Diagonal functor

In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} is given by  Δ ( a ) = ⟨ a , a ⟩ {\displaystyle \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 C {\displaystyle {\mathcal {C}}} : a product  a × b {\displaystyle a\times b} is a universal arrow from  Δ {\displaystyle \Delta } to  ⟨ a , b ⟩ {\displaystyle \langle a,b\rangle } . The arrow comprises the projection maps.

More generally, given a small index category J {\displaystyle {\mathcal {J}}} , one may construct the functor category C J {\displaystyle {\mathcal {C}}^{\mathcal {J}}} , the objects of which are called diagrams. For each object  a {\displaystyle a} in  C {\displaystyle {\mathcal {C}}} , there is a constant diagram Δ a : J → C {\displaystyle \Delta _{a}:{\mathcal {J}}\to {\mathcal {C}}} that maps every object in  J {\displaystyle {\mathcal {J}}} to  a {\displaystyle a} and every morphism in  J {\displaystyle {\mathcal {J}}} to  1 a {\displaystyle 1_{a}} . The diagonal functor  Δ : C → C J {\displaystyle \Delta :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}} assigns to each object  a {\displaystyle a} of  C {\displaystyle {\mathcal {C}}} the diagram  Δ a {\displaystyle \Delta _{a}} , and to each morphism  f : a → b {\displaystyle f:a\rightarrow b} in  C {\displaystyle {\mathcal {C}}} the natural transformation η {\displaystyle \eta } in  C J {\displaystyle {\mathcal {C}}^{\mathcal {J}}} (given for every object  j {\displaystyle j} of  J {\displaystyle {\mathcal {J}}} by  η j = f {\displaystyle \eta _{j}=f} ). Thus, for example, in the case that  J {\displaystyle {\mathcal {J}}} is a discrete category with two objects, the diagonal functor  C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} is recovered.

Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram F : J → C {\displaystyle {\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}} , a natural transformation  Δ a → F {\displaystyle \Delta _{a}\to {\mathcal {F}}} (for some object  a {\displaystyle a} of  C {\displaystyle {\mathcal {C}}} ) is called a cone for  F {\displaystyle {\mathcal {F}}} . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category ( Δ ↓ F ) {\displaystyle (\Delta \downarrow {\mathcal {F}})} , and a limit of  F {\displaystyle {\mathcal {F}}} is a terminal object in  ( Δ ↓ F ) {\displaystyle (\Delta \downarrow {\mathcal {F}})} , i.e., a universal arrow Δ → F {\displaystyle \Delta \rightarrow {\mathcal {F}}} . Dually, a colimit of  F {\displaystyle {\mathcal {F}}} is an initial object in the comma category  ( F ↓ Δ ) {\displaystyle ({\mathcal {F}}\downarrow \Delta )} , i.e., a universal arrow  F → Δ {\displaystyle {\mathcal {F}}\rightarrow \Delta } .

If every functor from  J {\displaystyle {\mathcal {J}}} to  C {\displaystyle {\mathcal {C}}} has a limit (which will be the case if  C {\displaystyle {\mathcal {C}}} is complete), then the operation of taking limits is itself a functor from  C J {\displaystyle {\mathcal {C}}^{\mathcal {J}}} to  C {\displaystyle {\mathcal {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  C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.

## See also

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* Diagram (category theory)
* Cone (category theory)
* Diagonal morphism

## References

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* Awodey, Steve (2006). "Functors and Naturality". Category Theory. pp. 125–158\. doi:10.1093/acprof:oso/9780198568612.003.0007. ISBN 978-0-19-856861-2.
* 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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Functor types

* Additive
* Adjoint
* Conservative
* Derived
* Diagonal
* Enriched
* Essentially surjective
* Exact
* Forgetful
* Full and faithful
* Logical
* Monoidal
* Representable
* Smooth

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