# Homotopy type of a topological category

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 . 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|]]. If $X$ is a normal scheme, then $\pi _{1} (X)$ coincides with the fundamental Grothendieck group scheme . 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 $X$ .

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