nLab balanced category

Definition

Definition

A category is balanced if every morphism which is both a monomorphism and an epimorphism is already an isomorphism.

Remark

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

Example

The category Set is balanced (Def. ).

More generally:

Example

Any topos and in fact any pretopos is balanced.

(eg. Johnstone 1977, Cor. 1.22)

Remark

Beware the counterexample: A quasitopos, need not be balanced.

Example

Any abelian category is balanced.

In particular categories Mod of modules and Vect of vector spaces are balanced.

Remark

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.

Example

The the category of groups is balanced (see here and here).

Remark

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.

Remark

Topological categories are rarely balanced. In Top, for example, the monic epimorphisms are the continuous bijections.

However:

Example

The category of compact Hausdorff spaces is balanced.

Remark

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.”

References

• 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]