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. Grp, the category of groups. An abelian category.§ 2.1. in Sandro M. Roch, A brief introduction to abelian categories, 2020 CHaus, the category of compact Hausdorff 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.
See also quasi-abelian category
References
Sources
Roy L. Crole, Categories for types, Cambridge University Press (1994)
Further reading