Balanced category

In mathematics, especially in category theory, a balanced category is a category in which every bimorphism (a morphism that is both a monomorphism and epimorphism) is an isomorphism.

The category of topological spaces is not balanced (since continuous bijections are not necessarily homeomorphisms), while a topos is balanced.^[1] This is one of the reasons why a topos is said to be nicer.^[2]

Examples

The following categories are balanced:

• Set, the category of sets.
• Grp, the category of groups.
• An abelian category.^[3]
• CHaus, the category of compact Hausdorff spaces (since a continuous bijection there is homeomorphic).

An additive category may not be balanced.^[4] Contrary to what one might expect, a balanced pre-abelian category may not be abelian.^[5]

A quasitopos is similar to a topos but may not be balanced.

See also

• quasi-abelian category