The generalization of the notion of functor category from category theory to (∞,1)-higher category theory.
Let and be (∞,1)-categories, taken in their incarnation as quasi-categories. Then
is the simplicial set of morphisms of simplicial sets between and (in the standard sSet-enrichment of ):
The objects in are the (∞,1)-functors from to , the morphisms are the corresponding natural transformations or homotopies, etc.
Using that sSet is a closed monoidal category the horn filling conditions
are equivalent to
Here the vertical map is inner anodyne for inner horn inclusions , and hence the lift exists whenever has all inner horn fillers, hence when is a quasi-category.
For the definition of -functors in other models for -categories see (∞,1)-functor.
The projective and injective global model structure on functors as well as the Reedy model structure if is a Reedy category presents -categories of -functors, at least when there exists a combinatorial simplicial model category model for the codomain.
Write for the homotopy coherent nerve. Since this is a right adjoint it preserves products and hence we have a canonical morphism
induced from the hom-adjunct of .
The fibrant-cofibrant objects of are enriched functors that in particular take values in fibrant cofibrant objects of . Therefore this restricts to a morphism
By the internal hom adjunction this corresponds to a morphism
Here is Kan complex enriched by the axioms of an - enriched model category, and so is a quasi-category, so that we may write this as
This canonical morphism
is an -equivalence in that it is a weak equivalence in the model structure for quasi-categories.
This is (Lurie, prop. 220.127.116.11).
The strategy is to show that the objects on both sides are exponential objects in the homotopy category of , hence isomorphic there.
That is an exponential object in the homotopy category is pretty immediate.
That the left hand is an isomorphic exponential follows from (Lurie, corollary A.3.4.12), which asserts that for and -enriched categories with cofibrant and as above, we have that composition with the evaluation map induces a bijection
Since this identifies also with the exponential object in question.
Limits and colimits
For an ordinary category that admits small limits and colimits, and for a small category, the functor category has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for -categories of -functors
Let and be quasi-categories, such that has all colimits indexed by .
Let be a small quasi-category. Then
The -category has all -indexed colimits;
A morphism is a colimiting cocone precisely if for each object the induced morphism is a colimiting cocone.
This is (Lurie, corollary 18.104.22.168).
A morphism in (that is, a natural transformation) is an equivalence if and only if each component is an equivalence in .
This is due to (Joyal, Chapter 5, Theorem C).
Between ordinary categories, it reproduces the ordinary category of functors.
Since the standard model structure on simplicial sets presents ∞ Grpd
the model structure on simplicial presheaves (more precisely and more generally the model structure on sSet-enriched presheaves) on the opposite (∞,1)-category presents the (∞,1)-category of (∞,1)-presheaves on :
The intrinsic definition is in section 1.2.7 of
The discussion of model category models is in A.3.4.
The theorem about equivalences is in