Definition:Commutative Diagram

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Definition

A commutative diagram is a graphical technique designed to illustrate the construction of composite mappings.

It is also a widespread tool in category theory, where it deals with morphisms instead of mappings.

$\begin{xy} \xymatrix@L+2mu@+1em{
S_1 \ar[r]^*{f_1}
    \ar@{-->}[rd]_*{f_2 \circ f_1}

& 

S_2 \ar[d]^*{f_2}

\\ & 

S_3

}\end{xy}$

It consists of:

$(1): \quad$ A collection of points representing the various domains and codomains of the mappings in question
$(2): \quad$ Arrows representing the mappings themselves.


The diagram is properly referred to as commutative if and only if all the various paths from the base of one arrow to the head of another represent equal mappings.

A mapping which is uniquely determined by the rest of the diagram may be indicated by a dotted arrow.

It is however generally advisable not to use more than one dotted arrow per diagram, so as to avoid confusion.


Relations

A similar technique can be applied to composition of relations in general, but this is rarely seen.


Also known as

Some sources refer to a diagram of this nature as a function diagram.

Some refer to it as just a diagram.


Examples

Arbitrary Example

Let $A, B, X, Y$ be arbitrary sets.

Let:

\(\ds f: \, \) \(\ds A\) \(\to\) \(\ds X\)
\(\ds g: \, \) \(\ds B\) \(\to\) \(\ds Y\)
\(\ds \alpha: \, \) \(\ds A\) \(\to\) \(\ds B\)
\(\ds \beta: \, \) \(\ds X\) \(\to\) \(\ds Y\)

be mappings such that:

$\beta \circ f = g \circ \alpha = k$

where $\circ$ denotes composition of mappings.


This can be depicted using the following commutative diagram:

$\begin{xy} \xymatrix@L+2mu@+1em{
A \ar[r]^*{\alpha} \ar[d]_*{f} \ar[rd]^*{k} & B \ar[d]^*{g}

\\ 

X \ar[r]^*{\beta} & Y

}\end{xy}$


Square Function with Square Root

Let $g$ and $h$ be the real functions defined as:

$\forall x \in \R: \map g x = x^2$
$\forall x \in \R_{\ge 0}: \map h x = \sqrt x$


The composition $h \circ g$ can be depicted using a commutative diagram as follows:

$\begin{xy} \xymatrix@L+2mu@+1em{
\R \ar[r]^*{g}
    \ar@{-->}[rd]_*{h \circ g}

& 

\R_{\ge 0} \ar[d]^*{h}

\\ & 

\R_{\ge 0}

}\end{xy}$


Also see

  • Results about commutative diagrams can be found here.


Sources

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