(infinity,1)-Yoneda extension

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

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.

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

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

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

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

We prove this in two steps.

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

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

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

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

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

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

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

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

preserves cofibrations and acyclic cofibrations.

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

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

given by

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

This is a standard argument.

We demonstrate the Hom-isomorphism that characterizes the adjunction:

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

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

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

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

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

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

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

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

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

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

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)