Difference between revisions of "Simplicial object in a category"

Revision as of 14:53, 7 June 2020

A contravariant functor (or, equivalently, a covariant functor ) from the category , whose objects are ordered sets , , and whose morphisms are non-decreasing mappings , into the category . A covariant functor (or, equivalently, a contravariant functor ) is called a co-simplicial object in .

The morphisms

of given by

generate all the morphisms of , so that a simplicial object is determined by the objects , (called the -fibres or -components of the simplicial object ), and the morphisms

(called boundary operators and degeneracy operators, respectively). In case is a category of structured sets, the elements of are usually called the -dimensional simplices of . The mappings and satisfy the relations

(*)

and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object can be identified with a system of objects , , of and morphisms and , , satisfying the relations

Similarly, a co-simplicial object can be identified with a system of objects , (-co-fibres) and morphisms , (co-boundary operators), and , (co-degeneracy operators), satisfying the relations (*) (with , ).

A simplicial mapping between simplicial objects (in the same category ) is a transformation (morphism) of functors from into , that is, a family of morphisms , , of such that

The simplicial objects of and their simplicial mappings form a category, denoted by .

A simplicial homotopy between two simplicial mappings between simplicial objects in a category is a family of morphisms , , of such that

On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category , for any category . 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 .

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=48710

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

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-$#C+1 = 77 : ~/encyclopedia/old_files/data/S085/S.0805380 Simplicial object in a category
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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" />.
-{{TEX|done}}
-'' $  {\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
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538012.png" /></td> </tr></table>
-\delta _ {i}  =  \delta _ {i}  ^ {n} :  [ n - 1]  \rightarrow  [ n],\ \
-\leq  i \leq  n,
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538013.png" /></td> </tr></table>
-\sigma _ {i}  =  \sigma _ {i}  ^ {n} :  [ n + 1]  \rightarrow  [ n],\  0 \leq  i \leq  n,
-$$
-of  $  \Delta $
+of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538014.png" /> given by
-given by
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538015.png" /></td> </tr></table>
-\delta _ {i}  ^ {n} ( j)  = \
-\left \{
-$$
+<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>
-\sigma _ {i}  ^ {n} ( j)  =  \left \{
-generate all the morphisms of  $  \Delta $,
+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
-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
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538026.png" /></td> </tr></table>
-d _ {i}  =  X ( \delta _ {i} ):  X _ {n}  \rightarrow  X _ {n - 1 }  \ \
-\textrm{ and } \ \
-s _ {i}  =  X ( \sigma _ {i} ):  X _ {n}  \rightarrow  X _ {n + 1 }
-$$
-(called boundary operators and degeneracy operators, respectively). In case  $  {\mathcal C} $
+(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
-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
-$$ \tag{* }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
-\left .
-and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object  $  X $
+and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538035.png" /> can be identified with a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538036.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538038.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538039.png" /> and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538042.png" />, satisfying the relations
-can be identified with a system  $  \{ X _ {n} , d _ {i} , s _ {i} \} $
-of objects  $  X _ {n} $,
-$  n \geq  0 $,
-of  $  {\mathcal C} $
-and morphisms  $  d _ {i} :  X _ {n} \rightarrow X _ {n - 1 }  $
-and  $  s _ {i} :  X _ {n} \rightarrow X _ {n + 1 }  $,
-$  0 \leq  i \leq  n $,
-satisfying the relations
-$$
+<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>
-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/s08538044.png" /></td> </tr></table>
-s _ {i} s _ {j}  =  s _ {j + 1 }  s _ {i} \  \textrm{ if }  i \leq  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/s08538045.png" /></td> </tr></table>
-d _ {i} s _ {j}  =  \left \{
-Similarly, a co-simplicial object  $  X $
+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" />).
-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 $
+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
-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
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538065.png" /></td> </tr></table>
-d _ {i} f _ {n + 1 }   =  f _ {n} d _ {i} ,\ \
-\leq  i \leq  n + 1,
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538066.png" /></td> </tr></table>
-s _ {i} f _ {n}  =  f _ {n + 1 }  s _ {i} ,\  0 \leq  i \leq  n.
-$$
-The simplicial objects of  $  {\mathcal C} $
+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" />.
-and their simplicial mappings form a category, denoted by  $  \Delta  ^ {0} {\mathcal C} $.
-A simplicial homotopy  $  h:  f \simeq g $
+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
-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
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538075.png" /></td> </tr></table>
-d _ {0} h _ {0}  =  f _ {n} ;
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538076.png" /></td> </tr></table>
-d _ {n} h _ {n}  =  g _ {n} ;
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085380/s08538077.png" /></td> </tr></table>
-d _ {i} h _ {j}  =  \left \{
-$$
+<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/s08538078.png" /></td> </tr></table>
-s _ {i} h _ {j}  =  \left \{
-On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category  $  \Delta  ^ {0} {\mathcal C} $,
+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.
-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.
-Every simplicial Abelian group can be made into a chain complex with boundary operator  $  d = \sum (- 1)  ^ {i} d _ {i} $.
+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" />.
 ====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>

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Difference between revisions of "Simplicial object in a category"

Revision as of 14:53, 7 June 2020

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