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. This is one of the reasons why a topos is said to be nicer.

Examples
The following categories are balanced
 * Set, the category of sets.
 * An abelian category.
 * The category of (Hausdorff) compact spaces (since a continuous bijection there is homeomorphic).

An additive category may not be balanced. Contrary to what one might expect, a balanced pre-abelian category may not be abelian.

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