Diagonal functor

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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 (given for every object of by ). 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 (for some object of ) 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 (which will be the case if is complete), 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 (which exists if the category is cocomplete) 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.

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