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

Definition

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

$$\mathrm{Fun}(C,D):=\mathrm{SSet}(C,D)$$Fun(C,D) := SSet(C,D)

is the simplicial set of morphisms of simplicial sets between $C$ and $D$ (in the standard SSet-enrichment of $\mathrm{SSet}$).

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

Proposition

$\mathrm{Fun}(C,D)$ is indeed a quasi-category.

Remarks

• In fact, $\mathrm{Fun}(C,D)$ is already a quasi-category if only $D$ is a quasi-category, where $C$ may be any simplicial set. This is is useful for instance in the context of the (∞,1)-category of (∞,1)-presheaves, as that makes sense already over a domain which is an arbitrary simplicial set.

• For definition of $(\mathrm{\infty }\mathrm{,1})$-functors in other models for $(\mathrm{\infty }\mathrm{,1})$-categories see (∞,1)-functor.

Reference

section 1.2.7 of

• Jacob Lurie, Higher Topos Theory

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