nLab
(infinity,1)-functor

Contents

Idea

An (,1)-functor is a morphism between (∞,1)-categories.

The collection of all (,1)-functors between two (,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

In terms of quasi-categories

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

Fun(C,D):=sSet(C,D).Fun(C,D) := sSet(C,D) \,.

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

References

sectrion 1.2.7 in

discusses morphisms of quasi-categories.