Diagonal functor

In category theory, a branch of mathematics, the diagonal functor $𝒞 \to 𝒞 \times 𝒞$ is given by $Δ (a) = ⟨ a, a ⟩$ , 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 $Δ$ 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 $Δ_{a} : 𝒥 \to 𝒞$ that maps every object in $𝒥$ to $a$ and every morphism in $𝒥$ to $1_{a}$ . The diagonal functor $Δ : 𝒞 \to 𝒞^{𝒥}$ assigns to each object $a$ of $𝒞$ the diagram $Δ_{a}$ , and to each morphism $f : a \to b$ in $𝒞$ the natural transformation $η$ in $𝒞^{𝒥}$ (given for every object $j$ of $𝒥$ by $η_{j} = f$ ). 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 $Δ_{a} \to ℱ$ (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 $(Δ ↓ ℱ)$ , and a limit of $ℱ$ is a terminal object in $(Δ ↓ ℱ)$ , i.e., a universal arrow $Δ \to ℱ$ . Dually, a colimit of $ℱ$ is an initial object in the comma category $(ℱ ↓ Δ)$ , i.e., a universal arrow $ℱ \to Δ$ .

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.

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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Diagonal functor

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References