nLab eso morphism (changes)

Showing changes from revision #6 to #7: Added | ~~Removed~~ | ~~Chan~~ged

Context

2-category theory

Definition Eso morphisms

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

Definition
Remarks

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

Definition

~~This~~ In ~~can~~ any be ~~rephrased~~ in ~~elementary~~ ~~terms,~~ ~~without~~ ~~the~~ ~~need~~ ~~for~~ a ~~category~~ ~~$Cat$~~ 2-category in ~~which~~ ~~the~~ ~~hom-categories~~ of $K$ , ~~live.~~ a morphism $f:A\to B$ is called eso, or strong 1-epic, if for any fully faithful morphism $m:C\to D$ , the following square is a (2-categorical) pullback in Cat:

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 $Cat$ .)

\array{K(B,C) & \to & K(B,D)\\ \downarrow & & \downarrow \\ K(A,C) & \to & K(A,D)}

~~Remarks~~

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

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

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

Remarks

If $K$ has finite limits, then $f:A\to B$ is eso if and only if whenever $f\cong m g$ where $m$ is ff, then $m$ is an equivalence.
Any coinserter, co-isoinserter, coinverter, coequifier, or (lax or oplax) codescent object 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).

Last revised on March 13, 2012 at 01:50:13. See the history of this page for a list of all contributions to it.