Homotopy type of a topological category

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 will be considered here, i.e. categories provided with a Grothendieck topology such that any one of their objects can be represented as a coproduct of indecomposable objects ; here, plays the role of the set of connected components of the topological space. The set of indices is uniquely determined, up to a bijection; it is denoted by . The assignment defines a functor from into the category of sets. An arbitrary covering of an object in the topology of defines a simplicial object in for which

and the simplicial set . The geometrical realization of the simplicial set yields the topological space . For any refinement of the covering ( factors through ) there exists a unique (up to a homotopy) continuous mapping . Thus, the object is mapped to the projective system of topological spaces , where is the family of all coverings of .

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 that were constructed above for the coverings . A hyper-covering again a simplicial object in the topological category with final object , satisfying the following conditions: is a covering of ; for any the canonical morphism , where is a functor of the -th co-skeleton, is a covering.

The assignment of the topological space to each hyper-covering 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 with final object . 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 on a scheme . In such a case the homotopy type of is a pro-object in the category of simplicial sets or in the category of finite CW-complexes. The homotopy groups , which can be defined for such objects, are pro-finite groups and are called the -th homotopy groups of the scheme [2]. If is a normal scheme, then coincides with the fundamental Grothendieck group scheme [3]. The homotopy type of the point , where is a field, coincides with the projective limit of Eilenberg–MacLane spaces , where is the Galois group of a finite Galois extension of . In the case of algebraic varieties over the field of complex numbers the following comparison theorem is valid: The groups are pro-finite completions of the ordinary homotopy groups of the complex space associated with .

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)

References

[a1]	D. Sulivan, "Genetics of homotopy theory and the Adams conjecture" Ann. of Math. , 100 (1974) pp. 1–79