Namespaces
Variants
Actions

Difference between revisions of "Quotient object"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
q0769001.png
 +
$#A+1 = 41 n = 0
 +
$#C+1 = 41 : ~/encyclopedia/old_files/data/Q076/Q.0706900 Quotient object
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''of an object in a category''
 
''of an object in a category''
  
 
A concept generalizing the notions of a quotient set, a quotient group, a quotient space, etc.
 
A concept generalizing the notions of a quotient set, a quotient group, a quotient space, etc.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769001.png" /> be some class of epimorphisms in a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769002.png" /> that contains all identity morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769003.png" /> and is closed under multiplication on the right by isomorphisms. In other words, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769005.png" /> and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769006.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769007.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769008.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q0769009.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690010.png" />. Two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690013.png" /> are said to be equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690014.png" /> for some isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690015.png" />. The equivalence class of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690016.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690018.png" />-quotient object of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690019.png" />, and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690020.png" /> is called a representative of the quotient object. A quotient object with representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690021.png" /> is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690023.png" /> or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690024.png" />.
+
Let $  {\mathcal E} $
 +
be some class of epimorphisms in a [[Category|category]] $  \mathfrak K $
 +
that contains all identity morphisms in $  \mathfrak K $
 +
and is closed under multiplication on the right by isomorphisms. In other words, for every $  X \in  \mathop{\rm Ob}  \mathfrak K $,  
 +
$  1 _ {X} \in {\mathcal E} $
 +
and for every $  \xi : B \rightarrow C $
 +
in $  \mathop{\rm Iso}  \mathfrak K $
 +
and every $  \epsilon : A \rightarrow B $
 +
in $  {\mathcal E} $
 +
the morphism $  \epsilon \xi \in {\mathcal E} $.  
 +
Two morphisms $  \epsilon : A \rightarrow B $
 +
and $  \epsilon _ {1} : A \rightarrow C $
 +
in $  {\mathcal E} $
 +
are said to be equivalent if $  \epsilon _ {1} = \xi \epsilon $
 +
for some isomorphism $  \xi $.  
 +
The equivalence class of a morphism $  \epsilon $
 +
is called an $  {\mathcal E} $-
 +
quotient object of the object $  A $,  
 +
and the pair $  ( \epsilon , B) $
 +
is called a representative of the quotient object. A quotient object with representative $  ( \epsilon , B ) $
 +
is sometimes denoted by $  [ \epsilon , B ] $,
 +
$  [ \epsilon , B) $
 +
or simply by $  [ \epsilon ] $.
  
Every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690025.png" /> has at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690026.png" />-quotient object, the improper quotient object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690027.png" />; other quotient objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690028.png" /> are called proper. A category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690029.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690031.png" />-locally small if for every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690033.png" /> the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690034.png" />-quotient objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690035.png" /> is a set.
+
Every object $  A $
 +
has at least one $  {\mathcal E} $-
 +
quotient object, the improper quotient object $  [ 1 _ {A} , A ] $;  
 +
other quotient objects of $  A $
 +
are called proper. A category $  \mathfrak K $
 +
is called $  {\mathcal E} $-
 +
locally small if for every object $  A $
 +
in $  \mathfrak K $
 +
the class of $  {\mathcal E} $-
 +
quotient objects of $  A $
 +
is a set.
  
If one takes as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690036.png" /> the subcategory of all epimorphisms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690038.png" />-quotient objects are simply called quotient objects. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690039.png" /> is part of a [[Bicategory(2)|bicategory]] structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690042.png" />-quotient objects are called admissible quotient objects. Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076900/q07690043.png" /> consists of all regular (strict, normal, etc.) epimorphisms, then the corresponding quotient objects are called regular (strict, normal, etc). For example, in the category of topological spaces quotient spaces correspond to regular quotient objects.
+
If one takes as $  {\mathcal E} $
 +
the subcategory of all epimorphisms, $  \mathop{\rm Epi}  \mathfrak K $,  
 +
then $  \mathop{\rm Epi}  \mathfrak K $-
 +
quotient objects are simply called quotient objects. If $  {\mathcal E} $
 +
is part of a [[Bicategory(2)|bicategory]] structure $  ( \mathfrak K , {\mathcal E} , \mathfrak M ) $
 +
on $  \mathfrak K $,  
 +
then $  {\mathcal E} $-
 +
quotient objects are called admissible quotient objects. Similarly, if $  {\mathcal E} $
 +
consists of all regular (strict, normal, etc.) epimorphisms, then the corresponding quotient objects are called regular (strict, normal, etc). For example, in the category of topological spaces quotient spaces correspond to regular quotient objects.
  
 
The concept of a quotient object of an object in a category is dual to that of a [[Subobject|subobject]].
 
The concept of a quotient object of an object in a category is dual to that of a [[Subobject|subobject]].
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:09, 6 June 2020


of an object in a category

A concept generalizing the notions of a quotient set, a quotient group, a quotient space, etc.

Let $ {\mathcal E} $ be some class of epimorphisms in a category $ \mathfrak K $ that contains all identity morphisms in $ \mathfrak K $ and is closed under multiplication on the right by isomorphisms. In other words, for every $ X \in \mathop{\rm Ob} \mathfrak K $, $ 1 _ {X} \in {\mathcal E} $ and for every $ \xi : B \rightarrow C $ in $ \mathop{\rm Iso} \mathfrak K $ and every $ \epsilon : A \rightarrow B $ in $ {\mathcal E} $ the morphism $ \epsilon \xi \in {\mathcal E} $. Two morphisms $ \epsilon : A \rightarrow B $ and $ \epsilon _ {1} : A \rightarrow C $ in $ {\mathcal E} $ are said to be equivalent if $ \epsilon _ {1} = \xi \epsilon $ for some isomorphism $ \xi $. The equivalence class of a morphism $ \epsilon $ is called an $ {\mathcal E} $- quotient object of the object $ A $, and the pair $ ( \epsilon , B) $ is called a representative of the quotient object. A quotient object with representative $ ( \epsilon , B ) $ is sometimes denoted by $ [ \epsilon , B ] $, $ [ \epsilon , B) $ or simply by $ [ \epsilon ] $.

Every object $ A $ has at least one $ {\mathcal E} $- quotient object, the improper quotient object $ [ 1 _ {A} , A ] $; other quotient objects of $ A $ are called proper. A category $ \mathfrak K $ is called $ {\mathcal E} $- locally small if for every object $ A $ in $ \mathfrak K $ the class of $ {\mathcal E} $- quotient objects of $ A $ is a set.

If one takes as $ {\mathcal E} $ the subcategory of all epimorphisms, $ \mathop{\rm Epi} \mathfrak K $, then $ \mathop{\rm Epi} \mathfrak K $- quotient objects are simply called quotient objects. If $ {\mathcal E} $ is part of a bicategory structure $ ( \mathfrak K , {\mathcal E} , \mathfrak M ) $ on $ \mathfrak K $, then $ {\mathcal E} $- quotient objects are called admissible quotient objects. Similarly, if $ {\mathcal E} $ consists of all regular (strict, normal, etc.) epimorphisms, then the corresponding quotient objects are called regular (strict, normal, etc). For example, in the category of topological spaces quotient spaces correspond to regular quotient objects.

The concept of a quotient object of an object in a category is dual to that of a subobject.

Comments

The terms "colocally small categorycolocally small" and "co-well-powered categoryco-well-powered" are often used instead of "locally small" .

References

[a1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221
[a2] B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. 7
How to Cite This Entry:
Quotient object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quotient_object&oldid=48408
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article