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Diagonal morphism

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated May 19, 2026

In category theory, a branch of mathematics, for every object A in every category {\mathcal {C}} where the product A\times A exists, there exists the diagonal morphism

\delta _{A}:A\rightarrow A\times A

satisfying

\pi _{k}\circ \delta _{A}=\operatorname {id} _{A} for k\in \{1,2\},

where \pi _{k} is the canonical projection morphism to the k-th component. The existence of this morphism is a consequence of the universal property that characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.

For concrete categories, the diagonal morphism can be simply described by its action on elements x of the object A. Namely, \delta _{A}(x)=\langle x,x\rangle, the ordered pair formed from x. The reason for the name is that the image of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism \mathbb {R} \rightarrow \mathbb {R} ^{2} on the real line is given by the line that is the graph of the equation y=x. The diagonal morphism into the infinite product X^{\infty } may provide an injection into the space of sequences valued in X; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions that the image of the diagonal map will fail to satisfy.

The dual notion of a diagonal morphism is a codiagonal morphism. For every object B in a category {\mathcal {C}} where the coproducts B\sqcup B exists, the codiagonal is the canonical morphism

\delta _{B}\colon B\sqcup B{\stackrel {[Id,Id]}{\to }}B

satisfying

\delta _{B}\circ \tau _{l}=\operatorname {id} _{B} for l\in \{1,2\}.

where \tau _{l} is the injection morphism to the l-th component.

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Sources and credits

This article is adapted from the Wikipedia article “Diagonal morphism”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Diagonal functor

In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} is given by Δ = ⟨ a , a ⟩ {\displaystyle \Delta (a)=\langle a,a\rangle } , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the category C {\displaystyle {\mathcal {C}}} : a product a × b {\displaystyle a\times b} is a universal arrow from Δ {\displaystyle \Delta } to ⟨ a , b ⟩ {\displaystyle \langle a,b\rangle } .

Diagonal morphism (algebraic geometry)

In algebraic geometry, given a morphism of schemes p : X → S {\displaystyle p:X\to S} , the diagonal morphism δ : X → X × S X {\displaystyle \delta :X\to X\times _{S}X} is a morphism determined by the universal property of the fiber product X × S X {\displaystyle X\times _{S}X} of p and p applied to the identity 1 X : X → X {\displaystyle 1_{X}:X\to X} and the identity 1 X {\displaystyle 1_{X}} . It is a special case of a graph morphism: given a morphism f : X → Y {\displaystyle f:X\to Y} over S, the graph morphism of it is X → X × S Y {\displaystyle X\to X\times _{S}Y} induced by f {\displaystyle f} and the identity 1 X {\displaystyle 1_{X}} .