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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