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A projective system of topological spaces that is associated with a [[Topologized category|topologized category]] and which makes it possible to define the homotopy groups of this category, the homology and cohomology groups with values in an Abelian group, etc.
 
A projective system of topological spaces that is associated with a [[Topologized category|topologized category]] and which makes it possible to define the homotopy groups of this category, the homology and cohomology groups with values in an Abelian group, etc.
  
Only locally connected topologized categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479501.png" /> will be considered here, i.e. categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479502.png" /> provided with a [[Grothendieck topology|Grothendieck topology]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479503.png" /> such that any one of their objects can be represented as a coproduct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479504.png" /> of indecomposable objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479505.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479506.png" /> plays the role of the set of connected components of the topological space. The set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479507.png" /> is uniquely determined, up to a bijection; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479508.png" />. The assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h0479509.png" /> defines a functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795010.png" /> into the category of sets. An arbitrary covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795011.png" /> of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795012.png" /> in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795013.png" /> defines a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795015.png" /> for which
+
Only locally connected topologized categories $  (C,\  \tau ) $
 +
will be considered here, i.e. categories $  C $
 +
provided with a [[Grothendieck topology|Grothendieck topology]] $  \tau $
 +
such that any one of their objects can be represented as a coproduct $  \cup _ {i \in I} X _{i} $
 +
of indecomposable objects $  X _{i} $ ;  
 +
here, $  I $
 +
plays the role of the set of connected components of the topological space. The set of indices $  I $
 +
is uniquely determined, up to a bijection; it is denoted by $  \pi _{0} (X) $ .  
 +
The assignment $  X \mapsto \pi _{0} (X) $
 +
defines a functor from $  C $
 +
into the category of sets. An arbitrary covering $  U \rightarrow X $
 +
of an object $  X $
 +
in the topology of $  (C,\  \tau ) $
 +
defines a simplicial object $  U $
 +
in $  C $
 +
for which$$
 +
U _{n}  =
 +
U _{X} \times \dots \times U _{X}    (n  \textrm{ factors } ),
 +
$$
 +
and the simplicial set $  \pi _{0} (U _ {\mathbf .} ) $ .  
 +
The geometrical realization of the simplicial set $  \pi _{0} (U _ {\mathbf .} ) $
 +
yields the topological space $  | U | = | \pi _{0} (U _ {\mathbf .} ) | $ .  
 +
For any refinement $  W \rightarrow X $
 +
of the covering $  U $ (
 +
$  W \rightarrow X $
 +
factors through $  U \rightarrow X $ )
 +
there exists a unique (up to a homotopy) continuous mapping $  | W | \rightarrow | U | $ .
 +
Thus, the object $  X $
 +
is mapped to the projective system of topological spaces $  \{ | U | \} _ {U \in  \mathop{\rm Cov}\nolimits (X)} $ ,
 +
where $  \mathop{\rm Cov}\nolimits (X) $
 +
is the family of all coverings of $  X $ .
  
<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/h/h047/h047950/h04795016.png" /></td> </tr></table>
 
  
and the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795017.png" />. The geometrical realization of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795018.png" /> yields the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795019.png" />. For any refinement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795020.png" /> of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795022.png" /> factors through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795023.png" />) there exists a unique (up to a homotopy) continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795024.png" />. Thus, the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795025.png" /> is mapped to the projective system of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795027.png" /> is the family of all coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795028.png" />.
+
This definition is analogous to the definition of Čech homology; it is known, however, that in the general case the Čech cohomology gives the "correct" cohomology in dimensions 0 and 1 only. For this reason the construction described above cannot be considered satisfactory. The concept of a  "hyper-coveringhyper-covering" has been introduced [[#References|[1]]]. It generalizes the simplicial objects $  U _ {\mathbf .} $
 +
that were constructed above for the coverings $  U \rightarrow X $ .  
 +
A hyper-covering again a simplicial object $  K _ {\mathbf .} $
 +
in the topological category $  (C,\  \tau ) $
 +
with final object $  X $ ,  
 +
satisfying the following conditions: $  K _{0} \rightarrow X $
 +
is a covering of $  X $ ;
 +
for any $  n $
 +
the canonical morphism $  K _{n+1} \rightarrow ( \cos \textrm{k} _{n} (K _ {\mathbf .} ) ) _{n+1} $ ,  
 +
where $  \cos \textrm{k} _{n} $
 +
is a functor of the $  n $ -
 +
th co-skeleton, is a covering.
  
This definition is analogous to the definition of Čech homology; it is known, however, that in the general case the Čech cohomology gives the  "correct"  cohomology in dimensions 0 and 1 only. For this reason the construction described above cannot be considered satisfactory. The concept of a  "hyper-coveringhyper-covering" has been introduced [[#References|[1]]]. It generalizes the simplicial objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795029.png" /> that were constructed above for the coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795030.png" />. A hyper-covering again a simplicial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795031.png" /> in the topological category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795032.png" /> with final object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795033.png" />, satisfying the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795034.png" /> is a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795035.png" />; for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795036.png" /> the canonical morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795038.png" /> is a functor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795039.png" />-th co-skeleton, is a covering.
+
The assignment of the topological space $ | \pi _{0} (K _ {\mathbf .} ) | $
 +
to each hyper-covering $ K _ {\mathbf .} $
 +
leads to a projective system of spaces, parametrized by hyper-coverings. This also defines the homotopy type (and, more accurately, the pro-homotopy type) of the topologized category $  (C,\  \tau ) $
 +
with final object $  X $ .  
 +
Homotopy, homology and cohomology groups are introduced by a standard procedure.
  
The assignment of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795040.png" /> to each hyper-covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795041.png" /> leads to a projective system of spaces, parametrized by hyper-coverings. This also defines the homotopy type (and, more accurately, the pro-homotopy type) of the topologized category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795042.png" /> with final object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795043.png" />. Homotopy, homology and cohomology groups are introduced by a standard procedure.
+
The homotopy type of a topologized category associated with a scheme makes it possible to determine the homotopy type of a scheme. The case which is most frequently considered is that of the [[Etale topology|étale topology]] $  X _ {\textrm et} $
 +
on a [[Scheme|scheme]] $  X $ .  
 +
In such a case the homotopy type of $  X $
 +
is a pro-object in the category of simplicial sets or in the category of finite CW-complexes. The homotopy groups $  \pi _{i} (X) $ ,
 +
which can be defined for such objects, are pro-finite groups and are called the $  i $ -
 +
th homotopy groups of the scheme $  X $ [[#
 +
References|[2]]]. If $  X $
 +
is a normal scheme, then $  \pi _{1} (X) $
 +
coincides with the fundamental Grothendieck group scheme [[#References|[3]]]. The homotopy type of the point $  X =  \mathop{\rm Spec}\nolimits \  k $ ,
 +
where $  k $
 +
is a field, coincides with the projective limit of Eilenberg–MacLane spaces $  K (G _{i} ,\  1) $ ,  
 +
where $  G _{i} $
 +
is the Galois group of a finite Galois extension $  K _{i} $
 +
of $  k $ .  
 +
In the case of algebraic varieties over the field $  \mathbf C $
 +
of complex numbers the following comparison theorem is valid: The groups $  \pi _{i} (X) $
 +
are pro-finite completions of the ordinary homotopy groups $  \pi _{n} ( X ^ {\textrm an} ) $
 +
of the complex space $  X ^ {\textrm an} $
 +
associated with $  X $ .
  
The homotopy type of a topologized category associated with a scheme makes it possible to determine the homotopy type of a scheme. The case which is most frequently considered is that of the [[Etale topology|étale topology]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795044.png" /> on a [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795045.png" />. In such a case the homotopy type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795046.png" /> is a pro-object in the category of simplicial sets or in the category of finite CW-complexes. The homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795047.png" />, which can be defined for such objects, are pro-finite groups and are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795048.png" />-th homotopy groups of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795049.png" /> [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795050.png" /> is a normal scheme, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795051.png" /> coincides with the fundamental Grothendieck group scheme [[#References|[3]]]. The homotopy type of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795053.png" /> is a field, coincides with the projective limit of Eilenberg–MacLane spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795055.png" /> is the Galois group of a finite Galois extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795057.png" />. In the case of algebraic varieties over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795058.png" /> of complex numbers the following comparison theorem is valid: The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795059.png" /> are pro-finite completions of the ordinary homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795060.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795061.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047950/h04795062.png" />.
 
  
 
====References====
 
====References====

Latest revision as of 13:07, 15 December 2019

h0479501.png 62 0 62 A projective system of topological spaces that is associated with a topologized category and which makes it possible to define the homotopy groups of this category, the homology and cohomology groups with values in an Abelian group, etc.

Only locally connected topologized categories $ (C,\ \tau ) $ will be considered here, i.e. categories $ C $ provided with a Grothendieck topology $ \tau $ such that any one of their objects can be represented as a coproduct $ \cup _ {i \in I} X _{i} $ of indecomposable objects $ X _{i} $ ; here, $ I $ plays the role of the set of connected components of the topological space. The set of indices $ I $ is uniquely determined, up to a bijection; it is denoted by $ \pi _{0} (X) $ . The assignment $ X \mapsto \pi _{0} (X) $ defines a functor from $ C $ into the category of sets. An arbitrary covering $ U \rightarrow X $ of an object $ X $ in the topology of $ (C,\ \tau ) $ defines a simplicial object $ U $ in $ C $ for which$$ U _{n} = U _{X} \times \dots \times U _{X} (n \textrm{ factors } ), $$ and the simplicial set $ \pi _{0} (U _ {\mathbf .} ) $ . The geometrical realization of the simplicial set $ \pi _{0} (U _ {\mathbf .} ) $ yields the topological space $ | U | = | \pi _{0} (U _ {\mathbf .} ) | $ . For any refinement $ W \rightarrow X $ of the covering $ U $ ( $ W \rightarrow X $ factors through $ U \rightarrow X $ ) there exists a unique (up to a homotopy) continuous mapping $ | W | \rightarrow | U | $ . Thus, the object $ X $ is mapped to the projective system of topological spaces $ \{ | U | \} _ {U \in \mathop{\rm Cov}\nolimits (X)} $ , where $ \mathop{\rm Cov}\nolimits (X) $ is the family of all coverings of $ X $ .


This definition is analogous to the definition of Čech homology; it is known, however, that in the general case the Čech cohomology gives the "correct" cohomology in dimensions 0 and 1 only. For this reason the construction described above cannot be considered satisfactory. The concept of a "hyper-coveringhyper-covering" has been introduced [1]. It generalizes the simplicial objects $ U _ {\mathbf .} $ that were constructed above for the coverings $ U \rightarrow X $ . A hyper-covering again a simplicial object $ K _ {\mathbf .} $ in the topological category $ (C,\ \tau ) $ with final object $ X $ , satisfying the following conditions: $ K _{0} \rightarrow X $ is a covering of $ X $ ; for any $ n $ the canonical morphism $ K _{n+1} \rightarrow ( \cos \textrm{k} _{n} (K _ {\mathbf .} ) ) _{n+1} $ , where $ \cos \textrm{k} _{n} $ is a functor of the $ n $ - th co-skeleton, is a covering.

The assignment of the topological space $ | \pi _{0} (K _ {\mathbf .} ) | $ to each hyper-covering $ K _ {\mathbf .} $ leads to a projective system of spaces, parametrized by hyper-coverings. This also defines the homotopy type (and, more accurately, the pro-homotopy type) of the topologized category $ (C,\ \tau ) $ with final object $ X $ . Homotopy, homology and cohomology groups are introduced by a standard procedure.

The homotopy type of a topologized category associated with a scheme makes it possible to determine the homotopy type of a scheme. The case which is most frequently considered is that of the étale topology $ X _ {\textrm et} $ on a scheme $ X $ . In such a case the homotopy type of $ X $ is a pro-object in the category of simplicial sets or in the category of finite CW-complexes. The homotopy groups $ \pi _{i} (X) $ , which can be defined for such objects, are pro-finite groups and are called the $ i $ - th homotopy groups of the scheme $ X $ [[# References|[2]]]. If $ X $ is a normal scheme, then $ \pi _{1} (X) $ coincides with the fundamental Grothendieck group scheme [3]. The homotopy type of the point $ X = \mathop{\rm Spec}\nolimits \ k $ , where $ k $ is a field, coincides with the projective limit of Eilenberg–MacLane spaces $ K (G _{i} ,\ 1) $ , where $ G _{i} $ is the Galois group of a finite Galois extension $ K _{i} $ of $ k $ . In the case of algebraic varieties over the field $ \mathbf C $ of complex numbers the following comparison theorem is valid: The groups $ \pi _{i} (X) $ are pro-finite completions of the ordinary homotopy groups $ \pi _{n} ( X ^ {\textrm an} ) $ of the complex space $ X ^ {\textrm an} $ associated with $ X $ .


References

[1] M. Artin, "The étale topology of schemes" , Proc. Internat. Congress Mathematicians (Moscow, 1966) , Kraus, reprint (1979) pp. 44–56
[2] "Théorie des toposes et cohomologie étale des schémas" A. Grothendieck (ed.) J.-L. Verdier (ed.) E. Artin (ed.) , Sem. Geom. Alg. 4 , 1–3 , Springer (1972)
[3] M. Artin, B. Mazur, "Etale homotopy" , Lect. notes in math. , 100 , Springer (1969)
[4] D. Sullivan, "Geometric topology" , I. Localization, periodicity, and Galois symmetry , M.I.T. (1971)
[5] A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971)


Comments

References

[a1] D. Sulivan, "Genetics of homotopy theory and the Adams conjecture" Ann. of Math. , 100 (1974) pp. 1–79
How to Cite This Entry:
Homotopy type of a topological category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homotopy_type_of_a_topological_category&oldid=44247
This article was adapted from an original article by V.I. DanilovI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article