Balanced category
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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.
See also
- quasi-abelian category
References
- ↑ Johnstone 1977
- ↑ "On a Topological Topos at The n-Category Café". https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html.
- ↑ § 2.1. in Sandro M. Roch, A brief introduction to abelian categories, 2020
- ↑ "Is an additive category a balanced category?". https://mathoverflow.net/questions/119821/is-an-additive-category-a-balanced-category.
- ↑ "Is every balanced pre-abelian category abelian?". https://mathoverflow.net/questions/41722/is-every-balanced-pre-abelian-category-abelian.
Sources
- Johnstone, P. T. (1977). Topos theory. Academic Press.
- Roy L. Crole, Categories for types, Cambridge University Press (1994)
Further reading
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