Balanced category explained

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.^[1]

Examples

The following categories are balanced

• Set, the category of sets.
• An abelian category.^[2]
• The category of (Hausdorff) compact spaces (since a continuous bijection there is homeomorphic).

An additive category may not be balanced.^[3] Contrary to what one might expect, a balanced pre-abelian category may not be abelian.^[4]

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

See also

• quasi-abelian category

Sources

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

Notes and References

1. Web site: On a Topological Topos at The n-Category Café. golem.ph.utexas.edu.
2. § 2.1. in Sandro M. Roch, A brief introduction to abelian categories, 2020
3. Web site: Is an additive category a balanced category?. MathOverflow.
4. Web site: Is every balanced pre-abelian category abelian?. MathOverflow.