Namespaces
Variants
Actions

Difference between revisions of "Quotient category"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A construct analogous to that of a quotient set or quotient algebra. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768701.png" /> be an arbitrary [[Category|category]], and suppose that an equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768702.png" /> is given on its class of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768703.png" />, satisfying the following conditions: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768704.png" />, then the sources and targets of the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768706.png" /> are the same; and 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768708.png" /> and if the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q0768709.png" /> is defined, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687011.png" /> denote the equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687012.png" />. The quotient category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687013.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687014.png" /> is the category (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687015.png" />) with the same objects as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687016.png" />, and for any pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687018.png" /> the set of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687020.png" /> consists of the equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687023.png" />; multiplication of two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687025.png" /> is defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687026.png" /> (when the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687027.png" /> is defined).
+
<!--
 +
q0768701.png
 +
$#A+1 = 28 n = 0
 +
$#C+1 = 28 : ~/encyclopedia/old_files/data/Q076/Q.0706870 Quotient category
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Every [[Small category|small category]] can be represented as a quotient category of the category of paths over an appropriate directed graph.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
A construct analogous to that of a quotient set or quotient algebra. Let  $  \mathfrak K $
 +
be an arbitrary [[Category|category]], and suppose that an equivalence relation  $  \sim $
 +
is given on its class of morphisms  $  \mathop{\rm Mor}  \mathfrak K $,
 +
satisfying the following conditions: 1) if  $  \alpha \sim \beta $,
 +
then the sources and targets of the morphisms  $  \alpha $
 +
and  $  \beta $
 +
are the same; and 2) if  $  \alpha \sim \beta $,
 +
$  \gamma \sim \delta $
 +
and if the product  $  \alpha \gamma $
 +
is defined, then  $  \alpha \gamma \sim \beta \delta $.
 +
Let  $  [ \alpha ] $
 +
denote the equivalence class of  $  \alpha $.
 +
The quotient category of  $  \mathfrak K $
 +
by  $  \sim $
 +
is the category (denoted by  $  \mathfrak K / \sim $)
 +
with the same objects as  $  \mathfrak K $,
 +
and for any pair of objects  $  A $,
 +
$  B $
 +
the set of morphisms  $  H ( A , B ) $
 +
in  $  \mathfrak K / \sim $
 +
consists of the equivalence classes  $  [ \alpha ] $,
 +
where  $  \alpha :  A \rightarrow B $
 +
in  $  \mathfrak K $;
 +
multiplication of two morphisms  $  [ \alpha ] $
 +
and  $  [ \beta ] $
 +
is defined by the formula  $  [ \alpha ] [ \beta ] = [ \alpha \beta ] $(
 +
when the product  $  \alpha \beta $
 +
is defined).
  
 +
Every [[Small category|small category]] can be represented as a quotient category of the category of paths over an appropriate directed graph.
  
 
====Comments====
 
====Comments====
Any equivalence relation satisfying the conditions above is commonly called a congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076870/q07687028.png" /> (cf. [[Congruence (in algebra)|Congruence (in algebra)]]).
+
Any equivalence relation satisfying the conditions above is commonly called a congruence on $  \mathfrak K $(
 +
cf. [[Congruence (in algebra)|Congruence (in algebra)]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)  pp. 4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)  pp. 4</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


A construct analogous to that of a quotient set or quotient algebra. Let $ \mathfrak K $ be an arbitrary category, and suppose that an equivalence relation $ \sim $ is given on its class of morphisms $ \mathop{\rm Mor} \mathfrak K $, satisfying the following conditions: 1) if $ \alpha \sim \beta $, then the sources and targets of the morphisms $ \alpha $ and $ \beta $ are the same; and 2) if $ \alpha \sim \beta $, $ \gamma \sim \delta $ and if the product $ \alpha \gamma $ is defined, then $ \alpha \gamma \sim \beta \delta $. Let $ [ \alpha ] $ denote the equivalence class of $ \alpha $. The quotient category of $ \mathfrak K $ by $ \sim $ is the category (denoted by $ \mathfrak K / \sim $) with the same objects as $ \mathfrak K $, and for any pair of objects $ A $, $ B $ the set of morphisms $ H ( A , B ) $ in $ \mathfrak K / \sim $ consists of the equivalence classes $ [ \alpha ] $, where $ \alpha : A \rightarrow B $ in $ \mathfrak K $; multiplication of two morphisms $ [ \alpha ] $ and $ [ \beta ] $ is defined by the formula $ [ \alpha ] [ \beta ] = [ \alpha \beta ] $( when the product $ \alpha \beta $ is defined).

Every small category can be represented as a quotient category of the category of paths over an appropriate directed graph.

Comments

Any equivalence relation satisfying the conditions above is commonly called a congruence on $ \mathfrak K $( cf. Congruence (in algebra)).

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. 4
How to Cite This Entry:
Quotient category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quotient_category&oldid=17571
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article