Diagonal morphism

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In category theory, a branch of mathematics, for any object in any category where the product exists, there exists the diagonal morphism

satisfying

for

where is the canonical projection morphism to the -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 of the object . Namely, , the ordered pair formed from . 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 on the real line is given by the line which is a graph of the equation . The diagonal morphism into the infinite product may provide an injection into the space of sequences valued in ; 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.

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