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A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathematics. An important special case of an inductive limit is the inductive limit of a directed family of mathematical structures of the same type. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507701.png" /> be a directed pre-ordered set, that is, a reflexive transitive relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507702.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507703.png" /> and for any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507704.png" /> there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507705.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507707.png" />. Suppose further that a structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507708.png" /> is associated with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i0507709.png" /> (for definiteness, suppose that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077010.png" /> are groups) and that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077011.png" /> homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077012.png" /> are given satisfying the two conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077013.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077015.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077017.png" />. An equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077018.png" /> is introduced on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077019.png" />: The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077020.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077023.png" />. The quotient set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077024.png" /> can then be endowed with a group structure: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077028.png" />, then the product of the equivalence classes represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077030.png" /> is defined to be the equivalence class with representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077031.png" />. The resulting group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077032.png" /> is called the inductive limit of the family of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077033.png" />. There exists for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077034.png" /> a natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077035.png" /> associating to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077036.png" /> its equivalence class. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077037.png" /> together with its homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077038.png" /> has the following property: For any system of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077040.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077042.png" />, there exists a unique homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077044.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077045.png" />.
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A generalization of the above construction of an inductive limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077046.png" /> of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077047.png" /> is called an inductive limit of the covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077048.png" /> if:
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1) there exist morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077050.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077051.png" /> for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077053.png" />; and
+
A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathematics. An important special case of an inductive limit is the inductive limit of a directed family of mathematical structures of the same type. Let  $  C $
 +
be a directed pre-ordered set, that is, a reflexive transitive relation  $  \prec $
 +
is defined on  $  C $
 +
and for any two elements  $  \alpha , \beta \in C $
 +
there exists an element  $  \gamma \in C $
 +
such that  $  \alpha \prec \gamma $
 +
and  $  \beta \prec \gamma $.  
 +
Suppose further that a structure  $  A _  \alpha  $
 +
is associated with each  $  \alpha \in C $(
 +
for definiteness, suppose that the  $  A _  \alpha  $
 +
are groups) and that for each  $  \alpha \prec \beta $
 +
homomorphisms  $  \phi _ {\alpha \beta }  :  A _  \alpha  \rightarrow A _  \beta  $
 +
are given satisfying the two conditions:  $  \phi _ {\alpha \alpha }  = 1 _ {A _  \alpha  } $
 +
for any  $  \alpha \in C $
 +
and  $  \phi _ {\alpha \beta }  \phi _ {\beta \gamma }  = \phi _ {\alpha \gamma }  $
 +
for any $  \alpha \prec \beta \prec \gamma $
 +
in  $  C $.
 +
An equivalence relation  $  \sim $
 +
is introduced on the set  $  \overline{A}\; = \cup _ {\alpha \in C }  A _  \alpha  $:
 +
The element  $  x \in A _  \alpha  $
 +
is equivalent to  $  y \in A _  \beta  $
 +
if  $  x \phi _ {\alpha \gamma }  = y \phi _ {\beta \gamma }  $
 +
for some  $  \gamma $.
 +
The quotient set  $  A = \overline{A}\; / \sim $
 +
can then be endowed with a group structure: If  $  x \in A _  \alpha  $,
 +
$  y \in A _  \beta  $
 +
and  $  \alpha \prec \gamma $,
 +
$  \beta \prec \gamma $,
 +
then the product of the equivalence classes represented by  $  x $
 +
and  $  y $
 +
is defined to be the equivalence class with representative  $  ( x \phi _ {\alpha \gamma }  ) ( y \phi _ {\beta \gamma }  ) $.  
 +
The resulting group  $  A $
 +
is called the inductive limit of the family of groups  $  A _  \alpha  $.  
 +
There exists for each  $  \alpha \in C $
 +
a natural homomorphism  $  \phi _  \alpha  :  A _  \alpha  \rightarrow A $
 +
associating to an element  $  x \in A _  \alpha  $
 +
its equivalence class. The group  $  A $
 +
together with its homomorphisms  $  \phi _  \alpha  $
 +
has the following property: For any system of homomorphisms  $  \psi _  \alpha  :  A _  \alpha  \rightarrow B $,
 +
$  \alpha \in C $,
 +
for which  $  \psi _  \alpha  = \phi _ {\alpha \beta }  \psi _  \beta  $
 +
for  $  \alpha \prec \beta $,
 +
there exists a unique homomorphism  $  \psi :  A \rightarrow B $
 +
such that  $  \psi _  \alpha  = \phi _  \alpha  \psi $
 +
for any  $  \alpha \in C $.
  
2) for any family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077055.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077056.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077058.png" />, there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077061.png" />.
+
A generalization of the above construction of an inductive limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object  $  A $
 +
of a category  $  \mathfrak K $
 +
is called an inductive limit of the covariant functor  $  F : \mathfrak D \rightarrow \mathfrak K $
 +
if:
  
An inductive limit is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077062.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077063.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077064.png" />. An inductive limit of a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077065.png" /> is defined as an inductive limit of the covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077066.png" /> from the dual category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077067.png" /> into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077068.png" />.
+
1) there exist morphisms  $  \phi _ {D} : F ( D) \rightarrow A $,
 +
where  $  D \in  \mathop{\rm Ob}  \mathfrak D $,
 +
such that  $  F ( \alpha ) \phi _ {D _ {1}  } = \phi _ {D} $
 +
for any morphism  $  \alpha : D \rightarrow D _ {1} $
 +
in  $  \mathfrak D $;
 +
and
  
Every pre-ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077069.png" /> can be regarded as a category whose objects are the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077070.png" /> and whose morphisms are all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077071.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077073.png" />, with the obvious law of composition. In an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077074.png" />, a family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077076.png" />, and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077078.png" />, can be regarded as the image of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077079.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077083.png" /> is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077084.png" /> coincides with the construction of the inductive limit defined above.
+
2) for any family of morphisms $  \psi _ {D} : F ( D) \rightarrow B $,  
 +
where $  D \in  \mathop{\rm Ob}  \mathfrak D $,
 +
such that  $  F ( \alpha ) \psi _ {D _ {1}  } = \psi _ {D} $
 +
for any  $  \alpha : D \rightarrow D _ {1} $
 +
in  $  \mathfrak D $,  
 +
there exists a unique morphism  $  \gamma : A \rightarrow B $
 +
such that  $  \psi _ {D} = \phi _ {D} \gamma $,  
 +
$  D \in  \mathop{\rm Ob}  \mathfrak D $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077085.png" /> is a small [[discrete category]], then an inductive limit of any functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077086.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077087.png" /> into an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077088.png" /> is a coproduct of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077090.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077091.png" /> is empty, then an inductive limit is a left null or an initial object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077092.png" />. Cokernels of pairs of morphisms of any category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077093.png" /> are inductive limits of functors defined on the category with two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077095.png" /> and four morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077097.png" />.
+
An inductive limit is denoted by  $  ( A , \phi _ {D} ) = \lim\limits _  \rightarrow  F $
 +
or  $  A = \lim\limits _  \rightarrow  F $
 +
or  $  A = \lim\limits _  \rightarrow  F ( D ) $.  
 +
An inductive limit of a contravariant functor $  F : \mathfrak D \rightarrow \mathfrak K $
 +
is defined as an inductive limit of the covariant functor  $  F ^ { * } $
 +
from the dual category $  \mathfrak D  ^ {*} $
 +
into the category $  \mathfrak K $.
  
Every covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077098.png" /> from an arbitrary small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077099.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770100.png" /> has an inductive limit if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770101.png" /> has coproducts and cokernels of pairs of morphisms.
+
Every pre-ordered set  $  C $
 +
can be regarded as a category whose objects are the elements of  $  C $
 +
and whose morphisms are all pairs  $  ( \alpha , \beta ) $
 +
for which  $  \alpha , \beta \in C $
 +
and  $  \alpha \prec \beta $,
 +
with the obvious law of composition. In an arbitrary category  $  \mathfrak K $,
 +
a family of objects  $  A _  \alpha  $,
 +
$  \alpha \in C $,
 +
and morphisms  $  \phi _ {\alpha \beta }  :  A _  \alpha  \rightarrow A _  \beta  $,
 +
where  $  \alpha \prec \beta $,
 +
can be regarded as the image of a functor $  F :  C \rightarrow \mathfrak K $
 +
if  $  \phi _ {\alpha \alpha }  = 1 $
 +
and  $  \phi _ {\alpha \beta }  \phi _ {\beta \gamma }  = \phi _ {\alpha \gamma }  $
 +
for  $  \alpha \prec \beta \prec \gamma $.  
 +
If  $  \mathfrak K $
 +
is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor  $  F :  C \rightarrow \mathfrak K $
 +
coincides with the construction of the inductive limit defined above.
 +
 
 +
If  $  \mathfrak D $
 +
is a small [[discrete category]], then an inductive limit of any functor  $  F $
 +
from $  \mathfrak D $
 +
into an arbitrary category $  \mathfrak K $
 +
is a coproduct of the objects  $  F ( D) $,
 +
$  D \in \mathfrak D $.  
 +
In particular, if  $  \mathfrak D $
 +
is empty, then an inductive limit is a left null or an initial object of  $  \mathfrak K $.  
 +
Cokernels of pairs of morphisms of any category  $  \mathfrak K $
 +
are inductive limits of functors defined on the category with two objects  $  X $
 +
and  $  Y $
 +
and four morphisms  $  1 _ {X} , 1 _ {Y} $
 +
and  $  \alpha , \beta :  X \rightarrow Y $.
 +
 
 +
Every covariant functor  $  F $
 +
from an arbitrary small category  $  \mathfrak D $
 +
into a category $  \mathfrak K $
 +
has an inductive limit if and only if $  \mathfrak K $
 +
has coproducts and cokernels of pairs of morphisms.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Topological duality theorems"  ''Trudy Mat. Inst. Steklov.'' , '''48'''  (1955)  pp. Chapt. 1  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Topological duality theorems"  ''Trudy Mat. Inst. Steklov.'' , '''48'''  (1955)  pp. Chapt. 1  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In the main article above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770102.png" /> stands for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770103.png" />: the value of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770104.png" /> on the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770105.png" />.
+
In the main article above, $  x \phi _ {\alpha \gamma }  $
 +
stands for $  \phi _ {\alpha \gamma }  ( x) $:  
 +
the value of the homomorphism $  \phi _ {\alpha \gamma }  $
 +
on the element $  x $.
  
Similarly, composition of homomorphisms is written backwards: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770106.png" /> means first apply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770107.png" /> and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770108.png" />.
+
Similarly, composition of homomorphisms is written backwards: $  \phi _ {\alpha \beta }  \psi _  \beta  $
 +
means first apply $  \phi _ {\alpha \beta }  $
 +
and then $  \psi _  \beta  $.
  
 
In English, the term  "inductive limit"  is usually restricted to limits over directed pre-ordered sets, the more general categorical concept being called a colimit.  "Cokernels of pairs of morphisms"  are commonly called coequalizers.
 
In English, the term  "inductive limit"  is usually restricted to limits over directed pre-ordered sets, the more general categorical concept being called a colimit.  "Cokernels of pairs of morphisms"  are commonly called coequalizers.

Latest revision as of 22:12, 5 June 2020


A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathematics. An important special case of an inductive limit is the inductive limit of a directed family of mathematical structures of the same type. Let $ C $ be a directed pre-ordered set, that is, a reflexive transitive relation $ \prec $ is defined on $ C $ and for any two elements $ \alpha , \beta \in C $ there exists an element $ \gamma \in C $ such that $ \alpha \prec \gamma $ and $ \beta \prec \gamma $. Suppose further that a structure $ A _ \alpha $ is associated with each $ \alpha \in C $( for definiteness, suppose that the $ A _ \alpha $ are groups) and that for each $ \alpha \prec \beta $ homomorphisms $ \phi _ {\alpha \beta } : A _ \alpha \rightarrow A _ \beta $ are given satisfying the two conditions: $ \phi _ {\alpha \alpha } = 1 _ {A _ \alpha } $ for any $ \alpha \in C $ and $ \phi _ {\alpha \beta } \phi _ {\beta \gamma } = \phi _ {\alpha \gamma } $ for any $ \alpha \prec \beta \prec \gamma $ in $ C $. An equivalence relation $ \sim $ is introduced on the set $ \overline{A}\; = \cup _ {\alpha \in C } A _ \alpha $: The element $ x \in A _ \alpha $ is equivalent to $ y \in A _ \beta $ if $ x \phi _ {\alpha \gamma } = y \phi _ {\beta \gamma } $ for some $ \gamma $. The quotient set $ A = \overline{A}\; / \sim $ can then be endowed with a group structure: If $ x \in A _ \alpha $, $ y \in A _ \beta $ and $ \alpha \prec \gamma $, $ \beta \prec \gamma $, then the product of the equivalence classes represented by $ x $ and $ y $ is defined to be the equivalence class with representative $ ( x \phi _ {\alpha \gamma } ) ( y \phi _ {\beta \gamma } ) $. The resulting group $ A $ is called the inductive limit of the family of groups $ A _ \alpha $. There exists for each $ \alpha \in C $ a natural homomorphism $ \phi _ \alpha : A _ \alpha \rightarrow A $ associating to an element $ x \in A _ \alpha $ its equivalence class. The group $ A $ together with its homomorphisms $ \phi _ \alpha $ has the following property: For any system of homomorphisms $ \psi _ \alpha : A _ \alpha \rightarrow B $, $ \alpha \in C $, for which $ \psi _ \alpha = \phi _ {\alpha \beta } \psi _ \beta $ for $ \alpha \prec \beta $, there exists a unique homomorphism $ \psi : A \rightarrow B $ such that $ \psi _ \alpha = \phi _ \alpha \psi $ for any $ \alpha \in C $.

A generalization of the above construction of an inductive limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object $ A $ of a category $ \mathfrak K $ is called an inductive limit of the covariant functor $ F : \mathfrak D \rightarrow \mathfrak K $ if:

1) there exist morphisms $ \phi _ {D} : F ( D) \rightarrow A $, where $ D \in \mathop{\rm Ob} \mathfrak D $, such that $ F ( \alpha ) \phi _ {D _ {1} } = \phi _ {D} $ for any morphism $ \alpha : D \rightarrow D _ {1} $ in $ \mathfrak D $; and

2) for any family of morphisms $ \psi _ {D} : F ( D) \rightarrow B $, where $ D \in \mathop{\rm Ob} \mathfrak D $, such that $ F ( \alpha ) \psi _ {D _ {1} } = \psi _ {D} $ for any $ \alpha : D \rightarrow D _ {1} $ in $ \mathfrak D $, there exists a unique morphism $ \gamma : A \rightarrow B $ such that $ \psi _ {D} = \phi _ {D} \gamma $, $ D \in \mathop{\rm Ob} \mathfrak D $.

An inductive limit is denoted by $ ( A , \phi _ {D} ) = \lim\limits _ \rightarrow F $ or $ A = \lim\limits _ \rightarrow F $ or $ A = \lim\limits _ \rightarrow F ( D ) $. An inductive limit of a contravariant functor $ F : \mathfrak D \rightarrow \mathfrak K $ is defined as an inductive limit of the covariant functor $ F ^ { * } $ from the dual category $ \mathfrak D ^ {*} $ into the category $ \mathfrak K $.

Every pre-ordered set $ C $ can be regarded as a category whose objects are the elements of $ C $ and whose morphisms are all pairs $ ( \alpha , \beta ) $ for which $ \alpha , \beta \in C $ and $ \alpha \prec \beta $, with the obvious law of composition. In an arbitrary category $ \mathfrak K $, a family of objects $ A _ \alpha $, $ \alpha \in C $, and morphisms $ \phi _ {\alpha \beta } : A _ \alpha \rightarrow A _ \beta $, where $ \alpha \prec \beta $, can be regarded as the image of a functor $ F : C \rightarrow \mathfrak K $ if $ \phi _ {\alpha \alpha } = 1 $ and $ \phi _ {\alpha \beta } \phi _ {\beta \gamma } = \phi _ {\alpha \gamma } $ for $ \alpha \prec \beta \prec \gamma $. If $ \mathfrak K $ is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor $ F : C \rightarrow \mathfrak K $ coincides with the construction of the inductive limit defined above.

If $ \mathfrak D $ is a small discrete category, then an inductive limit of any functor $ F $ from $ \mathfrak D $ into an arbitrary category $ \mathfrak K $ is a coproduct of the objects $ F ( D) $, $ D \in \mathfrak D $. In particular, if $ \mathfrak D $ is empty, then an inductive limit is a left null or an initial object of $ \mathfrak K $. Cokernels of pairs of morphisms of any category $ \mathfrak K $ are inductive limits of functors defined on the category with two objects $ X $ and $ Y $ and four morphisms $ 1 _ {X} , 1 _ {Y} $ and $ \alpha , \beta : X \rightarrow Y $.

Every covariant functor $ F $ from an arbitrary small category $ \mathfrak D $ into a category $ \mathfrak K $ has an inductive limit if and only if $ \mathfrak K $ has coproducts and cokernels of pairs of morphisms.

References

[1] P.S. Aleksandrov, "Topological duality theorems" Trudy Mat. Inst. Steklov. , 48 (1955) pp. Chapt. 1 (In Russian)
[2] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[3] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)

Comments

In the main article above, $ x \phi _ {\alpha \gamma } $ stands for $ \phi _ {\alpha \gamma } ( x) $: the value of the homomorphism $ \phi _ {\alpha \gamma } $ on the element $ x $.

Similarly, composition of homomorphisms is written backwards: $ \phi _ {\alpha \beta } \psi _ \beta $ means first apply $ \phi _ {\alpha \beta } $ and then $ \psi _ \beta $.

In English, the term "inductive limit" is usually restricted to limits over directed pre-ordered sets, the more general categorical concept being called a colimit. "Cokernels of pairs of morphisms" are commonly called coequalizers.

Dual to the notion of an inductive limit is that of a projective limit, also called an inverse limit.

References

[a1] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Inductive limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inductive_limit&oldid=47336
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article