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, in any functor category ${\mathcal {C}}^{{\mathcal {J}}}$ (here ${\mathcal {J}}$ should be thought of as a small index category), 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 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 $\eta _{j}=f$ ). 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 functors. The limit of any functor ${\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}$ is a universal arrow from $\Delta$ to ${\mathcal {F}}$ and a colimit is a universal arrow $F\rightarrow \Delta$ . 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.

References

Functor types

Additive Adjoint Conservative Derived Diagonal Enriched Essentially surjective Exact Faithful Forgetful Full Logical Monoidal Representable Smooth

This article is issued from Wikipedia - version of the 3/12/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.