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.
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References
M. Artin and B. Mazur: Etale homotopy, Lecture Notes in Maths, Vol. 100, Springer-Verlag, 1969.
H.J. Baues: Algebraic homotopy, Cambridge University Press, 1989.
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.
K. Hess: “Model categories in algebraic topology,” Appl. Categ. Struct., Vol. 10(3), (2002), pp. 195–220.
P.S. Hirschhorn: Model Categories and Their Localizations, Mathematical Surveys and Monographs, Vol. 99, Amer. Math. Soc, 2003.
M. Hovey: Model Categories, Mathematical Surveys and Monographs, Vol. 63, Amer. Math. Soc, 1999.
D.C. Isaksen: “Strict model structures for pro-categories, Categorical decomposition techniques in algebraic topology”, Prog. Math., Vol. 215, (2004), pp. 179–198.
D.G. Quillen: Homolopical Algebra, Lecture Notes in Maths, Vol. 43, Springer-Verlag, 1967.
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García-Calcines, J.M., García-Díaz, P.R. & Rodríguez-Machín, S. Non functorial cylinders in a model category. centr.eur.j.math. 4, 376–394 (2006). https://doi.org/10.2478/s11533-006-0021-x
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DOI: https://doi.org/10.2478/s11533-006-0021-x