Diagonal functor explained

In category theory, a branch of mathematics, the diagonal functor

l{C} → l{C} x l{C} is given by \Delta(a) = \langle a,a \rangle , 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 l{C} : a product a x b is a universal arrow from \Delta to \langle a,b \rangle . The arrow comprises the projection maps.l{J} , one may construct the functor category l{C} ^{l{J}} , the objects of which are called diagrams. For each object a in l{C} , there is a constant diagram \Delta _{a} : l{J} \to l{C} that maps every object in l{J} to a and every morphism in l{J} to 1 _{a} . The diagonal functor \Delta : l{C} → l{C} ^{l{J}} assigns to each object a of l{C} the diagram \Delta _{a} , and to each morphism f: a → b in l{C} the natural transformation η in l{C} ^{l{J}} (given for every object j of l{J} by η _{j} = f ). Thus, for example, in the case that l{J} is a discrete category with two objects, the diagonal functor l{C} → l{C} x l{C} is recovered.l{F} : l{J} → l{C} , a natural transformation \Delta _{a} \to l{F} (for some object a of l{C} ) is called a cone for l{F} . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category (\Delta\downarrow l{F}) , and a limit of l{F} is a terminal object in (\Delta\downarrow l{F}) , i.e., a universal arrow \Delta → l{F} . Dually, a colimit of l{F} is an initial object in the comma category ( l{F}\downarrow\Delta) , i.e., a universal arrow l{F} → \Delta .

If every functor from

l{J} to l{C} has a limit (which will be the case if l{C} is complete), then the operation of taking limits is itself a functor from l{C} ^{l{J}} to l{C} . 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 l{C} → l{C} x l{C} 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

• Book: 10.1093/acprof:oso/9780198568612.003.0007 . Functors and Naturality . Category Theory . 2006 . Awodey . Steve . 125–158 . 978-0-19-856861-2 .
• Book: Mac Lane, Saunders. Sheaves in geometry and logic a first introduction to topos theory. Moerdijk. Ieke. Springer-Verlag. 1992. 9780387977102. New York. 20–23.
• Book: May, J. P.. A Concise Course in Algebraic Topology. University of Chicago Press. 1999. 0-226-51183-9. 16.