Chapter II Model Categories

The closed model axioms have a list of basic abstract consequences, including an expanded notion of homotopy and a Whitehead theorem. The associated homotopy category is defined to be the result of of formally inverting the weak equivalences within the ambient closed model category, but can be constructed (in the CW-complex style) by taking homotopy classes of maps between objects which are fibrant and cofibrant. These topics are presented in the first section of this chapter.

The simplicial set category has rather more structure than just that of a closed model category: the set hom(X, Y) of maps between simplicial sets X and Y is the set of vertices of the function complex Hom(X, Y), and the collection of all such function complexes determines a simplicial category. We’ve already seen that the function complexes satisfy an exponential law and respect cofibrations and fibrations in a suitable sense. The existence of the function complexes and the interaction with the closed model structure can be abstracted to a definition of a simplicial model category, which is given in Section 2, along with various examples. Basic consequences of the additional simplicial structure are presented, with further examples, in Sections 3 and 4.

Sections 5, 6 and 7 are about detection principles for simplicial model structures. Generally speaking, a simplicial model category structure for the category sC of simplicial objects in a category C is induced from the simplicial model structure on the simplicial set category in the presence of an adjoint pair of functors

F : S ⇄ sC : G,

(or a collection of adjoint pairs) if G has a little extra structure (such as preservation of filtered colimits) in addition to being a right adjoint — this is Theorem 5.1. In one major stream of examples, the category C is some algebraic species, such as groups or abelian groups, and G is a forgetful functor. There is, however, an extra technical condition on Theorem 5.1, namely that every cofibration of sC having the left lifting property with respect to all fibrations should be a weak equivalence. This condition can often be verified by brute force, as can be done in the presence of a small object argument for the factorization axioms (eg. simplicial abelian groups), but there is a deeper criterion, namely the existence of a natural fibrant model (Lemma 6.1). The other major source of examples has to do with G being a representable functor of the form G = hom(Z, ), where Z is either small in the sense that hom_{C}(Z, ) respects filtered colimits, or is a disjoint union of small objects. In this setting, Kan’s Ex^{∞}-construction (see Section III.4) is used to construct the natural fibrant models required by Lemma 6.1. This line of argument is generalized significantly in Section 6, at the cost of the introduction of cofibrantly generated closed model categories and transfinite small object arguments.

Section 8 is an apparent return to basics. We develop a criterion for a pair of adjoint functors between closed model categories to induce adjoint functors on the homotopy category level, known as Quillen’s total derived functor theorem. Quillen’s result is, at the same time, a non-abelian version of the calculus of higher direct images, and a generalization of the standard result that cohomology is homotopy classes of maps taking values in Eilenberg-Mac Lane spaces.

The category of simplicial sets, finally, has even more structure: it is a proper simplicial model category, which means that, in addition to everything else, weak equivalences are preserved by pullback over fibrations and by pushout along cofibrations. This property is discussed in Section 9. Properness is the basis of the standard results about homotopy cartesian diagrams, as well as being of fundamental importance in stable homotopy theory. We discuss homotopy cartesian diagrams in the context of Gunnarsson’s axiomatic approach to the glueing and coglueing lemmas [40].

1. Homotopical algebra.

Recall that a closed model category C is a category which is equipped with three classes of morphisms, called cofibrations, fibrations and weak equivalences which together satisfy the following axioms:

CM1: The category C is closed under all finite limits and colimits.

CM2: Suppose that the following diagram commutes in C:

\begin{matrix}X w g_{N N N N P} h \\ Y ‚ ‚ ‚ ‚ “ f \end{matrix} ; Z.

If any two of f, g and h are weak equivalences, then so is the third.

CM3: If f is a retract of g and g is a weak equivalence, fibration or cofibration, then so is f.

CM4: Suppose that we are given a commutative solid arrow diagram

U  w u i  Xup

V w^{i i i ij} Y

where i is a cofibration and p is a fibration. Then the dotted arrow exists, making the diagram commute, if either i or p is also a weak equivalence.

CM5: Any map f : X → Y may be factored:

(a) f = p · i where p is a fibration and i is a trivial cofibration, and (b) f = q · j where q is a trivial fibration and j is a cofibration.

This set of axioms has a list of standard consequences which amplifies the interplay between cofibrations, fibrations and weak equivalences, giving rise to collection of abstract techniques that has been known as homotopical algebra since Quillen introduced the term in [75]. This theory is is really an older friend in modern dress, namely obstruction theory made axiomatic. The basic results, along with their proofs, are sketched in this section.

We begin with the original meaning of the word “closed”:

Lemma 1.1. Suppose that C is a closed model category. Then we have the following:

(1) A map i : U → V of C is a cofibration if and only if i has the left lifting property with respect to all trivial cofibrations.

(2) The map i is a trivial cofibration if and only if it has the right lifting property with respect to all fibrations.

(3) A map p : X → Y of C is a fibration if and only if it has the right lifting property with respect to all trivial cofibrations.

(4) The map p is a trivial fibration if and only if it has the right lifting property with respect to all cofibrations.

The point of Lemma 1.1 is that the various species of cofibrations and fibrations determine each other via lifting properties. Proof: We shall only give a proof of the first statement; the other proofs are similar.

Suppose that i is a cofibration, p is a trivial fibration, and that there is a commutative diagram

(1.2)  \begin{matrix}U  w α u i \\ Xup \end{matrix} ; {V  w β \atop Y}

Then there is a map θ : V → X such that pθ = β and θi = α, by CM4. Conversely suppose that i : U → V is a map which has the left lifting property with respect to all trivial fibrations. By CM5, i has a factorization

\begin{matrix}U w j_{’ ’ ’ ’)} i \\ Wuq \end{matrix} ; V

where j is a cofibration and q is a trivial fibration. But then there is a commutative diagram

U  w j u i  Wuq

V ^{h h hh j} V

and so i is a retract of j. CM3 then implies that i is a cofibration. ■

The proof of the Lemma 1.1 contains one of the standard tricks that is used to prove that the axiom CM4 holds in a variety of settings, subject to having an adequate proof of the factorization axiom CM5. Lemma 1.1 also immediately implies the following:

Corollary 1.3.

(1) The classes of cofibrations and trivial cofibrations are closed under composition and pushout. Any isomorphism is a cofibration.

(2) The classes of fibrations and trivial fibrations are closed under composition and pullback. Any isomorphism is a fibration.

The statements in Corollary 1.3 are axioms in Quillen’s original definition of a model category [75].

Quillen defines a cylinder object for an object A in a closed model category C to be a commutative triangle

(1.4)  A ⊔ A_{4 4 44 6} ∇ u i ; \begin{aligned}Ã & w & σ \\ & & A \end{aligned}

where ∇ : A⊔A → A is the canonical fold map which is defined to be the identity on A on each summand, i is a cofibration, and σ is a weak equivalence. Then a left homotopy of maps f, g : A → B is a commutative diagram

A ⊔ A_{4 4 44 6} (f, g) u i ; \begin{aligned}Ã & w & h \\ & & B \end{aligned}

where (f, g) is the map on the disjoint union which is defined by f on one summand and g on the other, and the data consisting of

i = (i_{0}, i_{1}) : A ⊔ A → Ã

comes from some choice of cylinder object for A.

There are many choices of cylinder object for a given object A of a closed model category C: any factorization of ∇ : A⊔A → A into a cofibration followed by a trivial fibration that one might get out of CM5 gives a cylinder object for A. In general, the object A needs to be cofibrant for its cylinder objects to be homotopically interesting:

Lemma 1.5.

(1) Suppose that A is a cofibrant object of a closed model category C, and that the diagram (1.4) is a cylinder object for A. Then the maps i_{0}, i_{1} : A → Ã are trivial cofibrations.

(2) Left homotopy of maps A → B in a closed model category C is an equivalence relation if A is cofibrant.

Proof: Denote the initial object of C by ∅.

For the first part, observe that the diagram

\begin{matrix}∅  w u \\ Auin_{R} \end{matrix} ; \begin{matrix}A  w in_{L} \\ A ⊔ A \end{matrix}

is a pushout since cofibrations are closed under pushout by Lemma 1.1, and the unique map ∅ → A is a cofibration by assumption. It follows that the inclusions in_{L} and in_{R} are cofibrations, so that the compositions i_{0} = (i_{0}, i_{1}) ∘ in_{L} and i_{1} = (i_{0}, i_{1}) ∘ in_{R} are cofibrations as well. Finally, the maps i_{0} and i_{1} are weak equivalences by CM2, since the map σ is a weak equivalence.

To prove the second statement, first observe that if τ : A ⊔ A → A ⊔ A is the automorphism which flips summands, then the diagram

A ⊔ A_{4 4 44 6} ∇ u i ∘ τ ; \begin{aligned}Ã & w & σ \\ & & A, \end{aligned}

which is constructed from (1.4) by twisting by τ, is a cylinder object for A. This implies that the left homotopy relation is symmetric.

Subject to the same definitions, the map fσ : A → B is clearly a left homotopy from f : A → B to itself, giving reflexivity.

Suppose given cylinder objects

A ⊔ A u (i_{0}^{ϵ}, i_{1}^{ϵ}) _{4 4 44 6} ∇ ; \begin{matrix}A_{ϵ} w σ^{ϵ} \\ A, \end{matrix}

where ϵ = 0, 1. Form the pushout

\begin{matrix}A  w i_{0}^{1} u i_{1}^{0} \\ A_{1} ui_{1∗} \end{matrix} ; A_{0}  w i_{0∗}  Ã

Then the map

A ⊔ A ⟶^{(i_{0∗}i_{0}^{0},i_{1∗}i_{1}^{1})} Ã

is a composite

A ⊔ A ⟶^{i_{0}^{0}⊔1} A_{0} ⊔ A ⟶^{(i_{0∗},i_{1∗}i_{1}^{1})} Ã.

The map i_{0}^{0} ⊔ 1 is a cofibration by the first statement of the lemma, and there is a pushout diagram

\begin{matrix}A ⊔ A  w i_{1}^{0} ⊔ 1 u (i_{0}^{1}, i_{1}^{1}) \\ A_{0} ⊔ A u(i_{0∗}, i_{1∗}i_{1}^{1}) \end{matrix} ; A_{1}  w i_{1∗}  Ã.

In particular, there is a cylinder object for A

A ⊔ A u (i_{0∗}i_{0}^{0}, i_{1∗}i_{1}^{1}) _{4 4 44 6} ∇ ; \begin{matrix}A_{ϵ} w σ_{∗} \\ A. \end{matrix}

It follows that if there are left homotopies h_{0} : A_{0} → B from f_{0} to f_{1} and h_{1} : A_{1} → B from f_{1} to f_{2}, then there is an induced left homotopy h_{∗} : Ã → B from f_{0} to f_{2}. ■

A path object for an object B of a closed model category C is a commutative triangle

(1.6)

B̂up = (p_{0}, p_{1})

B^{N N N NN P} s w Δ B × B

where Δ is the diagonal map, s is a weak equivalence, and p (which is given by p_{0} on one factor and by p_{1} on the other) is a fibration.

Once again, the factorization axiom CM5 dictates that there is an ample supply of path objects for each object of an arbitrary closed model category. If a simplicial set X is a Kan complex, then the function complex hom(Δ^{1}, X) is a path object for X, and the function space Y^{I} is a path object for each compactly generated Hausdorff space Y.

There is a notion of right homotopy which corresponds to path objects: two maps f, g : A → B are said to be right homotopic if there is a diagram

B̂u(p_{0}, p_{1})

A^{N N N NN P} h w (f, g) B × B

where the map (p_{0}, p_{1}) arises from some path object (1.6), and (f, g) is the map which projects to f on the left hand factor and g on the right hand factor.

Lemma 1.7.

(1) Suppose that B is a fibrant object of a closed model category C, and that B̂ is a path object for B as in (1.6). Then the maps p_{0} and p_{1} are trivial fibrations.

(2) Right homotopy of maps A → B in C is an equivalence relation if B is fibrant.

Lemma 1.7 is dual to Lemma 1.5 in a precise sense. If C is a closed model category, then its opposite C^{op} is a closed model category whose cofibrations (respectively fibrations) are the opposites of the fibrations (respectively cofibrations) in C. A map in C^{op} is a weak equivalence for this structure if and only if its opposite is a weak equivalence in C. Then Lemma 1.7 is an immediate consequence of the instance of Lemma 1.5 which occurs in C^{op}. This sort of duality is ubiquitious in the theory: observe, for example, that the two statements of Corollary 1.3 are dual to each other.

Left and right homotopies are linked by the following result:

Proposition 1.8. Suppose that A is cofibrant. Suppose further that

A ⊔ A_{4 4 44 6} (f, g) u (i_{0}, i_{1}) ; \begin{aligned}Ã & w & h \\ & & B \end{aligned}

is a left homotopy between maps f, g : A → B, and that

B̂up = (p_{0}, p_{1})

B^{N N N NN P} s w Δ B × B

is a fixed choice of path object for B. Then there is a right homotopy

B̂u(p_{0}, p_{1})

A^{N N N NN P} H w (f, g) B × B.

This result has a dual, which the reader should be able to formulate independently. Proposition 1.8 and its dual together imply

Corollary 1.9. Suppose given maps f, g : A → B, where A is cofibrant and B is fibrant. Then the following are equivalent:

(1) f and g are left homotopic.

(2) f and g are right homotopic with respect to a fixed choice of path object.

(3) f and g are right homotopic.

(4) f and g are left homotopic with respect to a fixed choice of cylinder object.

In other words, all possible definitions of homotopy of maps A → B are the same if A is cofibrant and B is fibrant. Proof of Proposition 1.8: The map i_{0} is a trivial cofibration since A is cofibrant, and (p_{0}, p_{1}) is a fibration, so that there is a commutative diagram

A  w sf u i_{0}  B̂u(p_{0}, p_{1})

Ã w (fσ, h)^{N N N NN P} K B × B

for some choice of lifting K. Then the composite K ∘ i_{1} is the desired right homotopy. ■

We can now, unambiguously, speak of homotopy classes of maps between objects X and Y of a closed model category C which are both fibrant and cofibrant. We can also discuss homotopy equivalences between such objects. The classical Whitehead Theorem asserts that any weak equivalence f : X → Y of CWcomplexes is a homotopy equivalence. CW-complexes are spaces which are both cofibrant and cofibrant. The analogue of this statement in an arbitrary closed model category is the following:

Theorem 1.10 (Whitehead). Suppose that f : X → Y is a morphism of a closed model category C such that the objects X and Y are both fibrant and cofibrant. Suppose also that f is a weak equivalence. Then the map f is a homotopy equivalence.

Proof: Suppose, first of all, that f is a trivial fibration, and that

X ⊔ X  w ∇ u (i_{0}, i_{1})  X

X̃^{N N N NN P} σ

is a cylinder object for X. Then one proves that f is a homotopy equivalence by finding, in succession, maps θ and h making the following diagrams commute:

\begin{matrix}∅  w u \\ Xuf \end{matrix} ; Y ^{h h hh j} θ  Y

X ⊔ X  w (θ ∘ f, 1) u (i_{0}, i_{1})  Xuf

X̃  w f ∘ σ^{A A A A AA C} h

Y.

Dually, if f is a trivial cofibration, then f is a homotopy equivalence.

Every weak equivalence f : X → Y between cofibrant and fibrant objects has a factorization

{X w f_{h hh j} i \atop Y} ; Z^{’ ’’)}p

in which i is a trivial cofibration and p is a trivial fibration. The object Z is both cofibrant and fibrant, so i and p are homotopy equivalences. ■

Suppose that X and Y are objects of a closed model category C which are both cofibrant and fibrant. Quillen denotes the set of homotopy classes of maps between such objects X and Y by π(X, Y). There is a category πC_{cf} associated to any closed model C: the objects are the cofibrant and fibrant objects of C, and the morphisms from X to Y in πC_{cf} are the elements of the set π(X, Y).

For each object X of C, use CM5 to choose, in succession, maps

∗ ⟶^{i_{X}} QX ⟶^{p_{X}} X

and

QX ⟶^{j_{X}} RQX ⟶^{q_{X}} ∗,

where i_{X} is a cofibration, p_{X} is a trivial fibration, j_{X} is a trivial cofibration, and q_{X} is a fibration. We can and will presume that π_{X} is the identity map if X is cofibrant, and that j_{X} is the identity map if QX is fibrant. Then RQX is an object which is both fibrant and cofibrant, and RQX is weakly equivalent to X, via the maps p_{X} and j_{X}.

Any map f : X → Y lifts to a map Qf : QX → QY, and then Qf extends to a map RQf : RQX → RQY. The map Qf is not canonically defined: it is any morphism which makes the following diagram commute:

∅  w u  QY uπ_{Y}

QX  w f ∘ π_{X}^{h h hh j} Qf  Y

Note, however, that any two liftings f_{1}, f_{2} : QX → QY of the morphism f ∘ π_{X} are left homotopic.

The argument for the existence of the morphism RQf : RQX → RQY is dual to the argument for the existence of Qf. If the maps f_{1}, f_{2} : QX → QY are liftings of f ∘ π_{X} and g_{i} : RQX → RQY is an extension of the map j_{Y} ∘ f_{i} for i = 1, 2, then f_{1} is left homotopic to f_{2} by what we’ve already seen, and so the composites j_{Y} ∘ f_{1} and j_{Y} ∘ f_{2} are right homotopic, by Lemma 1.8. Observe finally that any right homotopy between the maps j_{Y} f_{1}, j_{Y} f_{2} : QX → RQY can be extended to a right homotopy between the maps g_{1}, g_{2} : RQY → RQY. It follows that the assignment f ↦ RQf is well defined up to homotopy.

The homotopy category Ho(C) associated to a closed model category C can be defined to have the same objects as C, and with morphism sets defined by

hom_{Ho(C)}(X, Y) = π(RQX, RQY).

There is a functor

γ : C → Ho(C)

which is the identity on objects, and sends a morphism f : X → Y to the homotopy class [RQf] which is represented by any choice of map RQf : RQX → RQY defined as above. If f : X → Y is a weak equivalence of C, then RQf : RQX → RQY is a homotopy equivalence by the Whitehead Theorem, and so γ(f) is an isomorphism of Ho(C).

This functor γ is universal with respect to all functors F : C → D which invert weak equivalences:

Theorem 1.11. Suppose that F : C → D is a functor such that F (f) is an isomorphism of D for all weak equivalences f : X → Y of C. Then there is a unique functor F_{∗} : Ho(C) → D such that F_{∗} ∘ γ = F.

Proof: The functor F : C → D takes (left or right) homotopic maps of C to the same map of D, since it inverts weak equivalences. It follows that, if g : RQX → RQY represents a morphism from X to Y in Ho(C), one can specify a well-defined morphism F_{∗}([g]) of D by the assignment

(1.12)  F_{∗}([g]) = F (π_{Y})F (j_{Y})^{−1}F (g)F (j_{X})F (π_{X})^{−1}.

This assignment plainly defines a functor F_{∗} : Ho(C) → D such that F_{∗}γ = F.

Also, the morphisms γ(π_{X}) and γ(j_{X}) are both represented by the identity map on RQX, and so the composite

γ(π_{Y})γ(j_{Y})^{−1}γ(g)γ(j_{X})γ(j_{X})^{−1}

coincides with the morphism [g] : X → Y of Ho(C). The morphism F_{∗}([g]) must therefore have the form indicated in (1.12) if the composite functor F_{∗}γ is to coincide with F. ■

Remark 1.13. One can always formally invert a class Σ of morphisms of a category C to get a functor γ : C → C[Σ^{−1}] which is initial among functors F : C → D which invert all members of the class of morphisms Σ (see Schubert’s book [82]), provided that one is willing to construct C[Σ^{−1}] in some higher set theoretic universe. This means that the morphism “things” hom_{C[Σ^{−1}]}(X, Y) of C[Σ^{−1}] may no longer be sets. The whole point of Theorem 1.11 is that we have found an explicit way to formally invert the class W E of weak equivalences of a closed model category C to obtain the category Ho(C) without invoking a higher universe. After the fact, all models of C[W E^{−1}] must be isomorphic as categories to Ho(C) on account of the universal property of the functor γ : C → Ho(C), so that all possible constructions have small hom sets.

Let πC_{cf} denote the category whose objects are the cofibrant fibrant objects of the closed model category C, and whose sets of morphisms have the form

hom_{πC_{cf}} (X, Y) = π(X, Y).

It’s obvious that the functor γ : C → Ho(C) induces a fully faithful imbedding

γ_{∗} : πC_{cf} → Ho(C),

and that every object of Ho(C) is isomorphic to an object which is in the image of the functor γ_{∗}. In other words the category πC_{cf} of homotopy classes of maps between cofibrant fibrant objects of C is equivalent to the homotopy category Ho(C).

This observation specializes to several well-known phenomena. In particular, the category of homotopy classes of maps between CW-complexes is equivalent to the full homotopy category of topological spaces, and the homotopy category of simplicial sets is equivalent to the category of simplicial homotopy classes of maps between Kan complexes.

2. Simplicial categories.

A simplicial model category is, roughly speaking, a closed model category equipped with a notion of a mapping space between any two objects. This has to be done in such a way that it makes homotopy theoretic sense. Thus, besides the new structure, there is an additional axiom, which is called Axiom SM7 (See 2.1 below).

The initial property one wants is the following: let S be the category of simplicial sets and let C be a model category, and suppose A ∈ C is cofibrant and X ∈ C is fibrant. Then, the space of maps in C should be a functor to simplicial sets

Hom_{C} : C^{op} × C → S

with the property that

π_{0}Hom_{C}(A, X) ≅ [A, X]_{C} .

In addition, one would want to interpret π_{n}Hom_{C}(A, X) in C.

There are other desirable properties; for example, if A is cofibrant and X → Y a fibration in C, one would want

Hom_{C}(A, X) → Hom_{C}(A, Y)

to be a fibration of spaces – that is, of simplicial sets.

Before imposing the closed model category structure on C, let us make the following definition:

Definition 2.1. A category C is a simplicial category if there is a mapping space functor

Hom_{C}(·, ·) : C^{op} × C → S

with the properties that for A and B objects in C

(1) Hom_{C}(A, B)_{0} = hom_{C}(A, B);

(2) the functor Hom_{C}(A, ·) : C → S has a left adjoint

A ⊗ · : S → C

which is associative in the sense that there is a isomorphism

A ⊗ (K × L) ≅ (A ⊗ K) ⊗ L

natural in A ∈ C and K, L ∈ S;

(3) The functor Hom_{C}(·, B) : C^{op} → S has left adjoint

hom_{C}(·, B) : S → C^{op} .

Of course, the adjoint relationship in (3) is phrased

homS(K, Hom_{C}(A, B)) ≅ hom_{C}(A, hom_{C}(K, B)) .

Warning: The tensor product notation goes back to Quillen, and remains for lack of a better operator. But be aware that in this context we do not usually have a tensor product in the sense of algebra; that is, we don’t have a pairing arising out of bilinear maps. Instead (see the next lemma), we have an adjoint to an internal hom functor, and this is the sole justification for the notation.

Some immediate implications of the definition are the following.

Lemma 2.2.

(1) For fixed K ∈ S, there are adjoint functors

· ⊗ K : C → C

and

hom_{C}(K, ·) : C → C .

(2) For all K and L in S and B in C there is a natural isomorphism

hom_{C}(K × L, B) ≅ hom_{C}(K, hom_{C}(L, B)).

(3) For all n ≥ 0, Hom_{C}(A, B)_{n} ≅ hom_{C}(A ⊗ Δ^{n}, B).

Proof: For fixed K ∈ S and A ∈ C, the object A ⊗ K represents the functor

homS(K, Hom_{C}(A, ·)) : C → Sets.

A map A →^{f} A′ yields a natural transformation on representing objects

f ⊗ 1 : A ⊗ K → A′ ⊗ K

and we obtain the functor · ⊗ K. The functor hom_{C}(K, ·) is obtained similarly. The adjointness follows from 2.1.2 and 2.1.3. This proves part 1. Part 2 then follows from the associativity built into 2.1.2. Part 3 follows from 2.1.2 and the fact that homS(Δ^{n}, X) ≅ X_{n}. ■ Remark: A consequence of Lemma 2.2.1 is that there is a composition pairing of simplicial sets

Hom_{C}(A, B) × Hom_{C}(B, C) → Hom_{C}(A, C)

defined as follows. If f : A ⊗ Δ^{n} → B is an n-simplex of Hom_{C}(A, B) and g : B ⊗ Δ^{n} → C is an n-simplex of Hom_{C}(B, C) then their pairing in Hom_{C}(A, C) is the composition

A ⊗ Δ^{n} ⟶^{1⊗d} A ⊗ (Δ^{n} × Δ^{n}) ≅ A ⊗ Δ^{n} ⊗ Δ^{n} ⟶^{f⊗1} B ⊗ Δ^{n} ⟶^{g} C.

Here d : Δ^{n} → Δ^{n}×Δ^{n} is the diagonal. This pairing is associative in the obvious sense and reduces to the composition pairing in C in simplicial degree zero. It is also unital in the sense that if ∗ → Hom_{C}(A, A) is the vertex corresponding to the identity, then the following diagram commutes

∗ × Hom_{C}(A, B) u _{‚ ‚ ‚ ‚ ‚ ‚ ›} =

Hom_{C}(A, A) × Hom_{C}(A, B) w

Hom_{C}(A, B).

There is also a diagram using the identity of B. A shorthand way of encoding all this structure is to say that C is enriched over simplicial sets. ■

Another immediate consequence of the definition is the following result.

Lemma 2.3. For a simplicial category C then the following extended adjointness isomorphisms hold:

(1) HomS(K, Hom_{C}(A, B)) ≅ Hom_{C}(A ⊗ K, B).

(2) HomS(K, Hom_{C}(A, B)) ≅ Hom_{C}(A, hom(K, B)).

Proof: This is an easy exercise using Lemma 2.2. ■

Note that, in fact, Definition 2.1 implies that there are functors

· ⊗ · : C × S → C

and

hom_{C}(·, ·) : S^{op} × C → C

satisfying 2.1.1, 2.1.2 and 2.2.1. In order to produce examples of simplicial categories, we note the following:

Lemma 2.4. Let C be a category equipped with a functor

· ⊗ · : C × S → C .

Suppose the following three conditions hold:

(1) For fixed K ∈ S, · ⊗ K : C → C has a right adjoint hom_{C}(K, ·).

(2) For fixed A, the functor A ⊗ · : S → C commutes with arbitrary colimits and A ⊗ ∗ ≅ A.

(3) There is an isomorphism A ⊗ (K × L) ≅ (A ⊗ K) ⊗ L natural is A ∈ C and K, L ∈ S.

Then C is a simplicial category with Hom_{C}(A, B) defined by:

Hom_{C}(A, B)_{n} = hom_{C}(A ⊗ Δ^{n}, B)

Proof: We first prove 2.1.2 holds. If K ∈ S, write K as the coequalizer in a diagram

⨆_{q} Δ^{n_{q}} ⇉ ⨆_{p} Δ^{n_{p}} → K .

Then there is a coequalizer diagram

⨆_{q} A ⊗ Δ^{n_{q}} ⇉ ⨆_{p} A ⊗ Δ^{n_{p}} → A ⊗ K

Hence there is an equalizer diagram

hom_{C}(A ⊗ K, B) → hom_{C}(A ⊗ ( ⨆_{p} Δ^{n_{p}}), B) ⇉ hom_{C}(A ⊗ ( ⨆_{q} Δ^{n_{q}}), B) .

This, in turn, is equivalent to the assertion that the equalizer of the maps

homS( ⨆_{p} Δ^{n_{p}}, hom_{C}(A, B)) ⇉ homS( ⨆_{q} Δ^{n_{q}}, hom_{C}(A, B))

is the induced map

homS(K, hom_{C}(A, B)) → homS( ⨆_{p} Δ^{n_{p}}, hom_{C}(A, B))

so 2.1.2 holds. If we let hom_{C}(K, ·) be adjoint to · ⊗ K, as guaranteed by the hypotheses, 2.1.3 holds. Then finally, 2.1.1 is a consequence of the fact that A ⊗ ∗ = A. ■

We now give some examples. Needless to say S itself is a simplicial category with, for A, B, K ∈ S

A ⊗ K = A × K

and (a tautology)

HomS(A, B) = HomS(A, B)

and

homS(K, B) = HomS(K, B) .

Only slightly less obvious is the following: let S_{∗} denote the category of pointed (i.e., based) simplicial sets. Then S_{∗} is a simplicial category with

A ⊗ K = A ∧ K_{+} = A × K/ ∗ ×K

where ( )_{+} denote adding a disjoint basepoint

Hom_{S_{∗}}(A, B)_{n} = hom_{S_{∗}}(A ∧ Δ_{+}^{n}, B)

and

hom_{S_{∗}}(K, B) = HomS(K, B)

with basepoint given by the constant map

K → ∗ → B.

Note that Hom_{S_{∗}}(A, B) ∈ S, but hom_{S_{∗}}(K, B) ∈ S_{∗}.

This example can be radically generalized. Suppose C is a category that is co-complete; that is, C has all colimits. Let sC denote the simplicial objects in C. Then if K ∈ S, we may define, for A ∈ sC, an object A ⊗ K ∈ sC by

(A ⊗ K)_{n} = ⨆_{k∈K_{n}} A_{n}

where ⨆ denotes the coproduct in C, and if ϕ : n → m is an ordinal number map ϕ^{∗} : (A ⊗ K)_{m} → (A ⊗ K)_{n} is given by

⨆_{k∈K_{m}} A_{m} \underrightarrow{⨆ ϕ^{∗}} ⨆_{k∈K_{m}} A_{n} → ⨆_{k∈K_{n}} A_{n} .

The first map is induced by ϕ^{∗} : A_{m} → A_{n}, the second by ϕ^{∗} : K_{m} → K_{n}.

Theorem 2.5. Suppose that C is complete and complete. Then with this functor · ⊗ · : sC × S → sC, sC becomes a simplicial category with

Hom_{sC}(A, B)_{n} = hom_{sC}(A ⊗ Δ^{n}, B) .

Proof: This is an application of Lemma 2.3. First note that it follows from the construction that there is a natural isomorphism

A ⊗ (K × L) ≅ (A ⊗ K) ⊗ L.

And one has A ⊗ ∗ ≅ A. Thus, we need only show that, for fixed K ∈ S, the functor · ⊗ K : C → C has a right adjoint. To show this, one changes focus slightly. For Y ∈ sC, define a functor

F_{Y} : C^{op} → S

by

F_{Y} (A) = hom_{C}(A, Y) .

Then the functor C^{op} → Sets given by

A ↦ HomS(K, F_{Y} (A))_{n} = homS(K × Δ^{n}, hom_{C}(A, Y))

is representable. To see this, write K × Δ^{n} as a coequalizer

⨆_{q} Δ^{n_{q}} ⇉ ⨆_{p} Δ^{n_{p}} → K × Δ^{n}

then the representing object is defined by the equalizer diagram

∏_{q} Y_{n_{q}} ⇇ ∏_{q} Y_{m_{q}} ← hom_{sC}(K, Y)_{n} .

Letting the ordinal number vary yields an object hom_{sC}(K, Y) and a natural isomorphism of simplicial sets

(2.6)  hom_{C}(A, hom_{sC}(K, Y)) ≅ HomS(K, hom_{C}(A, Y)) ,

or a natural equivalence of functors

F_{hom_{SC}(K,Y)}(·) ≅ HomS(K, F_{Y} (·)) .

Now the morphisms X → Y in sC are in one-to-one correspondence with the natural transformations F_{X} → F_{Y}, by the Yoneda lemma. In formulas, this reads

hom_{sC}(X, Y) ≅ Nat(F_{X}, F_{Y}) .

Now if K ∈ S and X ∈ sC we can define a new functor

F_{X} ⊗ K : C^{op} → S

by

(F_{X} ⊗ K)(A) = F_{X}(A) × K

we will argue below that

Nat(F_{X⊗K}, F_{Y}) ≅ Nat(F_{X} ⊗ K, F_{Y}) .

Assuming this one has:

hom_{sC}(X, hom_{sC}(K, Y)) ≅ Nat(F_{X}, F_{hom_{sC}(K,Y)})
≅ Nat(F_{X}, HomS(K, F_{Y}))

by (2.6). Continuing, one has that this is isomorphic to

Nat(F_{X} ⊗ K, F_{Y}) ≅ Nat(F_{X⊗K}, F_{Y}) ≅ hom_{sC}(X ⊗ K, Y)

so that

hom_{sC}(X, hom_{sC}(K, Y)) ≅ hom_{sC}(X ⊗ K, Y) .

as required. Thus we are left with

Lemma 2.7. There is an isomorphism

Nat(F_{X⊗K}, F_{Y}) ≅ Nat(F_{X} ⊗ K, F_{Y}) .

Proof: It is easiest to show

Nat(F_{X} ⊗ K, F) ≅ hom_{sC}(X ⊗ K, Y) .

Given a natural transformation

Φ : F_{X} ⊗ K → F_{Y}

note that

(F_{X} ⊗ K)(X_{n})_{n} = ∏_{k∈K_{n}} hom_{C}(X_{n}, X_{n}) .

Thus, for each k ∈ K_{n}, there is a map

Φ(1)_{k} : X_{n} → Y_{n}

corresponding to the identity in the factor corresponding to k. These assemble into a map

f_{n} : (X ⊗ K)_{n} = ⨆_{k∈K_{n}} X_{n} → Y_{n} .

We leave it to the reader to verify that yields a morphism

f : X ⊗ K → Y

of simplicial objects, and that the assignment Φ → f yields the desired isomorphism. ■

Examples 2.8. One can now assemble a long list of simplicial categories: We note in particular

1) Let C be one of the following “algebraic” categories: groups, abelian groups, rings, commutative rings, modules over a ring R, algebras or commutative algebras over a commutative ring R, or Lie algebras. Then sC is a simplicial category.

2) Let C be the graded analog of one of the categories in the previous example. Then sC is a simplicial category. One might wish to examine the category of graded commutative algebras over a commutative ring R, for example.

3) Let C = CA be the category of coalgebras over a field 𝔽. Then sCA is a simplicial category.

4) Note that the hypotheses of C used Theorem 2.5 apply equally to C^{op}. Thus s(C^{op}) is also a simplicial category. But if s(C^{op}) is a simplicial category, so is (s(C)^{op})^{op}. But this is the category cC of cosimplicial objects in C. One must interpret the functors · ⊗ ·, hom_{cC}(·, ·), etc. in light of Theorem 2.5. Thus if K ∈ S,

hom_{cC}(K, A)^{n} = ∏_{k∈K_{n}} A^{n}

and

Hom_{cC}(A, B)_{n} = hom_{cC}(A, hom_{cC}(Δ^{n}, B))

and A ⊗ K is defined via Theorem 2.5.

To conclude this section, we turn to the following question: suppose given simplicial categories C and D and a functor G : D → C with left adjoint F. We want a criterion under which the simplicial structure is preserved.

Lemma 2.9. Suppose that for all K ∈ S and A ∈ C there is a natural isomorphism F (A ⊗ K) ≅ F (A) ⊗ K. Then

(1) the adjunction extends to a natural isomorphism

Hom_{D}(F A, B) ≅ Hom_{C}(A, GB);

(2) for all K ∈ S and B ∈ D, there is a natural isomorphism

Ghom_{D}(K, B) ≅ hom_{C}(K, GB) .

Proof: Part (1) uses that Hom_{D}(F A, B)_{n} ≅ hom_{D}(F A ⊗ Δ^{n}, B). Part (2) is an exercise in adjunctions. ■

We give some examples.

Examples 2.10.

1) Let G : D → C have a left adjoint F. Extend this to a pair of adjoint functors by prolongation:

G : sD → sC

with adjoint F. Thus G(X)_{n} = G(X)_{n}, and so on. Then, in the simplicial structure of Theorem 2.5, F (X ⊗ K) ≅ F (X) ⊗ K, since F commutes with colimits.

2) Let C be an arbitrary simplicial category and A ∈ C. Define

G : C → S

by G(B) = Hom_{C}(A, B). Then F (X) = A ⊗ X and the requirement on 2.9 is simply the formula

A ⊗ (X × K) ≅ (A ⊗ X) ⊗ K .

3. Simplicial model categories.

If a category C is at once a simplicial category and a closed model category, we would like the mapping space functor to have homotopy theoretic content. This is accomplished by imposing the following axiom.

3.1 Axiom SM7. Let C be a closed model category and a simplicial category. Suppose j : A → B is a cofibration and q : X → Y is a fibration. Then

Hom_{C}(B, X) ⟶^{(j^{∗},q_{∗})} Hom_{C}(A, X) ×_{Hom_{C}(A,Y)} Hom_{C}(B, Y)

is a fibration of simplicial sets, which is trivial if j or q is trivial.

A category satisfying this axiom will be called a simplicial model category. The next few sections will be devoted to producing a variety of examples, but in this section we will explore the consequences of this axiom.

Proposition 3.2. Let C be a simplicial model category and q : X → Y a fibration. Then if B is cofibrant

q_{∗} : Hom_{C}(B, X) → Hom_{C}(B, Y)

is a fibration in S. Similarly, if j : A → B is a cofibration and X is fibrant, then

j^{∗} : Hom_{C}(B, X) → Hom_{C}(A, X)

is a fibration.

Proof: One sets A to be the initial object and Y to be the final object, respectively, in Axiom SM7. ■

In other words, Hom_{C}(·, ·) has entirely familiar homotopical behavior. This is one way to regard this axiom. Another is that SM7 is a considerable strengthening of the lifting axiom CM4 of a closed model category.

Proposition 3.3. Axiom SM7 implies axiom CM4; that is, given a lifting problem in simplicial category C satisfying SM7

Au j wXu q

B^{i i i ij} wY

with j a cofibration and q a fibration, then the dotted arrow exists if either j or q is trivial.

Proof: Such a square is a zero-simplex in

Hom_{C}(A, X) ×_{Hom_{C}(A,Y)} Hom_{C}(B, Y)

and a lifting is a pre-image in the zero simplices of Hom_{C}(B, X). Since trivial fibrations are surjective, the result follows. ■

But more is true: Axiom SM7 implies that the lifting built in CM4 is unique up to homotopy. To explain that, however, requires a few words about homotopy. First we record

Proposition 3.4. Let C be a simplicial model category and j : K → L a cofibration of simplicial sets. If A ∈ C is cofibrant, then

1 ⊗ j : A ⊗ K → A ⊗ L

is a cofibration in C. If X ∈ C is fibrant

j^{∗} : hom_{C}(L, X) → hom_{C}(K, X)

is a fibration. If j is trivial, then so are 1 ⊗ j and j^{∗}.

Proof: For example, one needs to show 1 ⊗ j has the left lifting property with respect to all trivial fibrations q : X → Y in C. This is equivalent, by adjointness, to show j has the left lifting property with respect to

q_{∗} : Hom_{C}(A, X) → Hom_{C}(A, Y)

for all trivial fibrations q. But q_{∗} is a trivial fibration of simplicial sets by SM7. The other three claims are proved similarly. ■

Recall the definitions of left and right homotopy from the section 1. The following implies that if A is cofibrant, then A ⊗ Δ^{1} is a model for the the cylinder on A.

Lemma 3.5. Let C be a simplicial model category and let A ∈ C be cofibrant. Then if q : Δ^{1} → ∗ is the unique map

1 ⊗ q : A ⊗ Δ^{1} → A ⊗ ∗ ≅ A

is a weak equivalence. Furthermore,

d_{1} ⊔ d_{0} : A ⊔ A → A ⊗ Δ^{1}

is a cofibration and the composite

A ⊔ A ⟶^{d_{0}⊔d_{1}} A ⊗ Δ^{1} ⟶^{1⊗q} A

is the fold map.

Proof: The first claim follows from Proposition 3.4, since

d_{1} : A ≅ A ⊗ Δ^{0} → A ⊗ Δ^{1}

is a weak equivalence. The second claim follows from 3.4 also since d_{1} ⨆ d_{0} is equivalent to

1 ⊗ j : A ⊗ ∂Δ^{1} → A ⊗ Δ^{1}

where j : ∂Δ → Δ^{1} is inclusion of the boundary. For the third claim one checks that (1 ⊗ q) ∘ d_{1} = (1 ⊗ q) ∘ d_{0} = 1. ■

Thus, if C is a simplicial model category and A ∈ C is cofibrant, then two morphisms f, g : A → X are homotopic if and only if there is a factoring

(3.6)

A ⊔ A u f ⊔ g w d_{1} ⊔ d_{0} A ⊗ Δ^{1} N N N N N Q H

X

This, too, is no surprise. As a further exercise, note that if one prefers right homotopy for a particular application, one could require a factoring

(3.7)

hom_{C}(Δ[1], X) uj^{∗}

A^{4 4 4 4 4 4 4 6} G w f × g X × X ≅ hom_{C}(∂Δ[1], X)

In using this formulation, one wants X to be fibrant so that j^{∗} is a fibration.

To formulate the next notion, let A be cofibrant and j : A → B a cofibration. Given two maps f, g : B → X so that j ∘ f = j ∘ g, we say f and g are homotopic under A if there is a homotopy

H : B ⊗ Δ^{1} → X

so that H ∘ (j ⊗ 1) : A ⊗ Δ^{1} → X is the constant homotopy on j ∘ f. That is, h ⊗ (j ⊗ 1) is the composite

A ⊗ Δ^{1} ⟶^{1⊗q} A ⊗ ∗ ≅ A ⟶^{j∘f} X

where q : Δ^{1} → ∗ is the unique map. There is an obvious dual motion of homotopic over Y.

The following result says that in a simplicial model category, the liftings required by axiom CM4 are unique in a strong way.

Proposition 3.8. Let C be a simplicial model category and A a cofibrant object. Consider a commutative square

\begin{matrix}Au j  w \\ Xuq \end{matrix} ; {B  w \atop Y}

where j is a cofibration, q is a fibration and one of j or q is trivial. Then any two solutions f, g : B → X of the lifting problem are homotopic under A and over Y.

Proof: The commutative square is a zero-simplex α in

Hom_{C}(A, X) ×_{Hom_{C}(A,Y)} Hom_{C}(B, Y)

Let s_{0}α be the corresponding degenerate 1-simplex. Then s_{0}α is the commutative square

\begin{matrix}A ⊗ Δ^{1} u j ⊗ 1  w \\ Xuq \end{matrix} ; {B ⊗ Δ^{1}  w \atop Y}

where the horizontal maps are the constant homotopies. Let

f, g ∈ Hom_{C}(B, X)_{0} = hom_{C}(B, X)

be two solutions to the lifting problem. Thus (i^{∗}, q_{∗})f = (i^{∗}, q_{∗})g = α. Then by SM7, there is a 1-simplex

β ∈ Hom_{C}(B, X)_{1} ≅ hom_{C}(B ⊗ Δ^{1}, X)

so that d_{1}β = f, d_{0}β = g, and (i^{∗}, q_{∗})β = s_{0}α. Then

β : B ⊗ Δ^{1} → X

is the required homotopy. ■

We now restate a concept from Section 1. For a simplicial model category C, we define the homotopy category Ho(C) as follows: the objects are the objects of C and the morphisms are defined by

(3.9)  [A, X]_{C} = hom_{C}(B, Y)/ ∼

where q : B → A is a trivial fibration with B cofibrant, i : X → Y is a trivial cofibration with Y fibrant and f ∼ g if and only if f is homotopic to g. It is an exercise that ∼ is in fact an equivalence relation, and it is a consequence of Proposition 3.8 that [A, X]_{C} is well-defined.

There is some ambiguity in the notation: [A, X]_{C} depends not only on C, but on the particular closed model category structure.

In the sequel, [ , ] means [ , ]S.

The following result gives homotopy theoretic content to the functors · ⊗ K and hom_{C}(K, ·).

Proposition 3.10. Let C be a simplicial model category and A and B a cofibrant and a fibrant object of C, respectively. Then

[K, Hom_{C}(A, B)] ≅ [A ⊗ K, B]_{C}

and

[K, Hom_{C}(A, B)] ≅ [A, hom_{C}(K, B)]_{C} .

Proof: Note that Hom_{C}(A, B) is fibrant, by Proposition 3.2. Hence

[K, Hom_{C}(A, B)] = homS(K, Hom_{C}(A, B))/ ∼

where ∼ means “homotopy” as above. But, since

A ⊗ (K × Δ^{1}) = (A ⊗ K) ⊗ Δ^{1}

we have that

homS(K, Hom_{C}(A, B))/ ∼ ≅ hom_{C}(A ⊗ K, B)/ ∼
≅ [A ⊗ K, B]_{C}

where we use Proposition 3.4 to assert that A ⊗ K is cofibrant. ■

We now concern ourselves with developing a way of recognizing when SM7 holds.

Proposition 3.11. Let C be a closed model category and a simplicial category. Then the axiom SM7 holds if and only if for all cofibrations i : K → L in S and cofibrations j : A → B in C, the map

(j ⊗ 1) ∪ (1 ⊗ i) : (A ⊗ L) ∪_{(A⊗K)} (B ⊗ K) → B ⊗ L .

is a cofibration which is trivial if either j or i is.

Proof: A diagram of the form

Ku wHom_{C}(B, X) u

L w Hom_{C}(A, X) ×_{Hom_{C}(A,Y)} Hom_{C}(B, Y)

is equivalent, by adjointness, to a diagram

\begin{matrix}(A ⊗ L) ∪_{(A⊗K)} (B ⊗ K) u  w \\ Xu \end{matrix} ; \begin{matrix}B ⊗ L  w \\ Y . \end{matrix}

The result follows by using the fact that fibrations and cofibrations are determined by various lifting properties. ■

From this we deduce

Corollary 3.12. (Axiom SM7b) Let C be a closed model category and a simplicial category. The axiom SM7 is equivalent to the requirement that for all cofibrations j : A → B in C

(A ⊗ Δ^{n}) ∪_{(A⊗∂Δ^{n})} (B ⊗ ∂Δ^{n}) → B ⊗ Δ^{n}

is a cofibration (for n ≥ 0) that is trivial if j is, and that

(A ⊗ Δ^{1}) ∪_{(A⊗{e})} (B ⊗ {e}) → B ⊗ Δ^{1}

is the trivial cofibration for e = 0 or 1.

Proof: Let i : K → L be a cofibration of simplicial sets. Then, since i can be built by attaching cells to K, the first condition implies

(A ⊗ L) ∪_{(A⊗K)} (B ⊗ K) → (B ⊗ L)

is a cofibration which is trivial if j is. The second condition and proposition I.4.2 (applied to B_{2}) yields that (j ⊗ 1) ∪ (1 ⊗ i) is trivial if i is. ■

In the usual duality that arises in these situations, we also have

Proposition 3.13. Let C be a simplicial category and a model category and suppose i : K → L is a cofibration in S and q : X → Y a fibration in C. Then the following are equivalent:

(1) SM7,

(2) hom_{C}(L, X) → hom_{C}(K, X)×_{hom_{C}(K,Y)}hom_{C}(L, Y) is a fibration which is trivial if q or j is;

(3) (SM7a) hom_{C}(Δ^{n}, X) → hom_{C}(∂Δ^{n}, X)×_{hom_{C}(∂Δ^{n},Y)} hom_{C}(Δ^{n}, Y) is a fibration which is trivial if q is, and

hom_{C}(Δ^{1}, X) → hom_{C}(e, X) ×_{hom_{C}(e,Y)} hom_{C}(Δ^{1}, Y)

is a trivial fibration for e = 0, 1.

Example 3.14. A simplicial model category structure on CGHaus We can now show that the category CGHaus of compactly generated Hausdorff spaces is a simplicial model category. To supply the simplicial structure let X ∈ CGHaus and K ∈ S. Define

X ⊗ K = X ×_{Ke} |K|

where | · | denotes the geometric realization and ×_{Ke} the Kelly product, which is the product internal to the category CGHaus. Then if X and Y are in CGHaus, the simplicial set of maps between X and Y is given by

HomCGHaus(X, Y)_{n} = homCGHaus(X × Δ^{n}, Y)

regarded as a set. And the right adjoint to · ⊗ K is given by

homCGHaus(K, X) = F(|K|, X)

where F denotes the internal function space to CGHaus.

We have seen that CGHaus is a closed model category with the usual weak equivalences and Serre fibrations. In addition, Proposition 3.13 immediately implies that CGHaus is a simplicial model category.

It is worth pointing out that the realization functor | · | and its adjoint S(·) the singular set functor pass to the level of simplicial categories. Indeed, we’ve seen that if X ∈ S and K ∈ S, then

|X × K| = |X| ×_{Ke} |K|.

This immediately implies that if Y ∈ CGHaus, then

SF(|K|, Y) = HomS(K, SY)

and

HomCGHaus(|X|, Y) = HomS(X, SY).

4. Detecting weak equivalences.

Lemma 4.1. Suppose that f : X → Y is a morphism of a simplicial model category C which induces an isomorphism in the homotopy category Ho(C). Then f is a weak equivalence.

Proof: The map f induces an isomorphism in the homotopy category if and only if f induces a homotopy equivalence RQX → RQY of the associated cofibrant fibrant models. It suffices, therefore, to assume that X and Y are fibrant and cofibrant, and that f : X → Y is a homotopy equivalence.

Choose a factorization

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

where j is a trivial cofibration and p is a fibration. The map j is a homotopy equivalence, so it suffices to show that the fibration p : Z → Y of cofibrant fibrant objects is a weak equivalence if it is a homotopy equivalence.

Choose a homotopy inverse s : Y → Z for the map p, and choose a homotopy H : Z ⊗ Δ^{1} → Z from ps to the identity 1_{Y}. There is a diagram

Y  w s u d^{0}  Zup

Y ⊗ Δ^{1}  w H^{h h hh j} H_{∗}  Y.

The composite map H_{∗}d^{1} is a section of p, and this composite is homotopic to s. We can therefore assume that the homotopy inverse s is a section of p.

Choose a homotopy h : Z ⊗ Δ^{1} → Z from sp to 1_{Z}. The lifting γ exists in the diagram

Λ_{0}^{2} w ( , h, sph) y u Hom(Z, Z) up_{∗}

Δ^{2} w s_{1}(ph)^{A A A A AA C} γ

Hom(Z, Y)

since p_{∗} is a fibration. Write k : Z ⊗ Δ^{1} → Z for the 1-simplex d_{0}(γ) of Hom(X, X). Then k is a fibrewise homotopy from sp to 1_{Z}, meaning in particular that the following diagram commutes:

\begin{matrix}Z ⊗ Δ^{1}  w k u s^{0} \\ Zup \end{matrix} ; \begin{matrix}Z  w p \\ Y. \end{matrix}

Now suppose that the diagram

\begin{matrix}A  w α u j \\ Zup \end{matrix} ; {B  w β \atop Y}

commutes, where j is a cofibration. One can form the diagram

\begin{matrix}(A ⊗ Δ^{1}) ∪ (B ⊗ \{0\})  w (k(α ⊗ 1), sβ) u j_{∗} \\ Zup \end{matrix} ; {B ⊗ Δ^{1}  w s^{0}(β ⊗ 1)^{’ ’ ’ ’ ’ ’ ’ ’ ’’)} K \atop Y}

since the map j_{∗} is a trivial cofibration. Now check that the map θ defined by the composite

B ⟶^{d^{0}} B ⊗ Δ^{1} ⟶^{K} Z

satisfies pθ = β and θj = α. The map p therefore has the right lifting property with respect to all cofibrations, and is therefore a trivial fibration. ■

It may seem unlikely, but Lemma 4.1 is probably the the most important technical consequence of having a simplicial model structure in hand, as opposed to just a closed model structure. It will be used many times in the sequel. Here is a sample application:

Lemma 4.2. Suppose that f : A → B is a map between cofibrant objects in a simplicial model category C. Then f is a weak equivalence if and only if the induced map

f^{∗} : Hom(B, Z) → Hom(A, Z)

is a weak equivalence of simplicial sets for each fibrant object Z of C.

Proof: We use the fact, which appears as Lemma II.9.4 below, that a map f : A → B between cofibrant objects in a closed model category has a factorization

Xuq ; {A^{h h hh j} j w f \atop B}

such that j is a cofibration and the map q is left inverse to a trivial cofibration i : B → X.

If f : A → B is a weak equivalence, then the map j : A → X is a trivial cofibration, and hence induces a trivial fibration j^{∗} : Hom(X, Z) → Hom(A, Z) for all fibrant objects Z. Similarly, the trivial cofibration i induces a trivial fibration i^{∗}, so that the map p^{∗} : Hom(B, Z) → Hom(X, Z) is a weak equivalence.

Suppose that the map f^{∗} : Hom(B, Z) → Hom(A, Z) is a weak equivalence for all fibrant Z. To show that f is a weak equivalence, we can presume that the objects A and B are fibrant as well as cofibrant. In effect, there is a commutative diagram

\begin{matrix}A w f u \\ Bu \end{matrix} ; \overline{A} \overline{f} w ; \overline{B}

in which the objects \overline{A} and \overline{B} are fibrant, and the vertical maps are trivial cofibrations, and then one applies the functor Hom( , Z) for Z fibrant and invokes the previous paragraph.

Finally, suppose that A and B are fibrant as well as cofibrant, and presume that f^{∗} : Hom(B, Z) → Hom(A, Z) is a weak equivalence for all fibrant Z. We can assume further that f is a cofibration, by taking a suitable factorization. The map f^{∗} : Hom(B, A) → Hom(A, A) is therefore a trivial Kan fibration, and hence surjective in all degrees, so that there is a map g : B → A such that g · f = 1_{A}. The maps f · g and 1_{B} are both preimages of the vertex f under the trivial fibration f^{∗} : Hom(B, B) → Hom(A, B), so that there is a homotopy f · g ≃ 1_{B}. In particular, f is a homotopy equivalence and therefore a weak equivalence, by Lemma 4.1. ■

5. The existence of simplicial model category structures.

Here we concern ourselves with the following problem: Let C be a category and sC the category of simplicial objects over C. Then, does sC have the structure of a simplicial model category? We will assume that there is a functor G : sC → S with a left adjoint

F : S → sC .

Examples include algebraic categories such as the categories of groups, abelian groups, algebras over some ring R, commutative algebras, lie algebras, and so on. In these cases, G is a forgetful functor. See Example 2.10.

Define a morphism f : A → B is sC to be

a) a weak equivalence if Gf is a weak equivalence in S;

b) a fibration if Gf is a fibration in S;

c) a cofibration if it has the left lifting property with respect to all trivial fibrations in sC.

A final definition is necessary before stating the result. Let {X_{α}}_{α∈I} be a diagram in C. Then, assuming the category C has enough colimits, there is a natural map

\underrightarrow{lim}_{I} G(X_{α}) ⟶ G(\underrightarrow{lim}_{I} X_{α}).

This is not, in general, an isomorphism. We say that G commutes with filtered colimits if this is an isomorphism whenever the index category I is filtered.

Theorem 5.1. Suppose C has all limits and colimits and that G commutes with filtered colimits. Then with the notions of weak equivalence, fibration, and cofibration defined above, sC is a closed model category provided the following assumption on cofibrations holds: every cofibration with the left lifting property with respect to fibrations is a weak equivalence.

5. The existence of simplicial model category structures 103

We will see that, in fact, sC is a simplicial model category with the simplicial structure of Theorem 2.5.

The proof of Theorem 5.1 turns on the following observation. As we have seen, a morphism f : X → Y is a fibration of simplicial sets if and only if it has the right lifting property with respect to the inclusions for all n, k

Λ_{k}^{n} ↪ Δ^{n}

and f is a trivial fibration if and only if it has the right lifting property with respect to the inclusions ∂Δ^{n} → Δ^{n} of the boundary for all n. The objects Δ_{k}^{n}, ∂Δ^{n}, and Δ^{n} are small in the following sense: the natural map

\underrightarrow{lim}_{I} homS(Λ_{k}^{n}, X_{α}) ⟶ homS(Λ_{k}^{n}, \underrightarrow{lim}_{I} X_{α})

is an isomorphism for all filtered colimits in S. This is because Λ_{k}^{n} has only finitely many non-degenerate simplices. Similar remarks hold for ∂Δ^{n} and Δ^{n}.

Lemma 5.2. Any morphism f : A → B in sC can be factored

A→^{j}X→^{q}B

where the morphism j is a cofibration and q is a trivial fibration.

Proof: Coproducts of cofibrations are cofibrations, and given a pushout diagram

\begin{matrix}A_{0} u i  w \\ A_{2} uj \end{matrix} ; {A_{1}  w \atop B}

in sC, then i a cofibration implies j is a cofibration, and that if X → Y is a cofibration in S, then F X → F Y is a cofibration in sC. Inductively construct objects X_{n} ∈ sC with the following properties:

a) One has A = X_{0} and there is a cofibration j_{n} : X_{n} → X_{n+1}.

b) There are maps q_{n} : X_{n} → B so that q_{n} = q_{n+1} ∘ j_{n} and the diagram

\begin{matrix}A_{4 4 4 4 6} w \\ X_{n} uq_{n} \end{matrix} ; B

commutes, where A → X_{n} is the composite j_{n−1} ∘ ⋯ ∘ j_{0}.

c) Any diagram

\begin{matrix}F ∂Δ^{m} u  w φ \\ X_{n} uq_{n} \end{matrix} ; {F Δ^{m}  w ψ \atop B}

can be completed to a diagram

\begin{matrix}F ∂Δ^{m} u  w φ \\ X_{n} uj_{n}_{’ ’ ’ ’’)} q_{n} \end{matrix} ; F Δ^{m}  w ; X_{n+1}  w q_{n+1} ; B

where the bottom morphism is ψ.

Condition c) indicates how to construct X_{n+1} given X_{n}. Define j_{n} : X_{n} → X_{n+1} by the pushout diagrams

\begin{matrix}⨆ F ∂Δ^{m} u  w \\ X_{n} uj_{n} \end{matrix} ; \begin{matrix}⨆ F Δ^{m}  w \\ X_{n+1} \end{matrix}

where the coproduct is over all diagrams of the type presented in c).

Then condition c) automatically holds. Further, q_{n+1} : X_{n+1} → B is defined and satisfies condition b) by the universal property of pushouts. Lastly, condition a) holds by the remarks at the beginning of the proof.

Now define X = \underrightarrow{lim} X_{n} and notice that we have a factoring

A→^{j}X→^{q}B

of the original morphism. The morphism j is a cofibration since directed colimits of cofibrations are cofibrations. We need only show q : X → B is a trivial fibration. This amounts to showing that any diagram

\begin{matrix}∂Δ^{n}  w u \\ GX uGq \end{matrix} ; \begin{matrix}Δ^{n}  w \\ GB \end{matrix}

can be completed. But GX ≅ \underrightarrow{lim}_{n} GX_{n} by hypothesis on G, and the result follows by the small object argument. ■

The same argument, but using the trivial cofibrations in S

Λ_{k}^{n} ↪ Δ^{n}

for n, k ≥ 0, proves the following lemma.

Lemma 5.3. Any morphism f : A → B in sC can be factored

A ⟶^{j} X ⟶^{q} B

where q is a fibration and j is a cofibration which has the left lifting property with respect to all fibrations.

Proof of Theorem 5.1: The Axioms CM1–CM3 are easily checked. The axiom CM5b is Lemma 5.2; the axiom CM5a follows from Lemma 5.3 and the assumption on cofibrations. For axiom CM4, one half is the definition of cofibration. For the other half, one proceeds as follows. Let

i : A → B

be a trivial cofibration. Then by Lemma 5.3 we can factor the morphism i as

A→^{j}X→^{q}B

where j is a cofibration with the left lifting with respect to all fibrations, and q is a fibration. By the hypothesis on cofibrations, j is a weak equivalence. Since i is a weak equivalence, so is q. Hence, one can complete the diagram

\begin{matrix}Au i  w j \\ Xuq \end{matrix} ; {B  w = \atop B}

and finds that i is a retract of j. Hence i has the left lifting property with respect to fibrations, because j does. This completes the proof. ■

We next remark that, in fact, sC is a simplicial model category. For this, we impose the simplicial structure guaranteed by Theorem 2.5. Thus if X ∈ sC and K ∈ S, we have that

(A ⊗ K)_{n} = ⨆_{k∈K_{n}} A_{n}.

From this, one sees that if X ∈ S

F (X × K) ≅ F (X) ⊗ K.

This is because F, as a left adjoint, preserves coproducts. Thus Lemma 2.9 applies and

Ghom_{sC}(K, B) ≅ homS(K, GB).

Theorem 5.4. With this simplicial structure, sC becomes a simplicial model category.

Proof: Apply Proposition 3.13.1. If j : K → L is a cofibration in S and q : X → Y is a fibration in sC, the map

Ghom_{sC}(L, X) ⟶ G(hom_{sC}(K, X) ×_{hom_{sC}(K,Y)} hom_{sC}(L, Y)) is isomorphic to

homS(L, GX) ⟶ homS(K, GX) ×_{homS(K,GY)} homS(L, GY)

by the remarks above and the fact that G, as a right adjoint, commutes with pullbacks. Since S is a simplicial model category, the result holds. ■

5.5 A remark on the hypotheses. Theorem 5.1 and, by extension, Theorem 5.4 require the hypothesis that every cofibration with the left lifting property with respect to all fibrations is, in fact, a weak equivalence. This is so Lemma 5.3 produces the factoring of a morphism as a trivial cofibration followed by a fibration. In the next section we will give some general results about when this hypothesis holds; however, in a particular situation, one might be able to prove directly that the factoring produced in Lemma 5.3 actually yields a trivial cofibration. Then the hypothesis on cofibrations required by these theorems holds because any cofibration with the left lifting property with respect to all fibrations will be a retract of a trivial cofibration. Then one need say no more.

For example, in examining the proof of Lemma 5.3 (see Lemma 5.2), one sees that we would have a factorization of f : A → B as a trivial cofibration followed by a fibratiion provided one knew that 1.) F (Λ_{k}^{n}) → F (Δ^{n}) was a weak equivalence or, more generally, that F preserved trivial cofibrations, and 2.) trivial cofibrations in sC were closed under coproducts, pushouts, and colimits over the natural numbers.

6. Examples of simplicial model categories.

As promised, we prove that a variety of simplicial categories satisfy the hypotheses necessary for Theorem 5.4 of the previous section to apply.

We begin with a crucial lemma.

Lemma 6.1. Assume that for every A ∈ sC there is a natural weak equivalence

ε_{A} : A → QA

where QA is fibrant. Then every cofibration with the left lifting property with respect to all fibrations is a weak equivalence.

Proof: This is the argument given by Quillen, on page II.4.9 of [75]. Let j : A → B be the given cofibration. Then by hypothesis, we may factor

Au j w ε_{A} QA u

B^{i i i ij} w∗

to get a map u : B → QA so that uj = ε_{A}. Then we contemplate the lifting problem

Au j  w f  hom_{sC}(Δ^{1}, QB) uq

B^{5 5 5 5 5 5 5 6} w g hom_{sC}(∂Δ^{1}, QB) ≅ QB × QB

where q is induced by ∂Δ^{1} ⊆ Δ^{1}, f is the composite

AA ⟶^{j} B ⟶^{ε_{B}} QB = hom_{sC}(∗, QB) → hom_{sC}(Δ^{1}, QB)

and

g = (ε_{B}, Qj ∘ u) .

Note that f is adjoint to the constant homotopy on

ε_{B} ∘ j = Qj ∘ ε_{A} : A → QB .

Then q is a fibration since

Ghom_{sC}(K, X) = hom_{S}(K, GX) ,

and S is a simplicial model category. Hence, since j is a cofibration, there exists

H : B → hom_{sC}(Δ^{1}, QB)

making both triangles commute. Then H is a right homotopy from

ε_{B} : B → QB

to Qj ∘ u, and this homotopy restricts to the constant homotopy on ε_{B} ∘ j = Qj ∘ ε_{A} : A → QB. In other words, we have a diagram

A  w ϵ_{A} u j  QA uQj

B  w ϵ_{B}^{h h hh j} u  QB

such that the upper triangle commutes and the lower triangle commutes up to homotopy. Apply the functor G to this diagram. Then G preserves right homotopies, and one checks directly on the level of homotopy groups that Gj is a weak equivalence, which, by definition, implies j is a weak equivalence. ■

Example 6.2. Suppose every object of sC is fibrant. Then we may take ε_{A} : A → QA to be the identity. This happens, for example, if the functor G : sC → S factors through the sub-category of simplicial groups and simplicial group homomorphisms. Thus, Theorem 5.4 applies to

simplicial group s, simplicial abelian groups and simplicial R-modules, where G is the

forgetful functor;

(1) more generally to simplicial modules over a simplicial ring R, where G is the forgetful functor,

(2) for a fixed commutative ring R; simplicial R-algebras, simplicial commutative R-algebras and simplicial Lie algebras over R. Again G is the forgetful functor.

Another powerful set of examples arises by making a careful choice of the form the functor G can take.

Recall that an object A ∈ C is small if hom_{C}(A, ·) commutes with filtered colimits. Fix a small Z ∈ C and define

(6.3)  G : sC → S

by

G(X) = hom_{sC}(Z, X) .

Then G has left adjoint

F K = Z ⊗ K

and G(·) commutes with filtered colimits. Thus, to apply Theorem 5.4, we need to prove the existence of the natural transformation

ε : A → QA

as in Lemma 6.1. Let

Ex : S → S

be Kan’s Extension functor^{1}. Then for all K ∈ S there is a natural map

ε_{K} : K → Ex K

which is a weak equivalence. Furthermore, most crucially for the application here, Ex(·) commutes with all limits. This is because it’s a right adjoint. Finally, if Ex^{n} K is this functor applied n times and

Ex^{n} ε_{K} : Ex^{n} K ⟶ Ex^{n+1} K

the induced morphism, then Ex^{∞} K = \underrightarrow{lim} Ex^{n} K is fibrant in S and the induced map

K ⟶ Ex^{∞} K

is a trivial cofibration.

Lemma 6.4. Suppose the category C is complete and cocomplete. Fix n ≥ 0. Then there is a functor

Q_{0}(·)_{n} : sC ⟶ C

1 This construction is discussed in Section III.4 below.

so that, for all Z ∈ C, there is a natural isomorphism of sets

hom_{C}(Z, (Q_{0}A)_{n}) ≅ Ex(Hom_{sC}(Z, A))_{n} .

Proof: Recall that that the functor Ex on S is right adjoint to the the subdivision functor sd. Then one has a sequence of natural isomorphisms

Ex Hom_{sC}(Z, A)_{n} ≅ homS(Δ^{n}, Ex Hom_{sC}(Z, A))
≅ homS(sd Δ^{n}, Hom_{sC}(Z, A))
≅ hom_{sC}(Z ⊗ sd Δ^{n}, A)
≅ hom_{sC}(Z, hom_{sC}(sd Δ^{n}, A))
≅ hom_{C}(Z, hom_{sC}(sd Δ^{n}, A)_{0}).

The last isomorphism is due to the fact that Z is a constant simplicial object and maps out of a constant simplicial object are completely determined by what happens on zero simplices. Thus we can set

(Q_{0}A)_{n} = hom_{sC}(sd Δ^{n}, A)_{0}. ■

The simplicial object Q_{0}A defined by

n ↦ (Q_{0}A)_{n} = hom_{sC}(sd Δ^{n}, A)_{0}

is natural in A; that is, we obtain a functor Q_{0} : sC → sC. Since we regard Z ∈ C as a constant simplicial object in sC

Hom_{sC}(Z, Y)_{n} ≅ hom_{sC}(Z ⊗ Δ^{n}, Y)
≅ hom_{C}(Z, Y_{n})

one immediately has that

Hom_{sC}(Z, Q_{0}A) ≅ Ex Hom_{SC}(Z, A) .

Finally the natural transformation ε_{K} : K → Ex^{∞} K yields a natural map

ε_{A} : A ⟶ Q_{0}A

and, by iteration, maps

Q_{0}^{n}ε_{A} : Q_{0}^{n}A ⟶ Q_{0}^{n+1}A.

Define QA = \underrightarrow{lim} Q_{0}^{n}A. The reader will have noticed that Q_{0}A and QA are independent of Z.

Now fix a small object Z ∈ C and regard Z as a constant simplicial object in sC. Then we define a morphism A → B in sC to be a weak equivalence (or fibration) if and only if the induced map

Hom_{sC}(Z, A) ⟶ Hom_{sC}(Z, B)

is a weak equivalence (or fibration) of simplicial sets.

Proposition 6.5. If Z is small, the morphism ε_{A} : A → QA is a weak equivalence and QA is fibrant.

Proof: Since Z is small, we have that

Hom_{sC}(Z, QA) ≅ Ex^{∞} Hom_{sC}(Z, A).

The morphism ε_{A} is a weak equivalence if and only if

Hom_{sC}(Z, A) → Hom_{sC}(Z, QA) ≅ Ex^{∞} Hom_{sCa}(Z, A)

is a weak equivalence and QA → ∗ is a fibration if and only if

Hom_{sC}(Z, QA) ≅ Ex^{∞} Hom_{sC}(Z, A) → Hom_{sC}(Z, ∗) = ∗

is a fibration. Both of these facts follow from the properties of the functor Ex^{∞}(·). ■

Corollary 6.6. Let C be a complete and cocomplete category and Z ∈ C a small object. Then sC is a simplicial model category with A → B a weak equivalence (or fibration) if and only if

Hom_{sC}(Z, A) ⟶ Hom_{sC}(Z, B)

is a weak equivalence (or fibration) of simplicial sets.

In practice one wants an intrinsic definition of weak equivalence and fibration, in the manner of the following example.

Example 6.7. All the examples of 6.2 can be recovered from Corollary 6.6. For example, C be the category of algebras over a commutative ring. Then C has a single projective generator; namely A[x], the algebra on one generator. Then one sets Z = A[x], which is evidently small, and one gets a closed model category structure from the previous result. However, if B ∈ sC, then

Hom_{sC}(A[x], B) ≅ B

in the category of simplicial sets, so one recovers the same closed model category structure as in Example 6.2.

If C is a category satisfying 5.1 with a single small projective generator, then C is known as a category of universal algebras. Setting Z to be the generator, one immediately gets a closed model category structure on sC from Corollary 6.6. This is the case for all the examples of 6.2.

To go further, we generalize the conditions of Theorem 5.1 a little, to require the existence of a collection of functors G_{i} : sC → S, i ∈ I, each of which has a left adjoint F_{i} : S → sC. We now say that a morphism f : A → B of sC is

a) a weak equivalence if G_{i}f is a weak equivalence of simplicial sets for all i ∈ I;

b) a fibration if all induced maps G_{i}f are fibrations of S;

c) a cofibration if it has the left lifting property with respect to all trivial cofibrations of sC.

Then Theorem 5.1 and Theorem 5.4 together have the following analogue:

Theorem 6.8. Suppose that C has all small limits and colimits and that all of the functors G_{i} : C → S preserve filtered colimits. Then with the notions of weak equivalence, fibration and cofibration defined above, and if every cofibration with the left lifting property with respect to all fibrations is a weak equivalence, then sC is a simplicial model category.

Proof: The proof is the same as that of Theorem 5.1, except that the small object arguments for the factorization axiom are constructed from all diagrams of the form

\begin{matrix}F_{i}∂Δ^{m}  w u \\ Au \end{matrix} ; {F_{i}Δ^{m}  w \atop B} and \begin{matrix}F_{i}Λ_{k}^{n}  w u \\ Au \end{matrix} ; {F_{i}Δ^{n}  w \atop B}  ■

Theorem 6.8 will be generalized significantly in the next section — it is a special case of Theorem 7.8.

Now fix a set of small objects Z_{i} ∈ C, i ∈ I, and regard each Z_{i} as a constant simplicial object in sC. Then we define a morphism A → B in sC to be a weak equivalence (or fibration) if and only if the induced map

Hom_{sC}(Z_{i}, A) ⟶ Hom_{sC}(Z_{i}, B)

is a weak equivalence (or fibration) of simplicial sets. In the case where C is complete and cocomplete, we are still entitled to the construction of the natural map ϵ_{A} : A → QA in sC. Furthermore, each of the objects Z_{i} is small, so that Proposition 6.5 holds with Z replaced by Z_{i}, implying that the map ϵ_{A} is a weak equivalence and that QA is fibrant. Then an analogue of Lemma 6.1 holds for the setup of Theorem 6.8 (with G replaced by G_{i} in the proof), and we obtain the following result:

Theorem 6.9. Suppose that C is a small complete and cocomplete category, and let Z_{i} ∈ C, i ∈ I, be a set of small objects. Then sC is a simplicial model category with A → B a weak equivalence (respectively fibration) if and only if the induced map

Hom_{sC}(Z_{i}, A) → Hom_{sC}(Z_{i}, B)

is a weak equivalence (respectively fibration) for all i ∈ I

Example 6.10. Suppose that C is small complete and cocomplete, and has a set {P_{α}} of small projective generators. Theorem 6.9 implies that C has a simplicial model category structure, where A → B is a weak equivalence (or fibration) if

Hom_{sC}(P_{α}, A) → Hom_{sC}(P_{α}, B)

is a weak equivalence (or fibration) for all α.

Note that the requirement that the objects P_{α} are projective generators is not necessary for the existence of the closed model structure. However, if we also assume that the category C has sufficiently many projectives in the sense that there is an effective epimorphism P → C with P projective for all objects C ∈ C, then it can be shown that a morphism f : A → B of sC is a weak equivalence (respectively fibration) if every induced map

Hom_{sC}(P, A) → Hom_{sC}(P, B)

arising from a projective object P ∈ C is a weak equivalence (respectively fibration) of simplicial sets. This is a result of Quillen [75, II.4], and its proof is the origin of the stream of ideas leading to Theorem 6.9. We shall go further in this direction in the next section.

To be more specific now, let C be the category of graded A-algebras for some commutative ring A and let, for n ≥ 0,

P_{n} = A[x_{n}]

be the free graded algebra or an element of degree n. Then {P_{n}}_{n≥0} form a set of projective generators for C. Thus sC gets a closed model category structure and B → C in sC is a weak equivalence if and only if

(B)_{n} → (C)_{n}

is a weak equivalence of simplicial sets for all n. Here (·)_{n} denotes the elements of degree n. This is equivalent to the following: if M is a simplicial graded A-module, define

π_{∗}M = H_{∗}(M, ∂)

where ∂ is the alternating of the face operators. Then B → C in sC is a weak equivalence if and only if

π_{∗}B → π_{∗}C

is an isomorphism of bigraded A-modules.

This formalism works for graded groups, graded abelian groups, graded Amodules, graded commutative algebras, graded Lie algebras, and so on.

Example 6.11. Let 𝔽 be a field and let C = CA be the category of coalgebras over 𝔽. Then, by [87, p.46] every coalgebra C ∈ CA is the filtered colimit of its finite dimensional sub-coalgebras. Thus CA has a set of generators {C_{α}} where C_{α} runs over a set of representatives for the finite dimensional coalgebras. These are evidently small. Hence, sCA has a closed model category structure where A → B is a weak equivalence if and only if

Hom_{sCA}(C_{α}, A) → Hom_{sCA}(C_{α}, B)

is a weak equivalence for all C_{α}. The significance of this example is that the C_{α} are not necessarily projective.

7. A generalization of Theorem 5.1.

The techniques of the previous sections are very general and accessible to vast generalization. We embark some ways on this journey here. First we expand on what it means for an object in a category to be small. Assume for simplicity that we are considering a category C which has all limits and colimits. We shall use the convention that a cardinal number is the smallest ordinal number in a given bijection class.

Fix an infinite cardinal number γ, and let Seq(γ) denote the well-ordered set of ordinals less than γ. Then Seq(γ) is a category with hom(s, t) one element if s ≤ t and empty otherwise. A γ-diagram in C is a functor X : Seq(γ) → C. We will write \underrightarrow{lim}_{γ} X_{s} for the colimit. We shall say that X is a γ-diagram of cofibrations if each of the transition morphisms X_{s} → X_{t} is a cofibration of C.

Definition 7.1. Suppose that β is an infinite cardinal. An object A ∈ C is β-small if for all γ-diagrams of cofibrations X in C with γ ≥ β, the natural map

\underrightarrow{lim}_{γ} hom_{C}(A, X_{s}) → hom_{C}(A, \underrightarrow{lim}_{γ} X_{s})

is an isomorphism. A morphism A → B of C is said to be β-small if the objects A and B are both β-small.

Example 7.2. The small objects of the previous sections were ω-small, where ω is the first infinite cardinal. Compact topological spaces are also ω-small, but this assertion requires proof.

Supposethat X : Seq(γ) → Top is a γ-diagram of cofibrations. Then X is a retract of a γ-diagram of cofibrations \overline{X}, where each of the transition morphisms \overline{X}_{s} ↪ \overline{X}_{t} is a relative CW-complex. In effect, set \overline{X}_{0} = X_{0}, and set \overline{X}_{α} = \underrightarrow{lim}_{s<α} \overline{X}_{s} for limit ordinals α < γ. Suppose given maps

X_{s} ⟶^{r_{s}} \overline{X}_{s} ⟶^{π_{s}} X_{s}

with π_{s}r_{s} = 1. Then \overline{X}_{s+1} is defined by choosing a trivial fibration π_{s+1} and a relative CW-complex map j_{s+1} : \overline{X}_{s} ↪ \overline{X}_{s+1} (ie. \overline{X}_{s+1} is obtained from \overline{X}_{s} by attaching cells, and j_{s+1} is the corresponding inclusion), such that the following diagram commutes:

\begin{matrix}\overline{X} & s & w & j_{s+1} & u & π_{s} \\ \overline{X} & s+1 & u & π_{s+1} \end{matrix} ; \begin{matrix}X_{s} w i_{s+1} \\ X_{s+1}, \end{matrix}

where the map i_{s+1} : X_{s} → X_{s+1} is the cofibration associated to the relation s ≤ s + 1 by the functor X. Then there is a lifting in the diagram

X_{s}  w j_{s+1}r_{s} u i_{s+1}  \overline{X}_{s+1} uπ_{s+1}

X_{s+1}  w 1^{h h hh j} r_{s+1}  X_{s+1}

so that the section r_{s} extends to a section r_{s+1} of the trivial fibration π_{s+1}. The inclusion

X_{0} = \overline{X}_{0} ↪ \underrightarrow{lim}_{γ} \overline{X}_{s}

is a relative CW-complex map, and every compact subset of the colimit only meets finitely many cells outside of X_{0}. Every compact subset of \underrightarrow{lim}_{γ} \overline{X}_{s} is

therefore contained in some subspace \overline{X}_{s}. It follows that every compact subset of \underrightarrow{lim}_{γ} X_{s} is contained in some X_{s}.

We next produce an appropriate generalization of saturation.

Definition 7.3. A class M of morphisms in C is β-saturated if it is closed under

1) retracts: Suppose there is a commutative diagram in C

\begin{matrix}X  w u_{i} & X′  w u_{i′} & Xu_{i} \\ Y  w & Y′  w & Y \end{matrix}

with the horizontal composition the identity. Then if i′ ∈ M , then i ∈ M.

2) coproducts: if each j_{α} : X_{α} → Y_{α} is in M, then ⨆_{α} j_{α} : ⨆_{α} X_{α} → ⨆_{α} Y_{α} is in M;

3) pushouts: given a pushout diagram in C

\begin{matrix}A  w u_{i} \\ Xu_{j} \end{matrix} ; \begin{matrix}B  w \\ Y, \end{matrix}

if i is in M, then so is j.

4) colimits of β-sequences: Suppose we are given a β-sequence

X : Seq(β) → C

with the following properties: a) for each successor ordinal s+1 ∈ Seq(β), the map X_{s} → X_{s+1} is in M, and b) for each limit ordinal s ∈ Seq(β), the map \underrightarrow{lim}_{t<s} X_{t} → X_{s} is in M. Then

X_{s} → \underrightarrow{lim}_{β} X_{s}

is in M for all s ∈ Seq(β).

Up until now we have considered only saturated clesses of morphisms with β = ω, the cardinality of a countable ordinal. In this case, one doesn’t need the extra care required in making the definition of what it means to be closed under colimits.

Lemma 7.4. Let C be a closed model category. Then the class of cofibrations and the class of trivial cofibrations are both β-saturated for all β.

Proof: This is an exercise using the fact that cofibrations (or trivial cofibrations) are characterized by the fact that they have the left lifting property with respect to trivial fibrations (or fibrations). ■

The next step is to turn these concepts around.

Definition 7.5. Let M_{0} be a class of morphisms in C. Then the β-saturation of M_{0} is the smallest β-saturated class of morphisms in C containing M_{0}.

We now come to the crucial axiom.

Definition 7.6. A closed model category is cofibrantly generated with respect to a cardinal β if the class of cofibrations and the class of trivial cofibrations are the β-saturations of sets of β-small morphisms M_{0} and M_{1} respectively.

Remarks 7.7.

1) Suppose that β and γ are cardinals such that β ≤ γ. Then every γ-saturated class is β-saturated, because every sequence X : Seq(β) → C can be extended to a sequence X_{∗} : Seq(γ) → C having the same colimit. It follows that the β-saturation of any set of morphisms is contained in its γ-saturation. Observe also that every β-small object is γ-small, directly from Definition 7.1. The size of the cardinal β in Definition 7.6 therefore doesn’t matter, so long as it exists. One says that the closed model category C is cofibrantly generated in cases where the cardinal β can be ignored.

2) Until now, we’ve taken β to ω. Then the category of simplicial sets is cofibrantly generated, for example, by the usual small object argument. Similarly, modulo the care required for the assertion that finite CW-complexes are ω-small (Example 7.2), the category of topological spaces is cofibrantly generated with respect to ω. We will see larger cases later.

3) One could require one cardinal β_{0} for cofibrations and β_{1} for trivial cofibrations. However, β = max{β_{0}, β_{1}} would certainly work in either case, by 1).

To give the generalization of Theorem 5.1 we establish a situation. We fix a simplicial model category C and a simplicial category D. Suppose we have a set of functors G_{i} : D → C, indexed by the elements i in some set I, and suppose each G_{i} has a left adjoint F_{i} which preserves the simplicial structure in the sense that there is a natural isomorphism

F_{i}(X ⊗ K) ≅ F (X_{i}) ⊗ K

for all X ∈ C and K ∈ S. Define a morphism f : A → B in D to be a weak equivalence (or fibration) is

G_{i}f : G_{i}A → G_{i}B

is a weak equivalence (or fibration).

Theorem 7.8. Suppose the simplicial model category C is cofibrantly generated with respect to a cardinal β, and that

(1) all of the functors G_{i} commute with colimits over Seq(β), and

(2) the functors G_{i} take the β-saturation of the collection of all maps F_{j}A → F_{j}B arising from maps A → B in the generating family for the cofibrations of C and elements j of I to cofibrations of C.

Then if every cofibration in D with the left lifting property with respect to all fibrations is a weak equivalence, D is a simplicial model category.

Proof (Outline): There are no new ideas—only minor changes from the arguments of Section 5. The major difference is in how the factorizations are constructed. For example, to factor X → Y as X →^{j} Z →^{q} Y where j is a cofibration which has the left lifting property with respect to all fibrations and q is a fibration, one forms a β-diagram {Z_{s}} is D where

i) Z_{0} = X; ii) if s ∈ Seq(β) is a limit ordinal, Z_{s} = \underrightarrow{lim}_{t<s} Z_{t} and

iii) if s + 1 is a successor ordinal, there is a pushout diagram

\begin{matrix}⨆_{i} ⨆_{f} F_{i}(A)  w u \\ Z_{s} u \end{matrix} ; \begin{matrix}⨆_{i} ⨆_{f} F_{i}(B)  w \\ Z_{s+1} \end{matrix}

where f runs over all diagrams

\begin{matrix}Au  w \\ G_{i}(Z_{s}) u \end{matrix} ; \begin{matrix}B  w \\ G_{i}(Y) \end{matrix}

where A → B is in the set M_{1} of β-small cofibrations in C whose βsaturation is all trivial cofibrations. ■

Example 7.9. Suppose that C is a cofibrantly generated simplicial model category and I is a fixed small category. Write C^{I} for the category of functors X : I → C and natural transformations between them. There are i-section functors G_{i} : C^{I} → C defined by G_{i}X = X(i), i ∈ I, and each such G_{i} has a left adjoint F_{i} : C → C^{I} defined by

F_{i}D(j) = ⨆_{i→j in I} D.

Say that a map X → Y of C^{I} is a pointwise cofibration if each of the maps G_{i}X → G_{i}Y is a respectively cofibration of C. If A → B is a generating cofibration for C, the induced maps F_{i}A(j) → F_{i}B(j) are disjoint unions of cofibrations and hence are cofibrations of C. The induced maps maps F_{i}A → F_{i}B are therefore pointwise cofibrations of C^{I}. The functors G_{j} preserve all colimits, and so the collection of pointwise cofibrations of C^{I} is saturated (meaning β-saturated for some infinite cardinal β — similar abuses follow). The saturation of the collection of maps F_{i}A → F_{i}B therefore consists of pointwise cofibrations of C^{I}.

A small object argument for C^{I} produces a factorization

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

for an arbitrary map f : X → Y of C^{I}, with q a fibration, and for which j is in the saturation of the collection of maps F_{i}C → F_{i}D arising from the generating set C → D for the class of trivial cofibrations of C. But again, each induced map F_{i}C(j) → F_{i}D(j) is a trivial pointwise cofibration of C, and the j-section functors preserve all colimits. The collection of maps of C^{I} which are trivial cofibrations in sections is therefore saturated, and hence contains the saturation of the maps F_{i}C → F_{i}D. It follows that the map j is a weak equivalence as well as a cofibration. In particular, by a standard argument, every map of C^{I} which has the left lifting property with respect to all fibrations is a trivial cofibration.

It therefore follows from Theorem 7.8 that every diagram category C^{I} taking values in a cofibrantly generated simplicial model category has a simplicial model structure for which the fibrations and weak equivalences are defined pointwise. This result applies in particular to diagram categories Top^{I} taking values in topological spaces.

Here’s the analog of Lemma 6.1:

Proposition 7.10. Suppose there is a functor Q : D → D so that QX is fibrant for all X and there is a natural weak equivalence ϵ_{X} : X → QX. Then every cofibration with the left lifting property with respect to all fibrations is a weak equivalence.

Proof: The argument is similar to that of Lemma 6.1; in particular, it begins the same way.

The map j has the advertised lifting property, so we may form the diagrams

A  w ϵ_{A} u j  QA u

B w^{h h hh j} u ∗

and

A  w sϵ_{B}j u j  hom(Δ^{1}, QB) u

B w (ϵ_{B}, Qj ∘ u)^{N N N NN P} QB × QB

where sϵ_{B}j is the constant (right) homotopy on the composite

A ⟶^{j} B ⟶^{ϵ_{B}} QB.

The functors G_{i} preserve right homotopies, so the diagram

A  w ϵ_{A} u j  QA uQj

B  w ϵ_{B}^{h h hh j} u  QB

remains homotopy commutative after applying each of the functors G_{i}. It follows that the map G_{i}(j) is a retract of the map G_{i}(ϵ_{A}) in the homotopy category Ho(C), and is therefore an isomorphism in Ho(C). But then a map in a simplicial model category which induces an isomorphism in the associated homotopy category must itself be a weak equivalence: this is Lemma 4.1. ■

Example 7.11. The factorization axioms for a cofibrantly generated simplicial model category C can always be proved with a possibly transfinite small object argument (see, for example, the proofs of Proposition V.6.2 and Lemma IX.1.17). Such arguments necessarily produce factorizations which are natural in morphisms in C, so that there is a natural fibrant model X ↪ X̃ for all objects X of C. It follows that there are natural fibrant models for the objects of any diagram category C^{I} taking values in C. We therefore obtain a variation of the proof of the existence of the closed model structure for C^{I} of Example 7.9 which uses Proposition 7.10. This means that the requirement in Theorem 7.8 that every cofibration which has the left lifting property with respect to all fibrations should be a weak equivalence is not particularly severe.

Example 7.12. As an instance where the cofibrant generators are not ω-small, we point out that in Section IX.3 we will take the category of simplicial sets, with its usual simplicial structure and impose a new closed model category structure. Let E_{∗} be any homology theory and we demand that a morphism f : X → Y in S be a

1) E_{∗} equivalence if E_{∗}f is an isomorphism

2) E_{∗} cofibration if f is a cofibration as simplicial sets

3) E_{∗} fibration if f has the right lifting property with respect to all E_{∗} trivial cofibrations.

The E_{∗} fibrant objects are the Bousfield local spaces. In this case the E_{∗}-trivial cofibrations are the saturation of a set of E_{∗} trivial cofibrations f : A → B where B is β-small with β some infinite cardinal greater than the cardinality of E_{∗}(pt).

One has functorial factorizations, so Example 7.11 can be repeated to show that S^{I} has a simplicial model category structure with f : X → Y a weak equivalence (or fibration) if and only if X(i) → Y (i) is an E_{∗} equivalence (or E_{∗} fibration) for all i.

8. Quillen’s total derived functor theorem.

Given two closed model categories C and D and adjoint functors between them, we wish to know when these induce adjoint functors on the homotopy categories. This is Quillen’s Total Derived Functor Theorem. We also give criteria under which the induced adjoint functors give an equivalence of the homotopy categories.

The ideas of this section are a generalization to non-abelian settings of an old idea of Grothendieck which can be explained by the following example. If R is a commutative ring and M, N are two R-modules, one might want to compute Tor_{p}^{R}(M, N), p ≥ 0. However, there is a finer invariant, namely, the chain homotopy type of M ⊗_{R} P_{∗} where P_{∗} is a projective resolution of N. One calls the chain homotopy equivalence class of M ⊗_{R} P_{∗} by the name Tor^{R}(M, N). This is the total derived functor. The individual Tor terms can be recovered by taking homology groups.

For simplicity we assume we are working with simplicial model categories, although many of the results are true without this assumption.

Definition 8.1. Let C be a simplicial model category and A any category. Suppose F : C → A is a functor that sends weak equivalences between cofibrant objects to isomorphisms. Define the total left derived functor

LF : Ho(C) → A.

by LF (X) = F (Y) where Y → X is a trivial fibration with Y cofibrant.

It is not immediately clear that LF is defined on morphisms or a functor. If f : X → X′ is a morphism in C and Y → X and Y′ → X′ are trivial cofibrations with Y and Y′ cofibrant, then there is a morphism g making the following diagram commute:

(8.2)  \begin{matrix}Y  w^{g} u \\ Y′ u \end{matrix} ; \begin{matrix}X  w^{f} \\ X′ \end{matrix}

and we set LF (f) = LF (g).

Lemma 8.3. The objects LF (X) and morphisms LF (f) are independent of the choices and LF : Ho(C) → A is a functor.

Proof: First note that LF (f) is independent of the choice of g in diagram (8.2). This is because any two lifts g and g′ are homotopic and one has

\begin{matrix}F (Y) ⊔ F (Y)  w_{‚ ‚ ‚ ‚ ›}_{F (1)⊔F (1)} & F (Y ⊗ Δ^{1}) u_{≅}  w^{F H} & F (Y′) \\ F (Y) \end{matrix}

where H is the homotopy. Next, if we let f be the identity in (8.2), the same argument implies LF (X) is independent of the choice of Y. Finally, letting f = f_{1} ∘ f_{2} be a composite in diagram (8.2) the same argument shows LF (f_{1} ∘ f_{2}) = LF (f_{1}) ∘ LF (f_{2}). ■

Remark 8.4. For those readers attuned to category theory we note that LF is in fact a Kan extension in the following sense. Let γ : C → Ho(C) be the localization functor and

\begin{matrix}C w^{γ} u_{F} \\ Ho(C) ´ ´ ´ \end{matrix} ; ˜ A

the diagram of categories. There may or may not be a functor Ho(C) → A completing the diagram; however, one can consider functors T : Ho(C) → A equipped with a natural transformation

ε_{T} : T γ → F.

The Kan extension is the final such functor T, if it exists. If R denotes this Kan extension, final means that given any such T, there is a natural transformation σ : T → R so that ε_{T} = ε_{R}σγ. The Kan extension is unique if it exists. To see that it exists one applies Theorem 1, p. 233 of Mac Lane’s book [66]. This result, in this context reads as follows: one forms a category X ↓ γ consisting of pairs (Z, f) where Z ∈ C and f : X → Z is a morphism in Ho(C). Then if

R(X) = \underleftarrow{lim}_{X↓γ} F (Z)

exists for all X, then R exists. However, the argument of Lemma 6.3 says that the diagram F : (X ↓ γ) → A has a terminal object. In fact, X ↓ γ has a terminal object, namely X → Y where Y → X is a trivial fibration (which has an inverse in Ho(C)) with Y cofibrant. This shows that R = LF. ■

Corollary 8.5. Let X ∈ C. If X is cofibrant, then LF (X) ≅ F (X). If Y → X is any weak equivalence, with Y cofibrant, then LF X ≅ F Y.

Proof: The first statement is obvious and the second follows from

F Y ≅ LF (Y) →^{≅} LF (X)

since Y → X is an isomorphism in Ho(C). ■

Example 8.6. Let C = C_{∗}R be chain complexes of left modules over a ring R, and let A = nAb be graded abelian groups. Define

F (C) = H_{∗}(M ⊗_{R} C)

for some right module M. Then

LF (C) = H_{∗}(M ⊗_{R} D)

where D → C is a projective resolution of C. There is a spectral sequence

Tor_{p}^{R}(M, H_{q}C) ⇒ (LF (C))_{p+q}.

In particular, if H_{∗}C = N concentrated in degree 0,

LF (C) ≅ Tor_{∗}^{R}(M, N),

bringing us back to what we normally mean by derived functors.

If G : C → A sends weak equivalences between fibrant objects to isomorphisms, one also gets a total right derived functor

RG : Ho(C) → A.

It is also a Kan extension, suitably interpreted: it is initial among all functors S : Ho(C) → A equipped with a natural transformation η_{S} : F → Sγ.

Now suppose we are given two simplicial model categories C and D and a functor F : C → D with a right adjoint G. The following is one version of the total derived functor theorem:

Theorem 8.7. Suppose F preserves weak equivalences between cofibrant objects and G preserves weak equivalences between fibrant objects. Then LF : Ho(C) → Ho(D) and RG : Ho(D) → Ho(C) exists and RG is right adjoint to LF.

Proof: That LF and RG exist is a consequence of Lemma 8.3 and its analog for total right derived functors. We need only prove adjointness.

If X ∈ C, choose Y → X a trivial fibration with Y cofibrant. Hence LF (X) ≅ F (Y). Now choose F (Y) → Z, a trivial cofibration with Z fibrant. Then RG ∘ LF (X) ≅ G(Z) and one gets a unit

η : X → RG ∘ LF (X)

by X ← Y → GF (Y) → G(Z).

Similarly, let A ∈ D. Choose A → B a trivial cofibration with B fibrant. Then RG(A) ≅ G(B). Next choose C → G(B) a trivial fibration with C cofibrant. Then LF ∘ RG(A) ≅ F (C) and one gets a counit ε : LF ∘ RG(A) → A by

F (C) → F GB → B ← A.

We now wish to show

LF (X) ⟶^{LF η} LF ∘ RG ∘ LF (X) ⟶^{ε_{RG}} LF (X)

is an isomorphism. In evaluating ε_{RG} we set A = LF (X) = F (Y), so that B = Z. Factor Y → GF (Y) → G(Z) and Y →^{j} C →^{q} G(Z) where j is a cofibration (so C is cofibrant) and q is a trivial fibration. Then ε_{RG} is given by

F (C) ⟶^{F_{q}} F GZ → Z ← F Y.

Furthermore there is a commutative square

\begin{matrix}Yu_{j}  w \\ GF (Y) u \end{matrix} ; \begin{matrix}C  w \\ G(Z) \end{matrix}.

and F j ≅ P LF η_{X}. Expanding the diagram gives:

\begin{matrix}F (Y)  w u_{F j} \\ F GF Y u  w \\ F Y u_{≅} \end{matrix} ; F (C)  w^{F q} ; F GZ  w ; Z.

The line across the top is the identity and the indicated map is an isomorphism in the homotopy category. Hence we have proved the assertion.

The other assertion—that

RG(A) ⟶^{η_{ε}} RG ∘ LF ∘ RG(A) ⟶^{RG_{ε}} RG(A)

is an isomorphism—is proved similarly. The result now holds by standard arguments; e.g., [66] Thm. 2v), p. 81. ■

An immediate corollary used several times in the sequel is:

Corollary 8.8. Under the hypotheses of Theorem 8.7 assume further that for X ∈ C cofibrant and A ∈ D cofibrant

X → GA

is a weak equivalence if and only if its adjoint F X → A is a weak equivalence. Then LF and RG induce an adjoint equivalence of categories:

Ho(C) ≅ Ho(D).

Proof: We need to check that η : X → RG ∘ LF (X) is an isomorphism and ε : LF ∘ RG(A) → A is an isomorphism. Using the notation established in the previous argument, we have a sequence of arrows that define η:

X ← Y → GZ.

Now Y is cofibrant and Z is fibrant, and F Y → Z is a weak equivalence; so Y → GZ is a weak equivalence and this shows η is an isomorphism. The other argument is identical. ■

In practice one may not know a priori that F and G satisfy the hypotheses of Theorem 8.7. The following result is often useful. We shall assume that the model categories at hand are, in fact, simplicial model categories; however, it is possible to prove the result more generally.

Lemma 8.9. Let F : C → D be a functor between simplicial model categories, and suppose F has a right adjoint G. If G preserves fibrations and trivial fibrations, then F preserves cofibrations, trivial cofibrations and weak equivalences between cofibrant objects.

Proof: It follows from an adjointness argument that F preserves trivial cofibrations and cofibrations; for example, suppose j : X → Y is a cofibration in C. To show F j is a cofibration, one need only solve the lifting problem

\begin{matrix}F X u_{F j} w \\ Au_{q} \end{matrix} ; {F Y ^{\ \\ ]} w \atop B}

for every trivial fibration q in D. This problem is adjoint to

\begin{matrix}X w u \\ GA u_{Gq} \end{matrix} ; \begin{matrix}Y ^{\ \\ ]} w \\ GB \end{matrix}

which has a solution by hypothesis.

Now suppose f : X → Y is a weak equivalence between cofibrant objects. Factor f as X →^{j} Z →^{q} Y where j is a trivial cofibration and q a trivial fibration. We have just shown F j is a weak equivalence. Also q is actually a homotopy equivalence: there is a map s : Y → Z so that qs = 1_{Y} and sq ≃ 1_{Z}. Here X and Y are cofibrant. We claim that F q is a homotopy equivalence, so that it is a weak equivalence by Lemma 4.1.

To see that F q is a homotopy equivalence, note that F (q)F (s) = 1_{F Y}. Next note that since Z is cofibrant, Z ⊗ Δ^{1} is a cylinder object for Z and, since F preserves trivial cofibrations F (Z ⊗ Δ^{1}) is a cylinder object for F (Z). Hence F (s)F (q) ≃ 1_{F Z}. ■

Remark 8.10. As usual, the previous result has an analog that reverses the roles of F and G; namely, if F preserves cofibrations and trivial cofibrations, then G preserves fibrations, trivial fibrations, and weak equivalences between fibrant objects. The proof is the same, mutatis mutandis.

Example 8.11. Let I be a small category and S^{I} the category of I diagrams. Then S^{I} becomes a simplicial model category, where a morphism of diagrams X → Y is a weak equivalence or fibration if and only if X(i) → Y (i) is a weak equivalence of I-diagrams. The constant functor S → S^{I} preserves fibrations and weak equivalences, so (by Lemma 8.9), the left adjoint

F = \underrightarrow{lim}_{I} : S^{I} → S

preserves weak equivalences among cofibrant diagrams. Hence the total left derived functor

L \underrightarrow{lim}_{I} : Ho(S^{I}) → Ho(S)

exists. This functor is the homotopy colimit and we write L \underrightarrow{lim}_{I} = \underrightarrow{holim}. In a certain sense, made precise by the notion of Kan extensions in Remark 8.4, is the closest approximation to colimit that passes to the homotopy category. In any application it is useful to have an explicit formula for \underrightarrow{holim}_{I} X in terms of

the original diagram X; this is given by the coend formula

\underrightarrow{holim}_{I} X = ∫ ^{i} B(i ↓ I)^{op} ⊗ X(i).

These are studied in detail elsewhere in these notes.

This example, can be vastly generalized. If C is any cofibrantly generated simplicial model category, C^{I} becomes a simplicial model category and one gets

\underrightarrow{holim}_{I} : Ho(C^{I}) → Ho(C)

in an analogous manner. ■

9. Homotopy cartesian diagrams.

We return, in this last section, to concepts which are particular to the category of simplicial sets and its close relatives. The theory of homotopy cartesian diagrams of simplicial sets is, at the same time, quite deep and essentially axiomatic. The axiomatic part of the theory is valid in arbitrary categories of fibrant objects such as the category of Kan complexes, while the depth is implicit in the passage from the statements about Kan complexes to the category of simplicial sets as a whole. This passage is non-trivial, even though it is completely standard, because it involves (interchangeably) either Quillen’s theorem that the realization of a Kan fibration is a Serre fibration (Theorem I.10.10) or Kan’s Ex^{∞} construction (see III.4).

A proper closed model category C is a closed model category such that

P1 the class of weak equivalences is closed under base change by fibrations,

and P2 the class of weak equivalences is closed under cobase change by cofibra-

tions.

In plain English, axiom P1 says that, given a pullback diagram

\begin{matrix}X  w g_{∗} u \\ Yup \end{matrix} ; {Z  w g \atop W}

of C with p a fibration, if g is a weak equivalence then so is g_{∗}. Dually, axiom P2 says that, given a pushout diagram

\begin{matrix}A  w f u i \\ Bu \end{matrix} ; {C  w f_{∗} \atop D}

with i a cofibration, if f is a weak equivalence then so is f_{∗}.

The category of simplicial sets is a canonical example of a proper closed model category (in fact, a proper simplicial model category). Furthermore, this is the generic example: most useful examples of proper closed model categories inherit their structure from simplicial sets. The assertion that the category of simplicial sets satisfies the two axioms above requires proof, but this proof is in part a formal consequence of the fact that every simplicial set is cofibrant and every topological space is fibrant. The formalism itself enjoys wide applicability, and will be summarized here, now.

A category of cofibrant objects is a category D with all finite coproducts (including an initial object φ ), with two classes of maps, called weak equivalences and cofibrations, such that the following axioms are satisfied:

(A) Suppose given a commutative diagram

\begin{matrix}X w g_{N N N N P} h \\ Y ‚ ‚ ‚ ‚ “ f \end{matrix} ; Z

in D. If any two of f, g and h are weak equivalences, then so is the third. (B) The composite of two cofibrations is a cofibration. Any isomorphism is a

cofibration. (C) Pushout diagrams of the form

\begin{matrix}A  w u i \\ Bui_{∗} \end{matrix} ; {C  w \atop D}

exist in the case where i is a cofibration. Furthermore, i_{∗} is a cofibration which is trivial if i is trivial.

(D) For any object X there is at least one cylinder space X ⊗ I.

(E) For any object X, the unique map ∅ → X is a cofibration.

To explain, a trivial cofibration is a morphism of D which is both a cofibration and a weak equivalence. A cylinder object X ⊗I for X is a commutative diagram

{X ⊔ X w ∇ u i = (i_{0}, i_{1}) \atop X} ; X ⊗ I^{[ [ [ []} σ

in which i is a cofibration and σ is a weak equivalence, just like in the context of a closed model category (see Section 1 above). Each of the components i_{ϵ} of i must therefore be a trivial cofibration.

The definition of category of cofibrant objects is dual to the definition of category of fibrant objects given in Section I.9. All results about categories of fibrant objects therefore imply dual results for categories of cofibrant objects, and conversely. In particular, we immediately have the dual of one of the assertions of Proposition I.9.5:

Proposition 9.1. The full subcategory of cofibrant objects C_{c} in a closed model category C, together with the weak equivalences and cofibrations between them, satisfies the axioms (A) - (E) for a category of cofibrant objects.

Remark 9.2. One likes to think that a category of cofibrant objects structure (respectively a category of fibrant objects structure) is half of a closed model structure. This intuition fails, however, because it neglects the power of the axiom CM4.

Corollary 9.3.

(1) The category of simplicial sets is a category of cofibrant objects.

(2) The category of compactly generated Hausdorff spaces is a category of fibrant objects.

Lemma 9.4. Suppose that f : A → B is an arbitrary map in a category of cofibrant objects D. Then f has a factorization f = q · j, where j is a cofibration and q is left inverse to a trivial cofibration. In particular, q is a weak equivalence.

Proof: The proof of this result is the mapping cylinder construction. It’s also dual of the classical procedure for replacing a map by a fibration.

Choose a cylinder object

{A ⊔ A w ∇ u (i_{0}, i_{1}) \atop A} ; A ⊗ I^{h h hh j} σ

for A, and form the pushout diagram

\begin{matrix}A  w f u i_{0} \\ Bui_{0∗} \end{matrix} ; \begin{matrix}A ⊗ I  w f_{∗} \\ B_{∗}. \end{matrix}

Then (fσ)·i_{0} = f, and so there is a unique map q : B_{∗} → B such that q ·f_{∗} = fσ and q · i_{0∗} = 1_{B}. Then f = q · (f_{∗}i_{1}).

The composite map f_{∗}i_{1} is a cofibration, since the diagram

\begin{matrix}A ⊔ A  w f ⊔ 1_{A} u (i_{0}, i_{1}) \\ B ⊔ A u(i_{0∗}, f_{∗}i_{1}) \end{matrix} ; \begin{matrix}A ⊗ I  w f_{∗} \\ B_{∗} \end{matrix}

is a pushout. ■

Lemma 9.5. Suppose that

\begin{matrix}A  w u u i \\ Bu \end{matrix} ; {C  w u_{∗} \atop D}

is a pushout in a category of cofibrant objects D, such that i is a cofibration and u is a weak equivalence. Then the map u_{∗} is a weak equivalence.

Proof: Trivial cofibrations are stable under pushout, so Lemma 9.4 implies that it suffices to assume that there is a trivial cofibration j : B → A such that u · j = 1_{B}.

Form the pushout diagram

B  w j u j  Auj_{∗}
Au i
C  w j̃ D̃

Then j̃ is a trivial cofibration.

Let f : D̃ → C be the unique map which is determined by the commutative diagram

B  w j u j  Aui
Au i
C w 1_{C} C.

Form the prism

A  w u u j_{∗} _{’ ’ ’ ’ ’ ’ ’)} i  Bu_{’ ’ ’ ’ ’ ’ ’)}

D̃  w ũ_{N N NN P} f

B_{∗}_{N N NN P} f_{∗}

C  w u_{∗}  D

such that the front and back faces are pushouts (ie. push out the triangle on the left along u). Then ũ is a weak equivalence, since j̃ is a weak equivalence and u · j = 1_{B}. It therefore suffices to show that the map f_{∗} is a weak equivalence.

The bottom face

\begin{aligned}D̃ & w & ũ & u & f \\ & & B_{∗} uf_{∗} \end{aligned} ; {C w u_{∗} \atop D}

is a pushout, and the map f is a weak equivalence. The morphism

A h h h h k j_{∗} _{4 4 44 6} i ; \begin{aligned}D̃ & w & f \\ & & C \end{aligned}

is therefore a weak equivalence in the category A ↓ D, and the argument of Lemma 9.4 says that this map has a factorization in A ↓ D of the form f = q · j, where j is a cofibration and q is left inverse to a cofibration. It follows that pushing out along u preserves weak equivalences of A ↓ D, so that f_{∗} is a weak equivalence of D. ■

Corollary 9.6. The category S of simplicial sets is a proper simplicial model category.

Proof: Axiom P2 is a consequence of Lemma 9.5 and Corollary 9.3. The category CGHaus of compactly generated Hausdorff spaces is a category of fibrant objects, so the dual of Lemma 9.5 implies Axiom P1 for that category. One infers P1 for the simplicial set category from the exactness of the realization functor (Proposition I.2.4), the fact that the realization functor preserves fibrations (Theorem I.10.9), and the fact that the canonical map η : X → S|X| is a weak equivalence for all simplicial sets X (see the proof of Theorem I.11.4). ■

Remark 9.7. Axiom P1 for the category of simplicial sets can alternatively be seen by observing that Kan’s Ex^{∞} preserves fibrations and pullbacks (Lemma III.4.5), and preserves weak equivalences as well (Theorem III.4.6). Thus, given a pullback diagram

\begin{matrix}X  w g_{∗} u \\ Yup \end{matrix} ; {Z  w g \atop W}

with p a fibration and g a weak equivalence, if we want to show that g_{∗} is a weak equivalence, it suffices to show that the induced map Ex^{∞} g_{∗} in the pullback diagram

Ex^{∞} X w Ex^{∞} g_{∗} u

Ex^{∞} Y uEx^{∞} p

Ex^{∞} Z w Ex^{∞} g Ex^{infty} W

is a weak equivalence. But all of the objects in this last diagram are fibrant and the map Ex^{∞} g is a weak equivalence, so the desired result follows from the dual of Lemma 9.5.

The following result is commonly called the glueing lemma. The axiomatic argument for it that is given here is due to Thomas Gunnarsson [40].

Lemma 9.8. Suppose given a commutative cube

(9.9)

A_{1}  w j_{1}_{4 4 4 4 6} i_{1} u f_{A}  B_{1}_{4 4 4 4 6} uf_{B}

C_{1}  w u f_{C}  D_{1} uf_{D}

A_{2}  w j_{2}_{4 4 4 4 6} i_{2}  B_{2}_{4 4 4 4 6}

C_{2} wD_{2}

in a category of cofibrant objects D. Suppose further that the top and bottom faces are pushouts, that i_{1} and i_{2} are cofibrations, and that the maps f_{A}, f_{B} and f_{C} are weak equivalences. Then f_{D} is a weak equivalence.

Proof: It suffices to assume that the maps j_{1} and j_{2} are cofibrations. To see this, use Lemma 9.4 to factorize j_{1} and j_{2} as cofibrations followed by weak equivalences, and then use Lemma 9.5 to analyze the resulting map of cubes.

Form the diagram

A_{1}  w j_{1}_{‚ ‚ ‚‚ ›} i_{1} u f_{A}  B_{1}_{````}ˆuf_{A∗}

C_{1}  w j_{1∗} u f_{C}  D_{1} uf_{C∗}

B′ u n_{B} _{````}ˆ θ
D′ un_{D}

A_{2}^{A A A A A A A AA C} w j_{2}_{‚ ‚ ‚‚ ›} i_{2} B_{2}_{````}ˆ i_{2∗}

C_{2}^{A A A A A A A A A C}w

D_{2}.

by pushing out the top face along the left face of the cube (9.9). The square

\begin{matrix}A_{2}  w u i_{2} & B′ uθ \\ C_{2}  w & D′ \end{matrix}

is a pushout, so the square

\begin{matrix}B′  w n_{B} u θ \\ B_{2} ui_{2∗} \end{matrix} ; \begin{matrix}D′  w n_{D} \\ D_{2} \end{matrix}

is a pushout, and θ is a cofibration. The map f_{A∗} is a weak equivalence, since j_{1} is a cofibration and f_{A} is a weak equivalence. Similarly, f_{C∗} is a weak equivalence, since j_{1∗} is a cofibration and f_{C} is a weak equivalence. The map f_{B} = n_{B} · f_{A∗} is assumed to be a weak equivalence, so it follows that n_{B} is a weak equivalence. Then n_{D} is a weak equivalence, so f_{D} = n_{D} · f_{C∗} is a weak equivalence. ■

The dual of Lemma 9.8 is the co-glueing lemma for categories of fibrant objects:

Lemma 9.10. Suppose given a commutative cube

(9.11)

A_{1}  w_{4 4 4 4 6} u f_{A}  B_{1}_{4 4 4 4 6} p_{1} uf_{B}

C_{1}  w u f_{C}  D_{1} uf_{D}

A_{2}  w_{4 4 4 4 6}  B_{2}_{4 4 4 4 6} p_{2}

C_{2} wD_{2}

in a category of fibrant objects E. Suppose further that the top and bottom squares are pullbacks, that the maps p_{1} and p_{2} are fibrations, and that the maps f_{B}, f_{C} and f_{D} are weak equivalences. Then the map f_{A} is a weak equivalence.

Either of the two methods of proof of Corollary 9.6 given above (see also Remark 9.7), together with Lemma 9.10, now yield the following expanded consequence for the full category of simplicial sets:

Corollary 9.12. Suppose given a commutative cube

X_{1}  w_{4 4 4 4 6} u f_{X}  Y_{1}_{4 4 4 4 6} p_{1} uf_{Y}

W_{1}  w u f_{W}  Z_{1} uf_{Z}

X_{2}  w_{4 4 4 4 6}  Y_{2}_{4 4 4 4 6} p_{2}

W_{2} wZ_{2}

of morphisms in the category S of simplicial sets. Suppose further that the top and bottom squares are pullbacks, that the maps p_{1} and p_{2} are fibrations, and that the maps f_{Y}, f_{W} and f_{Z} are weak equivalences. Then the map f_{X} is a weak equivalence.

Corollary 9.12 is the basis for the theory of homotopy cartesian diagrams in the category of simplicial sets. We say that a commutative square of simplicial set maps

(9.13)  \begin{matrix}X  w u \\ Yuf \end{matrix} ; {W  w \atop Z}

is homotopy cartesian if for any factorization

(9.14)  {Y  w f_{h hh j} i \atop Z} ; Ỹ^{’ ’’)}p

of f into a trivial cofibration i followed by a fibration p the induced map

X ⟶^{i_{∗}} W ×_{Z} Ỹ

is a weak equivalence.

In fact (and this is the central point), for the diagram (9.13) to be homotopy cartesian, it suffices to find only one such factorization f = p ∘ i such that the map i_{∗} is a weak equivalence. This is a consequence of the following:

Lemma 9.15. Suppose given a commutative diagram

\begin{matrix}X  w u \\ Yuf \end{matrix} ; {W  w \atop Z}

of morphisms in the category of simplicial sets, and factorizations

{Y w f_{h hh j} i_{j} \atop Z} ; Y_{j}^{’ ’’)}p_{j}

of f as trivial cofibration i_{j} followed by a fibration p_{j} for j = 1, 2. Then the induced map i_{1∗} : X → W ×_{Z} Y_{1} is a weak equivalence if and only if the map i_{2∗} : X → W ×_{Z} Y_{2} is a weak equivalence.

Proof: There is a lifting θ in the diagram

Y  w i_{2} u i_{1}  Y_{2} up_{2}

Y_{1}  w p_{1}^{h h hh j} θ  Z,

by the closed model axioms. Form the commutative cube

W ×_{Z} Y_{1} w_{ȷ ȷ ȷ ȷȷ fi} u θ_{∗} Y_{1}_{4 4 4 4 6} p_{1} uθ

W  w u 1_{W}  Zu1_{Z}

W ×_{Z} Y_{2}  w_{ȷ ȷ ȷ ȷȷ fi}  Y_{2}_{4 4 4 4 6} p_{2}

W wZ.

Then the map θ_{∗} is a weak equivalence by Corollary 9.12. There is a commutative diagram

W ×_{Z} Y_{1} uθ_{∗}

X^{4 4 44 6} i_{1∗}_{````}ˆ i_{2∗}
W ×_{Z} Y_{2},

and the desired result follows. ■

Remark 9.16. The argument of Lemma 9.15 implies that the definition of homotopy cartesian diagrams can be relaxed further: the diagram (9.13) is homotopy cartesian if and only if there is a factorization (9.14) such that p is a fibration and i is a weak equivalence, and such that the induced map i_{∗} : X → W ×_{Z} Ỹ is a weak equivalence.

The way that the definition of homotopy cartesian diagrams has been phrased so far says that the diagram

\begin{matrix}X  w u \\ Yuf \end{matrix} ; {W  w g \atop Z}

is homotopy cartesian if a map induced by a factorization of the map f into a fibration following a trivial cofibration is a weak equivalence. In fact, it doesn’t matter if we factor f or g:

Lemma 9.17. Suppose given a commutative diagram

\begin{matrix}X  w u \\ Yuf \end{matrix} ; {W  w g \atop Z}

in the category of simplicial sets. Suppose also that we are given factorizations

{Y  w f_{4 4 44 6} i \atop Z} ; Ỹ^{h h hh j} p  {W  w g_{4 4 44 6} j \atop Z} ; W̃^{h h hh j} q

of f and g respectively such that i and j are trivial cofibrations and p and q are fibrations. Then the induced map i_{∗} : X → W ×_{Z} Ỹ is a weak equivalence if and only if the map j_{∗} : X → W̃ ×_{Z} Y is a weak equivalence.

Proof: There is a commutative diagram

\begin{matrix}X  w i_{∗} u j_{∗} \\ W ×_{Z} Ỹ uj × 1 \end{matrix} ; \begin{matrix}W̃ ×_{Z} Y  w 1 × i \\ W̃ ×_{Z} Ỹ . \end{matrix}

The map p is a fibration, so the map j × 1 is a weak equivalence, and q is a fibration, so 1 × i is a weak equivalence, all by Corollary 9.12. ■

The coglueing lemma also has the following general consequence for homotopy cartesian diagrams:

Corollary 9.18. Suppose given a commutative cube

X_{1}  w_{4 4 4 4 6} u f_{X}  Y_{1}_{4 4 4 4 6} uf_{Y}

W_{1}  w u f_{W}  Z_{1} uf_{Z}

X_{2}  w_{4 4 4 4 6}  Y_{2}_{4 4 4 4 6}

W_{2} wZ_{2}

of morphisms in the category S of simplicial sets. Suppose further that the top and bottom squares are homotopy cartesian diagrams, and that the maps f_{Y}, f_{W} and f_{Z} are weak equivalences. Then the map f_{X} is a weak equivalence.

A homotopy fibre sequence of simplicial sets is a homotopy cartesian diagram in S

\begin{matrix}X  w j u \\ Yuf \end{matrix} ; \begin{matrix}∗  w x \\ Z. \end{matrix}

In effect, one requires that the composite f ∘ j factor through the base point x of Z, and that if f = p ∘ i is a factorization of f into a trivial cofibration followed by a fibration, then the canonical map X → F is a weak equivalence, where F is the fibre of p over x. More colloquially (see also Remark 9.16), this means that X has the homotopy type of the fibre F of any replacement of the map f by a fibration up to weak equivalence. It is common practice to abuse notation and say that

X ⟶^{j} Y ⟶^{f} Z

is a homotopy fibre sequence, and mean that these maps are a piece of a homotopy cartesian diagram as above. Every fibration sequence

F → E → B

is plainly a homotopy fibre sequence.

Example 9.19. Suppose that

X ⟶^{j} Y ⟶^{f} Z

is a homotopy fibre sequence, relative to a base point x of Z, and that there is a vertex y ∈ Y such that f(y) = x. Suppose that the canonical map Y → ∗ is a weak equivalence. Then X is weakly equivalent to the loop space ΩZ̃ for some (and hence any) fibrant model Z̃ for Z. To see this, choose a trivial cofibration j : Z → Z̃, where Z̃ is a Kan complex, and use the factorization axioms to form the commutative square

\begin{matrix}Y  w j u f \\ Ỹ & u & p \end{matrix} ; Z  w j Z̃,

where both maps labelled j are trivial cofibrations and p is a fibration. Let F denote the fibre of the fibration p over the image of the base point x in Z̃. Then Corollary 9.18 implies that the induced map X → F is a weak equivalence. Now consider the diagram

\begin{matrix}Ỹ ×Z̃ P Z̃ w pr_{R} u pr_{L} \\ P Z̃ uπ \end{matrix} ; \begin{aligned}Ỹ & w & p \\ & & Z̃, \end{aligned}

where P Z̃ is the standard path space for the Kan complex Z̃ and the base point x, and π is the canonical fibration. Then the map y : ∗ → Ỹ is a weak equivalence, so that the inclusion ΩZ̃ → Ỹ ×Z̃ P Z̃ of the fibre of the fibration pr_{L} is a weak equivalence, by properness, as is the inclusion F → Ỹ ×Z̃ P Z̃ of the fibre of pr_{R}. In summary, we have constructed weak equivalences

X ⟶^{≃} F ⟶^{≃} Ỹ ×Z̃ P Z̃ ⟵^{≃} ΩZ̃.

This collection of ideas indicates that it makes sense to define the loop space of a connected simplicial set X to be the loops ΩX̃ of a fibrant model X̃ for X — the loop space of X is therefore an example of a total right derived functor, in the sense of Section II.7.

Here is a clutch of results that illustrates the formal similarities between homotopy cartesian diagrams and pullbacks:

Lemma 9.20.

(1) Suppose that

\begin{matrix}X  w α u \\ Yu \end{matrix} ; {Z  w β \atop W}

is a commutative diagram in the simplicial set category such that the maps α and β are weak equivalences. Then the diagram is homotopy cartesian.

(2) Suppose given a commutative diagram

X_{1}  w u  X_{2}  w u  X_{3} u

Y_{1}  w I  Y_{2}  w II Y_{3}

in S. Then

(a) if the diagrams I and II are homotopy cartesian then so is the composite diagram I + II

\begin{matrix}X_{1}  w u \\ X_{3} u \end{matrix} ; \begin{matrix}Y_{1}  w \\ Y_{3}, \end{matrix}

(b) if the diagrams I + II and II are homotopy cartesian, then I is homotopy cartesian.

The proof of this lemma is left to the reader as an exercise.

The stream of results and definitions beginning with Corollary 9.12 for simplicial sets holds much more generally for arbitrary proper simplicial model categories, in view of the following result (and its dual):

Lemma 9.21. Let C be a proper simplicial model category. Consider a diagram

\begin{matrix}D_{1} u & C_{1} u^{j_{1}}  w u & X_{1} u \\ D_{2} & C_{2}  w u_{j_{2}} & X_{2} \end{matrix}

where j_{1} and j_{2} are cofibrations and the three vertical maps are weak equivalences. Then the map D_{1} ∪_{C_{1}} X_{1} → D_{2} ∪_{C_{2}} X_{2} is a weak equivalence.

Proof: We use the proper hypothesis to reduce to the case C_{1} = C_{2} and X_{1} = X_{2} and then argue appropriately.

First notice that there is a pushout diagram

\begin{matrix}X_{1}  w u \\ X_{2} u \end{matrix} ; \begin{matrix}D_{1} ∪_{C_{1}} X_{1}  w \\ D_{1} ∪_{C_{1}} X_{2}. \end{matrix}

Since X_{1} → D_{1} ∪_{C_{1}} X_{1} is a cofibration (because j_{1} is) and X_{1} → X_{2} is a weak equivalence, properness implies D_{1} ∪_{C_{1}} X_{1} → D_{1} ∪_{C_{1}} X_{2} is a weak equivalence. Thus we may assume X_{1} = X_{2} = X.

Next notice that there is a pushout diagram

\begin{matrix}C_{1}  w u_{j_{1}} \\ C_{2} u \end{matrix} ; \begin{matrix}D_{1}  w \\ D_{1} ∪_{C_{1}} C_{2}. \end{matrix}

Since j_{1} is a cofibration and C_{1} → C_{2} is a weak equivalence, properness implies D_{1} → D_{1} ∪_{C_{1}} C_{2} is a weak equivalence. Now

D_{1} ∪_{C_{1}} X ≅ (D_{1} ∪_{C_{1}} C_{2}) ∪_{C_{2}} X

and we are reduced to considering the diagram

\begin{matrix}D_{1} ∪_{C_{1}} C_{2} u & C_{2} u  w u_{=} & Xu_{=} \\ D_{1} & C_{2}  w u & X. \end{matrix}

Note that C_{2} → D_{1} ∪_{C_{1}} C_{2} is a cofibration and D_{1} ∪_{C_{1}} C_{2} → D_{2} is a weak equivalence. Thus we may assume C_{1} = C_{2} = C.

Now factor D_{1} → D_{2} as

D_{1} ⟶^{j} Z ⟶^{q} D_{2}

where j is a trivial cofibration and q is a fibration. Then q is a trivial fibration, and the composite C → Z is a cofibration. There is a pushout diagram

\begin{matrix}D_{1}  w u_{j} \\ D_{1} ∪_{C} X u \end{matrix} ; \begin{matrix}Z  w \\ Z ∪_{C} X. \end{matrix}

Since j is a trivial cofibration, so is D_{1} ∪_{C} X → Z ∪_{C} X. We will argue that Z ∪_{C} X → D_{2} ∪_{C} X is a homotopy equivalence and we’ll be done. To see this last claim, note that there is a left s : D_{2} → Z solving the lifting problem

\begin{matrix}C w u \\ Zu_{q} \end{matrix} ; \begin{matrix}D_{2}^{O OO P}w_{=} \\ D_{2}. \end{matrix}

Then qs = 1_{D_{2}} and sq and 1_{Z} are both solutions to the lifting problem

\begin{matrix}C w u \\ Zu_{q} \end{matrix} ; \begin{matrix}Z^{O OO P}w_{q} \\ D_{2} \end{matrix}

so sq and 1_{Z} are homotopic under C. Thus Z ∪_{C} X → D_{2} ∪_{C} X has a section induced from s and is a homotopy equivalence. Now invoke Lemma 4.1. ■

The dual of Lemma 9.21 is an obvious generalization of the statement of Corollary 9.12. The proofs of the statements in 9.15 – 9.18 and Lemma 9.20 are formal consequences of properness for the simplicial set category and Corollary 9.12, and can therefore be promoted to arbitrary proper simplicial model categories C. Thus, we are entitled to say that a diagram

\begin{matrix}X  w u \\ Yuf \end{matrix} ; {W  w \atop Z}

in C is homotopy cartesian if for any factorization

{Y w f_{h hh j} i \atop Z} ; Ỹ^{’ ’’)}p

of f into a weak equivalence i followed by a fibration p the induced map

X ⟶^{i_{∗}} W ×_{Z} Ỹ

is a weak equivalence. Then the standard calculus of homotopy cartesian diagrams in simplicial sets obtains in this extended context.

We close with a further application of categories of cofibrant objects structures. Let C be a fixed choice of simplicial model category having an adequate supply of colimits. Suppose that β is a limit ordinal, and say that a cofibrant β-sequence in C is a functor X : Seq(β) → C, such that all objects X_{i} are cofibrant, each map X_{i} → X_{i+1} is a cofibration, and X_{t} = \underrightarrow{lim}_{i<t} X_{i} for all limit ordinals t < β. The cofibrant β-sequences, with ordinary natural transformations between them, form a category which will be denoted by C_{β}. Say that a map f : X → Y in C_{β} is a weak equivalence if all its components f : X_{i} → Y_{i} are weak equivalences of C, and say that g : A → B is a cofibration of C_{β} if the maps g : A_{i} → B_{i} are cofibrations of C, as are all induced maps B_{i} ∪_{A_{i}} A_{i+1} → B_{i+1}.

Lemma 9.22. Let C be a simplicial model category having all filtered colimits. With these definitions, the category C_{β} of cofibrant β-sequences in C satisfies the axioms for a category of cofibrant objects.

Proof: Suppose that A → B → C are cofibrations of C_{β}. To show that the composite A → C is a cofibration, observe that the canonical map C_{i}∪_{A_{i}} A_{i+1} → C_{i+1} has a factorization

C_{i} ∪_{A_{i}} A_{i+1}  w_{A A AA C}  C_{i+1}

C_{i} ∪_{B_{i}} B_{i+1}^{h h hh j}

and there is a pushout diagram

\begin{matrix}B_{i} ∪_{A_{i}} A_{i+1}  w u \\ B_{i+1} u \end{matrix} ; \begin{matrix}C_{i} ∪_{A_{i}} A_{i+1}  w \\ C_{i} ∪_{B_{i}} B_{i+1}. \end{matrix}

Suppose that

\begin{matrix}A  w u i \\ Bui_{∗} \end{matrix} ; {C  w \atop D}

is a pushout diagram of Seq(β)-diagrams in C, where A, B and C are cofibrant β-sequences and the map i is a cofibration of same. We show that D is a cofibrant β-sequence and that i_{∗} is a cofibration by observing that there are pushouts

\begin{matrix}C_{i} ∪_{A_{i}} A_{i+1}  w u \\ D_{i} ∪_{B_{i}} B_{i+1} u \end{matrix} ; \begin{matrix}C_{i+1}  w \\ D_{i+1}, \end{matrix}

and that the maps D_{i} → D_{i} ∪_{B_{i}} B_{i+1} are cofibrations since B is a cofibrant β-sequence.

Suppose that A is a cofibrant β-sequence, and let K be a simplicial set. Then the functor A ⊗ K : Seq(β) → C is defined by (A ⊗ K)_{i} = A_{i} ⊗ K. The functor X ↦ X ⊗ K preserves cofibrations and filtered colimits of C, so that A ⊗ K is a cofibrant β-sequence. Furthermore, if K → L is a cofibration of S then the induced map A ⊗ K → A ⊗ L is a cofibration of C_{β}: the proof is an instance of SM7. It follows that the obvious diagram

\begin{matrix}A ⊗ ∂Δ^{1} w u \\ A ⊗ Δ^{0} \end{matrix} ; A ⊗ Δ^{1}^{[ [ [ []}

is a candidate for the cylinder object required by the category of cofibrant objects structure for the category C_{β}. ■

Lemma 9.23. Suppose that C is a simplicial model category having all filtered colimits. Suppose that f : A → B is a cofibration and a weak equivalence of C_{β}. Then the induced map

f_{∗} : \underrightarrow{lim}_{i<β} A_{i} → \underrightarrow{lim}_{i<β} B_{i}

is a trivial cofibration of C.

Proof: Suppose given a diagram

\underrightarrow{lim}_{i<β} A_{i}  w u f_{∗}  Xup

\underrightarrow{lim}_{i<β} B_{i} wY

where p is a trivial fibration of C. We construct a compatible family of lifts

(9.24)

A_{i}  w u f_{i}  Xup

B_{i}  w^{h h hh j} θ_{i}  Y

as follows:

1) Let θ_{s} be the map induced by all θ_{i} for i < s at limit ordinals s < β.

2) Given a lifting θ_{i} as in diagram (9.24), form the induced diagram

B_{i} ∪_{A_{i}} A_{i+1}  w θ_{∗} u f_{∗}  Xup

B_{i+1}  w^{O O O OO P} θ_{i+1}  Y.

The map f_{∗} is a trivial cofibration of C, since f is a cofibration and a weak equivalence of C_{β}, so the indicated lift θ_{i+1} exists. ■

Corollary 9.25. Suppose that C is a simplicial model category having all filtered colimits, and that f : X → Y is a weak equivalence of cofibrant βsequences in C. Then the induced map

f_{∗} : \underrightarrow{lim}_{i<β} X_{i} → \underrightarrow{lim}_{i<β} Y_{i}

is a weak equivalence of C.

Proof: We have it from Lemma 9.22 that C_{β} is a category of cofibrant objects, and Lemma 9.4 says that f : X → Y has a factorization f = q · j, where j is a cofibration and q is left inverse to a trivial cofibration. Then j is a trivial cofibration since f is a weak equivalence, and so Lemma 9.23 implies that both j and p induce weak equivalences after taking filtered colimits. ■

The dual assertion for Corollary 9.25 is entertaining. Suppose, again, that β is a limit ordinal and that C is a simplicial model category having enough filtered inverse limits. Define a fibrant β-tower in C to be (contravariant) functor X :

Seq(β)^{op} → C such that each X_{i} is a fibrant object of C, each map X_{i+1} → X_{i} is a fibration of C, and X_{t} = \underleftarrow{lim}_{i<t} X_{i} for all limit ordinals t < β. Then the dual of Lemma 9.22 asserts that, for pointwise weak equivalences and a suitable definition of fibration, the category of fibrant β-towers in C has a category of fibrant objects structure. The dual of Lemma 9.23 asserts that the inverse limit functor takes trivial fibrations of fibrant β-towers to trivial fibrations of C, and then we have

Lemma 9.26. Suppose that C is a simplicial model category having all filtered inverse limits, and that f : X → Y is a weak equivalence of fibrant β-towers in C. Then the induced map

f_{∗} : \underleftarrow{lim}_{i<β} X_{i} → \underleftarrow{lim}_{i<β} Y_{i}

is a weak equivalence of C.

For fibrant β-towers X : Seq(β)^{op} → S taking values in simplicial sets, one can take a different point of view, in a different language. In that case, fibrant βtowers are globally fibrant Seq(β)^{op}-diagrams, and inverse limits and homotopy inverse limits coincide up to weak equivalence for globally fibrant diagrams, for all β. Homotopy inverse limits preserve weak equivalences, so inverse limits preserve weak equivalences of fibrant β-towers. Homotopy inverse limits and homotopy theories for categories of diagrams will be discussed in Chapters 6 and 7.