eso morphism

Eso morphisms


In any 2-category KK, a morphism f:ABf:A\to B is called eso, or strong 1-epic, if for any fully faithful morphism m:CDm:C\to D, the following square is a (2-categorical) pullback in Cat:

K(B,C) K(B,D) K(A,C) K(A,D)\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 CatCat in which the hom-categories of KK live.

One easily checks that when K=K= Cat, a functor ff 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 CatCat.)


  • If KK has finite limits, then f:ABf:A\to B is eso if and only if whenever fmgf\cong m g where mm is ff, then mm is an equivalence.

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

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

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

Revised on March 13, 2012 01:50:13 by Toby Bartels (