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).
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.
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.
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