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 18:24:33 by Urs Schreiber (131.211.241.101)