Non functorial cylinders in a model category

J. García-Calcines; P. García-Díaz; S. Rodríguez-Machín

Open Mathematics (2006)

  • Volume: 4, Issue: 3, page 376-394
  • ISSN: 2391-5455

Abstract

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Taking cylinder objects, as defined in a model category, we consider a cylinder construction in a cofibration category, which provides a reformulation of relative homotopy in the sense of Baues. Although this cylinder is not a functor we show that it verifies a list of properties which are very closed to those of an I-category (or category with a natural cylinder functor). Considering these new properties, we also give an alternative description of Baues’ relative homotopy groupoids.

How to cite

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J. García-Calcines, P. García-Díaz, and S. Rodríguez-Machín. "Non functorial cylinders in a model category." Open Mathematics 4.3 (2006): 376-394. <http://eudml.org/doc/269106>.

@article{J2006,
abstract = {Taking cylinder objects, as defined in a model category, we consider a cylinder construction in a cofibration category, which provides a reformulation of relative homotopy in the sense of Baues. Although this cylinder is not a functor we show that it verifies a list of properties which are very closed to those of an I-category (or category with a natural cylinder functor). Considering these new properties, we also give an alternative description of Baues’ relative homotopy groupoids.},
author = {J. García-Calcines, P. García-Díaz, S. Rodríguez-Machín},
journal = {Open Mathematics},
keywords = {55U35; 55P05},
language = {eng},
number = {3},
pages = {376-394},
title = {Non functorial cylinders in a model category},
url = {http://eudml.org/doc/269106},
volume = {4},
year = {2006},
}

TY - JOUR
AU - J. García-Calcines
AU - P. García-Díaz
AU - S. Rodríguez-Machín
TI - Non functorial cylinders in a model category
JO - Open Mathematics
PY - 2006
VL - 4
IS - 3
SP - 376
EP - 394
AB - Taking cylinder objects, as defined in a model category, we consider a cylinder construction in a cofibration category, which provides a reformulation of relative homotopy in the sense of Baues. Although this cylinder is not a functor we show that it verifies a list of properties which are very closed to those of an I-category (or category with a natural cylinder functor). Considering these new properties, we also give an alternative description of Baues’ relative homotopy groupoids.
LA - eng
KW - 55U35; 55P05
UR - http://eudml.org/doc/269106
ER -

References

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  1. [1] M. Artin and B. Mazur: Etale homotopy, Lecture Notes in Maths, Vol. 100, Springer-Verlag, 1969. 
  2. [2] H.J. Baues: Algebraic homotopy, Cambridge University Press, 1989. 
  3. [3] D.A. Edwards and H.M. Hastings: Cech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, Vol. 542, Springer Verlag, 1976. Zbl0334.55001
  4. [4] K. Hess: “Model categories in algebraic topology,” Appl. Categ. Struct., Vol. 10(3), (2002), pp. 195–220. http://dx.doi.org/10.1023/A:1015218106586 
  5. [5] P.S. Hirschhorn: Model Categories and Their Localizations, Mathematical Surveys and Monographs, Vol. 99, Amer. Math. Soc, 2003. 
  6. [6] M. Hovey: Model Categories, Mathematical Surveys and Monographs, Vol. 63, Amer. Math. Soc, 1999. 
  7. [7] D.C. Isaksen: “Strict model structures for pro-categories, Categorical decomposition techniques in algebraic topology”, Prog. Math., Vol. 215, (2004), pp. 179–198. 
  8. [8] D.G. Quillen: Homolopical Algebra, Lecture Notes in Maths, Vol. 43, Springer-Verlag, 1967. 

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