# nLab Dugger's theorem

model category

## Definitions

• category with weak equivalences

• weak factorization system

• homotopy

• small object argument

• resolution

• ## Universal constructions

• homotopy Kan extension

• Bousfield-Kan map

• ## Refinements

• monoidal model category

• enriched model category

• simplicial model category

• cofibrantly generated model category

• algebraic model category

• compactly generated model category

• proper model category

• stable model category

• ## Producing new model structures

• on functor categories (global)

• on overcategories

• Bousfield localization

• transferred model structure

• Grothendieck construction for model categories

• ## Presentation of $(\infty,1)$-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

• ## Model structures

• Cisinski model structure
• ### for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

• Strom model structure?
• Thomason model structure

• model structure on presheaves over a test category

• model structure on simplicial groupoids

• on cubical sets

• related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

• ### for $n$-groupoids

• for 1-groupoids

• ### for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

• ### for $\infty$-algebras

#### general

• on monoids

• on algebas over a monad

• on modules over an algebra over an operad

• #### specific

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

• model structure on dg-modules

• ### for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on presheaves of spectra

• ### for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

• ### for stable $(\infty,1)$-categories

• on dg-categories
• ### for $(\infty,1)$-operads

• on modules over an algebra over an operad

• ### for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

• ### for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

• model structure for (2,1)-sheaves/for stacks

• # Contents

## 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 01, theorem 1.1) building on results in (DuggerUniversalHomotopy).

###### 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

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 presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

## References

Dugger’s theorem is due to

based on results in

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

Last revised on July 11, 2018 at 11:13:07. See the history of this page for a list of all contributions to it.