Diagonal functor

In category theory, a branch of mathematics, the diagonal functor $𝒞\to 𝒞\times 𝒞$ is given by $\Delta (a)=⟨a,a⟩$, 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 $\Delta$ 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 ${\Delta }_a:𝒥\to 𝒞$ that maps every object in $𝒥$ to $a$ and every morphism in $𝒥$ to $1_a$. The diagonal functor $\Delta :𝒞\to {𝒞}^{𝒥}$ assigns to each object $a$ of $𝒞$ the diagram ${\Delta }_a$, 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 ${\Delta }_a\to ℱ$ (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 $(\Delta \downarrow ℱ)$, and a limit of $ℱ$ is a terminal object in $(\Delta \downarrow ℱ)$, i.e., a universal arrow $\Delta \to ℱ$. Dually, a colimit of $ℱ$ is an initial object in the comma category $(ℱ\downarrow \Delta )$, i.e., a universal arrow $ℱ\to \Delta$.

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 (PDF). University of Chicago Press. p. 16. ISBN 0-226-51183-9.

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