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, in any functor category (here should be thought of as a small index category), for each object in , there is a constant functor with fixed object : . The diagonal functor assigns to each object of the functor , and to each morphism in the obvious natural transformation in (given by ). 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 functors. The limit of any functor is a universal arrow from to and a colimit is 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.

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