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''inverse limit''
 
''inverse limit''
  
A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathematics. A common example of a projective limit is that of a family of mathematical structures of the same type indexed by the elements of a pre-ordered set. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752601.png" /> be a set endowed with a pre-order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752602.png" />, and suppose that with each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752603.png" /> a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752604.png" /> is associated and with each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752606.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752607.png" />, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752608.png" /> is associated, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526010.png" />, are identity mappings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526012.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526013.png" /> is called a projective limit of the family of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526014.png" /> and mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526015.png" /> if the following conditions are satisfied: a) there exists a family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526017.png" /> for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526018.png" />; b) for any family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526020.png" />, from an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526021.png" />, for which the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526022.png" /> hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526023.png" />, there exists a unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526025.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526026.png" />. The projective limit can be described explicitly as follows. One considers the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526027.png" /> and selects in it the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526028.png" /> for which the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526029.png" /> hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526030.png" />. This subset is the projective limit of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526031.png" />. If all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526032.png" /> are equipped with an additional structure of the same type and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526033.png" /> preserve it, then the same structure is induced in the projective limit. Therefore it is possible to speak of projective limits of groups, modules, topological spaces, etc.
+
A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathematics. A common example of a projective limit is that of a family of mathematical structures of the same type indexed by the elements of a pre-ordered set. Let $  I $
 +
be a set endowed with a pre-order relation $  \leq  $,  
 +
and suppose that with each element $  i \in I $
 +
a set $  X _ {i} $
 +
is associated and with each pair $  ( i , j ) $,
 +
$  i , j \in I $,  
 +
in which $  i \leq  j $,  
 +
a mapping $  \phi _ {ij} : X _ {i} \rightarrow X _ {j} $
 +
is associated, where the $  \phi _ {ii} $,  
 +
$  i \in I $,  
 +
are identity mappings and $  \phi _ {jk} \phi _ {ij} = \phi _ {ik} $
 +
for $  i \leq  j \leq  k $.  
 +
A set $  X $
 +
is called a projective limit of the family of sets $  X _ {i} $
 +
and mappings $  \phi _ {ij} $
 +
if the following conditions are satisfied: a) there exists a family of mappings $  \pi _ {i} : X \rightarrow X _ {i} $
 +
such that $  \phi _ {ij} \pi _ {i} = \pi _ {j} $
 +
for any pair $  i \leq  j $;  
 +
b) for any family of mappings $  \alpha _ {i} : Y \rightarrow X _ {i} $,  
 +
$  i \in I $,  
 +
from an arbitrary set $  Y $,  
 +
for which the equalities $  \phi _ {ij} \alpha _ {i} = \alpha _ {j} $
 +
hold for $  i \leq  j $,  
 +
there exists a unique mapping $  \alpha : Y \rightarrow X $
 +
such that $  \alpha _ {i} = \pi _ {i} \alpha $
 +
for every $  i \in I $.  
 +
The projective limit can be described explicitly as follows. One considers the direct product $  \prod _ {i \in I }  X _ {i} $
 +
and selects in it the set of all functions $  f : I \rightarrow \cup _ {i \in I }  X _ {i} $
 +
for which the equalities $  \phi _ {ij} ( f ( i) ) = f ( j) $
 +
hold for $  i \leq  j $.  
 +
This subset is the projective limit of the family $  X _ {i} $.  
 +
If all the $  X _ {i} $
 +
are equipped with an additional structure of the same type and the $  \phi _ {ij} $
 +
preserve it, then the same structure is induced in the projective limit. Therefore it is possible to speak of projective limits of groups, modules, topological spaces, etc.
  
A natural generalization of this concept of projective limit is that of the projective limit of a functor. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526034.png" /> be a functor from a [[Small category|small category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526035.png" /> to an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526036.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526037.png" /> together with morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526039.png" />, is called a projective limit (inverse limit, or simply limit) of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526040.png" /> if the following conditions are satisfied: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526041.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526042.png" /> for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526043.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526044.png" />) for any family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526045.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526046.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526047.png" /> there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526050.png" />. Notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526051.png" />.
+
A natural generalization of this concept of projective limit is that of the projective limit of a functor. Let $  F : \mathfrak D \rightarrow \mathfrak K $
 +
be a functor from a [[Small category|small category]] $  \mathfrak D $
 +
to an arbitrary category $  \mathfrak K $.  
 +
An object $  X \in  \mathop{\rm Ob}  \mathfrak K $
 +
together with morphisms $  \pi _ {D} : X \rightarrow F ( D) $,  
 +
$  D \in  \mathop{\rm Ob}  \mathfrak D $,  
 +
is called a projective limit (inverse limit, or simply limit) of the functor $  F $
 +
if the following conditions are satisfied: $  \alpha $)
 +
$  F ( \phi ) \pi _ {D} = \pi _ {D  ^  \prime  } $
 +
for any morphism $  \phi : D \rightarrow D  ^  \prime  $;  
 +
and $  \beta $)  
 +
for any family of morphisms $  \alpha _ {D} : Y \rightarrow F ( D) $
 +
satisfying $  F ( \phi ) \alpha _ {D} = \alpha _ {D  ^  \prime  } $
 +
for all $  \phi : D \rightarrow D  ^  \prime  $
 +
there exists a unique morphism $  \alpha : Y \rightarrow X $
 +
such that $  \alpha _ {D} = \phi _ {D  ^  \prime  } \alpha $
 +
for all $  D \in  \mathop{\rm Ob}  \mathfrak D $.  
 +
Notation: $  \lim\limits  F = ( X , \pi _ {D} ) $.
  
 
===Examples of projective limits.===
 
===Examples of projective limits.===
  
 +
1) Let  $  I $
 +
be a [[discrete category]]. Then for an arbitrary functor  $  F :  I \rightarrow \mathfrak K $
 +
the projective limit of the functor  $  F $
 +
coincides with the product of the family of objects  $  F ( i) $,
 +
$  i \in I $(
 +
cf. [[Product of a family of objects in a category|Product of a family of objects in a category]]).
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526052.png" /> be a [[discrete category]]. Then for an arbitrary functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526053.png" /> the projective limit of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526054.png" /> coincides with the product of the family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526056.png" /> (cf. [[Product of a family of objects in a category|Product of a family of objects in a category]]).
+
2) Let $  \mathfrak D $
 
+
be a category with two objects $  A , B $
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526057.png" /> be a category with two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526058.png" /> and two non-identity morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526059.png" />. Then the limit of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526060.png" /> is the [[equalizer]] of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526061.png" /> (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]).
+
and two non-identity morphisms $  \alpha , \beta : A \rightarrow B $.  
 +
Then the limit of a functor $  F : \mathfrak D \rightarrow \mathfrak K $
 +
is the [[equalizer]] of the pair of morphisms $  F ( \alpha ) , F ( \beta ) $(
 +
cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]).
  
 
If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories.
 
If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories.
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:08, 6 June 2020


inverse limit

A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathematics. A common example of a projective limit is that of a family of mathematical structures of the same type indexed by the elements of a pre-ordered set. Let $ I $ be a set endowed with a pre-order relation $ \leq $, and suppose that with each element $ i \in I $ a set $ X _ {i} $ is associated and with each pair $ ( i , j ) $, $ i , j \in I $, in which $ i \leq j $, a mapping $ \phi _ {ij} : X _ {i} \rightarrow X _ {j} $ is associated, where the $ \phi _ {ii} $, $ i \in I $, are identity mappings and $ \phi _ {jk} \phi _ {ij} = \phi _ {ik} $ for $ i \leq j \leq k $. A set $ X $ is called a projective limit of the family of sets $ X _ {i} $ and mappings $ \phi _ {ij} $ if the following conditions are satisfied: a) there exists a family of mappings $ \pi _ {i} : X \rightarrow X _ {i} $ such that $ \phi _ {ij} \pi _ {i} = \pi _ {j} $ for any pair $ i \leq j $; b) for any family of mappings $ \alpha _ {i} : Y \rightarrow X _ {i} $, $ i \in I $, from an arbitrary set $ Y $, for which the equalities $ \phi _ {ij} \alpha _ {i} = \alpha _ {j} $ hold for $ i \leq j $, there exists a unique mapping $ \alpha : Y \rightarrow X $ such that $ \alpha _ {i} = \pi _ {i} \alpha $ for every $ i \in I $. The projective limit can be described explicitly as follows. One considers the direct product $ \prod _ {i \in I } X _ {i} $ and selects in it the set of all functions $ f : I \rightarrow \cup _ {i \in I } X _ {i} $ for which the equalities $ \phi _ {ij} ( f ( i) ) = f ( j) $ hold for $ i \leq j $. This subset is the projective limit of the family $ X _ {i} $. If all the $ X _ {i} $ are equipped with an additional structure of the same type and the $ \phi _ {ij} $ preserve it, then the same structure is induced in the projective limit. Therefore it is possible to speak of projective limits of groups, modules, topological spaces, etc.

A natural generalization of this concept of projective limit is that of the projective limit of a functor. Let $ F : \mathfrak D \rightarrow \mathfrak K $ be a functor from a small category $ \mathfrak D $ to an arbitrary category $ \mathfrak K $. An object $ X \in \mathop{\rm Ob} \mathfrak K $ together with morphisms $ \pi _ {D} : X \rightarrow F ( D) $, $ D \in \mathop{\rm Ob} \mathfrak D $, is called a projective limit (inverse limit, or simply limit) of the functor $ F $ if the following conditions are satisfied: $ \alpha $) $ F ( \phi ) \pi _ {D} = \pi _ {D ^ \prime } $ for any morphism $ \phi : D \rightarrow D ^ \prime $; and $ \beta $) for any family of morphisms $ \alpha _ {D} : Y \rightarrow F ( D) $ satisfying $ F ( \phi ) \alpha _ {D} = \alpha _ {D ^ \prime } $ for all $ \phi : D \rightarrow D ^ \prime $ there exists a unique morphism $ \alpha : Y \rightarrow X $ such that $ \alpha _ {D} = \phi _ {D ^ \prime } \alpha $ for all $ D \in \mathop{\rm Ob} \mathfrak D $. Notation: $ \lim\limits F = ( X , \pi _ {D} ) $.

Examples of projective limits.

1) Let $ I $ be a discrete category. Then for an arbitrary functor $ F : I \rightarrow \mathfrak K $ the projective limit of the functor $ F $ coincides with the product of the family of objects $ F ( i) $, $ i \in I $( cf. Product of a family of objects in a category).

2) Let $ \mathfrak D $ be a category with two objects $ A , B $ and two non-identity morphisms $ \alpha , \beta : A \rightarrow B $. Then the limit of a functor $ F : \mathfrak D \rightarrow \mathfrak K $ is the equalizer of the pair of morphisms $ F ( \alpha ) , F ( \beta ) $( cf. Kernel of a morphism in a category).

If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories.

Comments

In most modern work in category theory, the unadorned name "limit" is used for this concept (and the dual concept is called a colimit). The terms "inverse limit" and its dual, direct limit (or inductive limit), are generally restricted to diagrams over directed pre-ordered sets (see Directed order); "projective limit" is best avoided, because of the danger of confusion with the notion of projective object of a category. Inverse and direct limits were first studied as such in the 1930's, in connection with topological concepts such as Čech cohomology; the general concept of limit was introduced in 1958 by D.M. Kan [a1].

References

[a1] D.M. Kan, "Adjoint functors" Trans. Amer. Math. Soc. , 87 (1958) pp. 294–329
[a2] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Projective limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_limit&oldid=48321
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article