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 $f\colon X \to Z$, $g\colon X \to Y$, $f'\colon Y \to W$, and $g'\colon Z \to W$. This is often pictured as a square

$$\array{& X & \overset{f}\rightarrow & Z & \\ g & \downarrow &&\downarrow & g'\\ &Y & \underset{f'}\rightarrow& W & \\ }$$

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

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:

$$\array{ \bullet & {\to} & \bullet & \\ \downarrow &&\downarrow \\ \bullet & {\to}& \bullet \\ \downarrow & & \downarrow\\ \bullet & \to & \bullet } \quad \quad \array{\bullet & {\to} & \bullet & \to & \bullet \\ \downarrow &&\downarrow && \downarrow \\ \bullet & {\to}& \bullet & \to & \bullet }$$

thus forming a (strict) double category, also written $\square C$, whose objects are those of $C$, whose horizontal and vertical 1-cells are given by morphisms in $C$, and whose 2-cells exhibit commutative squares. It contains the vertical category $\square_1 C$ and the horizontal category $\square _2 C$.

One can also form multiple compositions $[a_{ij}]$ of arrays $(a_{ij})$, $i=1, \ldots, m; j=1, \ldots , n$, 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 $Cat(2 \times 2, C)$.

If $D$ is a category, then $Cat(D, \square_1 C)$ 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 $Cat(D,\square _1 C)$ giving the functor category $CAT(D,C)$: 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

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

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 $CAT(2,C)$.