Homotopy Ends and Thomason Model Categories

Charles Weibel^{1}

Institute for Advanced Study Princeton, NJ 08540 weibel@math.ias.edu

In the last year of his life, Robert W. Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. The axioms for such a Thomason model category were published later in [WT]. In this paper we explain and prove Thomason’s results, based on his private notebooks [T78]–[T86].

Each of Thomason’s 180-page notebooks are spiral-bound with an index on the inside cover. We have chosen to follow their internal citation method: our citation [Tx, y] refers to page y of Thomason’s notebook x.

We first present Thomason’s ideas about homotopy ends and its generalizations (sections 1 and 2). These are first formulated for complete simplicial closed model categories in the sense of Quillen [QH], because of their usefulness in this setting, as demonstrated by Dwyer and Kan [DK].

Thomason’s axioms and examples come next, followed by a proof that the homotopy category ho𝒞 of a Thomason model category 𝒞 exists (sections 3-5). In the last two sections (6–7) we prove the main theorem: the functor category 𝒞^{K} inherits a Thomason model structure, at least when 𝒞 is right enriched over simplicial sets, and fibrations are preserved by products and inverse limits. The result was previously known when 𝒞 was the category of simplicial sets (Quillen, Joyal and Jardine) and also when K is any partially ordered set (Grothendieck).

1Partially supported by NSF grants

§1 Homotopy Ends

Recall from [Mac] that the end of a functor F: K^{op} × K → 𝒞 is an object e = ∫_{K} F (K, K) together with maps e → F (K, K) which are compatible in the sense that for each K_{1} → K_{2} the evident square commutes:

e  ⟶ F (K_{1}, K_{1})
↓ ↓
F (K_{2}, K_{2}) ⟶ F (K_{1}, K_{2}) .

Moreover, e is universal with respect to this property. If K is a small category and 𝒞 is complete, then ends always exist, and commute with limits [Mac, IX.5].

The “Tot” construction for cosimplicial objects is an example of an end in 𝒞. Its construction requires that for every finite simplicial set K there is a mapping object functor Map(K, −): 𝒞 → 𝒞 which is natural in K. Given a cosimplicial object X· in 𝒞, we have a functor Δ^{op} × Δ → 𝒞 defined by (p, q) ↦ Map(Δ[p], X^{q}). Definition 1.1. If X· is a cosimplicial object in a complete category 𝒞, and 𝒞 is equipped with a mapping object functor, we define the total object of X· to be the end

Tot(X·) = ∫_{Δ} Map(Δ[p], X^{p}).

In case 𝒞 is a simplicial closed model category in the sense of Quillen [QH], it is well known that this definition of Tot agrees with the Bousfield-Kan definition in [BK].

In a model category, it is not generally true that an end will preserve weak equivalences. To remedy this, we introduce the notion of a “derived end.”

Given F: K^{op} × K → 𝒞, with 𝒞 complete and K small, the cosimplicial replacement of F is the cosimplicial object ∏^{∗} F of 𝒞 defined by

p ⟼ ∏^{p} F = ∏_{K_{0}→···→K_{p}} F (K_{0}, K_{p}),

where the indexing set runs over all p-tuples K_{0} → K_{1} → ⋯ → K_{p} of composable maps in K, i.e., all functors p → K. This is equivalent to the Bousfield-Kan simplicial replacement of F [BK, XI.5], except for a change in orientation due to a different orientation of the nerve of K on p. 291 [BK]. Lemma 1.2. The end ∫_{K} F (K, K) is the equalizer of ∏^{0} F ⇉ ∏^{1} F.

Proof: ([T78, 115]) Let π^{0}F denote the equalizer. The projections π^{0}F → ∏^{0} F → F (K, K) have the property that for every map K_{0} → K_{1} in K the diagram

π^{0}F  ⟶ F (K_{0}, K_{0})
↓ ↓
F (K_{1}, K_{1}) ⟶ F (K_{0}, K_{1})

commutes. This gives a map from π^{0}F to the end ∫_{K} F (K, K). Since the end also equalizes ∏^{0} F ⇉ ∏^{1} F, this must be an isomorphism. □ Definition 1.3. If 𝒞 is a complete category equipped with a mapping object functor, the homotopy end ho∫ F of F: K^{op} × K → 𝒞 is the total object of ∏^{∗} F:

ho∫ F = Tot (∏^{∗} F ) = ∫_{Δ} Map (Δ[p], ∏^{p} F ) .

Compatibility 1.4. Suppose that F factors as the projection K^{op} ×K → K^{op} followed by a functor F̃: K^{op} → 𝒞. Then the homotopy end of F is just the Bousfield-Kan homotopy limit of F̃ as described in 4.5 and 5.2 of [BK, XI]:

ho∫ F = holim F̃ .

Since we have used K^{op}, there is no orientation problem. Functoriality 1.5. A natural transformation η: F ⇒ G of functors K^{op} × K → 𝒞 induces a map ∏^{∗} F → ∏^{∗} G, and so a natural map ho∫ F → ho∫ G. The homotopy end functor Cat(K^{op} × K, 𝒞) → 𝒞 commutes with all limits (because Tot and F ↦ ∏^{∗} F preserve limits). Given a functor Φ: L → K and F: K^{op} × K → 𝒞, the diagram

Cat(K^{op} × K, 𝒞) ⟶^{Φ} Cat(L^{op} × L, 𝒞)
ho∫^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{ho∫}^{.............................................................................................................................................}

𝒞 commutes up to a natural 2-cell φ which is induced by the “projection” ∏_{K}^{∗} F → ∏_{L}^{∗} F: the component indexed by L: p → L comes from the component indexed by ΦL.

We now introduce homotopy theory into the discussion. Suppose to fix ideas that 𝒞 is a simplicial closed model category in the sense of Quillen [QH, II.2.2], and that 𝒞 is complete. We first observe that 𝒞 is cotensored over all simplicial sets, not just finite ones, as in Quillen’s axiom (SM0).

Lemma 1.6 [T78, 58]. There is a mapping object functor Map(K, −): 𝒞 → 𝒞 for every simplicial set K, which is natural in K, and an isomorphism

Hom_{𝒞} (X, Map(K, Y)) ≅ Hom_{Δ^{op}Sets}(K, hom(X, Y)),

which is natural in K, X and Y.

Quillen’s axiom (SM7) holds in this setting. If i: K → L is a cofibration in Δ^{op}Sets (i.e., an injection) and f: X → Y is a fibration in 𝒞, then

Map(L, X) → Map(K, X) ×_{Map(K,Y)} Map(L, Y)

is a fibration in 𝒞, and is a weak equivalence if either i or f is. Moreover, the functor Map( , ): Δ^{op}Sets × 𝒞 → 𝒞 preserves limits in 𝒞 and converts colimits of simplicial sets to limits in 𝒞.

Proof: Write K = ⊔K_{α}, where K_{α} runs over all finite subspaces of K, and set Map(K, X) = lim Map(K_{α}, X). Checking the properties is routine. □

Let f: X· → Y · be a map of cosimplicial objects in 𝒞. We say that f is a fibration if each f^{p} is a fibration in 𝒞, and each X^{p+1} → Y^{p+1} ×_{M^{p}Y} M^{p}X is a fibration in 𝒞. Here M^{p}X ⊂ ∏_{n+1} X^{n} is the matching object of [BK, X.4.6]. Also recall from [BK, X.3.2] that Tot(X·) = hom(Δ·, X·).

Lemma 1.7 [T78, 92]. If f: X· → Y · is a fibration of cosimplicial objects in 𝒞, then Tot(X·) → Tot(Y ·) is a fibration in 𝒞. If in addition each f^{p} is a weak equivalence in 𝒞 then it is also a weak equivalence.

Proof: By [BK, X.5], axiom (SM7) for cosimplicial objects holds. The lemma is the case A = ∅ and B = Δ·: Tot(Y ·) = hom(Δ·, Y ·) ×_{∗} ∗, and hom(∅, Y ·) = ∗. □

Lemma 1.8 [T78, 110]. If η: F ⇒ G is such that F (K, K′) ↠ G(K, K′) is a fibration in 𝒞 for all K, K′ in K, then ho∫ F → ho∫ G is a fibration in 𝒞. If in addition each F (K, K′) ↠^{∼}G(K, K′) is a weak equivalence, so is ho∫ F → ho∫ G.

In particular, if each F (K, K′) is fibrant in 𝒞 then ho∫ F is fibrant in 𝒞.

Proof: If each η_{KK′} is a fibration (resp. trivial fibration) then so is each component of ∏^{∗} F → ∏^{∗} G. Indeed ∏^{∗} F → ∏^{∗} G is a fibration in Δ𝒞 [BK, XI.5.3] [T78, 106]. Hence this follows from lemma 1.7. □

Corollary 1.8.1. The homotopy end preserves (pointwise) homotopy fiber sequences.

Lemma 1.9 [T82, 83] [T83, 178]. Suppose that K has an initial object K_{0}. Then for any functor F: K → 𝒞, the natural map is a weak equivalence:

holim_{K} F ⟶^{∼} Map(Δ[0], F (K_{0})).

Proof: This is a special case of [BK, XI.4.1 (iii)]. □

We conclude this section with a generalization of the above results to a complete “right model category enriched over finite simplicial sets;” see definitions 1.11 and 1.12 below. The reader may easily verify that a complete simplicial closed model category 𝒞 has such a structure, using lemma 1.6 and the adjunction between Map(−, X) and \underline{Hom}_{𝒞} (−, X). We will use this generality in §7 below.

The following result is straightforward, once one checks (see [T78, 68–92]) that (SM7) holds, by the transfinite proof in [BK, X.5].

Proposition 1.10. If 𝒞 is a right model category enriched over finite simplicial sets, and 𝒞 is complete, then lemmas 1.6, 1.7, 1.8 and 1.9 hold in 𝒞.

Definition 1.11 [T85, 145]. A right model category is a category 𝒞 with a terminal object ∗ and two subcategories we(𝒞) and fib(𝒞) so that: every isomorphism of 𝒞 is in both we(𝒞) and fib(𝒞); we(𝒞) is saturated and closed under retractions; and (RM1) The pullback B ×_{A} C ↠ B of a fibration C ↠ A along any map B → A

exists and is a fibration. If either B → A or C → A is in we(𝒞), so is its pullback.

(RM5) Every map A → B factors as A ⟶^{∼} A′ ↠ B, the composite of a weak equivalence and a fibration. Note that the subcategory 𝒞_{fib} of fibrant objects A (those for which A → ∗

is a fibration) is a category of fibrant objects in the sense of Brown [Br], with the added property that weak equivalences are closed under retractions.

Let 𝒮_{f} denote the category of finite simplicial sets. Definition 1.12 [T85, 146]. We say that a complete right model category 𝒞 is right enriched over 𝒮_{f} if there is a “mapping object” functor

Map( , ): 𝒮_{f}^{op} × fib(𝒞) → fib(𝒞)

satisfying the following axioms. (RE1) For each K in 𝒮_{f}: Map(K, ∗) ≅ ∗; the functor Map(K, −) preserves both

fibrations and base change along fibrations in 𝒞. (RE2) For each C in 𝒞: Map(∅, C) ≅ ∗; the functor Map(−, C) sends cofibrations

in 𝒮_{f} to fibrations in 𝒞, and pushouts along cofibrations in 𝒮_{f} to pullbacks along fibrations in 𝒞.

(RE3) Given a cofibration K ↣^{i} L in 𝒮_{f} and a fibration B ↠^{p} C in 𝒞,

Map(L, B) → Map(K, B) ×_{Map(K,C)} Map(L, C)

is a fibration. It is a weak equivalence of either i or p is. (RE4) There is a weak equivalence ω: C ⟶^{∼} Map(Δ[0], C), natural in C, so that

∂_{0}ω = ∂_{1}ω: C → Map(Δ[1], C). (RE5) For each family {C_{i}}_{i∈I} of fibrant objects in 𝒞, and each K in 𝒮_{f}, the product

∏ C_{i} is fibrant and the canonical morphism

Map (K, ∏ C_{i}) → ∏ Map(K, C_{i})

is an isomorphism in 𝒞.

Remark. Axioms (RE4) and (RE5) are not needed to prove proposition 1.10. They are called (RWN) and (ℵEP), respectively, in [T85, 148–9].

Here is an elementary result we shall need for theorem 7.2. Lemma 1.13 [T85, 168]. Suppose given a functor F: K → fib(𝒞). Then there is a functor (holim F): K → 𝒞 defined by (holim F)(K) = holim_{(K/K)} F, and there is a natural (pointwise) weak equivalence

F ⟶^{∼} holim F.

Proof: For each K in K, K/K has an initial object (K=K) and a functor K/K → K sending K → K′ to K′. By assumption, F K → F K′ is a fibration for each K → K′ in K/K. Hence (RE4) and lemma 1.9 yield natural weak equivalences for all K:

F K ⟶_{RE4}^{∼} Map(Δ[0], F K) ⟶_{1.9}^{∼} holim_{(K/K)} F = (holim F)(K)F. □

Remark 1.14. It is easy to see that

holim_{K} F = \underleftarrow{lim}_{K} holim_{K/K} F = \underleftarrow{lim}_{K}(holim F)(K).

Definition 1.15. We say that 𝒞 is a left model category if 𝒞^{op} is a right model category. That is, 𝒞 has an initial object ∅ and two subcategories we(𝒞) and cof(𝒞), both containing all isomorphisms, so that we(𝒞) is saturated and closed under retractions, and (LM1) The pushout P of a cofibration c: A ↣ C with any map f: A → B exists,

and B → P is a cofibration. Moreover, if either f or c is a weak equivalence, so is its pushout.

(LM5) Every map A → B factors A ↣ B′ ⟶^{∼} B.

We say that 𝒞 is left enriched over 𝒮_{f} if 𝒞^{op} is right enriched by a functor

⊗: 𝒮_{f} × cof(𝒞) → cof(𝒞).

That is, ⊗ should satisfy axioms (LE1)–(LE5) dual to (RE1)–(RE5).

§2 Homotopy limits of natural systems

Let K be a category. The subdivision category Sd K is defined to be the category whose objects are morphisms in K, and whose morphisms from f: A → B to f′: A′ → B′ are factorizations of f′ through f:

B ⟶ B′
f↑ ↑^{f′}
A ⟵ A′

This construction is due to Quillen, and has many names in the literature. Baues calls Sd K the category of factorizations in K in [Baues, p. 232]. Dwyer and Kan call it the Twisted arrow category aK in [DK]. (It is not the “subdivision category” of [DK], or of [Mac]).

Following Baues, a natural system F on K with values in 𝒞 is a functor F: Sd K → 𝒞. If 𝒞 is complete and K is small we can form the products in 𝒞:

∏^{p} F = ∏_{K_{0}→···→K_{p}} F (K_{0} → K_{p}) .

Since the indexing is over all functors p → K, the naturality of F implies that ∏^{∗} F is a cosimplicial object of 𝒞. Indeed, for i: p → q in Δ there is a natural map ∏^{p} F → ∏^{q} F, induced by the morphism in Sd K from K_{i0} → K_{ip} to K_{0} → K_{q}. Part of the cosimplicial identities depend upon the following observation. Suppose that K_{0} → K_{1} and K_{p} → K_{p+1} are the identity. Then for every K_{1} → K_{p}:

F (K_{0} → K_{p}) = F (K_{1} → K_{p}) = F (K_{1} → K_{p+1}) .

We remark that ∏^{∗} F is the analogue of the chain complex used in [PW, 1.2] to define topological Hochshild homology; cf. [JP, 3.1] and [BW]. The Baues-Wirsching cohomology [Baues, IV.5.1] of K with coefficients in a natural system F: Sd K → Ab is the cohomology of the cochain complex associated to ∏^{∗} F by the Dold-Kan correspondence. ([Baues] uses the Bousfield-Kan orientation.)

Definition 2.1 [T78, 114]. Let F: Sd K → 𝒞 be a natural system on a small category K, with values in a complete category 𝒞 equipped with a mapping object functor. Set

holim^{BW} (F) = Tot (∏^{∗} F ) ,

where Tot is defined in 1.1 above. The name honors Baues and Wirsching, who introduced natural systems in [BW] as a way to study homotopy categories.

Compatibility 2.2. There is a natural functor Sd K → K^{op} × K sending A → B to (A, B). If F factors as Sd K → K^{op} × K⟶^{F′}𝒞 then ∏^{∗} F is the same cosimplicial object as ∏^{∗} F′ in the previous section, so holim^{BW} (F) is the homotopy end of F′:

ho∫ F′ = holim^{BW} (F) .

As we have seen in 1.4, the homotopy limit is also a special case of this construction. If F̃: K^{op} → 𝒞 and F is the composite of F̃ with the projection Sd(K) → K^{op} then

holim (F̃) = holim^{BW} (F) .

Functoriality 2.3. A natural transformation F ⇒ G of natural systems induces a map ∏^{∗} F → ∏^{∗} G, and so a natural map holim^{BW} (F) → holim^{BW} (G). The functor holim^{BW}: Cat(Sd K, 𝒞) → 𝒞 commutes with all limits. Given a functor Φ: L → K and F: Sd K → 𝒞, the diagram

Cat(Sd K, 𝒞) ⟶^{Φ} Cat(Sd L, 𝒞)
holim^{BW}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{holim^{BW}}^{.............................................................................................................................................}
𝒞

commutes up to a natural 2-cell φ induced by the projection ∏_{K}^{∗} F → ∏_{L}^{∗} F.

Lemma 2.4 [T78, 114] [T81, 164]. Let 𝒞 be a right model category enriched over 𝒮_{f}, or a complete closed model category. If F ⇒ G is such that F (k) ↠ G(k) is a fibration (resp. trivial fibration) in 𝒞 for every k in Sd K, then holim^{BW} (F) → holim^{BW} (G) is a fibration (resp. trivial fibration) in 𝒞.

In particular, if each F (k) is fibrant then so is holim^{BW} (F). Moreover, holim^{BW} preserves homotopy fiber sequences.

In fact, the proof of lemma 1.8 (or 1.10) goes through in this setting.

Now consider the comma category k/Sd K for a fixed morphism k in K, and write F_{k} for the restriction of F: Sd K → 𝒞 to k/Sd K. Since k is an initial object of the comma category k/Sd K we have a canonical weak equivalence (as F (k) is fibrant; cf. [BK, XI.2.5]):

F (k) ⟶^{∼} holim_{k/Sd K} F_{k} .

Remark [T78, 120]. If F: Sd K → 𝒞 takes fibrant values, then holim^{BW} (F) is weak equivalent to holim_{Sd K}(F). This result is suggested by [BW, 4.4] and [DK, 3.3]. In fact, if F′(K → K′) is defined to be holim_{K′/K} of (K′ → L) ↦ G(K → L), then holim^{BW} (F′) = holim (F).

Theorem 2.5 [T78, 117]. If F: Sd K → 𝒞 is pointwise fibrant, then

holim^{BW} (F) ≅ ker {∏_{K_{0}} holim_{K/Sd K} F ⇉ ∏_{k} holim_{k/Sd K} F_{k}} .

Since holim^{BW} preserves homotopy equivalences of such fibrant valued F, this shows that holim^{BW} (F) is the right derived functor of

F ⟼ ker {∏_{K_{0}} F (K_{0} = K_{0}) ⇉ ∏_{K_{0}→K_{1}} F (K_{0} → K_{1})}

for F: Sd K → 𝒞.

Proof: The right side is

ker ∏_{K}\{Tot (\begin{matrix}∏ & ∗ \\ K/Sd K \end{matrix} F ) ⇉ ∏_{k} Tot (\begin{matrix}∏ & ∗ \\ k/Sd K \end{matrix} F_{k})

= Tot\{ker ∏_{K} (\begin{matrix}∏ & ∗ \\ K/Sd K \end{matrix} F ) ⇉ ∏_{k} (\begin{matrix}∏ & ∗ \\ k/Sd K \end{matrix} F_{k})\} .

This is Tot of a cosimplicial object in 𝒞 which in degree p is an equalizer of ∏_{K} ∏_{λ} F (λ_{p}) ⇉ ∏_{k} ∏_{μ} F (μ_{p}), where λ and μ run over diagrams in K/Sd K and k/Sd K, respectively. This equalizer is easily seen to be ∏^{p} F, so the cosimplicial object is just the simplicial replacement ∏^{∗} F. We are done, since holim^{BW} (F) is defined to be Tot (∏^{∗} F ). □

Thomason defines an even more exotic generalization of a homotopy limit, which we christen holim^{T}. To define it, recall that the objects of the comma category Δ/K are pairs (p, K: p → K), i.e., diagrams K_{0} → K_{1} → ⋯ → K_{p} in K, while a morphism (p, K) → (q, L) consists of a simplicial map σ: p → q such that K = Lσ. It will be convenient to abbreviate (p, K) as K and write #K for the first entry p of (p, K).

Given a functor F: Δ/K → 𝒞, let ∏^{∗} F be the cosimplicial replacement defined by

∏^{p} F = ∏_{#K=p} F (K) .

If σ: q → p is a morphism in Δ, we define σ_{∗}: ∏^{q} F → ∏^{p} F by requiring that the component corresponding to K: p → K is the projection onto the component F (Kσ) followed by the map F (Kσ) → F (K) associated to the morphism Kσ → K induced by σ.

q⟶^{σ}p⟶^{K}K

Definition 2.6 [T85, 36]. Let F: Δ/K → 𝒞 be a functor, where 𝒞 is a complete category equipped with a mapping object functor. Then

holim^{T} (F) = Tot (∏^{∗} F ) = ∫_{p∈Δ} Map(Δ[p], ∏_{#K=p} F (K)) .

Lemma 2.7 [T83, 158]. There is a functor (K_{1}, K_{2}) ↦ Map(Δ[#K_{1}], F K_{2}) from (Δ/K)^{op} × (Δ/K) to 𝒞, and holim^{T} (F) = holim_{Δ/K}^{T}(F) is its end:

holim^{T} (F) ≅ ∫_{K∈Δ/K} Map(Δ[#K], F K) .

Proof: It suffices to check that the two ends have the same universal mapping property with respect to objects of 𝒞. The details are routine, and duly verified in [T83, 159]. □

There are forgetful functors

(Δ/K) ⟶ Sd K ⟶ K^{op} × K ⟶ K
(K_{0} → ⋯ → K_{p}) ⟼ (K_{0} → K_{p}) ⟼ (K_{0}, K_{p}) ⟼ K_{0} .

Proposition 2.8 [T85, 152]. For each F: Δ/K → 𝒞:

(a) If F factors through F′: Sd K → 𝒞 then holim^{T} (F) = holim^{BW} (F′);

(b) If F factors through F′′: K^{op} × K → 𝒞 then holim^{T} (F) = ho∫ F′′;

(c) If F factors through F′′′: K^{op} → 𝒞 then holim^{T} (F) = holim (F′′′). Indeed, (a) is routine since ∏^{∗} F is the same cosimplicial object as ∏^{∗} F′.

Parts (b) and (c) follow from Compatibility 2.2. Remark 2.8.1. Δ/(K^{op}) is not (Δ/K)^{op}, and Sd(K^{op}) is not (Sd K)^{op}.

Here is the generalization of lemma 1.8; again the proof of 1.8 goes through.

Theorem 2.9 [T85, 155]. Let 𝒞 be a complete right model category enriched over 𝒮_{f}. Suppose in addition that towers and products preserve fibrations and trivial fibrations in 𝒞. Then for every small category K and every pointwise fibrant functor

F: Δ/K → 𝒞 ,

the object holim^{T} (F) is fibrant in 𝒞. If η: F → G is a pointwise fibration between pointwise fibration functors then holim^{T} (η): holim^{T} (F) → holim^{T} (G) is a fibration. If in addition η is a weak equivalence, so is holim^{T} (η).

Variant 2.10 [T85, 6]. Given F: (Δ/K)^{op} → 𝒞_{cof}, we have a simplicial object in 𝒞_{cof} whose pth term is ∐_{#K=p} F (K). If 𝒞 is cocomplete and left enriched (1.15), one can

define hocolim^{T} (F) as the coend ∫^{Δ} Δ[#K]⊗∐_{∗} F. The dual assertions for 2.7–2.8 are explored in [T85]. The dual of theorem 2.9 is given in [T81, 10, 159] and on pp. 46–48 of [T86].

§3 Thomason’s Axioms for homotopy theory

Let 𝒞 be a category. If W is a distinguished family of morphisms, closed under composition and containing all identity maps, it is convenient to think of W as the morphisms of subcategory 𝒲 having the same objects as 𝒞. We say that 𝒲 is saturated if given any pair (f, g) of composable maps in 𝒞, whenever two of f, g and fg are in 𝒲 so is the third.

Let 𝒞 be a category equipped with three distinguished subcategories cof(𝒞), fib(𝒞) and we(𝒞) having the same objects as 𝒞. Morphisms in these subcategories will be called cofibrations (written ↣), fibrations (written ↠) and weak equivalences (written ⟶^{∼}), respectively.

A trivial cofibration, or equicofibration, (written ⟶^{>}^{∼} ) is a map which is both a cofibration and a weak equivalence; a trivial fibration, or equifibration, (written ↠^{∼}) is a map which is both a fibration and a weak equivalence.

We suppose that 𝒞 has an initial object ∅ and a terminal object ∗. Consider the following self-dual axioms [T78, 156]. (TM0) Every isomorphism of 𝒞 is in cof(𝒞), fib(𝒞) and we(𝒞). (TM1) The pushout P of a cofibration c: A ↣ C with any map f: A → B exists,

and B → P is a cofibration. Moreover, if either f or c is a weak equivalence, so is its pushout. The dual assertion (RM1) for the pullback B ×_{A} C of a fibration C ↠ A and any map B → A must also hold.

(TM2) (Quillen’s Axiom CM2) The subcategory we(𝒞) is saturated. (TM3) (Quillen’s Axiom CM3w)^{†} Any retract of a weak equivalence is a weak

equivalence. (TM4) (Quillen’s Lifting Axiom CM4) Suppose given a commutative square

\begin{matrix}A & ⟶ & X \\ ^{∨} ↓^{i} & & |↓ ↓^{p} \\ B & ⟶ & Y \end{matrix}

†Axiom (TM3) is not needed for cofibrations or fibrations, only for weak equivalences.

See [T79, 26] and [T83, 152].

in which i is a cofibration and p is a fibration. Then a map B → X exists, factoring both A → X and B → Y, provided that either i or p is a weak equivalence. Any such map B → X will be called a fill-in, or factorization.

(CM5) Any map A⟶^{f}B factors as A ↣ B′ ⟶^{∼} B, and also as A ⟶^{∼} A′ ↠ B. (TM5) The following factorizations of A ⟶^{∼} B can be made functorial in f:

A ↣ B′ ⟶^{∼} B (c)

A ⟶^{∼} A′ ↠ B (f)

The functoriality in (TM5) may be expressed as follows. Let 𝒞^{1} denote the category of arrows in 𝒞, i.e., functors 1⟶^{f}𝒞, and 𝒞^{2} the category of composable pairs of arrows in 𝒞, i.e., functors 2 → 𝒞. Then there are functors

T, M: 𝒞^{1} → 𝒞^{2}

sending the arrow f: A → B in 𝒞^{1} to the composable pairs (with composition f)

A ↣ T (f) ⟶^{∼} B  and  A ⟶^{∼} M(f) ↠ B

respectively. See [T79, 72-73] and [T78, 161].

We call an object C cofibrant if the canonical map ∅ → C is in cof(𝒞); we call C fibrant if the canonical map C → ∗ is in fib(𝒞).

Definition 3.1. A basic model category is a category 𝒞, equipped with distinguished subcategories cof(𝒞), fib(𝒞) and we(𝒞), an initial object ∅ and a terminal object ∗, satisfying axioms (TM0)-(TM4) and (CM5). If in addition it satisfies axiom (TM5), we call 𝒞 a Thomason model category.

Comparing definitions, we see that any basic model category is both a left model category and a right model category.

The following consequence of the axioms is of fundamental importance, so we include it here.

Glueing Lemma 3.2. Suppose that 𝒞 satisfies axiom (TM0), (TM1) and (TM2). Given a commutative diagram

B ⟵ A ⟶^{>} C
∼↓ ↓^{∼} ↓^{∼}
B′ ⟵ A′ ⟶^{>} C′

in which the vertical maps are weak equivalences, the pushout B ⊔_{A} C → B′ ⊔_{A′} C′ is a weak equivalence.

Proof: By (TM1) the map C → A′ ⊔_{A} C exists and is a weak equivalence. Its composition with g: A′ ⊔_{A} C → is C ⟶^{∼} C′, so g is a weak equivalence by (TM2). But then we may use (TM0) and (TM1) twice to get

B ⊔_{A} C ⟶^{∼} B′ ⊔_{A} C ≅ B′ ⊔_{A′} A′ ⊔_{A} C ⟶^{∼} B′ ⊔_{A′} C′ .

The composition is a weak equivalence, as desired. □

Remark 3.2.1. (TM0) is only used for we(𝒞) and (TM1) is only used for cofibrations.

§4 Examples

As usual, the paradigm of a Thomason model category is the category of topological spaces and continuous maps, where the fibrations are Serre fibrations and weak equivalences are homotopy equivalences; cofibrations are defined by the lifting property (TM4).

Definition 4.1. More generally, any proper model category in the sense of Quillen [QH] [QR] [BF] [GJ] is a basic model category; the adjective proper describes the hard part of axiom (TM1). The easy part of (TM1) is axiom (M4) in Quillen’s original definition of a model category [QH]; see [GJ, II.1.3].

I do not know of any cocomplete proper model category in which the factorizations required by axiom (CM5) are not functorial. We will see in 4.8 below that if 𝒞 admits a small object argument then 𝒞 satisfies (TM5) and is a Thomason model category. Compare [GJ, I.9.2].

Here is another difference between Quillen’s and Thomason’s axioms. Thomason’s Factorization Axiom (TM5) allows us to factor f as hi, where i is a cofibration and h is a weak equivalence; in Quillen’s Factorization Axiom (CM5) h must also be a fibration. Therefore if f is a map having the left lifting property with respect to all fibrations which are equimorphisms, Quillen’s axioms (CM3) and (CM5) imply that f must be a cofibration. This need not be the case in a Thomason model category, as the following example shows.

Example 4.2. Here is an example of a Thomason model category which is not a Quillen model category. Let Ch = Ch^{b}(R) denote the category of bounded chain complexes of modules over a ring R, with weak equivalences being quasi-isomorph-isms (maps inducing isomorphisms on homology).

Let cof(Ch) denote the category of injections A ↪ B whose cokernel B/A is a chain complex of projective modules, and let fib(Ch) denote the category of surjections whose kernel is a chain complex of injective modules. If R is a noetherian regular ring of finite global dimension, then Ch^{b}(R) is a Thomason model category but not a Quillen model category (unless R is semisimple).

Of course, Ch^{b}(R) has the same homotopy theory as the Quillen model structure of [QH, I.1.2], in which every surjection is a fibration. See also [QH, I.4.12]. Example 4.3. Baues’ cofibration category is another context in which one can do homotopy theory; this theory is developed in [Baues]. By definition, a cofibration category is a category 𝒞 equipped with two classes of morphisms (cof and we) such that Baues’ axioms (C1)–(C4) hold. Thus Baues’ cofibration categories lack all fibrations present in a Thomason model category.

Baues’ axioms (C1), (C2), and (C3) for (𝒞, cof, we) are axioms (TM0) , (TM1), (TM2) and (CM5c) for cofibrations and weak equivalences. Baues calls an object X a fibrant model if each trivial cofibration X ↣^{∼} Y admits a retraction, and Baues’ final axiom is: (C4) For each object A in 𝒞 there is a trivial cofibration A ↣^{∼} X where X is a

fibrant model. If 𝒞 is Thomason model category, then (𝒞, cof, we) is a left model category in the sense of 1.13 above. However, (𝒞, cof, we) may not satisfy Baues’ axiom (C4), so 𝒞 may not be a cofibration category in the sense of [Baues].

For purposes of comparison, recall from [Baues, I.2.6] that if (𝒞, cof, fib, we) is a model category in the sense of Quillen [QH], then the subcategory 𝒞_{cof} of all cofibrant objects in 𝒞 is a cofibration category. Example 4.4. Recall from [TT, 1.2.4] that a biWaldhausen category is a category 𝒞 with a zero object 0, together with subcategories cof(𝒞), fib(𝒞) and we(𝒞) so that:

(i) (𝒞, cof, we) and (𝒞^{op}, fib, we) are Waldhausen categories [TT, 1.2.3]; (ii) the canonical map A ⊔ B → A × B is an isomorphism; and (iii) the cofibration sequences in 𝒞 and 𝒞^{op} are dual to each other.

Note that every object of 𝒞 is both cofibrant and fibrant by axiom 1.2.1.2 of [TT]. A biWaldhausen category satisfies (TM0) and (TM1) by axioms 1.2.1.1, 1.2.1.3, 1.2.3.1 and 1.2.3.2 of [TT]. We say that 𝒞 is saturated if axiom (TM2) holds, i.e., if we(𝒞) is a saturated subcategory.

If 𝒞 has mapping cylinders satisfying the cylinder axiom, and mapping path spaces satisfying the path space axiom (see [TT, 1.3.1 and 1.3.2]) then 𝒞 satisfies (TM4) and (TM5). Thus only the retraction axiom (TM3) is needed to make 𝒞 a Thomason model category. Here is a partial converse.

Lemma 4.5. If 𝒞 satisfies (TM0) – (TM3), A⊔B ≃ A×B for all A and B, and every object is both fibrant and cofibrant, then 𝒞 is a saturated biWaldhausen category.

Proof: This is straightforward; since objects are fibrant and cofibrant, axiom (i) follows from (TM0), (TM1) and the gluing lemma 3.2. □

We caution the reader that the functoriality axiom (TM5) need not give a Waldhausen cylinder functor T, because the map A ⊔ B ↣ T (f) need not be in cof(𝒞).

We now give a set-theoretic condition under which a proper closed Quillen model category is a Thomason model category. It is based on II.3.4 of [QH].

Definition 4.6. Suppose that 𝒞 is a closed model category. We say that a set of trivial cofibrations {A_{α} ↣^{∼} B_{α}} is a set of test cofibrations if for each map E⟶^{p}F, p is a fibration if and only if every diagram of the following form admits a factorization B_{α} → E.

A_{α} ⟶ E
∨ ↓ ↓^{p}
B_{α} ⟶ F

For example, the anodyne extensions form a set of test cofibrations for the category 𝒞 of simplicial sets; see [GJ, I.4.3]. The horns {Λ_{k}^{n} ↣^{∼} Δ^{n}} form a countable set of test fibrations; see [GJ, p. 11].

Let κ be an infinite cardinal number. A poset is called κ-filtering if any subset of cardinality < κ has an upper bound.

Definition 4.7. An object A of a category 𝒞 is called κ-small if for every κ-filtering directed poset \{C_{i}\} for which \underrightarrow{lim} C_{i} exists we have

\underrightarrow{lim} Hom_{𝒞} (A, C_{i}) ≅ Hom_{𝒞} (A, \underrightarrow{lim} C_{i}) .

Theorem 4.8 (Small Object Argument). Let 𝒞 be a closed model category with arbitrary coproducts. Suppose 𝒞 has a set {A_{α} ↣^{∼} B_{α}} of test cofibrations in which each A_{α} is κ-small for some fixed κ. Then 𝒞 satisfies (TM5). If in addition 𝒞 is a proper closed model category, then 𝒞 is a Thomason model category.

Proof: We construct F_{μ}: 𝒞^{1} → 𝒞^{2} by induction on μ ≤ κ. Fix f: X → Y. The functor F_{0} sends f to X ↣^{id} X ⟶ Y. We define T_{μ+1} by

T_{μ+1}(f) = (∐_{S} B_{α}) ∐ ; (∐_{S} A_{α}) T_{μ}(f)

where S runs over all test diagrams

(∗)  \begin{matrix}A_{α} & ⟶ & T_{μ}(f) \\ ^{∨} ↓ & & | ; ↓ ↓ \\ B_{α} & ⟶ & Y . \end{matrix}

The map T_{μ} → T_{μ+1} is a trivial cofibration as in Lemma 3 of [QH, II.3.3] and the maps B_{α} → Y and T_{μ} ↠ Y induce a fibration T_{μ+1} → Y. We define F_{μ+1} by

X ↣^{∼} T_{μ} ↣^{∼} T_{μ+1} ↠ Y .

For a limit ordinal μ, we observe that X ↣ \underrightarrow{lim} T_{ν} = T_{μ} is a trivial cofibration. The map T_{μ} → Y is a fibration because for each test diagram (∗) there is a ν < μ so that A → T_{μ} factors through a map A → T_{ν}, ν < μ (A_{α} is μ-small as μ < κ), and the factorization B → T_{ν+1} → T_{μ} of (∗) exists by the construction of T_{ν+1}. □

§5 The Homotopy Category

Suppose that 𝒞 is a basic model category. The basic idea needed to construct the homotopy category Ho𝒞 is quite simple, and goes back to Quillen [QH]. Given A cofibrant and B fibrant, we show in 5.4 that there is a good equivalence relation ≃ on Hom_{𝒞} (A, B) and [A, B] is the set of equivalence classes. Given general A and B, axiom (CM5) gives factorizations ∅ ↣ A′ ⟶^{∼} A and B ⟶^{∼} B′ ↠ ∗; we set [A, B] = [A′, B′]. As usual, there are lots of technical details to check.

By a cylinder object for an object A of 𝒞, we mean an object C of 𝒞, together with a factorization A ⊔ A ↣ C ⟶^{∼} A of the fold map ∇: A ⊔ A → A (we assume A⊔A exists). Dually, a path object for B is an object M together with a factorization B ⟶^{∼} M ↠ B × B of the diagonal Δ (we assume B × B exists).

Definition 5.1. For f, g ∈ Hom_{𝒞} (A, B), define f ≃_{L} g to hold if the map f⊔g: A⊔A → B factors as A ⊔ A ↣^{∂} C⟶^{H}B, where there is a weak equivalence q: C ⟶^{∼} A such that q∂ is the fold map, i.e., C is a cylinder object for A.

Lemma 5.2 [T78, 166]. If A is cofibrant then ≃_{L} is an equivalence relation on the set Hom(A, B). We define π(A, B) to be Hom(A, B)_{≃_{L}}.

If A is cofibrant and B → B′ arbitrary, it is easy to see that Hom(A, B) → Hom(A, B′) induces a quotient map π(A, B) → π(A, B′). Proof: (Quillen [QH, I.1.8]) Axiom (CM5) implies that the fold map ∇ factors as A ⊔ A ↣ C ↠^{∼} A; the composition with f: A → B is clearly f ⊔ f, so f ≃_{L} f. To see that the relation is symmetric, precompose the factorization of f ⊔ g with the permutation τ; the composition

A ⊔ A⟶^{τ}A ⊔ A ⟶^{>}^{∂} C⟶^{H}B

will then be g ⊔f, and q∂τ = ∇τ = ∇. Finally, to see that ≃_{L} is transitive, suppose given factorizations of f ⊔ g and g ⊔ h:

A ⊔ A ↣ C_{1} ⟶ B  and  A ⊔ A ↣ C_{2} ⟶ B .

Since A is cofibrant, A = ∅ ⊔ A ↣ A ⊔ A ↣ C_{1} and A = A ⊔ ∅ ↣ A ⊔ A ↣ C_{2} are cofibrations. Hence we may apply the Glueing Lemma to

C_{1} ⟵^{<} A ⟶^{>} C_{2}
↓_{q_{1}}  ↓^{=}  ↓^{q_{2}}
A  A  A .

This shows that the pushout a: C_{1} ⊔_{A} C_{2} → A is a weak equivalence. Moreover, the composites A ↣ C_{i} → B are both g, so they induce a map h_{1} ⊔_{A} h_{2}: C_{1} ⊔_{A} C_{2} → B. If we let ∂ be the map

A ⊔ A = (A ⊔ ∅) ⊔ (∅ ⊔ A) ↣ C_{1} ⊔ C_{2} → C_{1} ⊔_{A} C_{2}

then clearly q∂ = ∇ and (h_{1} ⊔_{A} h_{2}) ∘ ∂ = f ⊔ h. Finally, ∂ is a cofibration because it is the composition of two cofibrations: ∂_{0} ⊔ A: A ⊔ A ↣ C_{1} ⊔ A, the pushout of A ↣ C_{1} by A = A ⊔ ∅ ↣ A ⊔ A, and the map C_{1} ⊔ A ↣ C_{1} ⊔_{A} C_{2}, which is the pushout of A ⊔ A ↣ C_{2} by ∂_{0} ⊔ A: A ⊔ A ↣ C_{1} ⊔ A. □

A ⟶ A ⊔ A ⟶^{>} C_{2}
∨ ↓ ^{∨} ↓^{∂_{0}⊔A} ^{∨} ↓
C_{1} ⟶ C_{1} ⊔ A ⟶^{>} C_{1} ⊔_{A} C_{2}

If A is cofibrant, the following argument shows that we can fix the choice of cylinder object in the definition of ≃_{L}. This fact allows us to simplify some set-theoretic considerations later on.

Lemma 5.3. If A is cofibrant, B is fibrant and C is a cylinder object for A, then whenever two maps f, g: A → B satisfy f ≃_{L} g there is a map C → B so that A ⊔ A ↣ C → B is f ⊔ g.

Proof: [T83, 77] We are given a factorization A ⊔ A ↣ C′⟶^{H′}B of f ⊔ g, where C′ is a different cylinder object for A. Let C′′ denote the pushout of C and C′ along A ⊔ A, and factor the pushout C′′ → A as C′′ ↣ C′′′ ⟶^{∼} A. Axiom (TM2) shows that the composite C′ ↣ C′′ ↣ C′′′ is a trivial cofibration. Since B is fibrant, axiom (TM4) applied to the right side of

A ⊔ A ⟶^{>} C′  ⟶^{H′} B
∨ ↓  ^{∨} ↓^{∼}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  | ; ↓ ↓
C  ⟶^{>} C′′′ ⟶ ∗

shows that H′ factors through C′′′. But then f ⊔ g factors as

A ⊔ A ↣ C ⟶ C′′′ ⟶ B . □

Dually, if B is fibrant, we get an equivalence relation ≃_{R} on Hom_{𝒞} (A, B) by declaring f ≃_{R} g if there is a diagram:

A ⟶ M ⟵^{∼} B
f×g^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  | ; ↓ ↓  Δ^{...................................................................................................}
B × B

Lemma 5.4. If A is cofibrant and B fibrant, then for all f, g ∈ Hom_{𝒞} (A, B) we have: f ≃_{L} g if and only if f ≃_{R} g. Proof: Suppose that f ≃_{L} g via A ⊔ A ↣ C⟶^{h}B. By (CM5) we can factor Δ as B ⟶^{∼} M ↠ B × B. By (TM4), there is a fill-in φ: C → M for the diagram:

A ⟶^{f} B ⟶^{∼} M
∨ ↓^{∂_{0}} | ; ↓ ↓
C ⟶_{(fg,h)} B × B

and φ∂_{1}: A → M is the map needed to show that f ≃_{R} g. By duality, f ≃_{R} g implies f ≃_{L} g. □ Proposition 5.4 [T78, 170]. Let 𝒞 be a basic model category. If A and A′ are cofibrant and B is fibrant, then for every weak equivalence w: A′ → A there is a bijection

w^{∗}: π(A, B) → π(A′, B)

Remark. This is [QH, I.1.10] if w is a trivial cofibration. Proof: The map w^{∗} is well defined since by definition f ≃_{R} g implies fw ≃_{R} gw. To see that w^{∗} is injective, suppose given f, g: A → B so that fw ≃_{L} gw. That is, fw ⊔ gw factors as A′ ⊔ A′ ↣ C → B. Form the pushout P of C and A ⊔ A along A′ ⊔ A′; the map f ⊔ g and C → B agree on A′ ⊔ A′ and induce a map G: P → B.

A′ ⊔ A′  ⟶^{>}^{∂′} C ⟶_{q′}^{∼} A′
w⊔w↓^{∼}  ↓  ↓^{w}
A ⊔ A ⟶^{>}^{∂} P ⟶^{q} A
f⊔g^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{↓^{G}}
B

Similarly ∇ and wq′ induce a map q: P → A so that q∂ = ∇. Since A and A′ are cofibrant, the glueing lemma shows that w ⊔ w: A′ ⊔ A′ → A ⊔ A is a weak equivalence. By axiom (TM1), its pushout C ⟶^{∼} P is a weak equivalence. Its composition with q is a weak equivalence, so q is a weak equivalence by (TM2). Thus P gives f ≃_{L} g.

To see that w^{∗} is onto we have to lift any A′⟶^{f}B to A. If w were a trivial cofibration, this would follow from (TM4), as in [QH, I.1.10]. In the general case, factor w ⊔ 1 as A′ ⊔ A ↣ A′′ ⟶^{∼} A. Since A and A′ are cofibrant, so are the evident maps A ↣ A′′ and A′ ↣ A′′. They are weak equivalences, since their composition with A′′ ⟶^{∼} A is. By the trivial cofibration case, we have the desired isomorphism. □

π(A′, B) ⟵^{≅} π(A′′, B) ⟶^{≅} π(A, B)

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π(A, B)

Lemma 5.5. Let 𝒞 be a basic model category, and A any object. Suppose given weak equivalences w′: A′ ⟶^{∼} A, w′′: A′′ ⟶^{∼} A with A′ and A′′ cofibrant, and a fibrant object B. Then for any two maps a_{1}, a_{2}: A′ → A′′ so that w′ = w′′a_{1} = w′′a_{2} we have

a_{1}^{∗} = a_{2}^{∗}: π(A′′, B) → π(A′, B).

Proof: Given f: A′′ → B, factor (f, w′′) as A′′ ⟶^{∼} P ↠ B × A. As B ↠ ∗ is a fibration and w′ is a weak equivalence, its pullback B × A′ → B × A exists and is a weak equivalence by (TM1). In turn, the pullback of this map along P ↠ B × A is a weak equivalence Q ⟶^{∼} P, and Q ↠ B × A′ is a fibration. We have a diagram in which the bottom square is a pullback, so the fold map on A′ factors through the dotted arrow.

A ⊔ A′ ⟶^{a_{1}⊔a_{2}} A′′
.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}  ^{^{∼}↓} f^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}
Q  ⟶^{∼} P  ↠ B
| ; ↓ ↓  | ; ↓ ↓
A′  ⟶^{∼} A

Factor the dotted arrow as A′⊔A′ ↣ C ⟶^{∼} Q. Now the map A′ = A′⊔∅ → A′⊔A′ → C is a weak equivalence, since its composite with C ⟶^{∼} P is A′ ⟶^{∼} A′′ ⟶^{∼} P. Its composite with C → A′ is the identity on A′, so C ⟶^{∼} A′ is also a weak equivalence. We have shown that the map fa_{1} ⊔ fa_{2} factors as

A ⊔ A′ ⟶^{>} C ⟶^{∼} Q ⟶^{∼} P ⟶ B
∇^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{↓^{∼}} A′

which shows that fa_{1} ≃_{L} fa_{2}, as required. □

In order to have an absolutely functorial set [A, B], it is useful to parametrize the cofibrant resolutions of A, and the fibrant resolutions of B.

Definition 5.6. Given A in 𝒞, let I_{A} denote the category whose objects are weak equivalences A′ ⟶^{∼} A with A′ cofibrant. There is a (unique) morphism from A′ ⟶^{∼} A to A′′ ⟶^{∼} A in I_{A} if there exists a (backwards!) arrow A′′ → A′ so that A′′ → A′ → A equals A′′ → A.

Dually, given B in 𝒞, let J_{B} denote the category of weak equivalences B ⟶^{∼} B′ with B′ fibrant, whose opposite category is I_{(A^{op})}.

Lemma 5.7. (a) I_{A} is a cofiltering category with at most one morphism between any two objects.

(b) Given A and B, there is an induced functor π( , ): I_{A}^{op} × J_{B} → Sets. Moreover, each morphism in I_{A}^{op} × J_{B} induces a bijection on π( , ).

Proof: (a) is obvious, once we note that for each A′ and A′′ we can factor A⊔A′′ ↣ A′′′ ⟶^{∼} A. Part (b) follows from 5.4 and 5.5. □

Definition 5.8. Given A and B in 𝒞, set [A, B] = lim_{I_{A}×J_{B}} π(A′, B′).

Lemma 5.9. If A and B are cofibrant, and C′ is fibrant, then a weak equivalence w: B ⟶^{∼} B′ determines a well-defined “composition” pairing

π(A, B′) × π(B, C′) → π(A, C′).

Proof: Suppose given f: A → B′ and g: B → C′. Since A and B are cofibrant, we can form f ⊔ w: A ⊔ B → B′. Factoring it as A ⊔ B ↣ B′′ ⟶^{∼} B′, axiom (TM2) shows that the composite B ↣ A ⊔ B ↣ B′′ is a trivial cofibration. Since C′ is fibrant, g lifts to g′′: B′′ → C′ by (TM4). Since π(B′′, C′) ≅ π(B, C′) by 5.4, g′′ is well-defined up to ≃_{L}. We let g ∘ f be the composite of A ↣ A ⊔ B ↣ B′′ and g′′.

B ⟶^{g} C′
∨ ↓^{∼} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} | ; ↓ ↓
A ⊔ B ⟶^{>} B′′ ⟶ ∗
↑_{∧}  ↓^{∼}
A  ⟶_{f} B′

Replacing A by a cylinder object in this argument shows that if f ≃_{L} f′ then g ∘ f ≃_{L} g ∘ f′. Dually, if g ≃_{R} g′ then g ∘ f ≃_{R} g′ ∘ f. □

Porism 5.10. Suppose given f′: A → B′ and g: B → C′, with w: B ⟶^{∼} B′.

(a) If f = wf_{0} for some f_{0}: A → B, then g ∘ f is the class of gf_{0} in π(A, C′).

(b) If g = g′w for some g′: B′ → C′, then g ∘ f is the class of g′f in π(A, C′). In particular, g ∘ w = g and w ∘ f = f. Hence the class 1_{B} of w in π(B, B) is a 2-sided unit for the composition pairing.

Lemma 5.11. The composition pairing is associative.

Proof: Suppose given f: A → B′, g: B → C′ and h: C → D′ with A, B cofibrant and B′, C′, D′ fibrant, and weak equivalences w: B ⟶^{∼} B′ and v: C ⟶^{∼} C′. The proof of 5.9 shows that we can replace B by B′′ and g by g′′ to assume that f = wf_{0} for some f_{0}: A → B. Similarly, we may replace C by C′′ and h by h′′ to assume that g = vg_{0} for some g_{0}: B → C. But then the formula

(h ∘ g) ∘ f = (hg_{0}) ∘ f = (hg_{0})f_{0} = h(g_{0}f_{0}) = h ∘ (gf_{0}) = h ∘ (g ∘ f) holds in π(A, D′), as required. □ Definition 5.12. Suppose that 𝒞 is a basic model category. Then Ho𝒞 is the category with the same objects as 𝒞, but the Hom-set of morphisms A → B in Ho𝒞 is [A, B]. Composition is defined in 5.9, and is associative with identity by 5.10 and 5.11.

Also, there is a canonical functor 𝒞 → Ho𝒞, defined by 5.10. Theorem 5.13. If 𝒞 is a basic model category then Ho𝒞 is the localization 𝒞[w^{−1}] of 𝒞 at the family w = we(𝒞) of weak equivalences. Proof: We first show that every weak equivalence f: A ⟶^{∼} B in 𝒞 becomes an isomorphism in Ho𝒞. Choosing weak equivalences A′ ⟶^{∼} A and B ⟶^{∼} B′ with A′ cofibrant and B′ fibrant, we see that π(A′, B′) is canonically isomorphic to [A, A], [A, B], [B, B] and [B, A]. The class of the composite map f′: A′ → B′ represents 1_{A}, [f], 1_{B} and a morphism g ∈ [B, A], respectively. The composition pairing

π(A′, B′) × π(A′, B′) → π(A′, B′)

satisfies f′ ∘ f′ = f′ by 5.10. By definition 5.8, this shows that [f] ∘ g = 1_{B} and g ∘ [f] = 1_{A}, i.e., [f] is an isomorphism in Ho𝒞.

Conversely, suppose that ϕ: 𝒞 → 𝒞′ is a functor sending weak equivalences to isomorphisms. If C is a cylinder object for A then the two composites i_{0}, i_{1}: A → A ⊔ A → C are both left inverses to q: C ⟶^{∼} A. Hence ϕ(i_{0}) = ϕ(i_{1}). Therefore if f ≃_{L} g via a map H: C → B (notation as in 5.1), then ϕ(f) = ϕ(g) because

ϕ(f) = ϕ(H)ϕ(i_{0}) = ϕ(H)ϕ(i_{1}) = ϕ(g).

It follows that ϕ factors uniquely through 𝒞 → Ho𝒞. □

§6 Functor Categories

If K is a small category and 𝒞 is a model category, we would like to make the functor category 𝒞^{K} into a model category. There are at least two model structures one could consider: a local structure and a global structure. We shall focus on the global structure.

Definition 6.1. Let K be a small category and 𝒞 a Thomason model catetory. We say that a natural transformation η: F → G of functors K → 𝒞 is a (global) cofibration (written F ↣ G) if for each K in K the map η_{K}: F (K) → G(K) is a cofibration in 𝒞; weak equivalence (written F ⟶^{∼} G) if for each K in K the map η_{K}: F (K) → G(K) is a weak equivalence in 𝒞; (global) fibration (written F ↠ G) for for each K in K the map η_{K} is a fibration in 𝒞, and if η has the right lifting property with respect to transformations A ↣^{∼} B which are simultaneously global cofibrations and weak equivalences. That is, every diagram of the following form admits a factorization B → F making the resulting diagram commute.

\begin{matrix}A & ⟶ & F \\ ^{∨} ↓^{∼} & & ^{η}↓ \\ B & ⟶ & G \end{matrix}

We shall say that F → G is a pointwise fibration (resp. has pointwise P) if each F (i) → G(i) is a fibration (resp. has property P) in 𝒞.

Proposition 6.2 [T85, 32]. If 𝒞 is a basic model category and K is a small category, then 𝒞^{K} with the global structure above satisfies axioms (TM0)–(TM3). If 𝒞 satisfies the factorization axiom (TM5c) then 𝒞^{K} satisfies (TM4) and (TM5c).

Proof: The constant functors F (K) = ∅ and F (K) = ∗ are the initial and terminal objects of 𝒞^{K}. Axioms (TM0), (TM2) and (TM3) for 𝒞^{K} are immediate from the pointwise definitions. Because limits and colimits in 𝒞^{K} are defined pointwise, there is only one nontrivial point to check for (TM1): if F ↣ G is a fibration in 𝒞^{K}, does the pullback η: F ×_{G} H → H have the lifting property with respect to a trivial cofibration A ↣^{∼} B? To see that it does, consider the diagram

A ⟶ F ×_{G} H ⟶ F
∨ ↓^{∼} ↓ |↓ ↓^{η}
B ⟶ H ⟶ G.

The lifting property for the global fibration η yields a fill-in B → F which must factor through a map to the pullback B → F ×_{G} H. So (TM1) holds for 𝒞^{K}.

If 𝒞 satisfies (TM5c), any η: A → B in 𝒞^{K} determines factorizations which are functorial in K:

A(K)^{>}⟶ T (η_{K}) ⟶^{∼} B(K).

This gives the functorial factorization A^{>}⟶ T ⟶^{∼} B in 𝒞^{K}.

To check axiom (TM4), consider a square in 𝒞^{I} of the form

\begin{matrix}A & ⟶ & X \\ ^{∨} ↓ & & | ; ↓ ↓ \\ B & ⟶ & Y. \end{matrix}

If A ⟶^{∼} B is a weak equivalence, the definition of global fibration yields a fill-in B → X. Suppose then that X ⟶^{∼} Y is a weak equivalence. By (TM1) the pushout X ↣ C of A ↣ B exists, and there is a canonical map C → Y. By (LM5) we can factor it as C ↣ D ⟶^{∼} Y. By saturation (TM2), the cofibration X ↣ D is also a weak equivalence since D ⟶^{∼} Y and X ⟶^{∼} Y are. But then the definition of global fibration yields a fill-in D → X of the right square in the diagram

A ⟶ X  X
∨ ↓ ^{∨} ↓^{∼} |↓ ↓^{∼}
B ⟶ D ⟶^{∼} Y,

and B → D → X provides the desired factorization. This completes the verification of (TM4). □

Remark 6.3. If 𝒞 is a Thomason model category, all that is missing is the factorization part (f) of (CM5). We do have a functorial factorization F ⟶^{∼} M ⟶ G from (RM5) on 𝒞, and M → G is a pointwise fibration, i.e., each M(K) ↠ G(K) is a fibration in 𝒞. However, there appears to be no reason for M → G to satisfy the right lifting property required of a global fibration.

Recall that an object C of 𝒞 is called fibrant if C ↠ ∗ is a fibration. We write 𝒞_{fib} for the full subcategory of fibrant objects in 𝒞 and 𝒞_{fib}^{K} for the category of all pointwise fibrant functors, i.e., all functors K → 𝒞_{fib}. Note that (𝒞^{K})_{fib} is strictly contained in 𝒞_{fib}^{K} = (𝒞_{fib})^{K}. The following argument allows us to reduce the verification of the fibration half (TM5f) of axiom (TM5) to pointwise fibrations in 𝒞_{fib}^{K}.

Lemma 6.4 [T85, 166]. Suppose that there is a functorial factorization in 𝒞^{K}

F′ ⟶^{∼} H′ ↠ G′

for every pointwise fibration F′ → G′ of pointwise fibrant functors. Then axiom (TM5f) holds in 𝒞^{K}, and 𝒞^{K} is a Thomason model category.

Proof: Suppose given F → G in 𝒞^{K}. Using functorial factorization in 𝒞, we have a factorization G ⟶^{∼} G′ ⟶ ∗, where G′ is pointwise fibrant, and then a factorization of F → G′

F ⟶ G
↓_{∼} ↓_{∼}
F′ ⟶ G′

in which F′ → G′ is pointwise fibrant. By assumption, it factors as F′ ⟶^{∼} H′ ↠ G′. By (TM1) the pullback H = F′ ×_{G′} G exists, H ↠ G is a fibration and H ⟶^{∼} H′ is a weak equivalence. By saturation (TM2), F ⟶^{∼} H is a weak equivalence, and we have the required functorial factorization F ⟶^{∼} H ↠ G. □

F ⟶ H ↠ G
∼↓ ↓^{∼} ^{∼}↓
F′ ⟶^{∼} H′ ↠ G′

§7 Enriched Functor Categories

If 𝒞 is a Thomason model category, we saw in the last section that 𝒞^{K} = Cat(K, 𝒞) satisfies all the axioms for a model category except for (TM5f), that every map F → G in 𝒞^{K} has a factorization F ⟶^{∼} H ↠ G. In order to do this we shall use homotopy limits and homotopy ends in 𝒞; this requires that 𝒞 be not only complete but enriched over finite simplicial sets. Recall from 1.12 that this means that 𝒞 has a mapping object functor satisfying axioms (RE1) – (RE5). We will also need 𝒞 to satisfy the following fibration conditions.

Definition 7.1 [T85, 147]. We say that towers preserve fibrations in 𝒞 if given a map between towers of fibrations in 𝒞

⋯ ↠ C_{n} ↠ C_{n−1} ↠ ⋯ ↠ C_{0} ↠ ∗
↓ ↓ | ; ↓ ↓
⋯ ↠ D_{n} ↠ D_{n−1} ↠ ⋯ ↠ D_{0} ↠ ∗

in which C_{0} → D_{0} is a fibration and each C_{n+1} ↠ C_{n} ×_{D_{n}} D_{n+1} is a fibration (n ≥ 0), then \underleftarrow{lim} C_{n} ↠ \underleftarrow{lim} D_{n} is a fibration. If in addition each C_{n} ⟶^{∼} D_{n} is a weak equivalence, so is \underleftarrow{lim} C_{n} ↠^{∼} \underleftarrow{lim} D_{n}. Note that this implies that \underleftarrow{lim} C_{n} is fibrant (take D_{n} = ∗).

We say that products preserve fibrations (and equifibrations) in 𝒞 if for every set I and every family {C_{i}}_{i∈I} of fibrant objects of 𝒞: (a) ∏ C_{i} exists in 𝒞, and (b) if B_{i} ↠ C_{i} are fibrations (resp. equifibrations) then ∏ B_{i} ↠ ∏ C_{i} is a fibration (resp. an equifibration) in 𝒞. Note that (b) applied to C_{i} ↠ ∗ implies that ∏ C_{i} is fibrant.

Theorem 7.2 [T85, 160]. Let 𝒞 be a complete Thomason model category with a right enrichment over finite simplicial sets (1.12). Suppose that products and towers in 𝒞 preserve fibrations and equifibrations. If K is a small category then 𝒞^{K} has the structure of a Thomason model category.

Moreover, 𝒞^{K} is right enriched, and products and towers in 𝒞^{K} preserve fibrations and equifibrations.

The proof of theorem 7.2 will be given after lemma 7.6 below. It requires a simplicial hom construction.

Definition 7.3. Given B and C in 𝒞, the simplicial set hom_{𝒞} (B, C) is defined to be [hom_{𝒞} (B, C)]_{p} = Hom_{𝒞} (B, Map(Δ[p], C)). Thus hom_{𝒞} is a functor from 𝒞^{op} × 𝒞 to simplicial sets, and we may take its homotopy end.

Lemma 7.4 [T85, 121]. If A ↣ B is a cofibration in 𝒞, then

hom_{𝒞} (B, C) ↠ hom_{𝒞} (A, C)

is a Kan fibration of simplicial sets.

Proof: For 0 ≤ k ≤ n, let V = V (n, k) denote the simplicial subcomplex of Δ[n] which is the union of all faces except the kth one. By axiom (RE2) the right vertical map is a fibration in 𝒞 in the diagram:

\begin{matrix}A & ⟶ & Map(Δ[n], C) \\ ^{∨} ↓ & ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} & | ; ↓ ↓ \\ B & ⟶ & Map(V (n, k), C). \end{matrix}

By (TM4), a fill-in exists. Thus the function

Hom_{𝒞} (B, Map(Δ[n], C)) → Hom_{𝒞} (A, Map(Δ[n], C)) ×_{Hom_{𝒞} (B,Map(V,C))} Hom_{𝒞} (A, Map(V, C))

is surjective. By (RE2), Map(−, C) sends the pushout V (m, k) of the cofibrations Δ[n − 2] ↣ Δ[n − 1] to the pullback in 𝒞 of the fibrations Map(Δ[n − 1], C) ↠ Map(Δ[n − 2], C). Applying Hom_{𝒞} (B, −), we see that Hom_{𝒞} (B, Map(V, C)) is the set-theoretic pullback of functions hom(B, C)_{n−1} → hom(B, C)_{n−2}. That is,

Hom_{𝒞} (B, Map(V, C)) ≅ Hom_{Δ^{op}Sets}(V, hom_{𝒞} (B, C)).

Since Hom_{𝒞} (B, Map(Δ[n], C)) = hom_{𝒞} (B, C)_{n} by definition, this shows that hom_{𝒞} (B, C) → hom_{𝒞} (A, C) is a Kan fibration. □.

If C is fibrant in 𝒞, then each Map(Δ[p], C) is fibrant by (RE1).

Theorem 7.5 [T85, 158]. For B in 𝒞 and F: K → 𝒞_{fib}, there is a natural bijection:

Hom_{𝒞} (B, holim_{K} F ) ≅ [holim_{K∈K} hom_{𝒞} (B, F K)]_{0} .

For B in 𝒞^{K}, there is a natural bijection of sets:

Hom_{𝒞^{K}} (B, holim F) ≅ [ho∫ hom_{𝒞} (BK′, F K)]_{0} .

Proof ([T85, 123–130]): By 1.4 and 1.14, part a) is a special case of b). By [Mac, p. 219], the left side is the end

∫_{K} Hom_{𝒞} (BK, holim_{K/K} F ) = ∫_{K} Hom_{𝒞} (BK, ∫_{Δ} Map(Δ[p], ∏_{I} F K_{p})) ,

where the indexing set I runs over all diagrams K → K_{0} → ⋯ → K_{p} in K. By the universal property of ends, this equals the double end

∫_{K} ∫_{Δ} Hom_{𝒞} (BK, Map(Δ[p], ∏_{I} F K_{p})) .

By the definition of hom_{𝒞}, this equals

∫∫_{K×Δ} hom_{𝒞} (BK, ∏_{I} F K_{p})_{p} = ∫∫_{K×Δ} Hom_{Δ^{op}Sets} (Δ[p], hom_{𝒞} (BK, ∏_{I} F K_{p})) .

Because Hom_{Δ^{op}Sets}(X, Y) is the set of 0-simplices in Map(X, Y), and limits in Δ^{op}Sets are formed pointwise, this equals

∫∫_{K×Δ}[Map(Δ[p], Y)]_{0} = [ ∫∫_{K×Δ} Map(Δ[p], Y)]_{0},

where Y = Y (K′, K) denotes hom_{𝒞} (BK, ∏_{I} F K_{p}). Hence it suffices to show that we have an isomorphism of simplicial sets between

∫∫_{K×Δ} Map(Δ[p], Y)  and  ho∫ hom_{𝒞} (BK′, F K).

Because Map(Δ[p], −) preserves limits of simplicial sets and ends commute, the left side equals

∫_{Δ} ∫_{K} Map(Δ[p], Y) = ∫_{Δ} Map (Δ[p], ∫_{K} Y ) .

Now axiom (RE5) in 𝒞 implies that (for each p and K)

Y (K′, K) = hom_{𝒞} (BK′, ∏_{K→···→K_{p}} F K_{p})≅ ∏_{K→···→K_{p}} hom_{𝒞} (BK′, F K_{p}).

But now ho∫ hom_{𝒞} (BK′, F K) is defined (1.3) to be the simplicial set

∫_{Δ} Map (Δ[p], ∏_{K_{0}→···→K_{p}} hom_{𝒞} (BK_{0}, F K_{p})) .

Hence it suffices to show that for all p

∏_{K_{0}→···→K_{p}} hom_{𝒞} (BK_{0}, F K_{p}) ≅ ∫_{K} ∏_{K→K_{0}→···→K_{p}} hom_{𝒞} (BK′, F K_{p}).

We can apply lemma 7.6 below to I = K^{op}, i_{0} = K, 𝒮 = simplicial sets and G(K′) = ∏_{K_{0}→···→K_{p}} hom_{𝒞} (BK′, F K_{p}), where the indexing set runs over diagrams of length p in K. The result is the required isomorphism:

G(K_{0}) ≅ ∫_{K^{op}}(K, K′) ⟼ ∏_{\begin{matrix}K→K_{0} \\ in K \end{matrix}} G(K′)

≅ ∫_{K}(K′, K) ⟼ ∏_{K→K_{0}} G(K′)

= ∫_{K}(K′, K) ⟼ ∏_{K→K_{0}→···→K_{p}} hom_{𝒞} (BK′, F K_{0}).  □

Lemma 7.6 [T85, 129]. Let I be a small category, i_{0} an object of I, and G: I → 𝒮 a functor to a complete category 𝒮. If F: I^{op} × I → 𝒮 denotes the functor F (i, j) = ∏_{i_{0}→i} G(j) then there is a natural isomorphism

G(i_{0}) ≅ ∫_{I} F = ∫_{I}(i, j) ⟼ ∏_{i_{0}→i} G(j).

Proof: The diagonal G(i_{0}) → ∏_{i_{0}→i} G(i_{0}) → ∏_{i_{0}→i} G(i) is dinatural, and induces a map G(i_{0}) → ∫_{I} F. Conversely, the projection onto the coordinate indexed by the identity of i_{0} yields a map ∫_{I} F → ∏_{i_{0}→i_{0}} G(i_{0}) → G(i_{0}). These maps are inverses to each other. □

Proof (of theorem 7.2): As remarked above, we are reduced by 6.2 and 6.4 to producing a functorial factorization of η: F → G under the assumption that each F K ↠ GK ↠ ∗ is a fibration. From the diagram using lemma 1.13:

F ⟶^{∼} holim F
η↓ ↓^{holim η}
G ⟶^{∼} holim G.

Now holim η is a pointwise fibration by lemma 1.8. Let H denote the pullback of G along holim η; axiom (TM1) in 𝒞 implies that H ⟶^{∼} holim F is a weak equivalence and H → G is a pointwise fibration. Saturation (TM2) gives F ⟶^{∼} H. We claim that in the resulting functorial factorization of η,

F ⟶^{∼} H ⟶ G,

the map H → G is a global fibration, i.e., it satisfies the right lifting property.

Let A ↣^{∼} B be a cofibration and weak equivalence. We need to show that a fill-in exists for any diagram

A ⟶ H ⟶ holim F
∨ ↓^{∼} ↓ ↓^{holim η}
B ⟶ G ⟶ holim G.

As the right square is a pullback, it suffices to find a fill-in B → holim F, i.e., to show that holim η is a global fibration. That is, it suffices to show that the following map is a surjection.

Hom_{𝒞^{K}} (B, holim F) → Hom_{𝒞^{K}} (A, holim F)] ×_{Hom(A,holim G)} Hom_{𝒞^{K}} (B, holim G).

By proposition 7.5, Hom(B, holim F) is naturally isomorphic to the degree zero part of the homotopy end ho∫_{K} hom_{𝒞} (BK′, F K), because each F K is fibrant.

As the homotopy end formed in Δ^{op}-Sets preserves limits, such as pullbacks along a fibration, the map we need to show surjective is identified with the degree zero part of the simplicial map

ho∫ hom_{𝒞} (BK′, F K)
→ [ho∫ hom(AK′, F K)] ×_{ho∫ hom(AK′,GK)} [ho∫ hom(BK′, GK)]
≅ ho∫ [hom(AK′, F K) ×_{hom(AK′,GK)} hom(BK′, GK)].

Because each AK′ ↣ BK′ is a cofibration, each simplicial map

hom_{𝒞} (BK′, GK) → hom_{𝒞} (AK′, GK)

is a Kan fibration by lemma 7.4. Since each F K ↠ GK ↠ ∗ is a fibration, the map

hom_{𝒞} (BK′, F K) → hom(AK′, F K) ×_{hom(AK′,GK)} hom(BK′, GK)

is a fibration and a weak equivalence of simplicial sets. By lemma 1.8, applying ho∫ yields a Kan fibration and weak equivalence of simplicial sets. But any such map is surjective on 0-simplices. This completes the proof that (TM5f) holds, and that 𝒞^{K} is a Thomason model category.

To see that products preserve fibrations and equifibrations in 𝒞^{K}, suppose given a family {F_{i} ↠ G_{i}}_{i∈I} of fibrations (resp. equifibrations). Since 𝒞 is complete, the maps ∏ F_{i}K ↠ ∏ G_{i}K exist and are fibrations (resp. equifibrations) in 𝒞 for all K. It suffices to show that ∏ F_{i} → ∏ G_{i} has the right lifting property. But for any A ↣^{∼} B, each square

A ⟶ ∏ F_{i}
∨ ↓ ↓
B ⟶ ∏ G_{i}

will admit a lift since lifts B → F_{i} exist for each i ∈ I and ∏ F_{i} is a product.

To see that towers preserve fibrations in Cat(K, 𝒞), suppose given a map between towers of fibrations in which F_{0} ↠ G_{0} and all F_{n+1} ↠ G_{n+1} ×_{G_{n}} F_{n} are fibrations.

⋯ ↠ F_{n} ↠ F_{n−1} ↠ ⋯ ↠ F_{0} ↠ ∗
↓ ↓ ↓
⋯ ↠ G_{n} ↠ G_{n−1} ↠ ⋯ ↠ G_{0} ↠ ∗

Since towers preserve fibrations in 𝒞, each \underleftarrow{lim} F_{n}K and \underleftarrow{lim} G_{n}K is fibrant and \underleftarrow{lim} F_{n}K → \underleftarrow{lim} G_{n}K is a fibration (and is a weak equivalence when each F_{n}K → G_{n}K is). To show it is a fibration it suffices to check the lifting property. Let A ↣^{∼} B be an equi-cofibration and suppose given a square

A ⟶ \underleftarrow{lim} F_{n} ⟶ F_{n}
∨ ↓^{∼}  ↓  ↓
B ⟶ \underleftarrow{lim} G_{n} ⟶ G_{n}

We need to find a lift B → \underleftarrow{lim} F_{n}.

We proceed by induction to produce a coherent tower of lifts B → F_{n}; passing to the inverse limit will yield the desired lift B → \underleftarrow{lim} F_{n}. For n = 0, a lift B → F_{0} exists because F_{0} → G_{0} is a fibration. Inductively, there is a lift B → F_{n}. It factors uniquely through the pullback G_{n+1} ×_{G_{n}} F_{n}, and we have a diagram

A ⟶  F_{n+1}  ⟶ F_{n}
∨ ↓  | ; ↓ ↓  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ↓
B ⟶ G_{n+1} ×_{G_{n}} F_{n} ⟶ G_{n}.

Since the middle vertical is a fibration, there is a lift B → F_{n+1} compatible with the preceeding lift B → F_{n}. This completes the inductive step, showing that the map \underleftarrow{lim} F_{n} → \underleftarrow{lim} G_{n} is a fibration, and that towers preserve fibrations in Cat(K, 𝒞).

Finally, we need to show that Cat(K, 𝒞) is right enriched over 𝒮_{f}. Because Cat(K, 𝒞)_{fib} ⊆ Cat(K, 𝒞_{fib}), there is a functor

Map( , ): 𝒮_{f}^{op} × Cat(K, 𝒞)_{fib} ⟶ Cat(K, 𝒞_{fib})

We need to show that this lands in Cat(K, 𝒞)_{fib} and makes Cat(K, 𝒞) right enriched over 𝒮_{f}. This is straightforward, and left to the reader. See [T86, 61].

This finishes the proof of Theorem 7.2 □

Acknowledgements

This work was done at the Institute for Advanced Study in Princeton, but the main sources were located at the Université de Paris 7. The author is deeply grateful to Liliane Barenghi, Bruno Kahn and Max Karoubi for their help in giving me access to Thomason’s private notebooks [T78]–[T86]. I am also grateful to Luca Barbieri-Viale, Marco Grandis and Ieke Moerdijk for providing me with their notes of seminar talks in which Thomason presented his ideas in Genova and Utrecht. Finally, I am grateful to Rick Jardine and Bill Dwyer for their comments on an earlier draft.

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