\begin{definition}[Definition:Commutative Diagram]
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''' {{iff}} 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.
\end{definition}