Commutative Diagram/Examples

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Examples of Commutative Diagrams

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:

$\begin{xy} \xymatrix@L+2mu@+1em{ \R \ar[r]^*{g} \ar@{-->}[rd]_*{h \circ g} & \R_{\ge 0} \ar[d]^*{h} \\ & \R_{\ge 0} }\end{xy}$