nLab strongly connected category

Definition

A category $C$ is strongly connected if it is inhabited and for any pair of objects $A,B$ there is a morphism $A \to B$. That is, each hom-set is inhabited.

This terminology is borrowed from graph theory: a directed graph is called strongly connected when it is inhabited and for every two vertices $u,v$ there is a directed path from $u$ to $v$. Hence, a category is strongly connected precisely when its underlying directed graph is strongly connected.

If we merely require that for all pairs of objects $A,B$ there is a morphism $A \to B$ or a morphism $B \to A$, we call $C$ semi-strongly connected. This is not standard terminology, though; in fact Bunge 1966 p 16 called this condition strongly connected.

In any case, both these conditions are indeed stronger than that on a connected category, where for each pair of objects there is only required to be a zig-zag of morphisms connecting them.

Properties

• Every strongly connected category is semi-strongly connected.

• Every semi-strongly connected category is a connected category.

• Every connected groupoid is strongly connected.

• Every inhabited category with zero morphisms is strongly connected.

Examples

Among semi-strongly connected categories are:

• the category of sets,

• every inhabited linear order, considered as a thin category.

Among strongly connected categories are:

• the category of abelian groups,

• the category of non-empty sets.

Counterexamples

Not semi-strongly connected are:

• the category of commutative rings,

• the empty category,

• the product category $Set \times Set$.

Related concepts

• connected category

References

The terminology was introduced in:

• Marta Bunge: Categories of set valued functors, PhD thesis, University of Pennsylvania (1966) [pdf]

  Reprints in Theory and Applications of Categories 30 (2024) 1-84 [tac:tr39abs, pdf]