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".