Schreiber (∞,1)-Yoneda extension

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 $\hat F : PSh(C) \to D$.

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

Statement

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.

Proposition

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 $F$.

Proof

We prove this in two steps.

Lemma

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

Proof

First notice that the Yoneda extension of $F : SPSh(C)_{proj}^{loc} \to \mathbf{D}$ 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.

Lemma

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

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

given by

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

Proof

This is a standard argument.

We demonstrate the Hom-isomorphism that characterizes the adjunction:

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

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

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.