial sets, for all objects A of A. Such maps will be called pointwise fibrations and pointwise weak equivalences, respectively. A cofibration of S^{A} is a map which has the left lifting property with respect to all trivial fibrations.

Proposition 1.4. With these definitions, the category S^{A} of simplicial setvalued simplicial functors defined on a small simplicial category A satisfies the axioms for a simplicial model category.

Proof: CM1 is a consequence of the completeness and cocompleteness of S^{A}, as described above. The weak equivalence axiom CM2 and the retract axiom CM3 are both trivial consequences of the corresponding axioms for simplicial sets.

Note that a map f : X → Y of S^{A} is a fibration if and only if f has the right lifting property with respect to all maps

Λ_{k}^{n} × Hom(A, ) ⟶^{i×1} Δ^{n} × Hom(A, ),

and f is a trivial fibration if and only if it has the right lifting property with respect to all maps

∂Δ^{n} × Hom(A, ) ⟶^{i′×1} Δ^{n} × Hom(A, ),

where i : Λ_{k}^{n} ↪ Δ^{n} and i′ : ∂Δ^{n} ↪ Δ^{n} denote the respective canonical inclusions. Both statements follow from the exponential law (1.1) and Lemma 1.2. The map i × 1 is a weak equivalence.

We may therefore apply standard small object arguments to prove the factorization axiom CM5 for S^{A}. In particular, any map f : X → Y has a factorization

\begin{matrix}X w j_{4 4 44 6} f \\ Zup \end{matrix} ; Y,

where p is a fibration and j is a filtered colimit of maps of the form j_{n} : X_{n} → X_{n+1}, each of which is defined by a pushout diagram

⨆_{r} Λ_{k_{r}}^{n_{r}} × Hom(A_{r}, )  w u ⨆_{r} (i × 1)  X_{n} uj_{n}

⨆_{r} Δ^{n_{r}} × Hom(A_{r}, ) wX_{n+1}.

It follows that each of the maps j_{n} is a pointwise weak equivalence, and has the left lifting property with respect to all fibrations, and so the map j has the same properties, giving one of the factorizations required by CM5 in particular. The other factorization has a similar construction.

The axiom CM4 is a standard consequence of the method of proof of the factorization axiom CM5, in that any map f : X → Y which is a cofibration and a pointwise weak equivalence has a factorization f = p · j, where p is a fibration and where j is a pointwise weak equivalence and has the right lifting property with respect to all fibrations. In particular, p is a pointwise weak equivalence, and so the indicated lifting exists in the diagram

\begin{matrix}X  w j u f \\ Zup \end{matrix} ; \begin{matrix}Y  w 1_{G}^{\ \ \ \ ]} \\ Y. \end{matrix}

The map f is therefore a retract of j, and therefore has the left lifting property with respect to all fibrations.

The axiom SM7 is a consequence of the corresponding statement for simplicial sets. ■

Remark 1.5. Proposition 1.4 is a special case of a result of Dwyer and Kan [24]. The proof given here is much simpler.

Remark 1.6. The Bousfield-Kan closed model structure for I-diagrams of simplicial sets given in Example II.7.11 and Section IV.3.1 is a special case of Proposition 1.4. Every category I having all coproducts and products is a simplicial category, for which Hom(X, Y) is the discrete simplicial set on the set of morphisms hom(X, Y) from X to Y in I. The required isomorphisms

hom(X ⊗ K, Y) ≅ hom_{S}(K, Hom(X, Y)) ≅ hom(X, hom(K, Y))

respectively force X ⊗ K to be a copy of the coproduct ⨆_{π_{0}K} X, and dictate that hom(K, Y) is the product ∏_{π_{0}K} Y. In the language of Section IV.3.1, the simplicial set K × Hom(A, ) is a copy of the Kan extension F_{A}K.

2. The Dwyer-Kan theorem.

A simplicial functor X : A → S can alternatively be described as a “rule” which associates to each object A a simplicial set X(A), and to each morphism α : A → B of A_{n} a function α_{∗} : X(A)_{n} → X(B)_{n} on the n-simplex level such that

(1) the assignment α ↦ α_{∗} is functorial in morphisms α of A_{n}, for all n ≥ 0, and

(2) for each morphism α : A → B of A_{n} and each ordinal number map θ : m → n, the following diagram commutes:

\begin{matrix}X(A)_{n}  w α_{∗} uθ^{∗} \\ X(B)_{n} uθ^{∗} \end{matrix} ; \begin{matrix}X(A)_{m}  w (θ^{∗}α)_{∗} \\ X(B)_{m}. \end{matrix}

Given a simplicial functor X, with induced maps α_{∗} : X(A) × Δ^{n} → X(B) as defined above for α : A → B, the corresponding morphism α_{∗} : X(A)_{n} → X(B)_{n} is the assignment x ↦ α_{∗}(x, ι_{n}). Conversely, given a system of maps α_{∗} : X(A)_{n} → X(B)_{n} having the properties described in (1) and (2) above, one defines a map α_{∗} : X(A) × Δ^{n} → X(B) on m-simplices (x, θ) by the assignment (x, θ) ↦ (θ^{∗}α)_{∗}(x). These two assignments are inverse to each other.

Example 2.1. Suppose that C is an object of a simplicial category A. Then, according to this new description, the representable simplicial functor Hom(C, ) : A → S associates to each object A the simplicial set Hom(C, A) and to each morphism α : A → B of A_{n} the function

Hom(C, A)_{n} ⟶^{α_{∗}} Hom(C, B)_{n}

defined by composition with α in A_{n}. It’s easier to make sense of this if one recalls that Hom(C, B)_{n} is precisely the set of morphisms from C to B in the category A_{n}.

From this new point of view, a natural transformation of simplicial functors f : X → Y is a collection of simplicial set maps f : X(A) → Y (A) such that, for each morphism α : A → B of A_{n} the diagram of functions

\begin{matrix}X(A)_{n}  w α_{∗} u f \\ X(B)_{n} uf \end{matrix} ; \begin{matrix}Y (A)_{n}  w α_{∗} \\ Y (B)_{n} \end{matrix}

commutes.

Suppose that f : A → B and X : A → S are simplicial functors and that B is an object of the simplicial category B, then the bisimplicial object f̃^{∗}X(B) defined by

f̃^{∗}X(B)_{n,m}

= ⨆ X(A)_{0,m} × Hom(A_{0}, A_{1})_{m} × ⋯ × Hom(A_{n−1}, A_{n})_{m} × Hom(fA_{n}, B)_{m}

The horizontal simplicial set f̃^{∗}X(B)_{∗,m} is the nerve of the translation category associated to the composite functor

f ↓ B ⟶^{Q} A_{m} ⟶^{X_{m}} Sets,

and each morphism ω : B → C of B_{m} determines a simplicial set map

ω_{∗} : f̃^{∗}X(B)_{∗,m} → f̃^{∗}X(C)_{∗,m}.

On the other hand, if τ : k → m is an ordinal number map, then there is a commutative diagram of simplicial set maps

(2.2)  \begin{matrix}f̃^{∗}X(B)_{∗,m}  w ω_{∗} u τ^{∗} \\ f̃^{∗}X(C)_{∗,m} uτ^{∗} \end{matrix} ; \begin{matrix}f̃^{∗}X(B)_{∗,k}  w (τ^{∗}ω)_{∗} \\ f̃^{∗}X(C)_{∗,k}. \end{matrix}

It follows that applying the diagonal simplicial set functor gives a simplicial functor B → S defined by (n, B) ↦ f̃^{∗}X(B)_{n,n}, which will also be denoted by f̃^{∗}X. The simplicial functor f̃^{∗}X is the homotopy left Kan extension of X along the simplicial functor f.

We’ve already noted that the simplicial set f̃^{∗}X(B)_{∗,m} is the nerve of a translation category. Furthermore, the simplicial set diagram (2.2) is induced by a commutative diagram of functors. Suppose, for example, that the simplicial functor X is the representable functor Hom(A, ). Then the corresponding simplicial set f̃^{∗}Hom(A, )(B)_{∗,m} can be identified with the nerve of the translation category associated to the composite contravariant functor

A ↓ A_{m} ⟶^{Q} A_{m} ⟶^{f} B_{m} ⟶^{Hom( ,B)_{m}} Sets.

The identity morphism on A is initial in A ↓ A_{m}, so the category

f̃^{∗}Hom(A, )(B)_{∗,m}

contracts canonically onto the discrete subcategory on the set of objects

Hom(f(A), B)_{m}.

In this way, we obtain maps

(2.3)  \{ \begin{matrix}Hom(f(A), B)_{m} ⟶^{s} f̃^{∗}Hom(A, )(B)_{∗,m}, \\ f̃^{∗}Hom(A, )(B)_{∗,m} ⟶^{r} Hom(f(A), B)_{m} \end{matrix}

such that rs = 1, as well as a natural transformation

f̃^{∗}Hom(A, )(B)_{∗,m} × 1 ⟶^{H} f̃^{∗}Hom(A, )(B)_{∗,m}

from the composite functor sr to the identity. All of this data is natural in B, and simplicial, and so we have proved Lemma 2.4. Suppose that f : A → B is a simplicial functor, and let A be an object of A. Then the simplicial functor Hom(f(A), ) is a strong deformation retract of f̃^{∗}Hom(A, ).

We shall also need

Lemma 2.5. The simplicial functors Hom(f(A), ) and f̃^{∗}Hom(A, ) are cofibrant.

Proof: The assertion about Hom(f(A), ) is a consequence of the Yoneda lemma and the observation that any trivial fibration p : X → Y of S^{B} consists of maps p : X(B) → Y (B) which are trivial fibrations of simplicial sets and are therefore surjective on vertices.

The object f̃^{∗}Hom(A, ) may also be interpreted as a simplicial object in the category S^{B} of simplicial functors defined on B.

Suppose, for the moment, that Z is an arbitrary simplicial object in S^{B}. Then Z consists of simplicial functors Z_{n}, n ≥ 0, and simplicial structure maps θ^{∗} : Z_{n} → Z_{m} in S^{B} corresponding to ordinal number maps θ : m → n. The object Z is, in other words, a type of diagram taking values in bisimplicial sets. The standard constructions on bisimplicial sets apply to Z. There is, in particular, an associated diagonal simplicial functor d(Z) which is defined by d(Z)_{n} = Z_{n,n}, and d(Z) is a co-end in the simplicial functor category for the diagrams of simplicial functor maps

\begin{matrix}Z_{n} × Δ^{m} w 1 × θ_{∗} u θ^{∗} × 1 \\ Z_{n} × Δ^{n} \end{matrix} ; Z_{m} × Δ^{n}

arising from ordinal number maps θ : m → n. This means that there are simplicial functor maps γ_{n} : Z_{n} × Δ^{n} → d(Z) defined “in sections” by (x, τ) ↦ τ^{∗}(x), and these maps assemble to give the universal arrow in a coequalizer diagram

⨆_{θ:m→n} Z_{n} × Δ^{m} ⇉ ⨆_{n} Z_{n} → d(Z).

Finally, d(Z) has a filtration d(Z)^{(p)} ⊂ d(Z)^{(p+1)} ⊂ …, where d(Z)^{(p)} is generated by the images of the maps γ_{r}, 0 ≤ r ≤ p. We are interested in proving that the simplicial functor d(Z) is cofibrant, so the key, for us, is the existence of the pushout diagram

(s_{[p]}Z_{p} × Δ^{p+1}) ∪ (Z_{p+1} × ∂Δ^{p+1})  w y u  d(Z)^{(p+1)} y u

Z_{p+1} × Δ^{p+1} wd(Z)^{(p+1)}.

in the category S^{B}. Here, the inclusion on the left is induced by the inclusion s_{[p]}Z_{p} ↪ Z_{p+1} of the “horizontally degenerate part” in Z_{p+1}. The central observation is that the simplicial model structure on S^{B} implies that d(Z) is cofibrant if each of the inclusions s_{[p]}Z_{p} ↪ Z_{p+1} is a cofibration.

The object at level n for the simplicial object f̃^{∗}Hom(A, ) is the disjoint union

⨆ Hom(A, A_{0}) × … Hom(A_{n−1}, A_{n}) × Hom(f(A_{n}), ),

which may in turn be written as a disjoint union

⨆_{C∈Ob(B)} B(A ↓ A)_{n}^{C} × Hom(f(C), ),

where B(A ↓ A)_{n}^{C} indicates the simplicial set of strings

A → A_{0} → ⋯ → A_{n−1} → A_{n} = C,

ending at C. The horizontal degeneracies preserve this decomposition, and so the inclusion of the degenerate part in

⨆_{C∈Ob(B)} B(A ↓ A)_{n+1}^{C} × Hom(f(C), ),

can be identified with the map

(2.6) ⨆_{C∈Ob(B)} DB(A↓A)_{n+1}^{C}×Hom(f(C), )↪⨆_{C∈Ob(B)} B(A↓A)_{n+1}^{C}×Hom(f(C), ),

which is induced by the simplicial set inclusions

DB(A ↓ A)_{n+1}^{C} ⊂ B(A ↓ A)_{n+1}^{C}.

Each of the simplicial functors Hom(f(C), ) is cofibrant, so the map (2.6) is a cofibration of simplicial functors, and the simplicial functor f̃^{∗}Hom(A, ) is cofibrant, as claimed. ■

Suppose now that Y : B → S is a simplicial functor. Then a simplicial functor f̃_{∗}Y : A → S is specified at an object A of A by

f̃_{∗}Y (A) = Hom(f̃^{∗}Hom(A, ), Y).

In other words, an n-simplex of f̃_{∗}(Y)(A), or rather a morphism

Δ^{n} × Hom(A, ) → f̃_{∗}Y

is defined to be a morphism

Δ^{n} × f̃^{∗}Hom(A, ) → Y.

There is an isomorphism

Δ^{n} × f̃^{∗}Hom(A, ) ≅ f̃^{∗}(Δ^{n} × Hom(A, )),

by formal nonsense. The resulting isomorphisms

Nat(Δ^{n} × Hom(A, ), f̃_{∗}Y) ≅ Nat(Δ^{n} × f̃^{∗}Hom(A, ), Y)
≅ Nat(f̃^{∗}(Δ^{n} × Hom(A, )), Y)

are natural with respect to maps

Δ^{m} × Hom(B, ) → Δ^{n} × Hom(A, ).

Furthermore, every simplicial functor X : A → S is a colimit of its simplices, as in (1.3), and the functor f̃^{∗} preserves colimits. It follows that there is an adjunction isomorphism

(2.7)  Nat(f̃^{∗}X, Y) ≅ Nat(X, f̃_{∗}Y).

The simplicial functor f : A → B also induces a functor f_{∗} : S^{B} → S^{A}, which is defined by f_{∗}Y (A) = Y (f(A)). The notation is supposed to remind one of a direct image functor.

Note in particular, that the adjoint of a n-simplex

Δ^{n} × Hom(A, ) ⟶^{x} f̃_{∗}Y

is the composite

f̃^{∗}(Δ^{n} × Hom(A, )) ≅ Δ^{n} × f̃^{∗}Hom(A, ) ⟶^{x} Y

It follows that the canonical map

η : Δ^{n} × Hom(A, ) → f̃_{∗}f̃^{∗}(Δ^{n} × Hom(A, ))

is the map which sends the classifying simplex (ι_{n}, 1_{A}) to the canonical isomorphism

Δ^{n} × f̃^{∗}Hom(A, ) ⟶^{≅} f̃^{∗}(Δ^{n} × Hom(A, )),

since η must be adjoint to the identity on f̃^{∗}(Δ^{n} × Hom(A, )). Thus, if x is an n-simplex of X(A), then η(x) ∈ f̃_{∗}f̃^{∗}X(A) is the simplex defined by the composite

(2.8)  Δ^{n} × f̃^{∗}Hom(A, ) ⟶^{≅} f̃^{∗}(Δ^{n} × Hom(A, )) ⟶^{f̃^{∗}(x)} f̃^{∗}X.

The adjunction (2.7) is easily promoted to a natural isomorphism of simplicial sets

(2.9)  Hom(f̃^{∗}X, Y) ≅ Hom(X, f̃_{∗}Y).

We have therefore proved the first part of the following result:

Proposition 2.10. Suppose that f : A → B is a functor between small simplicial categories. Then

(1) The functors

f̃^{∗} : S^{A} ↔ S^{B} : f̃_{∗}

are simplicially adjoint in the sense that there is a natural isomorphism of simplicial sets

Hom(f̃^{∗}X, Y) ≅ Hom(X, f̃_{∗}Y).

(2) the functor f̃_{∗} preserves weak equivalences and fibrations.

(3) the functor f̃^{∗} preserves weak equivalences and cofibrations.

Proof: To prove Part (2), observe that f̃^{∗}Hom(A, ) is cofibrant for every object A ∈ A, by Lemma 2.5, so that every map

Λ_{k}^{n} × f̃^{∗}Hom(A, ) → Δ^{n} × f̃^{∗}Hom(A, )

is a trivial cofibration on account of the simplicial model structure of the category S^{B}. It follows that the functor f̃_{∗} preserves fibrations.

The retraction map

r : f̃^{∗}Hom(A, ) → Hom(f(A), )

is natural in A, and therefore induces a map

r^{∗} : f_{∗}Y → f̃_{∗}Y

which is natural in Y ∈ S^{B}. It follows that f̃_{∗} preserves weak equivalences, if it can be shown that the map r^{∗} is a natural weak equivalence.

Recall that the map r has a section

s : Hom(f(A), ) → f̃^{∗}Hom(A, ),

and that there is a homotopy

H : f̃^{∗}Hom(A, ) × Δ^{1} → f̃^{∗}Hom(A, )

from the composite sr to the identity. The induced map

H^{∗} : Hom(f̃^{∗}Hom(A, ), Y) → Hom(f̃^{∗}Hom(A, ) × Δ^{1}, Y)

can be composed with the canonical map

Hom(f̃^{∗}Hom(A, ) × Δ^{1}, Y) → hom(Δ^{1}, Hom(f̃^{∗}Hom(A, ), Y))

to give a map which is adjoint to a simplicial homotopy

Hom(f̃^{∗}Hom(A, ), Y) × Δ^{1} → Hom(f̃^{∗}Hom(A, ), Y)

from the identity map to s^{∗}r^{∗}. This implies that r^{∗} is a pointwise weak equivalence, as required.

For Part (3), the functor f̃^{∗} preserves weak equivalences as a consequence of its definition as a type of homotopy colimit. Also, f̃^{∗} is left adjoint to the functor f̃_{∗}, and it is a consequence of Part (2) that the latter preserves trivial fibrations. Standard closed model category tricks therefore imply that the functor f̃^{∗} preserves cofibrations. ■

Corollary 2.11. The functor f_{∗} : S^{B} → S^{A} induces an equivalence

f_{∗} : Ho(S^{B}) → Ho(S^{A})

of homotopy categories if and only if the adjunction maps

η : X → f̃_{∗}f̃^{∗}X  and  ϵ : f̃^{∗}f̃_{∗}Y → Y

are weak equivalences.

Proof:

The functors f̃^{∗} and f̃_{∗} induce an adjoint pair of functors

f̃^{∗} : Ho(S^{A}) ↔ Ho(S^{B}) : f̃_{∗}

between the associated homotopy categories, by Proposition 2.10. If the induced functor f̃_{∗} is an equivalence of categories, then it is part of an adjoint equivalence (see [66, p.91]). The adjoint to f̃_{∗} of that adjoint equivalence is therefore naturally isomorphic to f̃^{∗} in Ho(S^{B}), and so the adjunction maps η and ϵ induce isomorphisms on the homotopy category level. It follows that η and ϵ are weak equivalences.

The converse is clear. ■ There is a simplicial map i : X(A) → f̃^{∗}X(f(A)) that is defined on the

bisimplicial set level to be the map

X(A) → ⨆ X(A_{0})×Hom(A_{0}, A_{1})×⋯×Hom(A_{n−1}, A_{n})×Hom(f(A_{n}), f(A))

by sending an n-simplex x to the element (x, 1_{A}, …, 1_{A}, 1_{f(A)}) in the summand

X(A) × Hom(A, A) × ⋯ × Hom(A, A) × Hom(f(A), f(A))

corresponding to A_{i} = A for 0 ≤ i ≤ n. It follows from the description of the adjunction map η given in (2.8) that i is the composite of the map

X(A) ⟶^{η} f̃_{∗}f̃^{∗}X(A) = Hom(f̃^{∗}Hom(A, ), f̃^{∗}X)

with the map

Hom(f̃^{∗}Hom(A, ), f̃^{∗}X) ⟶^{s^{∗}} Hom(Hom(f(A), ), f̃^{∗}X) ≅ f̃^{∗}X(f(A))

induced by precomposition with the section s : Hom(f(A), ) → f̃^{∗}Hom(A, ) given in (2.3). Neither s nor i is natural in A.

Corollary 2.12. The functor f_{∗} : S^{B} → S^{A} induces an equivalence

f_{∗} : Ho(S^{B}) → Ho(S^{A})

of homotopy categories if and only if the following conditions are satisfied:

(1) for every simplicial functor X ∈ S^{A} and every object A ∈ A the map

i : X(A) → f̃^{∗}X(f(A)) = f_{∗}f̃^{∗}X(A)

is a weak equivalence,

(2) a map α : Y_{1} → Y_{2} of S^{B} is a weak equivalence if and only if the induced map f_{∗}α : f_{∗}Y_{1} → f_{∗}Y_{2} is a weak equivalence of S^{A}.

Proof: If f_{∗} is an equivalence of homotopy categories, then (2) holds, by Lemma II4.1. Furthermore, the canonical map map η is a weak equivalence by Corollary 2.11. It follows from the proof of Proposition 2.10 that s^{∗} is a weak equivalence in general, and so the map i is a weak equivalence as claimed.

Conversely, if conditions (1) and (2) are satisfied, then η is a weak equivalence, and then f̃_{∗}(ϵ) is a weak equivalence on account of a triangle identity. The conditions for Corollary 2.11 are therefore met, since f_{∗}(ϵ) must also be a weak equivalence. ■

Dwyer and Kan say that a functor f : A → B between small simplicial categories is a weak r-equivalence if two conditions hold:

(a) for every pair of objects A_{1}, A_{2} of A, the functor f induces a weak equivalence of simplicial sets

Hom(A_{1}, A_{2}) → Hom(f(A_{1}), f(A_{2})),

(b) every object in the category of components π_{0}B is a retract of an object in the image of π_{0}f.

Every functor f : A → B which satisfies condition (a) and is surjective on objects is a weak r-equivalence. Most examples of weak r-equivalences that are encountered in nature have this form.

The following is the main result of this chapter:

Theorem 2.13 (Dwyer-Kan). Suppose that f : A → B is a functor between small simplicial categories. Then the induced functor

f_{∗} : Ho(S^{B}) → Ho(S^{A})

of homotopy categories is an equivalence of categories if and only if the functor f is a weak r-equivalence.

Proof: Suppose that the functor f is a weak r-equivalence. We shall verify the conditions of Corollary 2.12.

The instance of i corresponding to the identity functor 1_{A} on A has the form

i : X(A) → ˜1_{A}^{∗}X(A).

This map is a weak equivalence since 1_{A} induces an equivalence Ho(S^{A}) → Ho(S^{A}), and it fits into a commutative diagram

˜1_{A}^{∗}X(A) uf_{∗}

X(A)^{’ ’ ’ ’’)} i_{‚ ‚ ‚‚ ›} i

f̃^{∗}X(f(A)),

where the indicated map f_{∗} is induced by a bisimplicial set map given on summands by the simplicial set maps

X(A_{0}) × ⋯ × Hom(A_{n}, A) → X(A_{0}) × ⋯ × Hom(f(A_{n}), f(A))

defined by 1 × ⋯ × 1 × f. This map f_{∗} is a weak equivalence, by assumption, and so the map i : X(A) → f̃^{∗}X(f(A)) is a weak equivalence as well.

Suppose that α : Y_{1} → Y_{2} is a map of S^{B} such that f_{∗}α : f_{∗}Y_{1} → f_{∗}Y_{2} is a weak equivalence of S^{A}. Then the induced simplicial set maps α : Y_{1}(f(A)) → Y_{2}(f(A)) are weak equivalences for all objects A ∈ A. The assumption that f is a weak r-equivalence means, in part, that for each B ∈ B there is an object A ∈ A and maps j : B → f(A) and q : f(A) → B in B such that qj maps to the identity in π_{0}B. It follows that, for each Y ∈ S^{B}, there is a path of simplicial homotopies from the composite simplicial set map

Y (B) ⟶^{Y (j)} Y (f(A)) ⟶^{Y (q)} Y (B)

to the identity on Y (B). It follows that this composite is a weak equivalence for any Y. In particular, in the diagram

\begin{matrix}Y_{1}(B)  w Y_{1}(j) u α \\ Y_{1}(f(A))  w Y_{1}(q) uα \\ Y_{1}(B) uα \end{matrix} ; \begin{matrix}Y_{2}(B)  w Y_{2}(j) \\ Y_{2}(f(B))  w Y_{2}(q) \\ Y_{2}(B) \end{matrix}

both horizontal composites are weak equivalences, so that α : Y_{1}(B) → Y_{2}(B) is a weak equivalence as well.

Conversely, suppose that f : A → B induces the indicated equivalence of homotopy categories. The following diagram commutes:

Hom(B, A) w i_{A A AA C} f

f̃^{∗}Hom(B, )(A) ur

Hom(f(B), f(A))

The map i is a weak equivalence by assumption, and r is the weak equivalence of Lemma 2.4, and so the condition (a) for f to be weak r-equivalence is verified.

The simplicial functor f̃^{∗}f̃_{∗}Hom(B, ) is the colimit

\underrightarrow{lim}_{x:f̃^{∗}Hom(A, )×Δ^{n}→Hom(B, )} f̃^{∗}Hom(A, ),

and the canonical map

ϵ : f̃^{∗}f̃_{∗}Hom(B, ) → Hom(B, )

is the unique map having components given by the morphisms

x : f̃^{∗}Hom(A, ) × Δ^{n} → Hom(B, )

in the definition of the colimit. The natural map ϵ is a weak equivalence by assumption, and the path component functor commutes with colimits, so there is a map

f̃^{∗}Hom(A, ) × Δ^{n} → Hom(B, )

such that 1_{B} ∈ Hom(B, B)_{0} lifts to a vertex of f̃^{∗}Hom(A, B) × Δ^{n} up to (iterated) homotopy. The map

Hom(f(A), B) × Δ^{n} ⟶^{s×1} f̃^{∗}Hom(A, B) × Δ^{n}

is a weak equivalence, so that 1_{B} is in the image of the composite

Hom(f(A), B) × Δ^{n} ⟶^{s×1} f̃^{∗}Hom(A, B) × Δ^{n} → Hom(B, B)

up to homotopy. This composite classifies an n-simplex α of Hom(B, f(A)), and one verifies that this maps sends a 0-simplex (γ, v) of Hom(f(A), B) × Δ^{n} maps to the composite γ ∘ v^{∗}(α). ■

3. Homotopy coherence.

Homotopy coherence theory is more a collection of phenomena than theorems. The whole point of the theory is to determine when a diagram which commutes up to a system of higher homotopies can be replaced by a diagram which commutes on the nose. To this end, the game is either to recognize when systems of higher homotopies can be suitably defined to serve as input for the Dwyer-Kan theorem (Theorem 2.13), or to avoid that result altogether, as one does (preferably) with lax functors. Various examples of these phenomena will be described here. This material pertains to homotopy coherent diagrams of simplicial sets, or spaces. A discussion of homotopy coherence phenomena for other selected simplicial model categories appears in Section 4. 3.1. Classical homotopy coherence.

Suppose that I is a small category, and write UI for the underlying directed graph of I. The graph UI is pointed in the sense that there is a distinguished element, namely 1_{a}, in the set I(a, a) of arrows from a to itself, for all objects a ∈ I. There is a free category F X associated to each pointed directed graph X, which has the same objects as X, and all finite composable strings of non-identity arrows in X as morphisms. Composition in F X is given by concatenation. The free category functor F is left adjoint to the underlying graph functor U, with canonical maps η : 1 → UF and ϵ : F U → 1. The canonical maps can be used, along with various iterations of the composite functor F U to define a simplicial category F_{∗}I. The category F_{n}I of n-simplices of F_{∗}I has the form F_{n}I = (F U)^{n+1}I, and the faces and degeneracies of F_{∗}I are defined by

\{ \begin{matrix}d_{i} = (F U)^{i−1}ϵ(F U)^{n−i}, & and \\ s_{j} = (F U)^{j}F ηU(F U)^{n−j}. \end{matrix}

One can appeal to the dual of results of [66, p.134,171] to see that this definition works, or check the simplicial identities directly. The diagram

F UF U(I) w d_{0} w d_{1} F U(I) w ϵ I

is a coequalizer in the category of small categories. It follows that, for any pair of objects a, b of I, the diagram of functions between morphism sets

F UF U(I)(a, b) w d_{0} w d_{1} F U(I)(a, b) w ϵ I(a, b)

is a coequalizer. It can be shown directly that the underlying simplicial graph UF_{∗}I has an extra degeneracy (see Section III.5), given by the functors

s_{−1} = ηU(F U)^{n}(I) → U(F U)^{n+1}(I).

Now, UF_{∗}(I)(a, b) = F_{∗}(I)(a, b) as a simplicial set, so that F_{∗}(I)(a, b) has an extra degeneracy. The simplicial set map

F_{∗}I(a, b) → K(I(a, b), 0)

induced by ϵ is therefore a weak equivalence by Lemma III.5.1, so we have constructed a weak r-equivalence

ϵ : F_{∗}I → K(I, 0)

of simplicial categories called simplicial free resolution of the category I. It is a special case of a construction that is quite ubiquitous.

The simplicial free resolution F_{∗}I of a category I is the traditional basis of the definition of homotopy coherence. Explicitly, a homotopy coherent I-diagram is classically defined to be a simplicial functor X : F_{∗}I → S.

Homotopy coherent diagrams are notoriously difficult to interpret, much less construct. Intuitively, X : F_{∗}I → S associates a simplicial map α_{∗} : X(a) → X(b) to each non-identity morphism α : a → b of I (α_{∗} is the image of the string (α) ∈ F U(I)). Given β : b → c in I, it’s not the case that (βα)_{∗} = β_{∗}α_{∗}, but rather that the two maps are homotopic in a precise way, via the homotopy X((β, α)) : X(a) × Δ^{1} → X(c) which is the image of the arrow ((β, α)) ∈ F UF U(I). Given yet another I-morphism γ : c → d, one sees that the associativity relationship γ(βα) = γβα in I in the sense that the morphism (((γ), (β, α))) of F UF UF U(I) determines a map X(a) × Δ^{2} → X(d), which is a higher homotopy that has the homotopies

\begin{matrix}(γβα)_{∗} w X((γ, βα))_{‚ ‚ ‚ ‚ ‚ ‚ ›} X((γ, β, α)) \\ γ_{∗}(βα)_{∗} uX((γ), (β, α)) \end{matrix} ; γ_{∗}β_{∗}α_{∗}

appearing in its 1-skeleton. The higher, or iterated, associativity relation

ω(γ(αβ)) = ωγαβ

corresponds to the 3-simplex ((((ω)), ((γ), (β, α)))) of F_{∗}I. The homotopy coherent diagram X is determined by all higher associativities or iterated bracketings of strings α_{n}α_{n−1} ⋯ α_{1} of morphisms in I.

Homotopy coherent diagrams are very rarely constructed from scratch, although there are obstruction theoretic techniques for doing so [27]. They nevertheless appear quite naturally, usually as the output of large categorical machines — more will be said about this below.

Most of the point of having a homotopy coherent diagram in hand is that one can immediately replace it up to pointwise weak equivalence by a diagram that commutes on the nose. This is a consequence of the following realization theorem:

Theorem 3.1 (Realization). Suppose that X : F_{∗}I → S is a homotopy coherent diagram in the category of simplicial sets. Then X is naturally pointwise weakly equivalent to a diagram ϵ_{∗}Y, for some ordinary I-diagram Y : I → S.

Proof: This result follows easily from the Dwyer-Kan theorem 2.13 and its proof: the simplicial functor ϵ : F_{∗}I → K(I, 0) is a weak r-equivalence, so there are natural pointwise weak equivalences

X ⟶^{i} ϵ̃_{∗}ϵ̃^{∗}X ⟵^{r^{∗}} ϵ_{∗}ϵ̃^{∗}X.

Take Y to be ϵ̃^{∗}X. ■ 3.2. Homotopy coherence: an expanded version.

The classical definition of homotopy coherence is interesting and complicated, but not flexible enough for all applications. It does not, for example, take into account diagrams where images of identity maps could wiggle away from actual identities up to controlled homotopy The realization result, Theorem 3.1, is also obviously just a special case of a much broader statement.

Define a resolution of a category I to be a simplicial functor π : A → K(I, 0) such the π is a weak r-equivalence. A homotopy coherent diagram on the category I shall henceforth be defined to be a simplicial functor X : A → S, where π : A → K(I, 0) is a resolution of I. We now have an expanded form of the realization theorem:

Theorem 3.2. Suppose that X : A → S is a homotopy coherent diagram which is defined with respect to some realization π : A → K(I, 0) of I. Then X is naturally pointwise equivalent to a diagram of the form π_{∗}Y, for some I-diagram Y : I → S.

Proof: The proof is a copy of the proof of Theorem 3.1. ■

The rest of this section will be taken up with a description of a natural resolution BI_{s} → K(I, 0) of the category I which is different from the simplicial free resolution of the previous section. Homotopy coherent diagrams X : BI_{s} → S provide for variance of morphisms induced by the identities of I up to homotopy.

Let a and b be objects of the small category I. There is a category, denoted I_{s}(a, b), whose objects are the functors of the form θ : n → I, with θ(0) = a and θ(n) = b. The morphisms of I_{s}(a, b) are commutative diagrams

n_{0} u γ _{````}ˆ θ_{0}

I
n_{1}^{4 4 44 6} θ_{1}

where θ_{0} and θ_{1} are objects of I_{s}(a, b) and γ : n_{0} → n_{1} is an ordinal number map which is end point preserving in the sense that γ(0) = 0 and γ(n_{0}) = n_{1}.

Suppose that θ : n → I is an object of I_{s}(a, b) and ω : m → I is an object of I_{s}(b, c). The poset join n ∗ m may be identified up to isomorphism with the ordinal number n + m, in such a way that the inclusion n ↪ n ∗ m is identified with the ordinal number map n → n + m defined by j ↦ j + n for 0 ≤ j ≤ m. The objects θ and γ together determine a functor θ ∗ γ : n + m → I, defined by

θ ∗ γ(j) = \{ \begin{matrix}θ(j), & if 0 ≤ j ≤ n, \\ γ(j − n), & if n ≤ j ≤ m + n. \end{matrix}

This is plainly the object level description of a functor

I_{s}(a, b) × I_{s}(b, c) ⟶^{∗} I_{s}(a, c),

which is called the join functor. This operation is associative, and has a two-sided identity in each I_{s}(a, a) given by the object a : 0 → I.

We have constructed a category I_{s} which has the same objects as I, and is enriched in the category cat of small categories. Applying the nerve construction to each of the categories I_{s}(a, b) gives simplicial sets BI_{s}(a, b), which then form the morphism objects for a simplicial category BI_{s}, which has the same objects as I. These constructions are obviously natural: a functor f : I → J induces a functor f : I_{s} → J_{s} of categories enriched in cat, and hence determines a functor f : BI_{s} → BJ_{s} of of simplicial categories.

The category I_{s}(a, b) has the form

I_{s}(a, b) = ⨆_{f:a→b_{in I}} I_{s}(a, b)_{f},

where I_{s}(a, b)_{f} is the subcategory of those strings α : n → I_{s} whose composite is f : a → b. The category I_{s}(a, b)_{f} has an initial object, namely the functor f : 1 → I_{s} which is determined by f, so that, on the simplicial set level, we have a decomposition

BI_{s}(a, b) = ⨆_{f:a→b_{in I}} BI_{s}(a, b)_{f},

of BI_{s}(a, b) into connected components, each of which is contractible. It follows that the path component functor determines a resolution BI_{s} → K(I, 0) of the category I.

A homotopy coherent diagram X : BI_{s} → S exhibits the standard homotopy coherence phenomena. Suppose that

a_{0} ⟶^{α_{1}} a_{1} ⟶^{α_{2}} a_{2}

is a composable pair of morphisms of I, and let these morphisms canonically determine a functor α : 2 → I, which in turn is a 0-simplex of BI_{s}(a_{0}, a_{2}). Then X associates simplicial set morphisms (α_{1})_{∗} : X(a_{0}) → X(a_{1}) and (α_{2})_{∗} : X(a_{1}) → X(a_{2}) to the morphisms (aka. 0-simplices) α_{1} and α_{2} respectively, and associates the composite simplicial set map

(α_{2})_{∗}(α_{1})_{∗} : X(a_{0}) → X(a_{2})

to the simplex α, since X takes joins to composites. There is a 1-simplex σ of BI_{s}(a_{0}, a_{2}), defined by the picture

2_{````}ˆ α
I_{s},

1u d^{1} ^{4 4 44 6} α_{2}α_{1}

where 1 → 2 is the unique endpoint preserving ordinal number map, and this 1-simplex is mapped to a homotopy X(σ) : X(a_{0})×Δ^{1} → X(a_{2}) from (α_{2}α_{1})_{∗} to the composite (α_{2})_{∗}(α_{1})_{∗}. In the same way, the associativity relation α_{3}(α_{2}α_{1}) = α_{3}α_{2}α_{1} gives rise to a 2-simplex

1 w^{d^{1}}_{4 4 4 4 44 6}_{α_{3}α_{2}α_{1}} 2 w^{d^{2}} u_{(α_{3}α_{2},α_{1})} 3 h h h h h h k_{(α_{3},α_{2},α_{1})}

I

and hence to a higher homotopy X(a_{0}) × Δ^{2} → X(a_{3}).

If a is an object of I, then the identity map 1_{a} : a → a defines a 0-simplex 1_{a} : 1 → I, and hence gets mapped to the simplicial set map (1_{a})_{∗} : X(a) → X(a), whereas the 0-simplex a : 0 → I is an identity for BI_{s}, and is therefore sent to the identity 1_{X(a)} on X(a). There is a 1-simplex η of BI_{s}(a, a), of the form

\begin{aligned}0_{````}ˆ a \\ & I \\ 1u^{4 4 44 6} 1_{a} \end{aligned}

which is then mapped to a homotopy X(η) : X(a) × Δ^{1} → X(α) from (1_{a})_{∗} to 1_{X(a)}. 3.3. Lax functors.

A lax functor F : I ⇝ cat associates a category F (a) to each object a of I, and associates a functor α_{∗} : F (a) → F (b) to each morphism α : a → b, in such a way that there are natural transformations

θ(β, α) : (βα)_{∗} → β_{∗}α_{∗} and η_{a} : (1_{a})_{∗} → 1_{F (a)},

which together satisfy the cocycle conditions

(3.3)  \begin{matrix}(βαγ)_{∗}  w θ(βα, γ) u θ(β, αγ) \\ (βα)_{∗}γ_{∗} uθ(β, α)γ_{∗} \end{matrix} ; \begin{matrix}β_{∗}(αγ)_{∗}  w β_{∗}θ(α, γ) \\ β_{∗}α_{∗}γ_{∗} \end{matrix}

(3.4)  \begin{matrix}α_{∗}1_{F (a)} & α_{∗}(1_{a})_{∗} uα_{∗}η_{a} \\ α_{∗} & (α1_{a})_{∗} uθ(α, 1_{a}) \end{matrix}  \begin{matrix}(1_{b})_{∗}α_{∗}  w η_{b}α_{∗} \\ 1_{F (b)}α_{∗} \end{matrix} ; (1_{b}α)_{∗} u θ(1_{b}, α)  α_{∗}.

The category cat of categories is enriched in categories. Its morphism categories Hom(A, B) have the functors from A to B as objects with natural transformations as morphisms. The composition functor

Hom(A, B) × Hom(B, C) → Hom(A, C)

is defined on morphisms as follows: given natural transformations α_{1} and α_{2}

F_{1}  w  F_{2}  w
A  ↓ α_{1}  B  ↓ α_{2}  C
\begin{aligned}& w \\ G_{1} \end{aligned}  \begin{aligned}& w \\ G_{2} \end{aligned}

one defines the composition α_{2}α_{1} of α_{1} and α_{2} to be the diagonal transformation appearing in the commutative diagram

\begin{matrix}F_{2}F_{1}  w F_{2}α_{1} u α_{2}F_{1} _{4 4 4 4 6} \\ F_{2}G_{1} uα_{2}G_{1} \end{matrix} ; \begin{matrix}G_{2}F_{1}  w G_{2}α_{1} \\ G_{2}G_{1}. \end{matrix}

There is a one to one correspondence between functors

I_{s} → cat

and lax functors on I. In particular, given a lax functor F as above, associate to each string n → I in I of the form

a_{0} ⟶^{α_{1}} a_{1} ⟶^{α_{2}} … ⟶^{α_{n}} a_{n}

the composite functor

(α_{n})_{∗} … (α_{1})_{∗} : F (a_{0}) → F (a_{n}).

The requisite functors

I_{s}(a_{0}, a_{n}) → Hom(F (a_{0}), F (a_{n}))

are defined on morphisms of I_{s}(a_{0}, a_{n}) by first making a definition on cofaces and codegeneracies, which are completely determined up to join by the transformations θ and η, and then by showing that the relevant cosimplicial identities are satisfied. The non-trivial cosimplicial identities amount to the cocycle conditions which appear in the definition of the lax functor F.

The object cat has associated to it a simplicial category Bcat in the obvious way: one uses the nerves BHom(A, C) of the categories of morphisms Hom(A, C) appearing in cat. Note that BHom(A, C) is canonically isomorphic to the function complex Hom(BA, BC). The collection of such canonical isomorphisms respects the composition laws of Bcat and S and hence determine a functor Bcat → S. It follows in particular that any lax functor F : I_{s} → cat determines a homotopy coherent diagram

BI_{s} ⟶^{BF} Bcat → S

on I.

3.4. The Grothendieck construction.

Homotopy coherent diagrams arising from lax functors are most often realized (or rectified) by using a category theoretic method that is known as the Grothendieck construction in place of a result like Theorem 3.2. Suppose that F : I ⇝ cat is a lax functor. The Grothendieck construction associated to F is a category LF whose set of objects consists of all pairs (a, x), where a is an object of I and x is an object of the category F (a). A morphism (α, f) : (a, x) → (b, y) is a pair consisting of a morphism α : a → b of the base category I and a morphism α_{∗}(x) → y of the category F (b). The composite of (α, x) with the morphism (β, g) : (b, y) → (c, z) of LF is defined to be the map (βα, g ∗f), where g ∗ f is the composite

(βα)_{∗}(x) ⟶^{θ(β,α)} β_{∗}α_{∗}(x) ⟶^{β_{∗}(f)} β_{∗}(y) ⟶^{g} z

of F (c). The identity morphism on an element (a, x) of LF is the morphism (1_{a}, η_{a}). The associativity of the composition operation in LF and the fact that the morphisms (1_{a}, x) are two-sided identities are, respectively, consequences of the cocycle conditions (3.3) and (3.4).

Projection onto the first variable, for both objects and morphisms, defines a canonical functor π_{F} : LF → I. The comma category π_{F} ↓ c associated to an object c of I has all morphisms α : π_{F} (a, x) → c for objects. There is a functor f_{c} : π_{F} ↓ c → F (c) which associates the object α_{∗}(c) ∈ F (I) to the object α : π_{F} (a, x) → c. Also, there is a functor g_{c} : F (c) → π_{F} ↓ c which associates the object 1_{c} : π_{F} (c, x) → c to x ∈ F (c). The functor f_{c} is left adjoint to g_{c}, as can be seen by observing that the commutative diagram

π_{F} (a, x)_{````}ˆα u (α, h)
c
π_{F} (c, y)^{’ ’ ’’)} 1_{c}

is uniquely determined by the map h : α_{∗}(x) → y of F (c).

Any morphism β : c → d induces a functor β_{∗} : π_{F} ↓ c → π_{F} ↓ d in an obvious way, and this assignment determines a functor π_{F} ↓? : I → cat. The diagram of functors

\begin{matrix}F (c)  w g_{c} u β_{∗} \\ π_{F} ↓ c uβ_{∗} \end{matrix} ; \begin{matrix}F (c)  w g_{d} \\ π_{F} ↓ d \end{matrix}

commutes up to canonically determined natural transformation.

Generally, any functor f : D → E that has a left or right adjoint induces a homotopy equivalence f_{∗} : BD → BE of the associated nerves. This is a result of the fact that any natural transformation of functors gives rise to a homotopy of the respective induced maps of simplicial sets. The Grothendieck construction therefore gives rise to homotopy equivalences

g_{c∗} : BF (c) ⟶^{≃} B(π_{F} ↓ c)

such that the diagrams

\begin{matrix}BF (c)  w g_{c∗} u β_{∗} \\ B(π_{F} ↓ c) uβ_{∗} \end{matrix} ; \begin{matrix}BF (c)  w g_{d∗} \\ B(π_{F} ↓ d) \end{matrix}

commute up to homotopy. In other words, the I-diagram c ↦ B(π_{F} ↓ c) is a realization of the homotopy coherent diagram c ↦ BF (c).

4. Realization theorems.

Suppose that π : A → K(I, 0) is a resolution of a small category I, and let X : A → S be a homotopy coherent diagram on I. A realization of X is a simplicial set valued functor Y : I → S such that there is a pointwise weak equivalence X ≃ π_{∗}Y in the simplicial functor category S^{A}. Theorem 3.2 says that any homotopy coherent diagram X has a realization, and that the weak equivalence in S^{A} can be chosen canonically.

Given an arbitrary simplicial model category M and a resolution π : A → K(I, 0), it is certainly sensible to say that a homotopy coherent diagram on I is a simplicial functor X : A → M. One analogously defines a realization of X to be a functor Y : I → M such that there is a pointwise weak equivalence X ≃ π_{∗}Y. The purpose of this section is to show that all homotopy coherent diagrams in M admit realizations if the simplicial model category M has an adequate notion of homotopy colimit. This is done by giving a proof of Theorem 3.2 which does not depend on the Yoneda lemma 1.2. The proof is achieved by thinking about the functor π̃^{∗} in a different way; we lose the functor π̃_{∗} and along with it any notion of an equivalence of homotopy categories associated to the categories S^{A} and S^{I}. We begin with a weakened version of the Dwyer-Kan theorem.

Theorem 4.1. Suppose that a simplicial functor f : A → B is homotopically full and faithful in the sense that all induced simplicial set maps

f_{∗} : Hom(A, B) → Hom(f(A), f(B))

are weak equivalences, and let X : A → S be a simplicial functor. Then X is naturally pointwise weakly equivalent to the simplicial functor f_{∗}f̃^{∗}X.

Proof: The assumption on f implies that the map of simplicial objects in S^{B} defined by

⨆_{(A_{0},...,A_{n})} X(A_{0}) × Hom(A_{0}, A_{1}) × ⋯ × Hom(A_{n}, A) u 1 × f_{∗}

⨆_{(A_{0},...,A_{n})} X(A_{0}) × Hom(A_{0}, A_{1}) × ⋯ × Hom(f(A_{n}), f(A))

is a levelwise weak equivalence, which is natural in A ∈ A. The simplicial object

⨆_{(A_{0},...,A_{n})} X(A_{0}) × Hom(A_{0}, A_{1}) × ⋯ × Hom(A_{n}, A)

is, in n-simplices, the nerve of the translation category associated to the composite functor

A_{m} ↓ A ⟶^{Q} A_{m} ⟶^{X_{m}} Sets,

and the identity element 1_{A} is terminal in A_{m} ↓ A. It follows (see the development around (2.3)) that there is a canonical weak equivalence induced by the bisimplicial set map

r : ⨆_{(A_{0},...,A_{n})} X(A_{0}) × Hom(A_{0}, A_{1}) × ⋯ × Hom(A_{n}, A) → X(A)

which is induced on the translation category level by functors

(x, α : A′ → A) ↦ α_{∗}(x).

We therefore have natural pointwise weak equivalences

f_{∗}f̃^{∗}X ⟵^{≃} 1_{∗}˜1^{∗}X ⟶_{≃}^{r} X. ■

Suppose now that f : A → B is a simplicial functor, and that Y : A → M is a simplicial functor taking values in a simplicial model category M. We are entitled to an analogue of the functor f̃^{∗}; in particular, we define f̃^{∗}Y (B) to be the simplicial object having n-simplices

⨆_{(A_{0},...,A_{n})} Y (A_{0}) ⊗ Hom(A_{0}, A_{1}) ⊗ ⋯ ⊗ Hom(f(A_{n}), B).

In this way, we define a functor

f̃^{∗} : M^{A} → S(M^{B}),

where S(M^{B}) denotes the category of simplicial objects in M^{B}. We are also entitled to the following analogues of the maps appearing in the proof of Theorem 4.1 in the category S(M^{B}): (4.2)

⨆_{(A_{0},...,A_{n})} Y (A_{0}) ⊗ Hom(A_{0}, A_{1}) ⊗ ⋯ ⊗ Hom(A_{n}, A) w r u f_{∗} Y (A)

⨆_{(A_{0},...,A_{n})} Y (A_{0}) ⊗ Hom(A_{0}, A_{1}) ⊗ ⋯ ⊗ Hom(f(A_{n}), f(A)).

The trick, either for a given simplicial model category or for a particular class of objects of the simplicial functor category M^{A}, is to find a realization functor S(M) → M which takes the maps in this diagram to weak equivalences of M^{B}.

The naive realization d(Z) of a simplicial object Z in M is the coend, given by the coequalizer

⨆_{m→n} Z_{n} ⊗ Δ^{m} ⇉ ⨆_{n≥0} Z_{n} ⊗ Δ^{n} → d(Z).

The object d(Z) has a filtration

d(Z)^{(n)} ⊂ d(Z)^{(n+1)} ⊂ …,

where d(Z)^{(n)} is the epimorphic image of the restricted map

⨆_{p≤n} Z_{p} ⊗ Δ^{p} → d(Z).

There are pushout diagrams

\begin{matrix}(s_{[n]}Zn ⊗ Δ^{n+1}) ∪ (Z_{n+1} ⊗ ∂Δ^{n+1})  w u j_{∗} \\ d(Z)^{(n)} u \end{matrix} ; \begin{matrix}Z_{n+1} ⊗ Δ^{n+1}  w \\ d(Z)^{(n+1)}, \end{matrix}

in M, where the map j_{∗} is canonically induced by the monomorphism j : s_{[n]}Z_{n} ↪ Z_{n+1} given by taking the epimorphic image of the map

s : ⨆_{0≤i≤n} Z_{n} → Z_{n+1}

which applies the i^{th} degeneracy s_{i} on the i^{th} summand. The monomorphism d(Z)^{(n)} ↪ d(Z)^{(n+1)} is a cofibration of M if j : s_{[n]}Z_{n} → Z_{n+1} is a cofibration.

Say that a simplicial object Z in M is diagonally cofibrant if

(1) Z_{0} is a cofibrant object of M, and

(2) each morphism j : s_{[n]}Z_{n} → Z_{n+1} is a cofibration.

If Z is a diagonally cofibrant simplicial object of M in this sense, then

d(Z)^{(0)} = Z_{0}

is cofibrant and all of the maps d(Z)^{(n)} ↪ d(Z)^{(n+1)} are cofibrations, and so the realization d(Z) is a cofibrant object of M. It also follows that if f : Z → W is a map of diagonally cofibrant simplicial objects which is a levelwise weak equivalence in the sense that all of the maps f : Z_{n} → W_{n} are weak equivalences of M, then the induced map f_{∗} : d(Z) → d(W) is a weak equivalence as well.

Theorem 4.3. Suppose that f : A → B is a simplicial functor which is homotopically full and faithful. Suppose that Y : A → M is a simplicial functor taking values in simplicial model category M, such that Y (A) is a cofibrant object of M for all objects A ∈ A. Then the morphisms f_{∗} and r induce pointwise weak equivalences

f_{∗}(df̃^{∗}Y) ≅ d(f_{∗}f̃^{∗}Y) ⟵_{≃}^{f_{∗}} d(1_{∗}˜1^{∗}Y) ⟶_{≃}^{r} d(Y) = Y.

Proof: The assumptions imply that the simplicial object f̃^{∗}Y (B) is diagonally cofibrant, for all B ∈ B. This is seen by observing that the object

⨆_{(A_{0},...,A_{n})} Y (A_{0}) ⊗ Hom(A_{0}, A_{1}) ⊗ ⋯ ⊗ Hom(f(A_{n}), B)

can be rewritten in the form

⨆_{A_{0}} Y (A_{0}) ⊗ B(f ↓ B)_{n}^{A_{0}},

where

B(f ↓ B)_{n}^{A_{0}} = Hom(A_{0}, A_{1}) × ⋯ × Hom(f(A_{n}), B)

means strings of length n in the simplicial nerve B(f ↓ B) which begin at A_{0}. This decomposition is preserved by all degeneracies, and so we have

s_{[n]}f̃^{∗}(B) = ⨆_{A_{0}} X(A_{0}) ⊗ DB(f ↓ B)_{n+1}^{A_{0}},

where DB(f ↓ B)_{n+1}^{A_{0}} denotes the degenerate strings of length n + 1 which begin at A_{0}. Each map

Y (A_{0}) ⊗ DB(f ↓ B)_{n+1}^{A_{0}} → Y (A_{0}) ⊗ B(f ↓ B)_{n+1}^{A_{0}}

induced by the simplicial set inclusion DB(f ↓ B)_{n+1}^{A_{0}} ⊂ B(f ↓ B)_{n+1}^{A_{0}} is a cofibration, since Y (A_{0}) is cofibrant, and the claim is verified.

The maps

⨆_{(A_{0},...,A_{n})} Y (A_{0}) ⊗ Hom(A_{0}, A_{1}) ⊗ ⋯ ⊗ Hom(A_{n}, A) u 1 ⊗ f_{∗}

⨆_{(A_{0},...,A_{n})} Y (A_{0}) ⊗ Hom(A_{0}, A_{1}) ⊗ ⋯ ⊗ Hom(f(A_{n}), f(A))

are weak equivalences since all Y (A_{0}) are cofibrant and f_{∗} is a weak equivalence by assumption. It follows that the map

f_{∗} : d(1_{∗}˜1^{∗}Y (A)) → d(f_{∗}f̃^{∗}(f(A)))

is a weak equivalence, since the simplicial objects at issue are diagonally cofibrant.

The retraction map r : 1_{∗}˜1^{∗}Y (A) → Y (A) has a section s : Y (A) → 1_{∗}˜1^{∗}Y (A), along with an associated homotopy

h : 1_{∗}˜1^{∗}Y (A) ⊗ Δ^{1} → 1_{∗}˜1^{∗}Y (A)

from sr to the identity on 1_{∗}˜1^{∗}Y (A). Applying the coend functor to h gives a homotopy

d(h) : d(1_{∗}˜1^{∗}Y (A)) ⊗ Δ^{1} → d(1_{∗}˜1^{∗}Y (A)),

from s_{∗}r_{∗} to the identity, again since 1_{∗}˜1^{∗}Y (A) is diagonally cofibrant. It follows from Lemma II.4.1 that r_{∗} is a weak equivalence of M. ■

All pointed simplicial sets are cofibrant, so Theorem 4.3 immediately implies

Corollary 4.4. Suppose that a simplicial functor f : A → B is homotopically full and faithful, and let X : A → S_{∗} be a simplicial functor taking values in the category S_{∗} of pointed simplicial sets. Then X is naturally pointwise weakly equivalent to the simplicial functor f_{∗}d(f̃^{∗}X).

Spectra, and the Bousfield-Friedlander model for the stable category, are formally discussed in Section IX.6 below. Within that model, it is not true that all spectra are cofibrant, but homotopy colimits in the category of spectra are constructed levelwise within the category of pointed simplicial sets, giving

Corollary 4.5. Suppose that a simplicial functor f : A → B is homotopically full and faithful, and let X : A → Spt be a simplicial functor taking values in the category Spt of spectra. Then X is naturally pointwise strictly weakly equivalent to the simplicial functor f_{∗}d(f̃^{∗}X).

A simplicial functor X : A → Spt taking values in the category of spectra can be identified with a spectrum object in the category S_{∗}^{A} of pointed simplicial set valued simplicial functors. The object X therefore consists of simplicial functors X_{n} : A → S_{∗} and pointed transformations S^{1} ∧ X_{n} → X_{n+1}. A map X → Y in Spt^{A} is a pointwise strict weak equivalence if all of the maps X_{n}(A) → Y_{n}(A) are weak equivalences of pointed simplicial sets.

Despite all the huffing and puffing in its proof, Theorem 4.3 is a formal result which may not always be the best tool. To illustrate, suppose that Y : A → sAb is a simplicial functor taking values in the category of simplicial abelian groups, and consider the maps of bisimplicial abelian groups corresponding to the diagram (4.2). One uses spectral sequence arguments and the generalized Eilenberg-Zilber theorem (Theorem IV.2.5) to see that, if f : A → B is homotopically faithful, then the maps f_{∗} and r induce weak equivalences of the associated diagonal simplicial abelian groups, proving Lemma 4.6. Suppose that a simplicial functor f : A → B is homotopically full and faithful, and let Y : A → sAb be a simplicial functor taking values in the category sAb of simplicial abelian groups. Then Y is naturally pointwise weakly equivalent to the simplicial functor f_{∗}d(f̃^{∗}Y).

Here, d is the diagonal functor. Note that the simplicial abelian groups Y (A) do not have to be cofibrant.

The Dold-Kan equivalence (Corollary III.2.3) relating the categories sAb of simplicial abelian groups and the category Ch_{+} of ordinary chain complexes, gives a simplicial model structure to the chain complex category. It also immediately implies the following result:

Corollary 4.7. Suppose that a simplicial functor f : A → B is homotopically full and faithful, and let Y : A → Ch_{+} be a simplicial functor taking values in the category Ch_{+} of ordinary chain complexes. Then Y is naturally pointwise weakly equivalent to the simplicial functor f_{∗}d(f̃^{∗}Y).

In this case, d(f̃^{∗}Y) is the normalized chain complex Nd(f̃^{∗}(ΓY)), where ΓY is the diagram of simplicial abelian groups associated to the diagram of chain complexes Y.