PlanetMath: commutative diagram

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commutative diagram

(Definition)

Definition 1 Let $\mathcal{C}$ be a category. A diagram in $\mathcal{C}$ is a directed graph $\Gamma$ with vertex set

and edge set

, (“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 $m(e) \in$ 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 (respectively ), for example if $f\colon\thinspace A\to B$ is a morphism

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

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 $\mathcal{C}$

$\displaystyle \circ(\gamma ):=m(e_n)\circ\cdots\circ m(e_1)\,.$

We say that

is commutative or that it commutes if for any two objects in the image of

, say

and

, and any two paths $\gamma _1$ and $\gamma _2$ that connect

we have

$\displaystyle \circ(\gamma _1)=\circ(\gamma _2)\,.$

For example the commutativity of the triangle

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[rr]^{f}\ar[dr]_{h}&&{B}\ar[dl]^{g}\ &{C}& } } \end{xy}$

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

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\ {C}\ar[r]_{h}&{D} } } \end{xy}$

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

"commutative diagram" is owned by Dr_Absentius. [ full author list (2) ]

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Cross-references: square, translates, triangle, objects, commutative, composition, path, labels, morphism, images, vertices, labeling, graph, source, maps, edge, vertex, directed graph, category
There are 42 references to this entry.

This is version 9 of commutative diagram, born on 2003-02-02, modified 2006-10-15.
Object id is 3962, canonical name is CommutativeDiagram.
Accessed 6791 times total.

Classification:

AMS MSC:	18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
	18A10 (Category theory; homological algebra :: General theory of categories and functors :: Graphs, diagram schemes, precategories)

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