In category theory, a branch of mathematics, for any object a in any category C where the product a × a exists, there exists the diagonal morphism
δ a : a → a × a satisfying
π k ∘ δ a = i d a for
k ∈ { 1 , 2 } , where π k is the canonical projection morphism to the k -th component. The existence of this morphism is a consequence of the universal property which 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, δ a ( x ) = ⟨ x , x ⟩ , 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 R → R 2 on the real line is given by the line which is a graph of the equation y = x . The diagonal morphism into the infinite product X ∞ 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 which the image of the diagonal map will fail to satisfy.
