nLab
eso morphism

Definition

In any 2-category $K$, a morphism $f:A\to B$ is called eso if for any ff morphism $m:C\to D$, the following square is a (2-categorical) pullback in Cat:

$$\begin{array}{ccc}K(B,C)& \to & K(B,D)\\ \downarrow & & \downarrow \\ K(A,C)& \to & K(A,D)\end{array}$$\array{K(B,C) & \to & K(B,D)\\\downarrow & & \downarrow \\ K(A,C) & \to & K(A,D)}

This can be rephrased in elementary terms, without the need for a category $\mathrm{Cat}$ in which the hom-categories of $K$ live.

One easily checks that when $K=$ Cat, a functor $f$ is eso if and only if it is essentially surjective on objects in the usual sense. (This requires either the axiom of choice or the use of anafunctors in defining $\mathrm{Cat}$.)

Remarks

• If $K$ has finite limits, then $f:A\to B$ is eso if and only if whenever $f\cong mg$ where $m$ is ff, then $m$ is an equivalence.

• Any coinserter, co-isoinserter, coinverter, coequifier, or (lax or oplax) codescent morphism is eso.

• If $K$ has finite limits and $f:A\to B$ is eso, then for any $Z$ the functor $K(B,Z)\to K(A,Z)$ is faithful and conservative.

• If $K$ is a 1-category with finite limits, regarded as a 2-category with only identity 2-cells, then a morphism in $K$ is eso if and only if it is an extremal epimorphism (equivalently, a strong epimorphism).