nLab
eso morphism

Eso morphisms

Definition

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.)

Remarks

Revised on March 13, 2012 01:50:13 by Toby Bartels (75.88.85.16)