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homotopy Kan extension

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Homotopy theory

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model category

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(,1)(\infty,1)-Limits and colimits

Contents

Idea

Homotopy Kan extensions are models/presentations for (∞,1)-Kan extensions – i.e. Kan extensions in an (∞,1)-category theory – in terms of homotopical category theory and enriched category theory.

As a special case they reduce to homotopy limits and homotopy colimits, which in turn are models for (∞,1)-categorical limits and colimits.

In combinatorial simplicial model categories

The following describes homotopy Kan extensions in the context of combinatorial simplicial model categories, i.e. sSet QuillensSet_{Quillen}-enriched model categories whose underlying ordinary model category is combinatorial. The discussion goes through verbatim also with sSet QuillensSet_{Quillen} replaced by any excellent model category.

Recollection of ordinary global Kan extensions

Recall the global definition of ordinary Kan extensions: for AA a category and p:CCp : C \to C' a functor between small categories, we have the functor categories [C,A][C,A] and [C,A][C',A] and precomposition with pp induces a functor

p *:[C,A][C,A]. p^* : [C',A] \to [C,A] \,.

If AA has all limits and colimits, then this functor has a left adjoint Lan pLan_p and a right adjoint Ran pRan_p

(Lan pp *Ran p):=(p !p *p *):[C,A]Ran pp *Lan p[C,A]. (Lan_p \dashv p^* \dashv Ran_p) := (p_! \dashv p^* \dashv p_*) : [C,A] \stackrel{\overset{Lan_p}{\to}}{\stackrel{\overset{p^*}{\leftarrow}}{\underset{Ran_p}{\to}}} [C',A] \,.

These are the left and right Kan extension functors.

The following definition is the straightforward evident generalization of this from plain categories to simplicial model categories.

Definition

Let AA be a combinatorial simplicial model category. Let C,CC, C' be small sSet-enriched model categories. Write [C,A][C,A] and [C,A][C',A] for the corresponding enriched functor categories. Notice that these carry the injective and the projective model structure on functors [C,A] inj[C,A]_{inj} and [C,A] proj[C,A]_{proj}, which themselves are combinatorial simplicial model categories.

Let

f:CC f : C \to C'

be an sSet-enriched functor. Let

f *:[C.A][C,A] f^* : [C'.A] \to [C,A]

be the sSet-enriched functor induced by precomposition with ff.

Definition/Proposition

The functor f *f^* has both an sSet-left adjoint f !f_! as well as a right adjoint f *f_*

(f !f *f *):[C,A]f *f *f ![C,A] (f_! \dashv f^* \dashv f_*) \; : \; [C,A] \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C',A]

and

If f:CCf : C \to C' is a weak equivalence in the model structure on sSet-categories then these are Quillen equivalences.

Now

  • the right derived functor

    Rf *:([C,A] proj) ([C,A] proj) R f_* : ([C,A]_{proj})^\circ \to ([C',A]_{proj})^\circ

    is the homotopy right Kan extension functor;

  • the left derived functor

    Lf !:([C,A] inj) ([C,A] inj) L f_{!} : ([C,A]_{inj})^\circ \to ([C',A]_{inj})^\circ

    is the homotopy left Kan extension functor.

For the special case that C=*C' = * we have

  • the right derived functor

    Rf *:([C,A] proj) ([*,A] proj) =A R f_* : ([C,A]_{proj})^\circ \to ([*,A]_{proj})^\circ =\simeq A^\circ

    is the homotopy limit functor;

  • the left derived functor

    Lf !:([C,A] inj) ([*,A] inj) =A L f_{!} : ([C,A]_{inj})^\circ \to ([*,A]_{inj})^\circ = A^\circ

    is the homotopy colimit functor.

Proof

The statement of the Quillen adjunctions appears as HTT, prop A.3.3.7. The statement about the Quillen equivalences as HTT, prop A.3.3.8.

Definition

Since intrinsically Kan extensions, as every universal construciton, are supposed to be only defined up to weak equivalence, it is sometimes useful to make the extra freedom of choosing any weakly equivalent object explicit by the following definition.

Given F[C,A]F \in [C,A] and G[C,A]G \in [C',A] and a morphism η:Gf *F\eta : G \to f_* F, we say that η\eta exhibits GG as a homotopy right Kan extension of FF if for some injectively fibrant replacement FF^F \to \hat F the composite morphism Gf *Ff *F^G \to f_* F \to f_* \hat F is a weak equivalence.

So f *F^f_* \hat F here is a homotopy Kan extension as produced by the derived functor, while GG may be a more general object, weakly equivalent to it.

Properties

Derived-hom into a ho-Kan extension is a ho-Kan extention

Recall that for F:CAF : C \to A an ordinary functor between ordinary categories, its ordinary limit lim F\lim_\leftarrow F is characterized by the fact that for every object aAa \in A the set Hom(a,lim F)Hom(a, \lim_\leftarrow F) is the limit in Set of the functor CAHom A(a,)SetC \to A \stackrel{Hom_A(a,-)}{\to} Set. So all ordinary limits are determined by limits in Set.

The analogous statement here is that all homotopy limits are determined by homotopy limits in sSet QuillensSet_{Quillen}.

Proposition

Let F[C,A]F \in [C,A] and G[C,A]G \in [C',A] be fibrant in the projective model structure on functors. Then a morphism η:Gf *F\eta : G \to f_* F exhibits GG as a homotopy right Kan extension of FF precisely if for each cofibrant aAa \in A – equivalently for each fibrant-cofibrant aAa \in A – the morphism

η a:A(a,G())A(a,f *F()) \eta_a : A(a,G(-)) \to A(a, f_* F(-))

exhibits A(a,G())[C,sSet]A(a,G(-)) \in [C',sSet] as a homotopy right Kan extension of A(a,F())[C,sSet]A(a,F(-)) \in [C,sSet].

Proof

This appears as HTT, prop. A.3.3.12.

First notice that a replacement FF^F \stackrel{\simeq}{\to} \hat F in [C,A]{inj} by a fibrant F^\hat F induces a weak equivalence A(a,F())A(a,F^())A(a,F(-)) \to A(a,\hat F(-)) for all cofibrant aAa \in A, since FF is assumed projectively fibrant and using the properties of derived hom-spaces in an enriched model category.

Therefore we may assume without loss of generality that FF is already injectively fibrant. Then it also follows that for all cofibrant aAa \in A we have that A(a,F())[C,sSet] injA(a,F(-)) \in [C,sSet]_{inj} is fibrant: because A(a,())A(a,(-)) having right lifting property against all acyclic cofibrations HHH \to H' in [C,sSet] inj[C,sSet]_{inj}

H A(a,F()) H \array{ H &\to & A(a,F(-)) \\ \downarrow & \nearrow \\ H' }

is equivalent, by the sSet-tensor-adjunction in AA, to FF itself having the right lifting property against the map from AH:cH(c)AA \cdot H : c \mapsto H(c)\cdot A to HAH' \cdot A

Ha F Ha. \array{ H\cdot a &\to & F \\ \downarrow & \nearrow \\ H'\cdot a } \,.

But since tensoring in AA with sSet is a left Quillen bifunctor by definition of enriched model category, we have that tensoring the cofibrant aa with an acyclic cofibration of simplicial sets produces an acaclic cofibration in AA, so that HaHaH \cdot a \to H'\cdot a is an acyclic cofibration in [C,sSet] inj[C,sSet]_{inj}. But by the previous remark FF is (can assumed to be) injectively fibrant, hence the lift exists. Hence A(a,F())A(a,F(-)) is indeed injectively fibrant.

With this in hand, we have now the following equivalent restatement of the claim:

η\eta is a weak equivalence precisely if η a\eta_a is for all cofibrant (or cofibrant and fibrant) aa.

The implication (ηwe)(η awe)(\eta we) \Rightarrow (\eta_a we) follows because in the enriched model category AA, the functor A(a,F())A(a,F(-)) out of the cofibrant objectwise aa into the fibrant F()F(-) preserves weak equivalences.

Conversely, if η a:A(a,G())A(a,f *F())\eta_a: A(a,G(-)) \to A(a,f_* F(-)) is a weak equivalence for all fibrant and cofibrant aa, then for all cCc \in C η(c):G(c)f *F(c)\eta(c) : G(c) \to f_* F(c) is a weak equivalence for all cCc \in C by the Yoneda lemma, for instance in the Ho(sSet)Ho(sSet)-enriched homotopy category Ho(A)Ho(A) of AA: a morphism in Ho(A)Ho(A) is an iso if homming all other objects into it produces an isomorphism.

Remark

Notice that the statement makes sense in the full sSetsSet-subcategory A A^\circ on fibrant-cofibrant objects of AA, without needing any further mentioning on the model category structure on AA, only that on sSet QuillensSet_{Quillen} is involved. This allows to define homotopy Kan extensions in arbitrary Kan-complex enriched categories, which may or may not arise as A A^\circ for A a simplicial model category. This is discussed below.

In Kan-complex enriched categories

We obtain a notion of homotopy Kan extension that does not depend on any model category structure or even on weak equivalences anymore, but takes place entirely just in Kan-complex enriched categories.

Definition

The above characterization of homotopy Kan extensions in simplicial combinatorial model categories AA in terms of homotopy Kan extensions in sSetsSet only involves hom-objects of the form A(a,c)A(a,c), where aa is cofibrant and cc is fibrant. So it involves only the derived hom-spaces of AA, which are Kan complexes.

Accordingly, this characterization makes sense for AA any locally fibrant sSet QuillensSet_{Quillen}-enriched category, i.e. for every Kan-complex-enriched category:

Definition

For AA a Kan complex-enriched category and f:CCf : C \to C' an enriched functor of small sSet-enriched categories, given F[C,A]F \in [C,A] and G[C,A]G \in [C',A] we say a morphism η:Gf *F\eta : G \to f_* F, exhibits GG as a homotopy right Kan extension if for all aAa \in A the morphism

η a:A(a,G())A(a,f *F()) \eta_a : A(a,G(-)) \to A(a,f_* F(-))

exhibits A(a,G()):CsSet QuillenA(a,G(-)) : C' \to sSet_{Quillen} as a homotopy right Kan extension of A(a,F()):CsSet QuillenA(a,F(-)) : C \to sSet_{Quillen}.

Homotopy limits and colimits

If the diagram category CC' is the terminal sSetsSet-category, the left and right homotopy Kan extension along f:C*f : C \to {*} is the homotopy limit and homotopy colimit, respectively.

Characterization in terms of hom-adjuncts

In thae case that we are homotopy Kan extending to the point, if η:Gf *F\eta : G \to f_* F exhibits a right homotopy Kan extension, GAG \in A is a single object of AA and by adjunction this corresponds to a natural transformation f *G=constGFf^* G = const G \to F, whose components are a collection of morphisms

{η c:GF(c)} cC \{ \eta_c : G \to F(c) \}_{c \in C}

in AA. Then

Corollary

The fact that η\eta exhibits a right homotopy Kan extension is equivalent to the statement that for all aAa \in A the morphism

A(a,G)lim A(a,F()) A(a, G) \to \lim_{\leftarrow} A(a,F(-))

induced by composing with the {η c}\{\eta_c\} exhibits A(a,G)A(a,G) as a homotopy limit of A(a,F())A(a,F(-)) in sSet QuillensSet_{Quillen}, in the above sense.

Since lim A(a,F())=[C,sSet](consta,F)\lim_{\leftarrow} A(a,F(-)) = [C,sSet](const a,F) is an isomorphissm, this in turn is equivalent to the statemeent that

A(a,G)[C,sSet](consta,F) A(a, G) \to [C,sSet](const a, F)

exhibits that homotopy Kan extension.

Analogously for homotopy colimits.

Relation to quasi-categorical limits and colimits

The above considerations can be used to show that under the homotopy coherent nerve, homotopy colimits in a Kan-complex enriched categories as defined above are quasi-categorical colimits:

Proposition

For CC and AA Kan-complex-enriched categories and F[C,A]F \in [C,A], a morphism η:Fconst q\eta : F \to const_q exhibits qAq \in A as a homotopy colimit in AA in the above sense precisely if for N(f):N(C)N(A)N(f) : N(C) \to N(A) the corresponding morphism of quasi-categories under the homotopy coherent nerve and N(f) :N(C) N(A)N(f)^\triangleright : N(C)^\triangleright \to N(A) the extension to cones given by η\eta, N(f) N(f)^{\triangleright} is a quasi-categorical colimit diagram.

Proof

This is HTT, theorem 4.2.4.1. Some details on the proof are discussed at limit in a quasi-category.

In terms of derivators

The notion of derivator is largely a tool for handling homotopy Kan extensions. See there for details.

Properties

Pointwise homotopy Kan extensions

Under suitable conditions (but typically) homotopy Kan extensions may be computed pointwise by homotopy colimits.

Discussion of pointwise homotopy Kan extensions in cofibration categories is in (Radulescu-Banu 06, theorem 9.6.5). This is reviewed in the context of model categories in (Cisinski 09, prop. 1.14). In the more general context of relative categories discussion is in (Gonzales 11, section 4).

See also at (∞,1)-Kan extension – Properties – Pointwise.

References

General theory of homotopy Kan extensions is discussed in

A list of basic properties is in

Pointwise homotopy Kan extensions are discussed in

  • Andrei Radulescu-Banu, Cofibrations in Homotopy Theory (arXiv:0610009)

  • Denis-Charles Cisinski,

    Images directes cohomologiques dans les catégories de modèles, Ann. Math.Blaise Pascal 10 (2003), 195–244.

    Locally constant functors, Math. Proc. Camb. Phil. Soc. (2009), 147, 593 (pdf)

  • Beatriz Rodriguez Gonzalez, section 4 of Realizable homotopy colimits (arXiv:1104.0646)

Revised on July 14, 2014 23:04:56 by Urs Schreiber (89.204.138.244)