Applied Categorical Structures

, Volume 17, Issue 3, pp 287–301 | Cite as

S-types of Global Towers of Spaces and Exterior Spaces

  • A. Del Río
  • L. J. HernándezEmail author
  • M. T. Rivas


The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups \(\pi_k^{\mathbb{N}}\) for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak ℕ-S-equivalences is called the category of exterior ℕ-S-types. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories) given by the inversion of classes of week equivalences. The family of neighbourhoods ‘at infinity’ of an exterior space can be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of S-types of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models localized up to weak equivalences can be used to give some algebraic models of S-types.


Exterior space Proper homotopy S-types Global tower of spaces Global tower of groups Categorical group 

Mathematics Subject Classifications (2000)

55Q57 55Q70 54A05 


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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesUniversity of La RiojaLogroñoSpain

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