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# connected category

Let $\mathcal{C}$ be a category. Two objects $A,B$ in $\mathcal{C}$ are said to be *joined* if there is a morphism with domain one object and codomain the other. In other words, $\hom(A,B)\cup\hom(B,A)\neq\varnothing$. Two objects $A,B$ are said to be *connected* if there is a finite sequence of objects in $\mathcal{C}$

$A=C_{1},C_{2},\ldots,C_{n}=B$ |

such that $C_{i},C_{{i+1}}$ are joined for $i=1,\ldots,n-1$.

A category is said to be *connected* if every pair of objects are connected, and *strongly connected* if every pair of objects are joined.

For example, every category with either an initial object or a terminal object is connected. If a category has a zero object, it is strongly connected.

A small category may be viewed as a graph or a digraph. Then the underlying graph of a small connected category is connected, and the underlying digraph of a small strongly connected category is strongly connected. Conversely, the free category freely generated a connected graph is connected, and the free category freely generated by a strongly connected digraph is strongly connected.

The relation (on objects of $\mathcal{C}$) of being joined is in general not an equivalence relation (it is reflexive and symmetric, but not transitive). Let us call this relation $R$. The relation of being connected, on the other hand, is an equivalence relation, and is the transitive closure $R^{*}$ of $R$. Therefore, we may partition the class of objects in $\mathcal{C}$ by $R^{*}$. Furthermore, $R^{*}$ induces an equivalence relation $R^{{\prime}}$ on the class of all morphisms in $\mathcal{C}$: for morphisms $f,g$, set

$fR^{{\prime}}g\qquad\mbox{iff}\qquad\operatorname{dom}(f)R^{*}\operatorname{% dom}(g).$ |

If $A$ is an object of $\mathcal{C}$, denote $[A]$ the equivalence class containing $A$ under $R^{*}$, together with the equivalence class containing $1_{A}$ under $R^{{\prime}}$. Then $[A]$ is a connected full subcategory of $\mathcal{C}$. $[A]$ is called a connected component of $\mathcal{C}$. Every small category can be expressed as the disjoint union of its connected components.

# References

- 1 S. Mac Lane, Categories for the Working Mathematician, Springer, New York (1971).

## Mathematics Subject Classification

18A10*no label found*

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