nLab diagonal morphism

Definition

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

$$\Delta \;\colon\; X \stackrel{(Id,Id)}{\longrightarrow} X \times X$$

which is induced, via the universal property of the Cartesian product, by the span whose two legs each are both the identity morphism on $X$:

$$\array{ && X \\ & {}^{\mathllap{Id}}\swarrow & \downarrow {}^{\mathrlap{\Delta}} & \searrow^{\mathrlap{Id}} \\ X &\underset{pr_1}{\longleftarrow}& X \times X &\underset{pr_2}{\longrightarrow}& X }$$

The dual concept is codiagonal .

In the absence of Cartesian products, or when intentionally disregarding them, diagonal morphisms may still be considered in a generalized sense in monoidal categories with diagonals.

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 the morphism $\Delta_X: X \to X^2$ to the cartesian product of $X$ with itself given (using the universal property of the cartesian 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 $X$.

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 $Set$, this recovers the original notion of the diagonal subset of $X^2$.

In any category with binary pullbacks, the kernel pair of the identity morphism $id$ on an object $X$ is the diagonal morphism $(id,id)$ of $X$, and has a coequalizer isomorphic to $X$ itself.

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.

Related concepts

• fat diagonal

• monoidal category with diagonals

• formal neighbourhood of the diagonal

• diagonal of a bisimplicial set