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.
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.