Connected category
In category theory, a branch of mathematics, a connected category is a category in which, for every two objects X and Y there is a finite sequence of objects

with morphisms

or

for each 0 ≤ i < n (both directions are allowed in the same sequence). Equivalently, a category J is connected if each functor from J to a discrete category is constant. In some cases it is convenient to not consider the empty category to be connected. A stronger notion of connectivity would be to require at least one morphism f between any pair of objects X and Y. Any category with this property is connected in the above sense. A small category is connected if and only if its underlying graph is weakly connected, meaning that it is connected if one disregard the direction of the arrows. Each category J can be written as a disjoint union (or coproduct) of a collection of connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.

References

Categories in category theory