Diagonal functor

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

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

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

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

See also

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

References

• Awodey, Steve (2006). "Functors and Naturality". Category Theory. pp. 125–158. doi:10.1093/acprof:oso/9780198568612.003.0007. ISBN 978-0-19-856861-2. 
• Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in geometry and logic a first introduction to topos theory. New York: Springer-Verlag. pp. 20–23. ISBN 9780387977102. 
• May, J. P. (1999). A Concise Course in Algebraic Topology. University of Chicago Press. pp. 16. ISBN 0-226-51183-9. https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf.