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

$X=X_{0},X_{1},\ldots ,X_{{n-1}},X_{n}=Y$

with morphisms

$f_{i}:X_{i}\to X_{{i+1}}$

or

$f_{i}:X_{{i+1}}\to X_{i}$

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.

References

• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 ((2nd ed.) ed.). Springer-Verlag. ISBN 0-387-98403-8.