• Source: 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


,
…
,

X

n
−
1


,

X

n


=
Y


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


with morphisms





f

i


:

X

i


→

X

i
+
1




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


or





f

i


:

X

i
+
1


→

X

i




{\displaystyle 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. 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


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

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