Contents

category theory

# Contents

## Idea

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.

## Properties

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.

## References

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

Last revised on May 27, 2020 at 12:29:43. See the history of this page for a list of all contributions to it.