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Quillen adjunction

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Model category theory

model category

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                                      Contents

                                      Idea

                                      Quillen adjunctions are one convenient notion of morphisms between model categories. They present adjoint (∞,1)-functors between the (∞,1)-categories presented by the model categories.

                                      Definition

                                      Definition

                                      For CC and DD two model categories, a pair (L,R)(L,R)

                                      (LR):CLRD (L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D

                                      of adjoint functors (with LL left adjoint and RR right adjoint) is a Quillen adjunction if the following equivalent conditions are satisfied:

                                      1. LL preserves cofibrations and acyclic cofibrations;

                                      2. RR preserves fibrations and acyclic fibrations;

                                      3. LL preserves cofibrations and RR preserves fibrations;

                                      4. LL preserves acyclic cofibrations and RR preserves acyclic fibrations.

                                      Proposition

                                      The conditions in def. are indeed all equivalent.

                                      Proof

                                      Observe that

                                      We discuss statement (i), statement (ii) is formally dual. So let f:ABf\colon A \to B be an acyclic cofibration in 𝒟\mathcal{D} and g:XYg \colon X \to Y a fibration in 𝒞\mathcal{C}. Then for every commuting diagram as on the left of the following, its (LR)(L\dashv R)-adjunct is a commuting diagram as on the right here:

                                      A R(X) f R(g) B R(Y),L(A) X L(f) g L(B) Y. \array{ A &\longrightarrow& R(X) \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}} \\ B &\longrightarrow& R(Y) } \;\;\;\;\;\; \,, \;\;\;\;\;\; \array{ L(A) &\longrightarrow& X \\ {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}} \\ L(B) &\longrightarrow& Y } \,.

                                      If LL preserves acyclic cofibrations, then the diagram on the right has a lift, and so the (LR)(L\dashv R)-adjunct of that lift is a lift of the left diagram. This shows that R(g)R(g) has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if RR preserves fibrations, the same argument run from right to left gives that LL preserves acyclic fibrations.

                                      Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

                                      Remark

                                      Quillen adjunctions that are analogous to an equivalence of categories are called Quillen equivalences.

                                      In an enriched model category one speaks of enriched Quillen adjunction.

                                      Properties

                                      Derived adjunction

                                      Proposition

                                      (Ken Brown's lemma)

                                      Given a Quillen adjunction (LR)(L \dashv R) (def. ), then

                                      • the left adjoint LL preserves weak equivalences between cofibrant objects;

                                      • the right adjoint RR preserves weak equivalences between fibrant objects.

                                      Proof

                                      To show this for instance for RR, we may argue as in a category of fibrant objects and apply the factorization lemma which shows that every weak equivalence between fibrant objects may be factored, up to homotopy, as a span of acyclic fibrations.

                                      These weak equivalences are preserved by RR and hence by 2-out-of-3 the claim follows.

                                      For LL we apply the formally dual argument.

                                      Behaviour under Bousfield localization

                                      Proposition

                                      If

                                      (LR):CLRD (L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D

                                      is a Quillen adjunction, SMor(D)S \subset Mor(D) is a set of morphisms such that the left Bousfield localization of DD at SS exists, and such that the derived image 𝕃L(S)\mathbb{L}L(S) of SS lands in the weak equivalences of CC, then the Quillen adjunction descends to the localization D SD_S

                                      (LR):CLRD S. (L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D_S \,.

                                      This appears as (Hirschhorn, prop. 3.3.18)

                                      Of sSetsSet-enriched adjunctions

                                      Of particular interest are SSet-enriched adjunctions between simplicial model categories: simplicial Quillen adjunctions.

                                      These present adjoint (∞,1)-functors, as the first proposition below asserts.

                                      Proposition

                                      Let CC and DD be simplicial model categories and let

                                      (LR):CLRD (L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D

                                      be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let C C^\circ and D D^\circ be the (∞,1)-categories presented by CC and DD (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors

                                      (𝕃):C D . (\mathbb{L} \dashv \mathbb{R}) : C^\circ \stackrel{\leftarrow}{\to} D^{\circ} \,.

                                      On the decategorified level of the homotopy categories these are the total left and right derived functors, respectively, of LL and RR.

                                      Proof

                                      This is proposition 5.2.4.6 in HTT.

                                      The following proposition states conditions under which a Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).

                                      Proposition

                                      (recognition of simplicial Quillen adjunctions)

                                      If CC and DD are simplicial model categories and DD is a left proper model category, then an sSet-enriched adjunction

                                      (LR):CD (L \dashv R) : C \stackrel{\leftarrow}{\to} D

                                      is a Quillen adjunction already if LL preserves cofibrations and RR just fibrant objects.

                                      This appears as HTT, cor. A.3.7.2.

                                      See simplicial Quillen adjunction for more details.

                                      Associated (infinity,1)-adjunction

                                      Theorem

                                      Let F:CD:GF : C \rightleftarrows D : G be a Quillen adjunction between model categories (which are not assumed to admit functorial factorizations or infinite (co)limits). Then there is an induced adjunction of (infinity,1)-categories

                                      F:C[W C 1]D[W D 1]:G F : C[W_C^{-1}] \rightleftarrows D[W_D^{-1}] : G

                                      where C[W C 1]C[W_C^{-1}] and D[W D 1]D[W_D^{-1}] denote the respective simplicial localizations at the respective classes of weak equivalences.

                                      See (Mazel-Gee 16, Theorem 2.1). (This is also asserted as (Hinich 14, Proposition 1.5.1), but it is not completely proved there – see (Mazel-Gee 16, Remark 2.3).)

                                      For simplicial model categories with sSet-enriched Quillen adjunctions between them, this is also in (Lurie, prop. 5.2.4.6).

                                      See also at derived functor – As functors on infinity-categories

                                      References

                                      See the references at model category. For instance

                                      The proof that a Quillen adjunction of model categories induces an adjunction of (infinity,1)-categories is recorded in

                                      and this question is also partially addressed in

                                      The case for simplicial model categories is also in

                                      Last revised on July 12, 2018 at 07:49:05. See the history of this page for a list of all contributions to it.