category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a connected category is a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

in which, for every two objects ''X'' and ''Y'' there is a finite sequence of objects :

X = X_0, X_1, \ldots, X_, X_n = Y

with morphisms :

f_i : X_i \to X_

or :

f_i : X_ \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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

from ''J'' to a

discrete category In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: :hom''C''(''X'', ''X'') = {id''X''} for all objects ''X'' :hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ '' ...

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 In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...

is connected

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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 In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

) of a collection of connected categories, which are called the connected components of ''J''. Each connected component is a

full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

of ''J''.

References

* Categories in category theory {{categorytheory-stub