nLab (infinity,1)-category of (infinity,1)-functors

Contents

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Contents

Idea

The generalization of the notion of functor category from category theory to (∞,1)-higher category theory.

Definition

Let $C$ and $D$ be (∞,1)-categories, taken in their incarnation as quasi-categories. Then

$Func(C,D) := sSet(C,D)$

is the simplicial set of morphisms of simplicial sets between $C$ and $D$ (in the standard sSet-enrichment of $SSet$):

$sSet(C,D) := [C,D] := ([n] \mapsto Hom_{sSet}(\Delta[n]\times C,D)) \,.$

The objects in $Fun(C,D)$ are the (∞,1)-functors from $C$ to $D$, the morphisms are the corresponding natural transformations or homotopies, etc.

Proposition

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.

Proof

Using that sSet is a closed monoidal category the horn filling conditions

$\array{ \Lambda[n]_i &\to& [C,D] \\ \downarrow & \nearrow \\ \Delta[n] }$

are equivalent to

$\array{ C \times \Lambda[n]_i &\to& D \\ \downarrow & \nearrow \\ C \times \Delta[n] } \,.$

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.

Properties

Models

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

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

$N(C) \times N([C,A]) \simeq N(C \times [C,A]) \stackrel{N(ev)}{\to} N(A)$

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

$N(C) \times N([C,A]^\circ) \stackrel{N_{hc}(ev)}{\to} N(A^\circ) \,.$

By the internal hom adjunction this corresponds to a morphism

$N([C,A]^\circ) \stackrel{}{\to} sSet(N_{hc}(C), N(A^\circ)) \,.$

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

$\cdots = Func(N(C), N(A^\circ)) \,.$
Proposition

This canonical morphism

$N([C,A]^\circ) \stackrel{}{\to} Func(N(C), N(A^\circ))$

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).

Proof

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

$Hom_{Ho(sSet Cat)}(D, [C,A]^\circ) \stackrel{\simeq}{\to} Hom_{Ho(sSet Cat)}(C \times D, A^\circ) \,.$

Since $Ho(sSet Cat_{Bergner}) \simeq Ho(sSet_{Joyal})$ this identifies also $N([C,A]^\circ)$ with the exponential object in question.

Limits and colimits

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

Propositon

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).

Equivalences

Proposition

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).

Examples

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