category theory

# Contents

## Definition

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

$\Delta :X\stackrel{\left(\mathrm{Id},\mathrm{Id}\right)}{\to }X×X\phantom{\rule{thinmathspace}{0ex}}.$\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 $X×X$ 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

$\begin{array}{ccc}& & X\\ & ↙& {↓}_{{\Delta }_{X}}& ↘\\ X& & {X}^{2}& & X\\ {↓}_{{id}_{X}}& ↙& & ↘& {↓}_{{id}_{X}}\\ X& & & & X\end{array}$\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 $a↦\left(a,a\right)$ 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)