# Category with a natural cone

Francisco Díaz; Sergio Rodríguez-Machín

Open Mathematics (2006)

- Volume: 4, Issue: 1, page 5-33
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topFrancisco Díaz, and Sergio Rodríguez-Machín. "Category with a natural cone." Open Mathematics 4.1 (2006): 5-33. <http://eudml.org/doc/268844>.

@article{FranciscoDíaz2006,

abstract = {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.},

author = {Francisco Díaz, Sergio Rodríguez-Machín},

journal = {Open Mathematics},

keywords = {55U35; 18C; 55P05; 55P40; 55Q05},

language = {eng},

number = {1},

pages = {5-33},

title = {Category with a natural cone},

url = {http://eudml.org/doc/268844},

volume = {4},

year = {2006},

}

TY - JOUR

AU - Francisco Díaz

AU - Sergio Rodríguez-Machín

TI - Category with a natural cone

JO - Open Mathematics

PY - 2006

VL - 4

IS - 1

SP - 5

EP - 33

AB - 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.

LA - eng

KW - 55U35; 18C; 55P05; 55P40; 55Q05

UR - http://eudml.org/doc/268844

ER -

## References

top- [1] H.J. Baues: Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge-New York, 1989. Zbl0688.55001
- [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. Zbl0849.55009
- [3] F.J. Díaz and S. Rodríguez-Machín: “Homotopy theory induced by cones”, Extracta Math., Vol. 16(2), (2001), pp. 287–292. Zbl1001.55021
- [4] F.J. Díaz, J. Remedios and S. Rodríguez-Machín: “Generalized homotopy in C-categories”, Extracta Math., Vol. 16(3), (2001), pp 393–403. Zbl0997.55024
- [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 Zbl0907.55017
- [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 Zbl0099.17905
- [10] K.H. Kamps: “Note on normal sequences of chain complexes”, Colloq. Math., Vol 39(2), (1978), pp. 225–227. Zbl0403.55001
- [11] K.H. Kamps and T. Porter: Abstract Homotopy and Simple Homotopy, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. Zbl0890.55014
- [12] H. Kleisli: “Homotopy theory in Abelian Categories”, Canad. J. Math., Vol. 14, (1962), pp. 139–169. Zbl0108.02001
- [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 Zbl0138.01704
- [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. Zbl05508171
- [16] E. Padrón and S. Rodríguez-Machín: “Model additive categories”, Rend. Circ. Mat. Palermo, Suppl., Vol. 24, (1990), pp. 465–474. Zbl0726.18005
- [17] D.G. Quillen: Homotopical Algebra, Lecture Notes in Math, Vol. 43, Springer-Verlag, Berlin-New York, 1967. Zbl0168.20903
- [18] J.A. Seebach Jr.: “Injectives and homotopy”, Illinois J. Math., Vol. 16, (1972), pp. 446–453. Zbl0244.55015

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.