nLab
diagonal morphism

Definition

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

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

The dual concept is codiagonal .

Details

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

$$\begin{array}{ccc}& & X\\ & \swarrow & {\downarrow }_{{\Delta }_X}& \searrow \\ X& & X^2& & X\\ {\downarrow }_{{id}_X}& \swarrow & & \searrow & {\downarrow }_{{id}_X}\\ X& & & & X\end{array}$$\array { & & X \\ & \swarrow & \downarrow _ { \Delta _ X } & \searrow \\ X & & X ^ 2 & & X \\ \downarrow _ { \id _ X } & \swarrow & & \searrow & \downarrow _ { \id _ X } \\ X & & & & X }

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

Properties

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 $\mathrm{Set}$, this recovers the original notion of the diagonal subset of $X^2$.

Examples

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.