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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853801.png" />''
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A contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853802.png" /> (or, equivalently, a covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853803.png" />) from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853804.png" />, whose objects are ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853806.png" />, and whose morphisms are non-decreasing mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853807.png" />, into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853808.png" />. A covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s0853809.png" /> (or, equivalently, a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538010.png" />) is called a co-simplicial object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538011.png" />.
+
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 +
'' $  {\mathcal C} $''
 +
 
 +
A contravariant functor $  X: \Delta \rightarrow {\mathcal C} $(
 +
or, equivalently, a covariant functor $  X: \Delta  ^ {op} \rightarrow {\mathcal C} $)  
 +
from the category $  \Delta $,  
 +
whose objects are ordered sets $  [ n] = \{ 0 \dots n \} $,  
 +
$  n \geq  0 $,  
 +
and whose morphisms are non-decreasing mappings $  \mu : [ n] \rightarrow [ m] $,  
 +
into the category $  {\mathcal C} $.  
 +
A covariant functor $  X: \Delta \rightarrow {\mathcal C} $(
 +
or, equivalently, a contravariant functor $  X: \Delta  ^ {op} \rightarrow {\mathcal C} $)  
 +
is called a co-simplicial object in $  {\mathcal C} $.
  
 
The morphisms
 
The morphisms
  
<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/s085380/s08538012.png" /></td> </tr></table>
+
$$
 +
\delta _ {i}  = \delta _ {i}  ^ {n} :  [ n - 1]  \rightarrow  [ n],\ \
 +
0 \leq  i \leq  n,
 +
$$
 +
 
 +
$$
 +
\sigma _ {i}  = \sigma _ {i}  ^ {n} : [ n + 1]  \rightarrow  [ n],\  0 \leq  i \leq  n,
 +
$$
 +
 
 +
of  $  \Delta $
 +
given by
 +
 
 +
$$
 +
\delta _ {i}  ^ {n} ( j)  = \
 +
\left \{
 +
 
 +
\begin{array}{ll}
 +
j  &\textrm{ if }  j < i,  \\
 +
j + 1  &\textrm{ if }  j \geq  i,  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
$$
 +
\sigma _ {i}  ^ {n} ( j)  = \left \{
 +
\begin{array}{ll}
 +
j  &\textrm{ if }  j \leq  i,  \\
 +
j -
 +
1  &\textrm{ if }  j > i,  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
generate all the morphisms of  $  \Delta $,
 +
so that a simplicial object  $  X $
 +
is determined by the objects  $  X ([ n]) = X _ {n} $,
 +
$  n \geq  0 $(
 +
called the  $  n $-
 +
fibres or  $  n $-
 +
components of the simplicial object  $  X $),
 +
and the morphisms
 +
 
 +
$$
 +
d _ {i}  = X ( \delta _ {i} ): X _ {n}  \rightarrow  X _ {n - 1 }  \ \
 +
\textrm{ and } \ \
 +
s _ {i}  =  X ( \sigma _ {i} ):  X _ {n}  \rightarrow  X _ {n + 1 }
 +
$$
  
<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/s085380/s08538013.png" /></td> </tr></table>
+
(called boundary operators and degeneracy operators, respectively). In case  $  {\mathcal C} $
 +
is a category of structured sets, the elements of  $  X _ {n} $
 +
are usually called the  $  n $-
 +
dimensional simplices of  $  X $.  
 +
The mappings  $  \delta _ {i} $
 +
and  $  \sigma _ {i} $
 +
satisfy the relations
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538014.png" /> given by
+
$$ \tag{* }
 +
\left .
  
<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/s085380/s08538015.png" /></td> </tr></table>
+
\begin{array}{cll}
 +
\delta _ {j} \delta _ {i}  = \
 +
\delta _ {i} \delta _ {j - 1 }  \  \textrm{ if }  i < j,  &\sigma _ {j} \delta _ {i}  = \
 +
\left \{
  
<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/s085380/s08538016.png" /></td> </tr></table>
+
\begin{array}{}
 +
\delta _ {i} \sigma _ {j - 1 }  \\
 +
\end{array}
 +
  &\textrm{ if }  i < j,  \\
 +
\sigma _ {j} \sigma _ {i}  = \sigma _ {i} \sigma _ {j + 1 }  \ \
 +
\textrm{ if }  i \leq  j; & \mathop{\rm id}  &\textrm{ if }  i = j  \textrm{ or }  i = j + 1,  \\
 +
\sigma _ {j} \delta _ {i}  = \
 +
\left \{
  
generate all the morphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538017.png" />, so that a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538018.png" /> is determined by the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538020.png" /> (called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538022.png" />-fibres or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538024.png" />-components of the simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538025.png" />), and the morphisms
+
\begin{array}{}
 +
\delta _ {i} \sigma _ {j - 1 }  \\
 +
\end{array}
 +
  &\delta _ {i - 1 }  \sigma _ {j}  \\
 +
\end{array}
 +
\right \}
 +
$$
  
<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/s085380/s08538026.png" /></td> </tr></table>
+
and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object  $  X $
 +
can be identified with a system  $  \{ X _ {n} , d _ {i} , s _ {i} \} $
 +
of objects  $  X _ {n} $,
 +
$  n \geq  0 $,
 +
of  $  {\mathcal C} $
 +
and morphisms  $  d _ {i} : X _ {n} \rightarrow X _ {n - 1 }  $
 +
and  $  s _ {i} : X _ {n} \rightarrow X _ {n + 1 }  $,
 +
0 \leq  i \leq  n $,
 +
satisfying the relations
  
(called boundary operators and degeneracy operators, respectively). In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538027.png" /> is a category of structured sets, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538028.png" /> are usually called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538030.png" />-dimensional simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538031.png" />. The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538033.png" /> satisfy the relations
+
$$
 +
d _ {i} d _ {j}  = d _ {j - 1 }  d _ {i} \  \textrm{ if }  i < j;
 +
$$
  
<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/s085380/s08538034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
s _ {i} s _ {j}  = s _ {j + 1 }  s _ {i} \  \textrm{ if }  i \leq  j;
 +
$$
  
and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538035.png" /> can be identified with a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538036.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538038.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538039.png" /> and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538042.png" />, satisfying the relations
+
$$
 +
d _ {i} s _ {j}  = \left \{
 +
\begin{array}{ll}
 +
s _ {j - 1
 +
}  d _ {i}  & \textrm{ if }  i < j, \\
 +
\mathop{\rm id}  & \textrm{ if }  i = j \
 +
\textrm{ or }  i = j + 1, \\
 +
s _ {j} d _ {i - 1 }  & \textrm{ if }  i > j + 1. \\
 +
\end{array}
  
<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/s085380/s08538043.png" /></td> </tr></table>
+
\right .$$
  
<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/s085380/s08538044.png" /></td> </tr></table>
+
Similarly, a co-simplicial object  $  X $
 +
can be identified with a system  $  \{ X _ {n} , d  ^ {i} , s  ^ {i} \} $
 +
of objects  $  X  ^ {n} $,
 +
$  n \geq  0 $(
 +
$  n $-
 +
co-fibres) and morphisms  $  d _ {i} : X ^ {n - 1 } \rightarrow X  ^ {n} $,
 +
$  0 \leq  i \leq  n $(
 +
co-boundary operators), and  $  s  ^ {i} : X ^ {n + 1 } \rightarrow X  ^ {n} $,
 +
$  0 \leq  i \leq  n $(
 +
co-degeneracy operators), satisfying the relations (*) (with  $  \delta _ {i} = d  ^ {i} $,
 +
$  \sigma _ {i} = s ^ {i} $).
  
<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/s085380/s08538045.png" /></td> </tr></table>
+
A simplicial mapping  $  f: X \rightarrow Y $
 +
between simplicial objects (in the same category  $  {\mathcal C} $)
 +
is a transformation (morphism) of functors from  $  X: \Delta \rightarrow {\mathcal C} $
 +
into  $  Y: \Delta \rightarrow {\mathcal C} $,
 +
that is, a family of morphisms  $  f _ {n} :  X _ {n} \rightarrow Y _ {n} $,
 +
$  n \geq  0 $,
 +
of  $  {\mathcal C} $
 +
such that
  
Similarly, a co-simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538046.png" /> can be identified with a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538047.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538049.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538051.png" />-co-fibres) and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538053.png" /> (co-boundary operators), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538055.png" /> (co-degeneracy operators), satisfying the relations (*) (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538057.png" />).
+
$$
 +
d _ {i} f _ {n + 1 }  = f _ {n} d _ {i} ,\ \
 +
0 \leq  i \leq  n + 1,
 +
$$
  
A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538058.png" /> between simplicial objects (in the same category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538059.png" />) is a transformation (morphism) of functors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538060.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538061.png" />, that is, a family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538063.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538064.png" /> such that
+
$$
 +
s _ {i} f _ {n}  = f _ {n + 1 }  s _ {i} ,0 \leq  i \leq  n.
 +
$$
  
<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/s085380/s08538065.png" /></td> </tr></table>
+
The simplicial objects of  $  {\mathcal C} $
 +
and their simplicial mappings form a category, denoted by  $  \Delta  ^ {0} {\mathcal C} $.
  
<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/s085380/s08538066.png" /></td> </tr></table>
+
A simplicial homotopy  $  h: f \simeq g $
 +
between two simplicial mappings  $  f, g: X \rightarrow Y $
 +
between simplicial objects in a category  $  {\mathcal C} $
 +
is a family of morphisms  $  h _ {i} : X _ {n} \rightarrow Y _ {n + 1 }  $,
 +
0 \leq  i \leq  n $,
 +
of  $  {\mathcal C} $
 +
such that
  
The simplicial objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538067.png" /> and their simplicial mappings form a category, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538068.png" />.
+
$$
 +
d _ {0} h _ {0= f _ {n} ;
 +
$$
  
A simplicial homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538069.png" /> between two simplicial mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538070.png" /> between simplicial objects in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538071.png" /> is a family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538073.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538074.png" /> such that
+
$$
 +
d _ {n} h _ {n}  = g _ {n} ;
 +
$$
  
<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/s085380/s08538075.png" /></td> </tr></table>
+
$$
 +
d _ {i} h _ {j}  = \left \{
 +
\begin{array}{ll}
 +
h _ {j - 1 }  d _ {i}  &\textrm{ if }  i < j,  \\
 +
d _ {j} h _ {j - 1 }    &\textrm{ if }  i = j > 0,  \\
 +
h _ {j} d _ {i - 1
 +
}    &\textrm{ if }  i > j + 1;  \\
 +
\end{array}
  
<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/s085380/s08538076.png" /></td> </tr></table>
+
\right .$$
  
<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/s085380/s08538077.png" /></td> </tr></table>
+
$$
 +
s _ {i} h _ {j}  = \left \{
 +
\begin{array}{ll}
 +
h _ {j +
 +
1 }  s _ {i}  &\textrm{ if }  i \leq  j,  \\
 +
h _ {j} s _ {i - 1 }
 +
  &\textrm{ if }  i > j. \\
 +
\end{array}
  
<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/s085380/s08538078.png" /></td> </tr></table>
+
\right .$$
  
On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538079.png" />, for any category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538080.png" />. In the case of the category of sets or topological spaces, the geometric realization functor (see [[Simplicial set|Simplicial set]]) carries this  "simplicial"  theory into the usual one.
+
On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category $  \Delta  ^ {0} {\mathcal C} $,  
 +
for any category $  {\mathcal C} $.  
 +
In the case of the category of sets or topological spaces, the geometric realization functor (see [[Simplicial set|Simplicial set]]) carries this  "simplicial"  theory into the usual one.
  
 
Examples of simplicial objects are a simplicial set, a simplicial topological space, a simplicial algebraic variety, a simplicial group, a simplicial Abelian group, a simplicial Lie algebra, a simplicial smooth manifold, etc.
 
Examples of simplicial objects are a simplicial set, a simplicial topological space, a simplicial algebraic variety, a simplicial group, a simplicial Abelian group, a simplicial Lie algebra, a simplicial smooth manifold, etc.
  
Every simplicial Abelian group can be made into a chain complex with boundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538081.png" />.
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Every simplicial Abelian group can be made into a chain complex with boundary operator $  d = \sum (- 1)  ^ {i} d _ {i} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Gabriel,  M. Zisman,  "Calculus of fractions and homotopy theory" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.P. May,  "Simplicial objects in algebraic topology" , v. Nostrand  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Lamotke,  "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Gabriel,  M. Zisman,  "Calculus of fractions and homotopy theory" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.P. May,  "Simplicial objects in algebraic topology" , v. Nostrand  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Lamotke,  "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR></table>

Revision as of 14:55, 7 June 2020


$ {\mathcal C} $

A contravariant functor $ X: \Delta \rightarrow {\mathcal C} $( or, equivalently, a covariant functor $ X: \Delta ^ {op} \rightarrow {\mathcal C} $) from the category $ \Delta $, whose objects are ordered sets $ [ n] = \{ 0 \dots n \} $, $ n \geq 0 $, and whose morphisms are non-decreasing mappings $ \mu : [ n] \rightarrow [ m] $, into the category $ {\mathcal C} $. A covariant functor $ X: \Delta \rightarrow {\mathcal C} $( or, equivalently, a contravariant functor $ X: \Delta ^ {op} \rightarrow {\mathcal C} $) is called a co-simplicial object in $ {\mathcal C} $.

The morphisms

$$ \delta _ {i} = \delta _ {i} ^ {n} : [ n - 1] \rightarrow [ n],\ \ 0 \leq i \leq n, $$

$$ \sigma _ {i} = \sigma _ {i} ^ {n} : [ n + 1] \rightarrow [ n],\ 0 \leq i \leq n, $$

of $ \Delta $ given by

$$ \delta _ {i} ^ {n} ( j) = \ \left \{ \begin{array}{ll} j &\textrm{ if } j < i, \\ j + 1 &\textrm{ if } j \geq i, \\ \end{array} \right .$$

$$ \sigma _ {i} ^ {n} ( j) = \left \{ \begin{array}{ll} j &\textrm{ if } j \leq i, \\ j - 1 &\textrm{ if } j > i, \\ \end{array} \right .$$

generate all the morphisms of $ \Delta $, so that a simplicial object $ X $ is determined by the objects $ X ([ n]) = X _ {n} $, $ n \geq 0 $( called the $ n $- fibres or $ n $- components of the simplicial object $ X $), and the morphisms

$$ d _ {i} = X ( \delta _ {i} ): X _ {n} \rightarrow X _ {n - 1 } \ \ \textrm{ and } \ \ s _ {i} = X ( \sigma _ {i} ): X _ {n} \rightarrow X _ {n + 1 } $$

(called boundary operators and degeneracy operators, respectively). In case $ {\mathcal C} $ is a category of structured sets, the elements of $ X _ {n} $ are usually called the $ n $- dimensional simplices of $ X $. The mappings $ \delta _ {i} $ and $ \sigma _ {i} $ satisfy the relations

$$ \tag{* } \left . \begin{array}{cll} \delta _ {j} \delta _ {i} = \ \delta _ {i} \delta _ {j - 1 } \ \textrm{ if } i < j, &\sigma _ {j} \delta _ {i} = \ \left \{ \begin{array}{} \delta _ {i} \sigma _ {j - 1 } \\ \end{array} &\textrm{ if } i < j, \\ \sigma _ {j} \sigma _ {i} = \sigma _ {i} \sigma _ {j + 1 } \ \ \textrm{ if } i \leq j; & \mathop{\rm id} &\textrm{ if } i = j \textrm{ or } i = j + 1, \\ \sigma _ {j} \delta _ {i} = \ \left \{ \begin{array}{} \delta _ {i} \sigma _ {j - 1 } \\ \end{array} &\delta _ {i - 1 } \sigma _ {j} \\ \end{array} \right \} $$

and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object $ X $ can be identified with a system $ \{ X _ {n} , d _ {i} , s _ {i} \} $ of objects $ X _ {n} $, $ n \geq 0 $, of $ {\mathcal C} $ and morphisms $ d _ {i} : X _ {n} \rightarrow X _ {n - 1 } $ and $ s _ {i} : X _ {n} \rightarrow X _ {n + 1 } $, $ 0 \leq i \leq n $, satisfying the relations

$$ d _ {i} d _ {j} = d _ {j - 1 } d _ {i} \ \textrm{ if } i < j; $$

$$ s _ {i} s _ {j} = s _ {j + 1 } s _ {i} \ \textrm{ if } i \leq j; $$

$$ d _ {i} s _ {j} = \left \{ \begin{array}{ll} s _ {j - 1 } d _ {i} & \textrm{ if } i < j, \\ \mathop{\rm id} & \textrm{ if } i = j \ \textrm{ or } i = j + 1, \\ s _ {j} d _ {i - 1 } & \textrm{ if } i > j + 1. \\ \end{array} \right .$$

Similarly, a co-simplicial object $ X $ can be identified with a system $ \{ X _ {n} , d ^ {i} , s ^ {i} \} $ of objects $ X ^ {n} $, $ n \geq 0 $( $ n $- co-fibres) and morphisms $ d _ {i} : X ^ {n - 1 } \rightarrow X ^ {n} $, $ 0 \leq i \leq n $( co-boundary operators), and $ s ^ {i} : X ^ {n + 1 } \rightarrow X ^ {n} $, $ 0 \leq i \leq n $( co-degeneracy operators), satisfying the relations (*) (with $ \delta _ {i} = d ^ {i} $, $ \sigma _ {i} = s ^ {i} $).

A simplicial mapping $ f: X \rightarrow Y $ between simplicial objects (in the same category $ {\mathcal C} $) is a transformation (morphism) of functors from $ X: \Delta \rightarrow {\mathcal C} $ into $ Y: \Delta \rightarrow {\mathcal C} $, that is, a family of morphisms $ f _ {n} : X _ {n} \rightarrow Y _ {n} $, $ n \geq 0 $, of $ {\mathcal C} $ such that

$$ d _ {i} f _ {n + 1 } = f _ {n} d _ {i} ,\ \ 0 \leq i \leq n + 1, $$

$$ s _ {i} f _ {n} = f _ {n + 1 } s _ {i} ,\ 0 \leq i \leq n. $$

The simplicial objects of $ {\mathcal C} $ and their simplicial mappings form a category, denoted by $ \Delta ^ {0} {\mathcal C} $.

A simplicial homotopy $ h: f \simeq g $ between two simplicial mappings $ f, g: X \rightarrow Y $ between simplicial objects in a category $ {\mathcal C} $ is a family of morphisms $ h _ {i} : X _ {n} \rightarrow Y _ {n + 1 } $, $ 0 \leq i \leq n $, of $ {\mathcal C} $ such that

$$ d _ {0} h _ {0} = f _ {n} ; $$

$$ d _ {n} h _ {n} = g _ {n} ; $$

$$ d _ {i} h _ {j} = \left \{ \begin{array}{ll} h _ {j - 1 } d _ {i} &\textrm{ if } i < j, \\ d _ {j} h _ {j - 1 } &\textrm{ if } i = j > 0, \\ h _ {j} d _ {i - 1 } &\textrm{ if } i > j + 1; \\ \end{array} \right .$$

$$ s _ {i} h _ {j} = \left \{ \begin{array}{ll} h _ {j + 1 } s _ {i} &\textrm{ if } i \leq j, \\ h _ {j} s _ {i - 1 } &\textrm{ if } i > j. \\ \end{array} \right .$$

On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category $ \Delta ^ {0} {\mathcal C} $, for any category $ {\mathcal C} $. In the case of the category of sets or topological spaces, the geometric realization functor (see Simplicial set) carries this "simplicial" theory into the usual one.

Examples of simplicial objects are a simplicial set, a simplicial topological space, a simplicial algebraic variety, a simplicial group, a simplicial Abelian group, a simplicial Lie algebra, a simplicial smooth manifold, etc.

Every simplicial Abelian group can be made into a chain complex with boundary operator $ d = \sum (- 1) ^ {i} d _ {i} $.

References

[1] P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)
[2] J.P. May, "Simplicial objects in algebraic topology" , v. Nostrand (1967)
[3] K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)
How to Cite This Entry:
Simplicial object in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_object_in_a_category&oldid=49427
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article