End (category theory)

In the mathematical branch of category theory , an end is a special limit .

definition

Let there be categories , the category that is too dual and finally a functor . ${\displaystyle {\mathcal {C}};{\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}^{\mathrm {op} }}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F\colon {\mathcal {C}}^{\text{op}}\times {\mathcal {C}}\to {\mathcal {D}}}$

One end of is a pair consisting of an object and an indexed family of arrows , called projections , such that for all objects and morphisms the diagram ${\displaystyle F}$ ${\displaystyle (E;\pi )}$ ${\displaystyle E\in \operatorname {Ob} ({\mathcal {D}})}$ ${\displaystyle \operatorname {Ob} ({\mathcal {C}})}$ ${\displaystyle \pi _{X}\colon E\to F(X,X)}$ ${\displaystyle X,Y\in \operatorname {Ob} ({\mathcal {C}})}$ ${\displaystyle h\in {\mathcal {C}}(X,Y)}$

${\displaystyle {\begin{array}{ccc}\quad E&{\xrightarrow {\quad \pi _{X}\quad }}&\!F(X,X)\\\scriptstyle \pi _{Y}{\Big \downarrow }&&\quad {\Big \downarrow }\scriptstyle F(X,h)\\F(Y,Y)&{\xrightarrow[{F(h,Y)}]{}}&F(X,Y)\end{array}}}$

commutes. (In short: is a dinatural transformation .) ${\displaystyle \pi }$ ${\displaystyle \Delta E\to F}$

An end is also universal , that is, for each alternative with corresponding projections there is a clearly defined arrow , so that applies to all . ${\displaystyle E'}$ ${\displaystyle \pi '_{X}\colon E'\to F(X,X)}$ ${\displaystyle k\colon E'\to E}$ ${\displaystyle \pi _{X}\circ k=\pi '_{X}}$ ${\displaystyle X\in \operatorname {Ob} ({\mathcal {C}})}$

notation

A common notation for the end of is is ${\displaystyle F\colon {\mathcal {C}}^{\text{op}}\times {\mathcal {C}}\to {\mathcal {D}}}$

${\displaystyle E\cong \int _{X\in {\mathcal {C}}}F(X,X)}$.

example

Let functors be given for locally small categories . The amount of natural transformations of to is precisely one end of the functor , by explained, wherein the hom functor of call. ${\displaystyle {\mathcal {C}},{\mathcal {D}}}$ ${\displaystyle F,G\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle T\colon {\mathcal {C}}^{\text{op}}\times {\mathcal {C}}\to \mathbf {Set} }$ ${\displaystyle T(X,Y)={\mathcal {D}}(FX,GY)}$ ${\displaystyle {\mathcal {D}}(-,-)}$ ${\displaystyle {\mathcal {D}}}$

The above diagram is here

${\displaystyle {\begin{array}{ccc}\quad E&{\xrightarrow {\quad \pi _{X}\quad }}&\!{\mathcal {D}}(FX,GX)\\\scriptstyle \pi _{Y}{\Big \downarrow }&&\quad {\Big \downarrow }\scriptstyle {\mathcal {D}}(FX,Gh)\\{\mathcal {D}}(FY,GY)&{\xrightarrow[{{\mathcal {D}}(Fh,GY)}]{}}&{\mathcal {D}}(FX,GY).\end{array}}}$

The projections of the end assign a component to every natural transformation . So at the element level of , the diagram states that for components and ${\displaystyle \psi \in E}$ ${\displaystyle \psi _{X}=\pi _{X}(\psi )\in {\mathcal {D}}(FX,GX)}$ ${\displaystyle E}$ ${\displaystyle \psi _{X}}$ ${\displaystyle \psi _{Y}}$

${\displaystyle Gh\circ \psi _{X}=\psi _{Y}\circ Fh}$

applies. The universality ensures that it all contains natural transformations. ${\displaystyle E}$

This example can also be interpreted as a definition of natural transformations. In this form the definition can easily be generalized to enriched categories and functors.

literature

• GM Kelly: Basic Concepts of Enriched Category Theory . In: Lecture Notes in Mathematics 64 . Cambridge University Press, 1982 ( mta.ca [accessed March 8, 2014]).

Web links

• end , entry in the nLab . (English)