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.
An abelian category.^[3]
The category of (Hausdorff) compact 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.

References

^ Johnstone 1977
^ "On a Topological Topos at The n-Category Café". golem.ph.utexas.edu.
^ § 2.1. in Sandro M. Roch, A brief introduction to abelian categories, 2020
^ "Is an additive category a balanced category?". MathOverflow.
^ "Is every balanced pre-abelian category abelian?". MathOverflow.

Sources

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

Balanced category

Examples

See also

References

Sources

Further reading