nLab
commutative square

Definition and notation

Let $C$ be a category. A square of morphisms of $C$ consists of objects $X,Y,Z,W$ of $C$ and morphisms $f:X\to Z$, $g:X\to Y$, $f\prime :Y\to W$, and $g\prime :Z\to W$. This is often pictured as a square

$$\begin{array}{ccccc}& X& \stackrel{f}{\to }& Z& \\ g& \downarrow & & \downarrow & g\prime \\ & Y& \underset{f\prime }{\to }& W& \\ \end{array}$$\array{& X & \overset{f}\rightarrow & Z & \\ g & \downarrow &&\downarrow & g'\\ &Y & \underset{f'}\rightarrow& W & \\ }

The square is commutative if $y\prime \circ f=f\prime \circ g$.

The class of commutative squares in $C$ is written $\square C$.

Structure

This class has partial compositions ${\circ }_1$ and ${\circ }_2$ which are vertical and horizontal:

$$\begin{array}{cccc}•& \to & •& \\ \downarrow & & \downarrow \\ •& \to & •\\ \downarrow & & \downarrow \\ •& \to & •\end{array}\begin{array}{ccccc}•& \to & •& \to & •\\ \downarrow & & \downarrow & & \downarrow \\ •& \to & •& \to & •\end{array}$$\array{ \bullet & {\to} & \bullet & \\ \downarrow &&\downarrow \\ \bullet & {\to}& \bullet \\ \downarrow & & \downarrow\\ \bullet & \to & \bullet } \quad \quad \array{\bullet & {\to} & \bullet & \to & \bullet \\ \downarrow &&\downarrow && \downarrow \\ \bullet & {\to}& \bullet & \to & \bullet }

thus forming a (strict) double category, also written $\square C$. It contains the vertical category ${\square }_{1}C$ and the horizontal category ${\square }_{2}C$.

One can also form multiple compositions $[a_{\mathrm{ij}}]$ of arrays $(a_{\mathrm{ij}})$, $i=1,\dots ,m;j=1,\dots ,n$, of commutative squares provided that in the obvious sense adjacent squares are composible. One checks by induction that:

any composition of commutative squares is commutative.

Applications

Let $2$ denote the walking arrow: the category with two objects $0,1$ and one arrow $0\to 1$. This has the structure of cocategory. Then the class of commutative squares in $C$ can also be described as $\mathrm{Cat}(2\times 2,C)$.

If $D$ is a category, then $\mathrm{Cat}(D,{\square }_{1}C)$ can be regarded as the class of natural transformations of functors $D\to C$. Then the category structure ${\square }_{2}C$ induces a category structure on $\mathrm{Cat}(D,{\square }_{1}C)$ giving the functor category $\mathrm{CAT}(D,C)$: the category of functors and natural transformations. (This account is due to Charles Ehresmann.)

One deduces that if also $E$ is a category then there is a natural bijection

$$\mathrm{Cat}(E\times D,C)\cong (E,\mathrm{CAT}(D,C)),$$Cat(E \times D, C) \cong (E, CAT(D,C)),

which thus states that the category of (small if you like!) categories is cartesian closed.

The commutative squares serve as the morphisms in the arrow category of $C$, which is the functor category $\mathrm{CAT}(2,C)$.