In

category theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a branch of

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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

functorIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

from

*J* to a

discrete categoryIn mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of 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 ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

its underlying graph is weakly connected.

Each category

*J* can be written as a disjoint union (or

coproductIn category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

) of a connected categories, which are called the

**connected components** of

*J*. Each connected component is a full subcategory of

*J*.