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''direct and inverse system in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919301.png" />''
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A direct system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919302.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919303.png" /> consists of a collection of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919304.png" />, indexed by a directed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919305.png" />, and a collection of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919306.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919307.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s0919309.png" />, such that
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{{TEX|auto}}
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{{TEX|done}}
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193011.png" />;
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''direct and inverse system in a category  $  C $''
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193014.png" />.
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A direct system  $  \{ Y  ^  \alpha  , f _  \alpha  ^ { \beta } \} $
 +
in  $  C $
 +
consists of a collection of objects  $  \{ Y  ^  \alpha  \} $,
 +
indexed by a directed set  $  \Lambda = \{ \alpha \} $,
 +
and a collection of morphisms  $  \{ f _  \alpha  ^ { \beta } : Y  ^  \alpha  \rightarrow Y  ^  \beta  \} $
 +
in  $  C $,
 +
for $  \alpha \leq  \beta $
 +
in $  \Lambda $,
 +
such that
  
There exists a category, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193015.png" />, whose objects are indexed collections of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193019.png" /> and whose morphisms with domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193020.png" /> and range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193021.png" /> are morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193024.png" />. An [[initial object]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193025.png" /> is called a direct limit of the direct system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193026.png" />. The direct limits of sets, topological spaces, groups, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193027.png" />-modules are examples of direct limits in their respective categories.
+
a)  $  f _  \alpha  ^ { \alpha } = 1 _ {Y  ^  \alpha  } $
 +
for $  \alpha \in \Lambda $;
  
Dually, an inverse system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193029.png" /> consists of a collection of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193030.png" />, indexed by a directed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193031.png" />, and a collection of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193033.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193035.png" />, such that
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b)  $  f _  \alpha  ^ { \gamma } = f _  \beta  ^ { \gamma } f _  \alpha  ^ { \beta } : Y  ^  \alpha  \rightarrow Y  ^  \gamma  $
 +
for $  \alpha \leq  \beta \leq  \gamma $
 +
in $  \Lambda $.
  
a<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193036.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193038.png" />;
+
There exists a category,  $  \mathop{\rm dir} \{ Y  ^  \alpha  , f _  \alpha  ^ { \beta } \} $,
 +
whose objects are indexed collections of morphisms  $  \{ g _  \alpha  :  Y  ^  \alpha  \rightarrow Z \} _ {\alpha \in \Lambda }  $
 +
such that  $  g _  \alpha  = g _  \beta  f _  \alpha  ^ { \beta } $
 +
if  $  \alpha \leq  \beta $
 +
in  $  \Lambda $
 +
and whose morphisms with domain  $  \{ g _  \alpha  :  Y  ^  \alpha  \rightarrow Z \} $
 +
and range  $  \{ g _  \alpha  ^  \prime  :  Y  ^  \alpha  \rightarrow Z  ^  \prime  \} $
 +
are morphisms  $  h: Z \rightarrow Z  ^  \prime  $
 +
such that  $  hg _  \alpha  = g _  \alpha  ^  \prime  $
 +
for $  \alpha \in \Lambda $.  
 +
An [[initial object]] of  $  \mathop{\rm dir} \{ Y  ^  \alpha  , f _  \alpha  ^ { \beta } \} $
 +
is called a direct limit of the direct system  $  \{ Y  ^  \alpha  , f _  \alpha  ^ { \beta } \} $.  
 +
The direct limits of sets, topological spaces, groups, and  $  R $-
 +
modules are examples of direct limits in their respective categories.
  
b<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193039.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193040.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193042.png" />.
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Dually, an inverse system  $  \{ Y _  \alpha  , f _  \alpha  ^ { \beta } \} $
 +
in  $  C $
 +
consists of a collection of objects  $  \{ Y _  \alpha  \} $,
 +
indexed by a directed set  $  \Lambda = \{ \alpha \} $,
 +
and a collection of morphisms  $  \{ f _  \alpha  ^ { \beta } : Y _  \beta  \rightarrow Y _  \alpha  \} $
 +
in  $  C $,
 +
for $  \alpha \leq  \beta $
 +
in $  \Lambda $,
 +
such that
  
There exists a category, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193043.png" />, whose objects are indexed collections of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193045.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193047.png" /> and whose morphisms with domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193048.png" /> and range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193049.png" /> are morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193052.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193053.png" />. A [[terminal object]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193054.png" /> is called an inverse limit of the inverse system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193055.png" />. The inverse limits of sets, topological spaces, groups, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091930/s09193056.png" />-modules are examples of inverse limits in their respective categories.
+
a $  {}  ^  \prime  $)
 +
$  f _  \alpha  ^ { \alpha } = 1 _ {Y _  \alpha  } $
 +
for  $  \alpha \in \Lambda $;
 +
 
 +
b $  {}  ^  \prime  $)
 +
$  f _  \alpha  ^ { \gamma } = f _  \alpha  ^ { \beta } f _  \beta  ^ { \gamma } :  Y _  \gamma  \rightarrow Y _  \alpha  $
 +
for  $  \alpha \leq  \beta \leq  \gamma $
 +
in  $  \Lambda $.
 +
 
 +
There exists a category, $  \mathop{\rm inv} \{ Y _  \alpha  , f _  \alpha  ^ { \beta } \} $,  
 +
whose objects are indexed collections of morphisms $  \{ g _  \alpha  : X \rightarrow Y _  \alpha  \} _ {\alpha \in \Lambda }  $
 +
such that $  g _  \alpha  = f _  \alpha  ^ { \beta } g _  \beta  $
 +
if $  \alpha \leq  \beta $
 +
in $  \Lambda $
 +
and whose morphisms with domain $  \{ g _  \alpha  : X \rightarrow Y _  \alpha  \} $
 +
and range $  \{ g _  \alpha  ^  \prime  : X  ^  \prime  \rightarrow Y _  \alpha  \} $
 +
are morphisms $  h: X \rightarrow X  ^  \prime  $
 +
of $  C $
 +
such that $  g _  \alpha  ^  \prime  h = g _  \alpha  $
 +
for $  \alpha \in \Lambda $.  
 +
A [[terminal object]] of $  \mathop{\rm inv} \{ Y _  \alpha  , f _  \alpha  ^ { \beta } \} $
 +
is called an inverse limit of the inverse system $  \{ Y _  \alpha  , f _  \alpha  ^ { \beta } \} $.  
 +
The inverse limits of sets, topological spaces, groups, and $  R $-
 +
modules are examples of inverse limits in their respective categories.
  
 
The concept of an inverse limit is a categorical generalization of the topological concept of a [[Projective limit|projective limit]].
 
The concept of an inverse limit is a categorical generalization of the topological concept of a [[Projective limit|projective limit]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:24, 6 June 2020


direct and inverse system in a category $ C $

A direct system $ \{ Y ^ \alpha , f _ \alpha ^ { \beta } \} $ in $ C $ consists of a collection of objects $ \{ Y ^ \alpha \} $, indexed by a directed set $ \Lambda = \{ \alpha \} $, and a collection of morphisms $ \{ f _ \alpha ^ { \beta } : Y ^ \alpha \rightarrow Y ^ \beta \} $ in $ C $, for $ \alpha \leq \beta $ in $ \Lambda $, such that

a) $ f _ \alpha ^ { \alpha } = 1 _ {Y ^ \alpha } $ for $ \alpha \in \Lambda $;

b) $ f _ \alpha ^ { \gamma } = f _ \beta ^ { \gamma } f _ \alpha ^ { \beta } : Y ^ \alpha \rightarrow Y ^ \gamma $ for $ \alpha \leq \beta \leq \gamma $ in $ \Lambda $.

There exists a category, $ \mathop{\rm dir} \{ Y ^ \alpha , f _ \alpha ^ { \beta } \} $, whose objects are indexed collections of morphisms $ \{ g _ \alpha : Y ^ \alpha \rightarrow Z \} _ {\alpha \in \Lambda } $ such that $ g _ \alpha = g _ \beta f _ \alpha ^ { \beta } $ if $ \alpha \leq \beta $ in $ \Lambda $ and whose morphisms with domain $ \{ g _ \alpha : Y ^ \alpha \rightarrow Z \} $ and range $ \{ g _ \alpha ^ \prime : Y ^ \alpha \rightarrow Z ^ \prime \} $ are morphisms $ h: Z \rightarrow Z ^ \prime $ such that $ hg _ \alpha = g _ \alpha ^ \prime $ for $ \alpha \in \Lambda $. An initial object of $ \mathop{\rm dir} \{ Y ^ \alpha , f _ \alpha ^ { \beta } \} $ is called a direct limit of the direct system $ \{ Y ^ \alpha , f _ \alpha ^ { \beta } \} $. The direct limits of sets, topological spaces, groups, and $ R $- modules are examples of direct limits in their respective categories.

Dually, an inverse system $ \{ Y _ \alpha , f _ \alpha ^ { \beta } \} $ in $ C $ consists of a collection of objects $ \{ Y _ \alpha \} $, indexed by a directed set $ \Lambda = \{ \alpha \} $, and a collection of morphisms $ \{ f _ \alpha ^ { \beta } : Y _ \beta \rightarrow Y _ \alpha \} $ in $ C $, for $ \alpha \leq \beta $ in $ \Lambda $, such that

a $ {} ^ \prime $) $ f _ \alpha ^ { \alpha } = 1 _ {Y _ \alpha } $ for $ \alpha \in \Lambda $;

b $ {} ^ \prime $) $ f _ \alpha ^ { \gamma } = f _ \alpha ^ { \beta } f _ \beta ^ { \gamma } : Y _ \gamma \rightarrow Y _ \alpha $ for $ \alpha \leq \beta \leq \gamma $ in $ \Lambda $.

There exists a category, $ \mathop{\rm inv} \{ Y _ \alpha , f _ \alpha ^ { \beta } \} $, whose objects are indexed collections of morphisms $ \{ g _ \alpha : X \rightarrow Y _ \alpha \} _ {\alpha \in \Lambda } $ such that $ g _ \alpha = f _ \alpha ^ { \beta } g _ \beta $ if $ \alpha \leq \beta $ in $ \Lambda $ and whose morphisms with domain $ \{ g _ \alpha : X \rightarrow Y _ \alpha \} $ and range $ \{ g _ \alpha ^ \prime : X ^ \prime \rightarrow Y _ \alpha \} $ are morphisms $ h: X \rightarrow X ^ \prime $ of $ C $ such that $ g _ \alpha ^ \prime h = g _ \alpha $ for $ \alpha \in \Lambda $. A terminal object of $ \mathop{\rm inv} \{ Y _ \alpha , f _ \alpha ^ { \beta } \} $ is called an inverse limit of the inverse system $ \{ Y _ \alpha , f _ \alpha ^ { \beta } \} $. The inverse limits of sets, topological spaces, groups, and $ R $- modules are examples of inverse limits in their respective categories.

The concept of an inverse limit is a categorical generalization of the topological concept of a projective limit.

References

[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)

Comments

There is a competing terminology, with "direct limit" replaced by "colimit" , and "inverse limit" by "limit" .

References

[1a] B. Mitchell, "Theory of categories" , Acad. Press (1965)
How to Cite This Entry:
System (in a category). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=System_(in_a_category)&oldid=42577
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article