Diagonal functor

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

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

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

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