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# commutative diagram

###### Definition 1.

Let $\mathcal{C}$ be a category. A *diagram* in $\mathcal{C}$ is a
directed graph $\Gamma$ with vertex set $V$ and edge set $E$, (“loops”
and “parallel edges” are allowed) together with two maps
$o\colon\thinspace V\to\mathrm{Obj}(\mathcal{C})$, $m\colon\thinspace E\to\mathrm{Morph}(\mathcal{C})$ such that
if $e\in E$ has source $s(e)\in V$ and target $t(e)\in V$ then
$m(e)\in\text{Hom}_{{\mathcal{C}}}\left(o\left(s(e)\right),o\left(t(e)\right)\right)$.

Usually diagrams are denoted by drawing the corresponding graph and labeling its vertices (respectively edges) with their images under $o$ (respectively $m$), for example if $f\colon\thinspace A\to B$ is a morphism

$\xymatrix@1{{A}\ar[r]^{f}&{B}}$ |

is a diagram. Often (as in the previous example) the vertices themselves are not drawn since their position can be deduced by the position of their labels.

###### Definition 2.

Let $D=(\Gamma,o,m)$ be a diagram in the category $\mathcal{C}$ and $\gamma=(e_{1},\ldots,e_{n})$
be a path in $\Gamma$. Then the *composition along* $\gamma$ is the following
morphism of $\mathcal{C}$

$\circ(\gamma):=m(e_{n})\circ\cdots\circ m(e_{1})\,.$ |

We say that $D$ is
*commutative* or that it *commutes* if for any two objects in
the image of $o$, say $A=o(v_{1})$ and $B=o(v_{2})$, and any two paths $\gamma_{1}$
and $\gamma_{2}$ that connect $v_{1}$ to $v_{2}$ we have

$\circ(\gamma_{1})=\circ(\gamma_{2})\,.$ |

For example the commutativity of the triangle

$\xymatrix{{A}\ar[rr]^{{f}}\ar[dr]_{{h}}&&{B}\ar[dl]^{{g}}\\ &{C}&}$ |

translates to $h=g\circ f$, while the commutativity of the square

$\xymatrix{{A}\ar[r]^{{f}}\ar[d]_{{k}}&{B}\ar[d]^{{g}}\\ {C}\ar[r]_{{h}}&{D}}$ |

translates to $g\circ f=h\circ k$.

## Mathematics Subject Classification

18A10*no label found*18A05

*no label found*

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