A functor $F:C\to D$ is **surjective on objects** if for each object $y\in D$ there is an object $x\in C$ such that $F(x)=y$.

This is of course not an invariant notion; the corresponding invariant notion is essentially surjective functor. However, surjective-on-objects functors are sometimes useful when doing strict 2-category theory.

Surjective-on-objects functors are the left class in an orthogonal factorization system on Cat whose right class consists of the injective-on-objects and fully faithful functors.

**basic properties of…**

- Ross Street,
*Two-dimensional sheaf theory*refers to surjective-on-objects functors as*acute*and their corresponding right class as*chronic*.

Last revised on November 3, 2021 at 17:21:37. See the history of this page for a list of all contributions to it.