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.

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

See also

• quasi-abelian category

References

• P.T. Johnstone,"Topos theory", Academic Press (1977)
• Roy L. Crole, Categories for types, Cambridge University Press (1994)

Further reading

• https://ncatlab.org/nlab/show/balanced+category
• https://mathoverflow.net/questions/41722/is-every-balanced-pre-abelian-category-abelian