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

satisfying

is the canonical projection morphism-th component. The existence of this morphism is a consequence of the universal property that characterizess 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 ordered pair. The reason for the name is that the image on the real line is given by the line that is the graph. The diagonal morphism into the infinite product may provide an injection into the space of sequences; 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 where the coproducts exists, the codiagonal is the canonical morphism

satisfying

-th component.

See also

• Diagonal functor

• Diagonal embedding

• wikibooks:Category Theory/(Co-)cones and (co-)limits

References

Bibliography

External links