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.

The category of rings is not balanced; $Z\hookrightarrow Q$ is monic and epic but not an isomorphism.

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.”

Revised on October 7, 2014 10:14:53
by Ingo Blechschmidt
(137.250.162.16)