nLab semipresheaf

Definition

A semipresheaf on a semicategory $C$ is a semifunctor

$$F : C^{op} \to Set$$

from the opposite semicategory $C^{op}$ of $C$ to the category Set of sets.

The category of semipresheaves on $C$, usually denoted $[C^{op},Set]_{semi}$, or just $[C^{op},Set]$ when the context is clear has:

• semifunctors $F : C^{op} \to Set$ as objects;

• natural transformations between such semifunctors as morphisms.