Diagonal morphism

In category theory, a branch of mathematics, for every object

is the canonical projection morphism to the

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

, 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

The diagonal morphism into the infinite product

may provide an injection into the space of sequences valued in

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 co-diagonal morphism.

exists, the co-diagonal[3][2][7][5][6] is the canonical morphism satisfying where

with the pushout is an epimorphism if and only if the codiagonal is an isomorphism.

[8] This category theory-related article is a stub.