Diagonal functor

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

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

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

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