Categorical models of N-types for pro-crossed complexes and ℑn-prospaces

Hernández, L. J.; Porter, Timothy

doi:10.1007/bfb0087509

Cited by 5 publications

(4 citation statements)

References 14 publications

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“…The notion of n-type introduced by Whitehead [26,27] has a clear geometric meaning and can also be established in the context of non compact spaces and proper maps, see Geoghegan [14]. A program for the study of proper n-types was initiated by Hernández and Porter in [19], where some of the Whitehead's results about n-types were generalized to the context of pro-pointed spaces and some applications were given for proper n-types and shape theory. One advantage of our formulation is that the new result gives a category equivalence instead of the category embeddding given in [18].…”

Section: Theorem 1 Suppose That S Is a Set Of Non Negative Integer Numentioning

confidence: 98%

S-types of Global Towers of Spaces and Exterior Spaces

Río

Hernández

Rivas

2008

Appl Categor Struct

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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 π N k 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-Stypes. 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 Stypes 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.The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431.

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Section: Theorem 1 Suppose That S Is a Set Of Non Negative Integer Numentioning

confidence: 98%

S-types of Global Towers of Spaces and Exterior Spaces

Río

Hernández

Rivas

2008

Appl Categor Struct

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“…Remark 2-2. The first notion of' ^-structure' appears in the thesis of the first author for the category of spaces [4] and in the joint paper [6] for the category of crossed complexes. In these cases, we had the same notion of weak n-equivalence, but different notions of n-cofibration and w-fibration were considered.…”

Section: Definition 2-2 a Map F:x-*-y In Ss Is Said To Be A Weak N-ementioning

confidence: 99%

“…We can change Top by SS in Proposition 25. PROPOSITION [2][3][4][5][6]. (GM4) A trivial n-cofibration has the LLP with respect to all n-fibrations.…”

mentioning

confidence: 99%

Closed model categories for the n-type of spaces and simplicial sets

Elvira-Donazar

Hernandez-Paricio

1995

Math. Proc. Camb. Phil. Soc.

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For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

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“…This article and [9] view n-equivalences and co-n-equivalences from a particular perspective involving model structures. We warn the reader that our definitions are different than those in [11], which also deals with model structures and a notion of n-equivalence. Our approach is analogous to the so-called good truncation of chain complexes, while the approach of [11] is analogous to the so-called brutal truncation (see [20, 1.2.7], for example).…”

Section: Introductionmentioning

confidence: 99%