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Quillen adjunction in nLab

Contents

Context

Model category theory

model category, model ∞ \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)(\infty,1)-categories

Model structures

for ∞\infty-groupoids

for ∞-groupoids

for equivariant ∞\infty-groupoids

for rational ∞\infty-groupoids

for rational equivariant ∞\infty-groupoids

for nn-groupoids

for ∞\infty-groups

for ∞\infty-algebras

general ∞\infty-algebras

specific ∞\infty-algebras

for stable/spectrum objects

for (∞,1)(\infty,1)-categories

for stable (∞,1)(\infty,1)-categories

for (∞,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (∞,1)(\infty,1)-sheaves / ∞\infty-stacks

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

Proposition

The conditions in def. are indeed all equivalent.

Proof

Observe that

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

Properties

Derived adjunction

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 the right Quillen property of RR and hence the claim follows by 2-out-of-3.

For LL we apply the formally dual argument.

(Quillen 67, I.4 theorem 3)

Behaviour under Bousfield localization

Proposition

Given a Quillen adjunction

(L⊣R):𝒞⊥ Qu⟶R⟵L𝒟 (L \dashv R) \;\;\;\colon\;\;\; \mathcal{C} \underoverset {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{{}_{Qu}}}} \mathcal{D}

and S⊂Mor(𝒟)S \subset Mor(\mathcal{D}) is a set of morphisms such that

  1. the left Bousfield localization of 𝒟\mathcal{D} at SS exists,

  2. the derived image 𝕃L(S)\mathbb{L}L(S) of SS lands in the weak equivalences of 𝒞\mathcal{C},

then the Quillen adjunction descends to the Bousfield localization 𝒟 S\mathcal{D}_S

(L⊣R):𝒞⊥ Qu⟶R⟵L𝒟 S. (L \dashv R) \;\;\;\colon\;\;\; \mathcal{C} \underoverset {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{{}_{Qu}}}} \mathcal{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.

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).

This appears as HTT, cor. A.3.7.2.

See simplicial Quillen adjunction for more details.

Associated (infinity,1)-adjunction

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 May 13, 2023 at 13:38:57. See the history of this page for a list of all contributions to it.