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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.
The following describes homotopy Kan extensions in the context of combinatorial simplicial model categories, i.e. $sSet_{Quillen}$-enriched model categories whose underlying ordinary model category is combinatorial. The discussion goes through verbatim also with $sSet_{Quillen}$ replaced by any excellent model category.
Recall the global definition of ordinary Kan extensions: for $A$ a category and $p : C \to C'$ a functor between small categories, we have the functor categories $[C,A]$ and $[C',A]$ and precomposition with $p$ induces a functor
If $A$ has all limits and colimits, then this functor has a left adjoint $Lan_p$ and a right adjoint $Ran_p$
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.
Let $A$ be a combinatorial simplicial model category. Let $C, C'$ be small sSet-enriched model categories. Write $[C,A]$ and $[C',A]$ for the corresponding enriched functor categories. Notice that these carry the injective and the projective model structure on functors $[C,A]_{inj}$ and $[C,A]_{proj}$, which themselves are combinatorial simplicial model categories.
Let
be an sSet-enriched functor. Let
be the sSet-enriched functor induced by precomposition with $f$.
The functor $f^*$ has both an sSet-left adjoint $f_!$ as well as a right adjoint $f_*$
and
$(f_! \dashv f^*)$ is a Quillen adjunction for the projective model structure on functors
$(f^* \dashv f_*)$ is a Quillen adjunction for the injective model structure on functors
If $f : C \to C'$ is a weak equivalence in the model structure on sSet-categories then these are Quillen equivalences.
Now
the right derived functor
is the homotopy right Kan extension functor;
the left derived functor
is the homotopy left Kan extension functor.
For the special case that $C' = *$ we have
the right derived functor
is the homotopy limit functor;
the left derived functor
is the homotopy colimit functor.
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.
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 $F \in [C,A]$ and $G \in [C',A]$ and a morphism $\eta : G \to f_* F$, we say that $\eta$ exhibits $G$ as a homotopy right Kan extension of $F$ if for some injectively fibrant replacement $F \to \hat F$ the composite morphism $G \to f_* F \to f_* \hat F$ is a weak equivalence.
So $f_* \hat F$ here is a homotopy Kan extension as produced by the derived functor, while $G$ may be a more general object, weakly equivalent to it.
Recall that for $F : C \to A$ an ordinary functor between ordinary categories, its ordinary limit $\lim_\leftarrow F$ is characterized by the fact that for every object $a \in A$ the set $Hom(a, \lim_\leftarrow F)$ is the limit in Set of the functor $C \to A \stackrel{Hom_A(a,-)}{\to} Set$. 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 $sSet_{Quillen}$.
Let $F \in [C,A]$ and $G \in [C',A]$ be fibrant in the projective model structure on functors. Then a morphism $\eta : G \to f_* F$ exhibits $G$ as a homotopy right Kan extension of $F$ precisely if for each cofibrant $a \in A$ – equivalently for each fibrant-cofibrant $a \in A$ – the morphism
exhibits $A(a,G(-)) \in [C',sSet]$ as a homotopy right Kan extension of $A(a,F(-)) \in [C,sSet]$.
This appears as HTT, prop. A.3.3.12.
First notice that a replacement $F \stackrel{\simeq}{\to} \hat F$ in [C,A]{inj} by a fibrant $\hat F$ induces a weak equivalence $A(a,F(-)) \to A(a,\hat F(-))$ for all cofibrant $a \in A$, since $F$ 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 $F$ is already injectively fibrant. Then it also follows that for all cofibrant $a \in A$ we have that $A(a,F(-)) \in [C,sSet]_{inj}$ is fibrant: because $A(a,(-))$ having right lifting property against all acyclic cofibrations $H \to H'$ in $[C,sSet]_{inj}$
is equivalent, by the sSet-tensor-adjunction in $A$, to $F$ itself having the right lifting property against the map from $A \cdot H : c \mapsto H(c)\cdot A$ to $H' \cdot A$
But since tensoring in $A$ with sSet is a left Quillen bifunctor by definition of enriched model category, we have that tensoring the cofibrant $a$ with an acyclic cofibration of simplicial sets produces an acaclic cofibration in $A$, so that $H \cdot a \to H'\cdot a$ is an acyclic cofibration in $[C,sSet]_{inj}$. But by the previous remark $F$ is (can assumed to be) injectively fibrant, hence the lift exists. Hence $A(a,F(-))$ is indeed injectively fibrant.
With this in hand, we have now the following equivalent restatement of the claim:
$\eta$ is a weak equivalence precisely if $\eta_a$ is for all cofibrant (or cofibrant and fibrant) $a$.
The implication $(\eta we) \Rightarrow (\eta_a we)$ follows because in the enriched model category $A$, the functor $A(a,F(-))$ out of the cofibrant objectwise $a$ into the fibrant $F(-)$ preserves weak equivalences.
Conversely, if $\eta_a: A(a,G(-)) \to A(a,f_* F(-))$ is a weak equivalence for all fibrant and cofibrant $a$, then for all $c \in C$ $\eta(c) : G(c) \to f_* F(c)$ is a weak equivalence for all $c \in C$ by the Yoneda lemma, for instance in the $Ho(sSet)$-enriched homotopy category $Ho(A)$ of $A$: a morphism in $Ho(A)$ is an iso if homming all other objects into it produces an isomorphism.
Notice that the statement makes sense in the full $sSet$-subcategory $A^\circ$ on fibrant-cofibrant objects of $A$, without needing any further mentioning on the model category structure on $A$, only that on $sSet_{Quillen}$ is involved. This allows to define homotopy Kan extensions in arbitrary Kan-complex enriched categories, which may or may not arise as $A^\circ$ for A a simplicial model category. This is discussed below.
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.
The above characterization of homotopy Kan extensions in simplicial combinatorial model categories $A$ in terms of homotopy Kan extensions in $sSet$ only involves hom-objects of the form $A(a,c)$, where $a$ is cofibrant and $c$ is fibrant. So it involves only the derived hom-spaces of $A$, which are Kan complexes.
Accordingly, this characterization makes sense for $A$ any locally fibrant $sSet_{Quillen}$-enriched category, i.e. for every Kan-complex-enriched category:
For $A$ a Kan complex-enriched category and $f : C \to C'$ an enriched functor of small sSet-enriched categories, given $F \in [C,A]$ and $G \in [C',A]$ we say a morphism $\eta : G \to f_* F$, exhibits $G$ as a homotopy right Kan extension if for all $a \in A$ the morphism
exhibits $A(a,G(-)) : C' \to sSet_{Quillen}$ as a homotopy right Kan extension of $A(a,F(-)) : C \to sSet_{Quillen}$.
If the diagram category $C'$ is the terminal $sSet$-category, the left and right homotopy Kan extension along $f : C \to {*}$ is the homotopy limit and homotopy colimit, respectively.
In thae case that we are homotopy Kan extending to the point, if $\eta : G \to f_* F$ exhibits a right homotopy Kan extension, $G \in A$ is a single object of $A$ and by adjunction this corresponds to a natural transformation $f^* G = const G \to F$, whose components are a collection of morphisms
in $A$. Then
The fact that $\eta$ exhibits a right homotopy Kan extension is equivalent to the statement that for all $a \in A$ the morphism
induced by composing with the $\{\eta_c\}$ exhibits $A(a,G)$ as a homotopy limit of $A(a,F(-))$ in $sSet_{Quillen}$, in the above sense.
Since $\lim_{\leftarrow} A(a,F(-)) = [C,sSet](const a,F)$ is an isomorphissm, this in turn is equivalent to the statemeent that
exhibits that homotopy Kan extension.
Analogously for homotopy 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:
For $C$ and $A$ Kan-complex-enriched categories and $F \in [C,A]$, a morphism $\eta : F \to const_q$ exhibits $q \in A$ as a homotopy colimit in $A$ in the above sense precisely if for $N(f) : N(C) \to N(A)$ the corresponding morphism of quasi-categories under the homotopy coherent nerve and $N(f)^\triangleright : N(C)^\triangleright \to N(A)$ the extension to cones given by $\eta$, $N(f)^{\triangleright}$ is a quasi-categorical colimit diagram.
This is HTT, theorem 4.2.4.1. Some details on the proof are discussed at limit in a quasi-category.
The notion of derivator is largely a tool for handling homotopy Kan extensions. See there for details.
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.
General theory of homotopy Kan extensions is discussed in
Jacob Lurie, section A.3.3 of Higher Topos Theory
William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories , volume 113 of Mathematical Surveys and Monographs
Bruno Kahn, Georges Maltsiniotis, Structures de Dérivabilité
Jean-Marc Cordier and Tim Porter, Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.
A list of basic properties is in
Pointwise homotopy Kan extensions are discussed in
Andrei Radulescu-Banu, Cofibrations in Homotopy Theory (arXiv:0610009)
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)