Category with a natural cone

Francisco Díaz; Sergio Rodríguez-Machín

Open Mathematics (2006)

  • Volume: 4, Issue: 1, page 5-33
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Francisco Díaz, and Sergio Rodríguez-Machín. "Category with a natural cone." Open Mathematics 4.1 (2006): 5-33. <http://eudml.org/doc/268844>.

@article{FranciscoDíaz2006,
abstract = {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.},
author = {Francisco Díaz, Sergio Rodríguez-Machín},
journal = {Open Mathematics},
keywords = {55U35; 18C; 55P05; 55P40; 55Q05},
language = {eng},
number = {1},
pages = {5-33},
title = {Category with a natural cone},
url = {http://eudml.org/doc/268844},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Francisco Díaz
AU - Sergio Rodríguez-Machín
TI - Category with a natural cone
JO - Open Mathematics
PY - 2006
VL - 4
IS - 1
SP - 5
EP - 33
AB - 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.
LA - eng
KW - 55U35; 18C; 55P05; 55P40; 55Q05
UR - http://eudml.org/doc/268844
ER -

References

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  14. [14] E.G. Minian: “Generalized cofibration categories and global actions”, Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part I. K-Theory, Vol 20(1), (2000), pp. 37–95. 
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