Difference between revisions of "System (in a category)"
From Encyclopedia of Mathematics
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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" />. | 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" />. | ||
| − | 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. | + | 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. |
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 | 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<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" />. | 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" />. | ||
| − | 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. | + | 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. |
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]]. | ||
Revision as of 21:05, 21 December 2017
direct and inverse system in a category 
A direct system 
in 
consists of a collection of objects 
, indexed by a directed set 
, and a collection of morphisms 
in 
, for 
in 
, such that
a) 
for 
;
b) 
for 
in 
.
There exists a category, 
, whose objects are indexed collections of morphisms 
such that 
if 
in 
and whose morphisms with domain 
and range 
are morphisms 
such that 
for 
. An initial object of 
is called a direct limit of the direct system 
. The direct limits of sets, topological spaces, groups, and 
-modules are examples of direct limits in their respective categories.
Dually, an inverse system 
in 
consists of a collection of objects 
, indexed by a directed set 
, and a collection of morphisms 
in 
, for 
in 
, such that
a
) 
for 
;
b
) 
for 
in 
.
There exists a category, 
, whose objects are indexed collections of morphisms 
such that 
if 
in 
and whose morphisms with domain 
and range 
are morphisms 
of 
such that 
for 
. A terminal object of 
is called an inverse limit of the inverse system 
. The inverse limits of sets, topological spaces, groups, and 
-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=18745
System (in a category). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=System_(in_a_category)&oldid=18745
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article