# Contents

## Idea

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 $\stackrel{^}{F}:\mathrm{PSh}\left(C\right)\to D$.

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

###### Proposition

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

$\stackrel{^}{F}:\mathrm{SPSh}\left(C{\right)}_{\mathrm{proj}}\stackrel{←}{\to }D:R\phantom{\rule{thinmathspace}{0ex}}.$\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 $Q:D\to D$ is given, the Yoneda extension $\stackrel{^}{QF}$ of $QF$ is an $\left(\infty ,1\right)$-extension up to weak equivalence of $F$.

### Proof

We prove this in two steps.

###### Lemma

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

###### Proof

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

$\stackrel{^}{F}:X↦{\int }^{U\in C}F\left(U\right)\cdot X\left(U\right)\phantom{\rule{thinmathspace}{0ex}},$\hat F : X \mapsto \int^{U \in C} F(U) \cdot X(U) \,,

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

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

$\int \left(-\right)\cdot \left(-\right):\left[C,D{\right]}_{\mathrm{inj}}\cdot \left[{C}^{\mathrm{op}},\mathrm{SSet}{\right]}_{\mathrm{proj}}\to D$\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 $F\left(U\right)$ is cofibrant for all $U$ we have that $\stackrel{^}{F}$ itself is cofibrant regarded as an object of $\left[C,D{\right]}_{\mathrm{inj}}$. From the definition of Quillen bifunctors it follows that

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

preserves cofibrations and acyclic cofibrations.

###### Lemma

The functor $\stackrel{^}{F}$ has an enriched right adjoint

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

given by

$R\left(A\right)=D\left(F\left(-\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\stackrel{^}{F}$

$D\left(\stackrel{^}{F}\left(X\right),A\right)\simeq D\left({\int }^{U\in S}F\left(U\right)\cdot X\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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:

$\cdots \simeq {\int }_{U\in S}D\left(\stackrel{^}{F}\left(U\right)\cdot X\left(U\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\simeq {\int }_{U\in S}\mathrm{SSet}\left(X\left(U\right),D\left(F\left(U\right),A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\simeq \int_{U \in S} SSet( X(U), \mathbf{D}(F(U),A)) \,.

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

$\cdots \simeq \left[{C}^{\mathrm{op}},\mathrm{SSet}\right]\left(X,D\left(\Pi \left(-\right),A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq [C^{op},SSet](X, \mathbf{D}(\Pi(-), A)) \,.

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

$\cdots =\left[{C}^{\mathrm{op}},\mathrm{SSet}\right]\left(X,R\left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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)