nLab
diagonal morphism

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Contents

Definition

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

Δ:X(Id,Id)X×X \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 XX:

X Id Δ Id X pr 1 X×X pr 2 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 2X^2. If XX is now an object in some cartesian monoidal category CC, then the diagonal of XX is now a subobject of its categorial square X 2X^2. (Actually, CC need not be cartesian monoidal, as long as the product X×XX \times X exists.)

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

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

If CC is SetSet (the category of sets), then this diagonal morphism is precisely the diagonal function of XX.

Properties

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

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

Examples

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

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

In Cat the diagonal morphisms are diagonal functors.

Last revised on July 10, 2021 at 11:36:10. See the history of this page for a list of all contributions to it.