• Source: 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. This is one of the reasons why a topos is said to be nicer.




Examples


The following categories are balanced:

Set, the category of sets.
Grp, the category of groups.
An abelian category.
CHaus, the category of compact Hausdorff spaces (since a continuous bijection there is homeomorphic).
An additive category may not be balanced. Contrary to what one might expect, a balanced pre-abelian category may not be abelian.
A quasitopos is similar to a topos but may not be balanced.




See also


quasi-abelian category




References






Sources


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




Further reading


balanced category at the nLab

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