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.<ref>{{harvnb|Johnstone|1977}} </ref> This is one of the reasons why a topos is said to be nicer.<ref>{{Cite web|url=https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html|title=On a Topological Topos at The n-Category Café|website=golem.ph.utexas.edu}}</ref>

== Examples == The following categories are balanced: *'''Set''', the category of sets. *'''Grp''', the category of groups. *An abelian category.<ref>§ 2.1. in Sandro M. Roch, [https://page.math.tu-berlin.de/~roch/files/abelian_categories.pdf A brief introduction to abelian categories], 2020</ref> *'''CHaus''', the category of compact Hausdorff spaces (since a continuous bijection there is homeomorphic).

An additive category may not be balanced.<ref>{{Cite web|url=https://mathoverflow.net/questions/119821/is-an-additive-category-a-balanced-category|title=Is an additive category a balanced category?|website=MathOverflow}}</ref> Contrary to what one might expect, a balanced pre-abelian category may not be abelian.<ref>{{Cite web|url=https://mathoverflow.net/questions/41722/is-every-balanced-pre-abelian-category-abelian|title=Is every balanced pre-abelian category abelian?|website=MathOverflow}}</ref>

A quasitopos is similar to a topos but may not be balanced.

== See also == *quasi-abelian category

== References == {{reflist}}

==Sources== * {{cite book| first = P. T. | last = Johnstone | title = Topos theory |publisher = Academic Press | date = 1977}} * Roy L. Crole, Categories for types, Cambridge University Press (1994)

== Further reading == * {{nlab|id=balanced+category|title=balanced category}}

Category:Category theory