Namespaces
Variants
Actions

Difference between revisions of "Small object"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A concept which singles out objects in a [[Category|category]] that have intrinsically the properties of a mathematical structure with a finite number of generators (finite-dimensional linear space, finitely-generated group, etc.). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858101.png" /> be a category with coproducts. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858102.png" /> is called small if for any morphism
+
<!--
 +
s0858101.png
 +
$#A+1 = 20 n = 0
 +
$#C+1 = 20 : ~/encyclopedia/old_files/data/S085/S.0805810 Small object
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858103.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858105.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858106.png" /> is the imbedding of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858107.png" />-th summand in the coproduct, there is a finite subset of the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858108.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s0858109.png" /> factors through the morphism
+
A concept which singles out objects in a [[Category|category]] that have intrinsically the properties of a mathematical structure with a finite number of generators (finite-dimensional linear space, finitely-generated group, etc.). Let  $  \mathfrak N $
 +
be a category with coproducts. An object  $  U \in  \mathop{\rm Ob}  \mathfrak N $
 +
is called small if for any morphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581010.png" /></td> </tr></table>
+
$$
 +
\phi : U  \rightarrow \
 +
\sum _ {i \in I } U _ {i}  $$
  
induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581011.png" />. Sometimes a stronger definition is given in which it is not assumed that all summands in the coproduct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581012.png" /> coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581013.png" />.
+
where  $  U _ {i} = U $,
 +
$  i \in I $,
 +
and  $  \sigma _ {i} $
 +
is the imbedding of the  $  i $-
 +
th summand in the coproduct, there is a finite subset of the indices  $  1 \dots n $
 +
such that $  \phi $
 +
factors through the morphism
  
In varieties of finitary universal algebras the following conditions on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581014.png" /> are equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581015.png" /> is a small object of the category; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581016.png" /> has a finite number of generators; and c) the covariant hom-functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581017.png" /> commutes with colimits (direct limits) of directed families of monomorphisms. Property c) is often taken as the definition of a finitely-generated object of an arbitrary category.
+
$$
 +
U _ {1} + \dots + U _ {n}  \rightarrow  \sum _ {i \in I } U _ {i}  $$
  
 +
induced by  $  \sigma _ {1} \dots \sigma _ {n} $.
 +
Sometimes a stronger definition is given in which it is not assumed that all summands in the coproduct  $  \sum _ {i \in I }  U _ {i} $
 +
coincide with  $  U $.
  
 +
In varieties of finitary universal algebras the following conditions on an algebra  $  A $
 +
are equivalent: a)  $  A $
 +
is a small object of the category; b)  $  A $
 +
has a finite number of generators; and c) the covariant hom-functor  $  H _ {A} ( X) = H ( A, X) $
 +
commutes with colimits (direct limits) of directed families of monomorphisms. Property c) is often taken as the definition of a finitely-generated object of an arbitrary category.
  
 
====Comments====
 
====Comments====
In an [[Additive category|additive category]], an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581018.png" /> is small if and only if the Abelian-group-valued functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581019.png" /> preserves coproducts. Some authors take this as the definition of smallness in non-additive categories: it produces a more restrictive condition than the one above, equivalent to the requirement that every morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085810/s08581020.png" /> to a coproduct should factor through a unique summand. In practice the term is rarely used outside the context of additive categories.
+
In an [[Additive category|additive category]], an object $  U $
 +
is small if and only if the Abelian-group-valued functor $  H( U , -) $
 +
preserves coproducts. Some authors take this as the definition of smallness in non-additive categories: it produces a more restrictive condition than the one above, equivalent to the requirement that every morphism from $  U $
 +
to a coproduct should factor through a unique summand. In practice the term is rarely used outside the context of additive categories.

Latest revision as of 08:14, 6 June 2020


A concept which singles out objects in a category that have intrinsically the properties of a mathematical structure with a finite number of generators (finite-dimensional linear space, finitely-generated group, etc.). Let $ \mathfrak N $ be a category with coproducts. An object $ U \in \mathop{\rm Ob} \mathfrak N $ is called small if for any morphism

$$ \phi : U \rightarrow \ \sum _ {i \in I } U _ {i} $$

where $ U _ {i} = U $, $ i \in I $, and $ \sigma _ {i} $ is the imbedding of the $ i $- th summand in the coproduct, there is a finite subset of the indices $ 1 \dots n $ such that $ \phi $ factors through the morphism

$$ U _ {1} + \dots + U _ {n} \rightarrow \sum _ {i \in I } U _ {i} $$

induced by $ \sigma _ {1} \dots \sigma _ {n} $. Sometimes a stronger definition is given in which it is not assumed that all summands in the coproduct $ \sum _ {i \in I } U _ {i} $ coincide with $ U $.

In varieties of finitary universal algebras the following conditions on an algebra $ A $ are equivalent: a) $ A $ is a small object of the category; b) $ A $ has a finite number of generators; and c) the covariant hom-functor $ H _ {A} ( X) = H ( A, X) $ commutes with colimits (direct limits) of directed families of monomorphisms. Property c) is often taken as the definition of a finitely-generated object of an arbitrary category.

Comments

In an additive category, an object $ U $ is small if and only if the Abelian-group-valued functor $ H( U , -) $ preserves coproducts. Some authors take this as the definition of smallness in non-additive categories: it produces a more restrictive condition than the one above, equivalent to the requirement that every morphism from $ U $ to a coproduct should factor through a unique summand. In practice the term is rarely used outside the context of additive categories.

How to Cite This Entry:
Small object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_object&oldid=48736
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article