Ind objects and Pro objects

Ind objects and pro objects are a substitute for inductive or projective limits in mathematics if these do not exist in a category .

definition

Let be a category and the covariant Yoneda embedding ( is the category of contravariant functors of in the category of sets). The category of ind objects in is a full subcategory of . An object is in if and only if there is a small filtering index category and a functor such that is isomorphic to . ( Is called filtering, if the following applies: is not empty; for exists with morphisms and ; for two morphisms in there exists a morphism such that . If there is a partial order , then it is filtering if and only if it is directed .) The objects of are also ind- called representable functors, cf. Representability (category theory) . ${\displaystyle C}$ ${\displaystyle h\colon C\to C^{\wedge },\ X\mapsto h_{X}=C({-},X)}$ ${\displaystyle C^{\wedge }={\text{Set}}^{C^{\text{op}}}}$ ${\displaystyle C}$ ${\displaystyle {\text{Set}}}$ ${\displaystyle {\text{Ind}}(C)}$ ${\displaystyle C}$ ${\displaystyle C^{\wedge }}$ ${\displaystyle X\in C^{\wedge }}$ ${\displaystyle {\text{Ind}}(C)}$ ${\displaystyle Y\colon I\to C}$ ${\displaystyle X}$ ${\displaystyle \varinjlim _{i\in I}h_{Y(i)}}$ ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle i,j\in I}$ ${\displaystyle k\in I}$ ${\displaystyle i\to k}$ ${\displaystyle j\to k}$ ${\displaystyle f,g\colon i\to j}$ ${\displaystyle I}$ ${\displaystyle h\colon j\to k}$ ${\displaystyle hf=hg}$ ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle {\text{Ind}}(C)}$

Continue to be the contravariant Yoneda embedding. Then is the full sub-category of consisting of objects that are isomorphic to for a projective system . The objects of are also called pro-representable functors. It is . ${\displaystyle h\colon C\to ({\text{Set}}^{C})^{\text{op}},\ X\mapsto h^{X}=C(X,{-})}$ ${\displaystyle {\text{Pro}}(C)}$ ${\displaystyle ({\text{Set}}^{C})^{\text{op}}}$ ${\displaystyle \varprojlim _{i\in I}h^{Y(i)}}$ ${\displaystyle Y\colon I^{\text{op}}\to C}$ ${\displaystyle {\text{Pro}}(C)}$ ${\displaystyle {\text{Pro}}(C)={\text{Ind}}(C^{\text{op}})^{\text{op}}}$

Instead of the generic designation Ind- or Pro-Object, one speaks specifically of Pro-Groups or Ind-Schemes etc.

Alternative description

Let be the category of pairs consisting of a small filtering category and a functor , with the morphisms from to through ${\displaystyle {\text{Ind'}}(C)}$ ${\displaystyle (I,Y)}$ ${\displaystyle I}$ ${\displaystyle Y\colon I\to C}$ ${\displaystyle (I_{1},Y_{1})}$ ${\displaystyle (I_{2},Y_{2})}$

${\displaystyle \varprojlim _{i_{1}\in I_{1}}\varinjlim _{i_{2}\in I_{2}}C(Y_{1}(i_{1}),Y_{2}(i_{2}))}$

given are. The functor is completely true , its essential image is by definition . The fact that there is no element in the identity can serve as a memory aid for the order ${\displaystyle {\text{Ind'}}(C)\to {\text{Set}}^{C^{\text{op}}}}$ ${\displaystyle {\text{Ind}}(C)}$

${\displaystyle \varinjlim _{i_{2}\in I}\varprojlim _{i_{1}\in I}C(Y(i_{1}),Y(i_{2}))}$

Are defined.

Because of this description, objects are often written in the form or, if it is clear that the inductive limit is not meant in, simply . ${\displaystyle {\text{Ind}}(C)}$ ${\displaystyle {\text{``}}{\varinjlim _{i\in I}}{\text{''}}Y(i)}$ ${\displaystyle C}$ ${\displaystyle \varinjlim _{i\in I}Y(i)}$

Analog is equivalent to the category of projective systems with morphisms ${\displaystyle {\text{Pro}}(C)}$ ${\displaystyle (I,Y)}$

${\displaystyle \varprojlim _{i_{2}\in I_{2}}\varinjlim _{i_{1}\in I_{1}}C(Y_{1}(i_{1}),Y_{2}(i_{2}))}$

Pro objects are also noted as. ${\displaystyle {\text{``}}{\varprojlim _{i\in I}}{\text{''}}Y(i)}$

Remarks

• For and an inductive system is ${\displaystyle X\in C}$ ${\displaystyle Y\colon I\to C}$

${\displaystyle {\text{Ind}}(C)\left(X,{\text{``}}{\varinjlim _{i\in I}}{\text{''}}Y\right)=\varinjlim _{i\in I}C(X,Y(i))}$

• For and a projective system is ${\displaystyle X\in C}$ ${\displaystyle Y\colon I^{\text{op}}\to C}$

${\displaystyle {\text{Pro}}(C)\left({\text{``}}{\varprojlim _{i\in I}}{\text{''}}Y,X\right)=\varinjlim _{i\in I}C(Y(i),X)}$

• If in is isomorphic to for one , then there is an inductive limit of in . However, the reverse of this statement does not apply. Example: If the inductive system is the finitely generated subgroups of , then (see above) ${\displaystyle {\text{``}}{\varinjlim _{i\in I}}{\text{''}}Y}$ ${\displaystyle C^{\wedge }}$ ${\displaystyle h_{X}}$ ${\displaystyle X\in C}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle C}$ ${\displaystyle (I,Y)}$ ${\displaystyle \mathbb {Q} }$

${\displaystyle {\text{Ind}}({\text{Ab}})\left(\mathbb {Q} ,{\text{``}}{\varinjlim _{i\in I}}{\text{''}}Y\right)=\varinjlim _{i\in I}{\text{Ab}}(\mathbb {Q} ,Y(i))=0\neq {\text{Ab}}(\mathbb {Q} ,\varinjlim _{i\in I}Y(i))=\mathbb {Q} }$

• The prerequisites for the index categories are essential because each object is an inductive limit of the system (the so-called Grothendieck construction ; see comma category for the notation). ${\displaystyle X\in C^{\wedge }}$ ${\displaystyle (C\downarrow X)\to C^{\wedge }}$
• If there is a small category in which finite limits exist, then a functor is ind-representable if and only if it is left exact . The left-hand precision means that the category in the Grothendieck construction is filtering. ${\displaystyle C}$ ${\displaystyle X\colon C\to {\text{Set}}}$ ${\displaystyle C\downarrow X}$

Examples

• Is the category of finite sets, is equivalent to the category of all sets and equivalent to the category of Boolean spaces (i.e. totally disconnected compact Hausdorff spaces ). The equivalence is given in both cases by the evaluation of the inductive or projective limit in the larger category, whereby in the second case finite sets are equipped with the discrete topology in order to understand them as Boolean spaces. ${\displaystyle C}$ ${\displaystyle {\text{Ind}}(C)}$ ${\displaystyle {\text{Pro}}(C)}$
• Is the category of finite groups is equivalent to the category of pro- finite groups . ${\displaystyle C}$ ${\displaystyle {\text{Pro}}(C)}$
• If the category of the finitely presented modules is over a ring , is canonically equivalent to the category of all modules. ${\displaystyle C}$ ${\displaystyle R}$ ${\displaystyle {\text{Ind}}(C)}$ ${\displaystyle R}$

literature

• Masaki Kashiwara , Pierre Schapira : Categories and Sheaves . Springer, Berlin 2006, ISBN 978-3-540-27949-5 .
• Michael Artin , Alexander Grothendieck , Jean-Louis Verdier : Séminaire de géométrie algébrique du Bois-Marie. Théorie des topos et cohomologie étale des schémas. (SGA 4) 1963-1964. Lecture notes in mathematics 269. Springer Berlin 1972 ISBN 978-3-540-05896-0
• Saunders Mac Lane , Ieke Moerdijk: Sheaves in Geometry and Logic. A First Introduction to Topos Theory . Springer, Berlin 1992, ISBN 3-540-97710-4 .

Footnotes

1. ↑ The term “pro-representability” is used differently in deformation theory , see Remark 6.2.2 in: Barbara Fantechi, Lothar Göttsche et al .: Fundamental Algebraic Geometry. Grothendieck's FGA explained . AMS, Providence 2005, ISBN 978-0-8218-4245-4 .
2. ↑ The essential proof step is carried out in Lemma 1.1.16 (a) in: Luis Ribes, Pavel Zalesskii: Profinite Groups. Springer Berlin 2000 ISBN 978-3-540-66986-9