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
[edit]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.
See also
[edit]References
[edit]- ^ 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
[edit]- Johnstone, P. T. (1977). Topos theory. Academic Press.
- Roy L. Crole, Categories for types, Cambridge University Press (1994)
Further reading
[edit]- balanced category at the nLab
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