Diagonal functor

In category theory, a branch of mathematics, the diagonal functor $\mathcal \rightarrow \mathcal \times \mathcal$ 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 $\mathcal$: 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 $\mathcal$, one may construct the functor category $\mathcal^\mathcal$, the objects of which are called diagrams. For each object $a$ in $\mathcal$, there is a constant diagram $\Delta_a : \mathcal \to \mathcal$ that maps every object in $\mathcal$ to $a$ and every morphism in $\mathcal$ to $1_a$. The diagonal functor $\Delta : \mathcal \rightarrow \mathcal^\mathcal$ assigns to each object $a$ of $\mathcal$ the diagram $\Delta_a$, and to each morphism $f: a \rightarrow b$ in $\mathcal$ the natural transformation $\eta$ in $\mathcal^\mathcal$ (given for every object $j$ of $\mathcal$ by $\eta_j = f$). Thus, for example, in the case that $\mathcal$ is a discrete category with two objects, the diagonal functor $\mathcal \rightarrow \mathcal \times \mathcal$ is recovered.

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

If every functor from $\mathcal$ to $\mathcal$ has a limit (which will be the case if $\mathcal$ is complete), then the operation of taking limits is itself a functor from $\mathcal^\mathcal$ to $\mathcal$. 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 $\mathcal \rightarrow \mathcal \times \mathcal$ described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor. Other well-known examples include the pushout, which is the limit of the span, and the terminal object, which is the limit of the empty category.

See also

* Diagram (category theory)
* Cone (category theory)
* Diagonal morphism

References

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Category theory

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