equivalences in/of $(\infty,1)$-categories
The generalization of the notion of functor category from category theory to (∞,1)-higher category theory.
Let $C$ and $D$ be (∞,1)-categories, taken in their incarnation as quasi-categories. Then
is the simplicial set of morphisms of simplicial sets between $C$ and $D$ (in the standard sSet-enrichment of $SSet$):
The objects in $Fun(C,D)$ are the (∞,1)-functors from $C$ to $D$, the morphisms are the corresponding natural transformations or homotopies, etc.
The simplicial set $Fun(C,D)$ is indeed a quasi-category.
In fact, for $C$ and $D$ any simplicial sets, $Fun(C,D)$ is a quasi-category if $D$ is a quasi-category.
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 $\Lambda[n]_i \hookrightarrow \Delta[n]$, and hence the lift exists whenever $D$ has all inner horn fillers, hence when $D$ is a quasi-category.
For the definition of $(\infty,1)$-functors in other models for $(\infty,1)$-categories see (∞,1)-functor.
The projective and injective global model structure on functors as well as the Reedy model structure if $C$ is a Reedy category presents $(\infty,1)$-categories of $(\infty,1)$-functors, at least when there exists a combinatorial simplicial model category model for the codomain.
Let
$C$ be a small sSet-enriched category;
$A$ a combinatorial simplicial model category and $A^\circ$ its full sSet-subcategory of fibrant cofibrant objects;
$[C,A]$ the sSet-enriched functor category equipped with either the injective or projective global model structure on functors – here: the injective or injective model structure on sSet-enriched presheaves – and $[C,A]^\circ$ its full sSet-subcategory on fibrant-cofibrant objects.
Write $N : sSet Cat \to sSet$ 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 $Id : [C,A] \to [C,A]$.
The fibrant-cofibrant objects of $[C,A]$ are enriched functors that in particular take values in fibrant cofibrant objects of $A$. Therefore this restricts to a morphism
By the internal hom adjunction this corresponds to a morphism
Here $A^\circ$ is Kan complex enriched by the axioms of an $sSet_{Quillen}$- enriched model category, and so $N(A^\circ)$ is a quasi-category, so that we may write this as
This canonical morphism
is an $(\infty,1)$-equivalence in that it is a weak equivalence in the model structure for quasi-categories.
This is (Lurie, prop. 4.2.4.4).
The strategy is to show that the objects on both sides are exponential objects in the homotopy category of $sSet_{Joyal}$, hence isomorphic there.
That $Func(N(C), N(A^\circ)) \simeq (N(A^\circ))^{N(C)}$ 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 $C$ and $D$ $sSet$-enriched categories with $C$ cofibrant and $A$ as above, we have that composition with the evaluation map induces a bijection
Since $Ho(sSet Cat_{Bergner}) \simeq Ho(sSet_{Joyal})$ this identifies also $N([C,A]^\circ)$ with the exponential object in question.
For $C$ an ordinary category that admits small limits and colimits, and for $K$ a small category, the functor category $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for $(\infty,1)$-categories of $(\infty,1)$-functors
Let $K$ and $C$ be quasi-categories, such that $C$ has all colimits indexed by $K$.
Let $D$ be a small quasi-category. Then
The $(\infty,1)$-category $Func(D,C)$ has all $K$-indexed colimits;
A morphism $K^\triangleright \to Func(D,C)$ is a colimiting cocone precisely if for each object $d \in D$ the induced morphism $K^\triangleright \to C$ is a colimiting cocone.
This is (Lurie, corollary 5.1.2.3).
A morphism $\alpha$ in $Func(D,C)$ (that is, a natural transformation) is an equivalence if and only if each component $\alpha_d$ is an equivalence in $C$.
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 $C^{op}$ presents the (∞,1)-category of (∞,1)-presheaves on $C$:
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