nLab
Dugger's theorem

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Model category theory

model category

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                                      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 CC is Quillen equivalent to a left Bousfield localization L SSPSh(K) projL_S SPSh(K)_{proj} of the global projective model structure on simplicial presheaves SPSh(K) projSPSh(K)_{proj} on a small category KK

                                      L SSPSh(K) proj QuillenC. 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[C^{op}, sSet_{Quillen}]_{proj} is in some precise sense the homotopy- free cocompletion of CC. This means that every functor γ:CB\gamma : C \to B from a plain category CC to a model category BB factors in an essentially unique way through the Yoneda embedding j:C[C op,sSet]j : C \to [C^{op},sSet] by a Quillen adjunction

                                        (γ^R):BRγ^[C op,sSet Quillen] proj. (\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 BB, choose C:=B λ cofC := B_\lambda^{cof} the full subcategory on a small set (guaranteed to exist since BB is locally presentable) of cofibrant λ\lambda-compact objects, for some regular cardinal λ\lambda, and show that the induced Quillen adjunction

                                        BRi^[(B λ cof) op,sSet] proj 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 λ cofBi : B_\lambda^{cof} \hookrightarrow B exhibits BB as a homotopy-reflective subcategory in that the derived adjunction counit i^QRId \hat i \circ Q \circ R \stackrel{\simeq}{\to} Id (QQ 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 SS to be the set of morphisms in [(C λ cof) op,sSet][(C_\lambda^{cof})^{op}, sSet] that the left derived functor i^Q\hat i \circ Q of i^\hat i (here QQ is some cofibrant replacement functor) sends to weak equivalences in BB. Then form the left Bousfield localization L S[(C λ cof) op,sSet] projL_S [(C_\lambda^{cof})^{op},sSet]_{proj} with respect to this set of morphisms and prove that this is Quillen equivalent to BB.

                                      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 (i^R)(\hat i \dashv R) from ii, we are to choose a cofibrant resolution functor

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

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

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

                                      Lemma (Dugger, prop. 4.2)

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

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

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

                                      is a homotopy-reflective embedding precisely if the canonical morphisms

                                      hocolim(C×Δb)b hocolim (C \times \Delta \downarrow b) \to b

                                      are weak equivalences for every fibrant object bBb \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)Funct(C^{op}, SSet) is an SSet-enriched category and its projective global model structure on functors Func(C op,SSet) projFunc(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.

                                      A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\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.