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.
@article{bwmeta1.element.doi-10_1007_s11533-005-0002-5,
author = {Francisco D\'\i az and Sergio Rodr\'\i guez-Mach\'\i n},
title = {Category with a natural cone},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {5-33},
zbl = {1104.55009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1007_s11533-005-0002-5}
}
Francisco Díaz; Sergio Rodríguez-Machín. Category with a natural cone. Open Mathematics, Tome 4 (2006) pp. 5-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1007_s11533-005-0002-5/
[1] H.J. Baues: Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge-New York, 1989. | Zbl 0688.55001
[2] H.J. Baues and A. Quintero: “On the locally finite chain algebra of a proper homotopy type”, Bull. Belg. Math. Soc. Simon Stevin, Vol. 3(2), (1996), pp. 161–175. | Zbl 0849.55009
[3] F.J. Díaz and S. Rodríguez-Machín: “Homotopy theory induced by cones”, Extracta Math., Vol. 16(2), (2001), pp. 287–292. | Zbl 1001.55021
[4] F.J. Díaz, J. Remedios and S. Rodríguez-Machín: “Generalized homotopy in C-categories”, Extracta Math., Vol. 16(3), (2001), pp 393–403. | Zbl 0997.55024
[5] F.J. Díaz, J. García-Calcines and S. Rodríguez-Machín: Homotopía algebraica: descripción e interrelación de las principales teorías, Monografías de la Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza 5, 1994.
[6] J. García-Calcines, M. García-Pinillos and L.J. Hernández-Paricio: “A closed simplicial model category for proper homotopy and shape theories” B. Aust. Math. Soc., Vol. 57, (1998), pp. 221–242. http://dx.doi.org/10.1017/S0004972700031610 | Zbl 0907.55017
[7] L.J. Hernández: Un ejemplo de Teoría de homotopía en los grupos abelianos, Departamento de Geometría y Topología, Universidad de Zaragoza, 1980.
[8] P.J. Hilton: Homotopy theory and duality, Gordon and Breach Science Publishers, New York-London-Paris, 1965.
[9] P.J. Huber: “Homotopy theory in general categories”, Math. Ann., Vol. 144, (1961), pp. 361–385. http://dx.doi.org/10.1007/BF01396534 | Zbl 0099.17905
[10] K.H. Kamps: “Note on normal sequences of chain complexes”, Colloq. Math., Vol 39(2), (1978), pp. 225–227. | Zbl 0403.55001
[11] K.H. Kamps and T. Porter: Abstract Homotopy and Simple Homotopy, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. | Zbl 0890.55014
[12] H. Kleisli: “Homotopy theory in Abelian Categories”, Canad. J. Math., Vol. 14, (1962), pp. 139–169. | Zbl 0108.02001
[13] H. Kleisli: “Every Standard construction is induced by a pair of Adjoint Functors”, Proc. Am. Math. Soc., Vol 16(3), (1965), pp. 544–546. http://dx.doi.org/10.2307/2034693 | Zbl 0138.01704
[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.
[15] T. Porter: “Abstract Homotopy Theory: The Interaction of Category Theory and Homotopy”, Cubo Mat. Educ., Vol 5(1), (2003), pp. 115–165. | Zbl 05508171
[16] E. Padrón and S. Rodríguez-Machín: “Model additive categories”, Rend. Circ. Mat. Palermo, Suppl., Vol. 24, (1990), pp. 465–474. | Zbl 0726.18005
[17] D.G. Quillen: Homotopical Algebra, Lecture Notes in Math, Vol. 43, Springer-Verlag, Berlin-New York, 1967. | Zbl 0168.20903
[18] J.A. Seebach Jr.: “Injectives and homotopy”, Illinois J. Math., Vol. 16, (1972), pp. 446–453. | Zbl 0244.55015