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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