nLab
(infinity,1)-Yoneda extension

Contents

Idea

In ordinary category theory the Yoneda extension of a functor F:CDF : C \to D is its left Kan extension through the Yoneda embedding of its domain to a functor F^:PSh(C)D\hat F : PSh(C) \to D.

In higher category theory there should be a corresponding version of this construction.

In particular with categories replaced by (∞,1)-catgeories there should be a version with the category of presheaves replaced by a (∞,1)-category of (∞,1)-presheaves, corresponding to the Yoneda lemma for (∞,1)-categories.

This in turn should have a presentation in terms of the global model structure on simplicial presheaves.

Statement

Urs Schreiber: this here is something I thought about. Check. Even to the extent that this is right, it is clearly not yet a full answer, but at best a step in the right direction.

Let CC be a category and write [C op,SSet] proj=SPSh(C) proj[C^{op}, SSet]_{proj} = SPSh(C)_{proj} for the projective model structure on simplicial presheaves on CC. Let D\mathbf{D} be any combinatorial simplicial model category.

Proposition

If FF takes values in cofibrant objects of D\mathbf{D} then the SSet-enriched Yoneda extension F^\hat F of FF is the left adjoint part of an SSet-Quillen adjunction

F^:SPSh(C) projD:R. \hat F : SPSh(C)_{proj} \stackrel{\leftarrow}{\to} \mathbf{D} : R \,.

Accordingly, if FF does not take values in cofibrant objects but where a cofibrant replacement functor Q:DDQ : \mathbf{D} \to \mathbf{D} is given, the Yoneda extension QF^\widehat{Q F} of QFQ F is an (,1)(\infty,1)-extension up to weak equivalence of FF.

Proof

We prove this in two steps.

Lemma

The Yoneda extension F:SPSh(C) projDF : SPSh(C)_{proj} \to \mathbf{D} preserves cofibrations and acyclic cofibrations.

Proof

Recall that the Yoneda extension of F:SPSh(C) projDF : SPSh(C)_{proj} \to \mathbf{D} is given by the coend formula

F^:X UCF(U)X(U), \hat F : X \mapsto \int^{U \in C} F(U) \cdot X(U) \,,

where in the integrand we have the tensoring of the object F(U)DF(U) \in \mathbf{D} by the simplicial set X(U)X(U).

The lemma now rests on the fact that this coend over the tensor

()():[C,D] inj[C op,SSet] projD \int (-)\cdot (-) : [C,\mathbf{D}]_{inj} \cdot [C^{op}, SSet]_{proj} \to \mathbf{D}

is a Quillen bifunctor using the injective and projective global model structure on functors as indicated. This is HTT prop. A.2.9.26 rmk. A.2.9.27 and recalled at Quillen bifunctor.

Since by assumption F(U)F(U) is cofibrant for all UU we have that F^\hat F itself is cofibrant regarded as an object of [C,D] inj[C,\mathbf{D}]_{inj}. From the definition of Quillen bifunctors it follows that

F^= UF(U)()(U):SPSh(C) projD \hat F = \int^U F(U) \cdot (-)(U) : SPSh(C)_{proj} \to \mathbf{D}

preserves cofibrations and acyclic cofibrations.

Lemma

The functor F^\hat F has an enriched right adjoint

R:DSPSh(C) R : \mathbf{D} \to \mathrm{SPSh}(C)

given by

R(A)=D(F(),A). R(A) = \mathbf{D}(F(-), A) \,.
Proof

This is a standard argument.

We demonstrate the Hom-isomorphism that characterizes the adjunction:

Start with the above coend description of F^\hat F

D(F^(X),A)D( USF(U)X(U),A). \mathbf{D}({\hat F}(X), A) \simeq \mathbf{D}( \int^{U \in S} F(U) \cdot X(U) , A ) \,.

Then use the continuity of the enriched Hom-functor to pass it through the coend and obtain the following end:

USD(F^(U)X(U),A). \cdots \simeq \int_{U \in S} \mathbf{D}({\hat F}(U) \cdot X(U), A) \,.

The defining property of the tensoring operation implies that this is equivalent to

USSSet(X(U),D(F(U),A)). \simeq \int_{U \in S} SSet( X(U), \mathbf{D}(F(U),A)) \,.

But this is the end-formula for the SSetSSet-object of natural transformations between simplicial presheaves:

[C op,SSet](X,D(Π(),A)). \cdots \simeq [C^{op},SSet](X, \mathbf{D}(\Pi(-), A)) \,.

By definition this is the desired right hand of the hom isomorphism

=[C op,SSet](X,R(A)). \cdots = [C^{op}, SSet](X, R(A)) \,.

These two lemmas together constitute the proof of the proposition.

Created on November 9, 2009 at 18:09:16. See the history of this page for a list of all contributions to it.