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. The diagonal functor is one particular diagram: 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 ). 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 the functor , the natural transform from this functor to any other diagram is called a cone. Among all such cones is a universal cone; this cone is the limit of the diagram . That is, the limit of any functor is a universal arrow ; the 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. 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.

See also

References

  • Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in geometry and logic a first introduction to topos theory. New York: Springer-Verlag. pp. 20–23. ISBN 9780387977102. 
  • May, J. P. (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. p. 16. ISBN 0-226-51183-9. 
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