Connected category explained

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 morphismsf _{i} : X _{i} \to X _{i+1} orf _{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. 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 disregards 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

• Book: Mac Lane, Saunders . Saunders Mac Lane

. Saunders Mac Lane . 1998 . . Graduate Texts in Mathematics 5 . 2nd . Springer-Verlag . 0-387-98403-8.