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.
@article{bwmeta1.element.doi-10_2478_s11533-006-0021-x,
author = {J. Garc\'\i a-Calcines and P. Garc\'\i a-D\'\i az and S. Rodr\'\i guez-Mach\'\i n},
title = {Non functorial cylinders in a model category},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {376-394},
zbl = {1134.55015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0021-x}
}
J. García-Calcines; P. García-Díaz; S. Rodríguez-Machín. Non functorial cylinders in a model category. Open Mathematics, Tome 4 (2006) pp. 376-394. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0021-x/
[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. | Zbl 0334.55001
[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.