In category theory, a branch of mathematics, the diagonal functor, 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. The arrow comprises the projection maps.

More generally, given a small index category, one may construct the functor category, the objects of which are called diagrams, there is a constant diagram the natural transformation is a discrete category is recovered.

Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram) is called a cone. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category, i.e., a universal arrow. Dually, a colimit.

is complete. The limit functor is the right-adjoint described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.

See also

• Diagram (category theory)

• Cone (category theory)

• Diagonal morphism

References