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.
- Grp, the category of groups.
- An abelian category.3
- CHaus, the category of compact Hausdorff 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
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
Sources
- Johnstone, P. T. (1977). Topos theory. Academic Press.
- Roy L. Crole, Categories for types, Cambridge University Press (1994)