In ordinary category theory the Yoneda extension of a functor is its left Kan extension through the Yoneda embedding of its domain to a functor .
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 be a category and write for the projective model structure on simplicial presheaves on . Let be any combinatorial simplicial model category.
Accordingly, if does not take values in cofibrant objects but where a cofibrant replacement functor is given, the Yoneda extension of is an -extension up to weak equivalence of .
We prove this in two steps.
The Yoneda extension preserves cofibrations and acyclic cofibrations.
Recall that the Yoneda extension of is given by the coend formula
where in the integrand we have the tensoring of the object by the simplicial set .
The lemma now rests on the fact that this coend over the tensor
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 is cofibrant for all we have that itself is cofibrant regarded as an object of . From the definition of Quillen bifunctors it follows that
preserves cofibrations and acyclic cofibrations.
The functor has an enriched right adjoint
This is a standard argument.
We demonstrate the Hom-isomorphism that characterizes the adjunction:
Start with the above coend description of
Then use the continuity of the enriched Hom-functor to pass it through the coend and obtain the following end:
The defining property of the tensoring operation implies that this is equivalent to
But this is the end-formula for the -object of natural transformations between simplicial presheaves:
By definition this is the desired right hand of the hom isomorphism
These two lemmas together constitute the proof of the proposition.