In category theory
, a branch of mathematics
, a connected category
is a category
in which, for every two objects X
there is a finite sequence
for each 0 ≤ i
(both directions are allowed in the same sequence). Equivalently, a category J
is connected if each functor
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.