# 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

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topJ. 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

top- [1] M. Artin and B. Mazur: Etale homotopy, Lecture Notes in Maths, Vol. 100, Springer-Verlag, 1969.
- [2] H.J. Baues: Algebraic homotopy, Cambridge University Press, 1989.
- [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] 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] P.S. Hirschhorn: Model Categories and Their Localizations, Mathematical Surveys and Monographs, Vol. 99, Amer. Math. Soc, 2003.
- [6] M. Hovey: Model Categories, Mathematical Surveys and Monographs, Vol. 63, Amer. Math. Soc, 1999.
- [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] D.G. Quillen: Homolopical Algebra, Lecture Notes in Maths, Vol. 43, Springer-Verlag, 1967.

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