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