commutative square

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 $\square C$.

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.*

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 $CAT(2,C)$.