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.

Examples and non-examples

Any topos (in fact, any pretopos) is balanced. Of course, this includes Set. A quasitopos, however, need not be.

Any abelian category is balanced. An additive category need not be; for example in the category of torsion subgroup-free abelian groups, each nonzero homomorphism $\mathbb{Z} \to \mathbb{Z}$ is both monic and epic.

The category of rings is not balanced; $Z\hookrightarrow Q$ 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. However, the category of compact Hausdorff spaces is balanced.

In a poset or a quiver, or more generally any thin category, every morphism is both monic and epic; thus such categories are “as far as possible from being balanced.”

Last revised on November 5, 2021 at 10:14:27.
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