Diagonal functor
In category theory, a branch of mathematics, the diagonal functor is given by, 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 : a product is a universal arrow from to. 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. For each object in, there is a constant diagram that maps every object in to and every morphism in to. The diagonal functor assigns to each object of the diagram, and to each morphism in the natural transformation in . Thus, for example, in the case that is a discrete category with two objects, the diagonal functor is recovered.
Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram, a natural transformation is called a cone for. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category, and a limit of is a terminal object in, i.e., a universal arrow. Dually, a colimit of is an initial object in the comma category, i.e., a universal arrow.
If every functor from to has a limit, then the operation of taking limits is itself a functor from to. The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor is the left-adjoint of the diagonal functor.
For example, the diagonal functor described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor. Other well-known examples include the pushout, which is the limit of the span, and the terminal object, which is the limit of the empty category.
