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Balanced category

From Wikipedia, the free encyclopedia

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

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The following categories are balanced

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

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References

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  1. ^ Johnstone 1977
  2. ^ "On a Topological Topos at The n-Category Café". golem.ph.utexas.edu.
  3. ^ § 2.1. in Sandro M. Roch, A brief introduction to abelian categories, 2020
  4. ^ "Is an additive category a balanced category?". MathOverflow.
  5. ^ "Is every balanced pre-abelian category abelian?". MathOverflow.

Sources

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

Further reading

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