> [!NOTE] Definition (Commutative Square) > A *commutative square* is a collection of four [[Sets|sets]] and four [[Function|maps]] $ \begin{CD} U_{1} @> \varphi_{1} >> U_{2}\\ @VV\chi_{1} V @VV \chi_{2} V\\ V_{1} @>\varphi_{2}>> V_{2} \end{CD} $so that $\chi_{2} \circ \varphi_{1} = \varphi_{2 } \circ \chi_{1}$. # Properties > [!NOTE] Lemma (Commutative squares can be glued together if they have a map in common) > If $\begin{CD} U_{1} @> \varphi_{1} >> U_{2}\\ @VV\chi_{1} V @VV \chi_{2} V\\ V_{1} @>\varphi_{2}>> V_{2} \end{CD} \quad \quad \quad > \begin{CD} U_{1} @> \varphi_{3} >> U_{2}\\ @VV\chi_{2} V @VV \chi_{3} V\\ V_{1} @>\varphi_{4}>> V_{2} \end{CD}$are two commutative square then $\begin{CD} U_{1} @> \varphi_{3}\, \circ \, \varphi_{1} >> U_{2}\\ @VV\chi_{1} V @VV \chi_{3} V\\ V_{1} @>\varphi_{4} \, \circ \, \varphi_{2}>> V_{2} \end{CD}$is also a commutative square. ^812a94 >Proof. Check that $\chi_{3} \circ \varphi_{3}\circ \varphi_{1} = \varphi_{4} \circ \varphi_{2} \circ \chi_{1}$. # Examples - See [[Linear maps]].