Commutative Diagram/Examples/Arbitrary Example 1

Example of Commutative Diagram

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:

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

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): commutative diagram
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): commutative diagram