# Diagonal functor

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

More generally, in any functor category ${\displaystyle {\mathcal {C}}^{\mathcal {J}}}$ (here ${\displaystyle {\mathcal {J}}}$ should be thought of as a small index category), for each object ${\displaystyle a}$ in ${\displaystyle {\mathcal {C}}}$, there is a constant functor with fixed object ${\displaystyle a}$: ${\displaystyle \Delta (a)\in {\mathcal {C}}^{\mathcal {J}}}$. The diagonal functor ${\displaystyle \Delta :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}}$ assigns to each object of ${\displaystyle {\mathcal {C}}}$ the functor ${\displaystyle \Delta (a)}$, and to each morphism ${\displaystyle f:a\rightarrow b}$ in ${\displaystyle {\mathcal {C}}}$ the obvious natural transformation ${\displaystyle \eta }$ in ${\displaystyle {\mathcal {C}}^{\mathcal {J}}}$ (given by ${\displaystyle \eta _{j}=f}$). In the case that ${\displaystyle {\mathcal {J}}}$ is a discrete category with two objects, the diagonal functor ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}$ is recovered.

Diagonal functors provide a way to define limits and colimits of functors. The limit of any functor ${\displaystyle {\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}}$ is a universal arrow ${\displaystyle \Delta \rightarrow {\mathcal {F}}}$ and a colimit is a universal arrow ${\displaystyle {\mathcal {F}}\rightarrow \Delta }$. If every functor from ${\displaystyle {\mathcal {J}}}$ to ${\displaystyle {\mathcal {C}}}$ has a limit (which will be the case if ${\displaystyle {\mathcal {C}}}$ is complete), then the operation of taking limits is itself a functor from ${\displaystyle {\mathcal {C}}^{\mathcal {J}}}$ to ${\displaystyle {\mathcal {C}}}$. 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 ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}$ described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.

## 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.