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.

Contents

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

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

  1. Johnstone 1977
  2. "On a Topological Topos at The n-Category Café". golem.ph.utexas.edu.
  3. https://page.math.tu-berlin.de/~roch/files/abelian_categories.pdf
  4. "Is an additive category a balanced category?". MathOverflow.
  5. "Is every balanced pre-abelian category abelian?". MathOverflow.

Further reading