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. Clearly, any category which this property is connected in the above sense.
A small category is connected if and only if its underlying graph is weakly connected.
Each category J can be written as a disjoint union (or coproduct) of a connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.
Famous quotes containing the words connected and/or category:
“Religious fervor makes the devil a very real personage, and anything awe-inspiring or not easily understood is usually connected with him. Perhaps this explains why, not only in the Ozarks but all over the State, his name crops up so frequently.”
—Administration in the State of Miss, U.S. public relief program (1935-1943)
“I see no reason for calling my work poetry except that there is no other category in which to put it.”
—Marianne Moore (1887–1972)