A category is balanced if every morphism which is both a monomorphism and an epimorphism is already an isomorphism.
The possibility of monic epics that are not isomorphisms does not survive any strengthening of “monic” or “epic.” Any monic extremal epimorphism is necessarily an isomorphism, and therefore so is any monic strong epimorphism or regular epimorphism (and dually). It follows that if all epics, or all monos, are extremal, then the category is automatically balanced.
In an “unbalanced” category it is frequently the case that the monomorphisms, the epimorphisms, or both, are not the “right” notion to consider and should be replaced by their extremal, strong, or regular counterparts.
The category Set is balanced (Def. ).
Beware the counterexample: A quasitopos, need not be balanced.
Any abelian category is balanced.
In particular categories Mod of modules and Vect of vector spaces are balanced.
An additive category need not be balanced: A counterexample is the category of torsion subgroup-free abelian groups, each nonzero homomorphism $\mathbb{Z} \to \mathbb{Z}$ is both monic and epic.
The the category of groups is balanced (see here and here).
Not all categories of algebraic structures are balanced.
As a counterexample, the category of rings is not balanced: $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is monic and epic but not an isomorphism.
On similar grounds, the category of commutative monoids is not balanced, as the inclusion $\mathbb{N} \hookrightarrow \mathbb{Z}$ is both monic and epic.
Topological categories are rarely balanced. In Top, for example, the monic epimorphisms are the continuous bijections.
The category of compact Hausdorff spaces is balanced.
In a free category on a directed graph, and also in any poset and generally in any thin category, every morphism is both monic and epic while only the identity morphisms are invertible; thus such categories are “as far as possible from being balanced.”
Peter Johnstone, Cor. 1.22 in: Topos theory, London Math. Soc. Monographs 10, Acad. Press (1977), Dover (2014)
Roy L. Crole, p. 115 of: Categories for types, Cambridge University Press (1994) [doi:10.1017/CBO9781139172707]
Last revised on January 17, 2024 at 14:33:30. See the history of this page for a list of all contributions to it.