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, given a small index category $$\mathcal{J}$$, one may construct the functor category $$\mathcal{C}^\mathcal{J}$$, the objects of which are called diagrams. For each object $$a$$ in $$\mathcal{C}$$, there is a constant diagram $$\Delta_a : \mathcal{J} \to \mathcal{C}$$ that maps every object in $$\mathcal{J}$$ to $$a$$ and every morphism in $$\mathcal{J}$$ to $$1_a$$. The diagonal functor $$\Delta : \mathcal{C} \rightarrow \mathcal{C}^\mathcal{J}$$ assigns to each object $$a$$ of $$\mathcal{C}$$ the diagram $$\Delta_a$$, and to each morphism $$f: a \rightarrow b$$ in $$\mathcal{C}$$ the natural transformation $$\eta$$ in $$\mathcal{C}^\mathcal{J}$$ (given for every object $$j$$ of $$\mathcal{J}$$ by $$\eta_j = f$$). Thus, for example, 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 diagrams. Given a diagram $$\mathcal{F} : \mathcal{J} \rightarrow \mathcal{C}$$, a natural transformation $$\Delta_a \to \mathcal{F}$$ (for some object $$a$$ of $$\mathcal{C}$$) is called a cone for $$\mathcal{F}$$. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category $$(\Delta\downarrow\mathcal{F})$$, and a limit of $$\mathcal{F}$$ is a terminal object in $$(\Delta\downarrow\mathcal{F})$$, i.e., a universal arrow $$\Delta \rightarrow \mathcal{F}$$. Dually, a colimit of $$\mathcal{F}$$ is an initial object in the comma category $$(\mathcal{F}\downarrow\Delta)$$, i.e., a universal arrow $$\mathcal{F} \rightarrow \Delta$$.

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.