Diagonal functor

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

Diagonal functors provide a way to define limits and colimits of diagrams. Given the functor $\Delta (a)$ , the natural transform from this functor to any other diagram ${\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}$ is called a cone. Among all such cones is a universal cone; this cone is the limit of the diagram ${\mathcal {F}}$ . That is, the limit of any functor ${\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}$ is a universal arrow $\Delta \rightarrow {\mathcal {F}}$ ; the colimit is a universal arrow ${\mathcal {F}}\rightarrow \Delta$ .

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

References

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