diagonal morphism

The **diagonal** of an object $X$ in a category with products is the canonical morphism

$\Delta : X \stackrel{(Id,Id)}{\to} X \times X
\,.$

The dual concept is *codiagonal* .

Recall that the diagonal of a set is a subset of its cartesian square $X^2$. If $X$ is now an object in some cartesian monoidal category $C$, then the diagonal of $X$ is now a subobject of its categorial square $X^2$. (Actually, $C$ need not be cartesian monoidal, as long as the product $X \times X$ exists.)

Specifically, the **diagonal morphism** of $X$ is a morphism $\Delta_X: X \to X^2$ given (using the universal property of the product) by the identity morphism from $X$ to itself, taken twice. That is, $\Delta_X$ is the universal solution to

$\array {
& & X \\
& \swarrow & \downarrow _ { \Delta _ X } & \searrow \\
X & & X ^ 2 & & X \\
\downarrow _ { \id _ X } & \swarrow & & \searrow & \downarrow _ { \id _ X } \\
X & & & & X
}$

If $C$ is $Set$ (the category of sets), then this diagonal morphism is precisely the diagonal function of $X$.

The diagonal morphism is always a regular monomorphism, since it is the equaliser of the two projection maps $X^2 \to X$. (In fact, it is a split monomorphism, since it is also a section of either projection map.) Thus, it makes $X$ into a regular subobject of $X^2$, the **diagonal subobject**. When $C$ is the $Set$, this recovers the original notion of the diagonal subset of $X^2$.

In the category Set the diagonal $\Delta_X$ is the function $a \mapsto (a,a)$ for all $a \in X$. See diagonal subset.

In the category Top of topological spaces, an object $X$ is a Hausdorff space if and only if its diagonal subobject is a closed subspace of $X^2$; this fact can be generalised to other notions of space.

In Cat the diagonal morphisms are diagonal functors.

Revised on December 13, 2011 10:12:29
by Urs Schreiber
(82.169.65.155)