Diagonal functor

In category theory, a branch of mathematics, the diagonal functor [math]\displaystyle{ \mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C} }[/math] is given by [math]\displaystyle{ \Delta(a) = \langle a,a \rangle }[/math], 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 [math]\displaystyle{ \mathcal{C} }[/math]: a product [math]\displaystyle{ a \times b }[/math] is a universal arrow from [math]\displaystyle{ \Delta }[/math] to [math]\displaystyle{ \langle a,b \rangle }[/math]. The arrow comprises the projection maps. More generally, given a small index category [math]\displaystyle{ \mathcal{J} }[/math], one may construct the functor category [math]\displaystyle{ \mathcal{C}^\mathcal{J} }[/math], the objects of which are called diagrams. For each object [math]\displaystyle{ a }[/math] in [math]\displaystyle{ \mathcal{C} }[/math], there is a constant diagram [math]\displaystyle{ \Delta_a : \mathcal{J} \to \mathcal{C} }[/math] that maps every object in [math]\displaystyle{ \mathcal{J} }[/math] to [math]\displaystyle{ a }[/math] and every morphism in [math]\displaystyle{ \mathcal{J} }[/math] to [math]\displaystyle{ 1_a }[/math]. The diagonal functor [math]\displaystyle{ \Delta : \mathcal{C} \rightarrow \mathcal{C}^\mathcal{J} }[/math] assigns to each object [math]\displaystyle{ a }[/math] of [math]\displaystyle{ \mathcal{C} }[/math] the diagram [math]\displaystyle{ \Delta_a }[/math], and to each morphism [math]\displaystyle{ f: a \rightarrow b }[/math] in [math]\displaystyle{ \mathcal{C} }[/math] the natural transformation [math]\displaystyle{ \eta }[/math] in [math]\displaystyle{ \mathcal{C}^\mathcal{J} }[/math] (given for every object [math]\displaystyle{ j }[/math] of [math]\displaystyle{ \mathcal{J} }[/math] by [math]\displaystyle{ \eta_j = f }[/math]). Thus, for example, in the case that [math]\displaystyle{ \mathcal{J} }[/math] is a discrete category with two objects, the diagonal functor [math]\displaystyle{ \mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C} }[/math] is recovered.

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

If every functor from [math]\displaystyle{ \mathcal{J} }[/math] to [math]\displaystyle{ \mathcal{C} }[/math] has a limit (which will be the case if [math]\displaystyle{ \mathcal{C} }[/math] is complete), then the operation of taking limits is itself a functor from [math]\displaystyle{ \mathcal{C}^\mathcal{J} }[/math] to [math]\displaystyle{ \mathcal{C} }[/math]. 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 [math]\displaystyle{ \mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C} }[/math] 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.