nLab Dugger's theorem

Idea

Dugger’s theorem identifies combinatorial model categories as the model category-presentations of locally presentable (infinity,1)-categories.

Statement

Theorem

(Dugger’s theorem)

Every combinatorial model category $C$ is Quillen equivalent to a left Bousfield localization $L_S SPSh(K)_{proj}$ of the global projective model structure on simplicial presheaves $SPSh(K)_{proj}$ on a small category $K$

$$L_S SPSh(K)_{proj} \stackrel{\simeq_{Quillen}}{\to} C \,.$$

This is Dugger (2001), theorem 1.1 building on results in Dugger (2001b).

Proof

The proof proceeds (the way Dugger presents it, at least) in roughly three steps:

1. Use that $[C^{op}, sSet_{Quillen}]_{proj}$ is in some precise sense the homotopy- free cocompletion of $C$. This means that every functor $\gamma : C \to B$ from a plain category $C$ to a model category $B$ factors in an essentially unique way through the Yoneda embedding $j : C \to [C^{op},sSet]$ by a Quillen adjunction

   $$(\hat \gamma \dashv R) : B \stackrel{\overset{\hat \gamma}{\leftarrow}} {\underset{R}{\to}} [C^{op}, sSet_{Quillen}]_{proj} \,.$$

   The detailed definitions and detailed proof of this are discussed at (∞,1)-category of (∞,1)-presheaves.

2. For a given combinatorial model category $B$, choose $C := B_\lambda^{cof}$ the full subcategory on a small set (guaranteed to exist since $B$ is locally presentable) of cofibrant $\lambda$-compact objects, for some regular cardinal $\lambda$, and show that the induced Quillen adjunction

   $$B \stackrel{\overset{\hat i}{\leftarrow}}{\underset{R}{\hookrightarrow}} [(B_\lambda^{cof})^{op}, sSet]_{proj}$$

   induced by the above statement from the inclusion $i : B_\lambda^{cof} \hookrightarrow B$ exhibits $B$ as a homotopy-reflective subcategory in that the derived adjunction counit $\hat i \circ Q \circ R \stackrel{\simeq}{\to} Id$ ($Q$ some cofibrant replacement functor) is a natural weak equivalence on fibrant objects (recall from adjoint functor the characterization of adjoints to full and faithful functors).

3. Define $S$ to be the set of morphisms in $[(C_\lambda^{cof})^{op}, sSet]$ that the left derived functor $\hat i \circ Q$ of $\hat i$ (here $Q$ is some cofibrant replacement functor) sends to weak equivalences in $B$. Then form the left Bousfield localization $L_S [(C_\lambda^{cof})^{op},sSet]_{proj}$ with respect to this set of morphisms and prove that this is Quillen equivalent to $B$.

Carrying this program through requires the following intermediate results.

First recall from the discussion at (∞,1)-category of (∞,1)-presheaves that to produce the Quilen adjunction $(\hat i \dashv R)$ from $i$, we are to choose a cofibrant resolution functor

$$I : C \to [\Delta,B]$$

of $i : C= B_\lambda^{cof} \to B$.

The adjunct of this is a functor $\tilde I : C \times \Delta \to B$. For each object $b \in B$ write $(C \times \Delta \downarrow b)$ for the slice category induced by this functor.

Lemma (Dugger, prop. 4.2)

For every fibrant object $b \in B$ we have that the homotopy colimit $hocolim (C \times \Delta \downarrow b) \to B)$ is weakly equivalent to $\hat i \circ Q\circ R (b)$.

Corollary (Dugger, cor. 4.4) The induced Quillen adjunction

$$B \stackrel{\leftarrow}{\to} [C^{op}, sSet]$$

is a homotopy-reflective embedding precisely if the canonical morphisms

$$hocolim (C \times \Delta \downarrow b) \to b$$

are weak equivalences for every fibrant object $b \in B$.

…

Notice that the theorem just mentions plain combinatorial model categories, not simplicial model categories. But of course by basic facts of enriched category theory $Funct(C^{op}, SSet)$ is an SSet-enriched category and its projective global model structure on functors $Func(C^{op}, SSet)_{proj}$ is compatibly a simplicial model category, as are all its Bousfield localizations. (See model structure on simplicial presheaves for more details.) Therefore an immediate but very useful corollary of the above statement is

Corollary

Every combinatorial model category is Quillen equivalent to one which is

• a simplicial model category

• a left proper model category.

Related concepts

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

Dugger’s theorem is due to

• Daniel Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 1 177-201 [arXiv:math/0007068, doi:10.1006/aima.2001.2015]

based on results in

• Dan Dugger, Universal homotopy theories, Advances in Mathematics 164 (2001)

  144-176 [arXiv:math/0007070]

The interpretation in terms of locally presentable (infinity,1)-categories is due to

• Jacob Lurie, Higher Topos Theory