# 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

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*. Any category with this property is connected in the above sense.

A small category is connected if and only if its underlying graph is weakly connected, meaning that it is connected if one disregard the direction of the arrows.

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*.

## References[edit]

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