Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
In any 2-category , a morphism is called eso, or strong 1-epic, if for any fully faithful morphism , the following square is a (2-categorical) pullback in Cat:
This can be rephrased in elementary terms, without the need for a category in which the hom-categories of live.
One easily checks that when Cat, a functor 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 .)
If has finite limits, then is eso if and only if whenever where is ff, then is an equivalence.
Any coinserter, co-isoinserter, coinverter, coequifier, or (lax or oplax) codescent object? is eso.
If has finite limits and is eso, then for any the functor is faithful and conservative.
If is a 1-category with finite limits, regarded as a 2-category with only identity 2-cells, then a morphism in 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