# nLab essentially surjective functor

Essentially surjective functors

category theory

# Essentially surjective functors

## Idea

A functor $F\colon C \to D$ is essentially surjective, or essentially surjective on objects (sometimes abbreviated to eso), if it is surjective on objects “up to isomorphism”.

## Definition

$F\colon C \to D$ is essentially surjective if for every object $y$ of $D$, there exists an object $x$ of $C$ and an isomorphism $F(x) \cong y$ in $D$.

### In homotopy type theory

A functor $F : C \to D$ is essentially surjective if for all $y:D$ there merely exists an $x:C$ such that $F(x) \cong y$.

## Properties

• Strengthening the last example, there is an orthogonal factorization system (in the up-to-isomorphism strict sense) on $Cat$, in which eso functors are the left class and fully faithful functors are the right class.

This is an “up-to-isomorphism” version of the bo-ff factorization system, which is a 1-categorical orthogonal factorization system on $Cat$ in which the left class consists of bijective-on-objects functors. Thus the notion of essential surjectivity is a version of “bijective on objects” which does respect the principle of equivalence, i.e. the version which views Cat as a bicategory.

In particular, while a functor factors uniquely-up-to-isomorphism as a b.o. functor followed by a fully faithful one, it factors only uniquely-up-to-equivalence as an e.s.o. functor followed by a fully faithful one. Since b.o. functors are also e.s.o., any (eso,ff) factorization of some functor is equivalent to its (bo,ff) factorization.

• In any 2-category there is a notion of eso morphism which generalizes the essentially surjective functors in Cat. In a regular 2-category, these form a factorization system in a 2-category together with the ff morphisms.

Last revised on May 1, 2023 at 11:04:04. See the history of this page for a list of all contributions to it.