nLab
(infinity,1)-functor

Idea

An $(\mathrm{\infty }\mathrm{,1})$-functor is a morphism between (∞,1)-categories.

The collection of all $(\mathrm{\infty }\mathrm{,1})$-functors between two $(\mathrm{\infty }\mathrm{,1})$-categories form an (∞,1)-category of (∞,1)-functors.

Definition

The details of the definition depend on the model chosen for (∞,1)-categories.

1. quasi-category

2. simplicially enriched category

3. Segal category

4. complete Segal space

Definition in terms of quasi-categories

For $C$ and $D$ quasi-categories the simplicial set of simplicial maps ${\mathrm{Hom}}_{\mathrm{SSet}}(C,D)$ from $C$ to $D$ is itself a quasi-category (for that it is sufficient that $D$ is a quasi-category). Therefore

$$\mathrm{Fun}(C,D):={\mathrm{Hom}}_{\mathrm{SSet}}(C,D).$$Fun(C,D) := Hom_{SSet}(C,D) \,.

These form the (infinity,1)-category of (infinity,1)-functors.

Definition in terms of simplicial sets

Definition in terms of Segal categories

Definition in terms of complete Segal spaces

References

sectrion 1.2.7 in

• Jacob Lurie, Higher Topos Theory

discusses morphisms of quasi-categories.