#
nLab

surjective on objects functor

Contents
### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# 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.

## Related pages

## 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.
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