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.