Category with a natural cone

Generally, in homotopy theory a cylinder object (or, its dual, a path object) is used to define homotopy between morphisms, and a cone object is used to build exact sequences of homotopy groups. Here, an axiomatic theory based on a cone functor is given. Suspension objects are associated to based ob...

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Bibliographic Details
Published inOpen mathematics (Warsaw, Poland) Vol. 4; no. 1; pp. 5 - 33
Main Authors Díaz, Francisco, Rodríguez-Machín, Sergio
Format Journal Article
LanguageEnglish
Published Warsaw Versita 01.03.2006
De Gruyter Poland
De Gruyter
Subjects
18C
55P05
55P40
55Q05
55U35
algebraic homotopy theory
Category
cone construction
Group theory
Homotopy theory
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Summary:Generally, in homotopy theory a cylinder object (or, its dual, a path object) is used to define homotopy between morphisms, and a cone object is used to build exact sequences of homotopy groups. Here, an axiomatic theory based on a cone functor is given. Suspension objects are associated to based objects and cofibrations, obtaining homotopy groups referred to an object and relative to a cofibration, respectively. Exact sequences of these groups are built. Algebraic and particular examples are given. We point out that the main results of this paper were already stated in [3], and the purpose of this article is to give full details of the foregoing.
ISSN:2391-5455
2391-5455
DOI:10.1007/s11533-005-0002-5