Published 2021-09-27
Keywords
- Simplicial sets,
- simplicial abelian groups,
- homotopy,
- homology,
- homotopic system
- singular functor,
- admissible categories,
- model object ...More
How to Cite
Copyright (c) 2021 Revista Integración, temas de matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
In this paper, we present the construction of a general homology theory in the category of abelian groups in the sense of the textbook Introducción a la Teoría de Homología General by R. Ruiz [22], to general categories, meeting axioms similar to those presented by Eilenberg and Steenrod in homology theories for admissible categories of topological pairs (X, A) [3]. We built this general homology theory in abelian groups through what we called Homoto-Homological Skeleton, showing the connections with monoidal categories and simplicial categories.
Downloads
References
- Artin E. and Braun H., Introduction to Algebraic Topology, Merrill Publishing Company, 1st ed., 1969.
- Bauer F.W. and Dugundji J., “Categorical homotopy and fibrations”, Trans. Amer. Math. Soc., 140 (1969), 239–256. doi:10.2307/1995135.
- Eilenberg S. and Steenrod N.E., “Axiomatic Approach to Homology Theory”, Proc. Natl. Acad. Sci. USA., 31 (1945), No. 4, 117–120. doi: 10.1073/pnas.31.4.117.
- Etingof P., et. al., Tensor Categories, American Mathematical Soc, vol. 205, Providence, 2016.
- Gabriel P. and Zisman M., Calculus of Fractions and Homotopy Theory, Springer Science & Business Media, vol. 35, New York, 2012.
- Goerss P.G. and Jardine J.F., Simplicial Homotopy Theory, Springer Science & Business Media, 2009.
- Gaitan R., “Esqueleto Homoto-Homológico En La Categoría De Los Grupos Abelianos”, Tesis (Ph.D.), Universidad del Valle, 2018, 112p.
- Hernández L.J., “Un ejemplo de teoría de homotopía en los grupos abelianos”, Collect. Math., 33, 161-176, 1982.
- Hatcher A., Algebraic topology, Cambridge University Press, 1st ed., 2002.
- Hovey M., Model categories, Mathematical Surveys and Monographs, vol. 63, 1999.
- Hungerford T.W., Algebra, Springer-Verlag, 1st ed., vol. 73, New York, 1974.
- Hu S.T., Homology Theory: a first course in Algebraic Topology, Holden-Day, 1st ed., 1966.
- Hernández L.J., “Un Ejemplo de Teoría de Homotopía en los Grupos Abelianos”, Tesis (Ph.D.), Universidad de Zaragoza, España, 1980, 232 p.
- Kan D.M., “Abstract Homotopy. I”, Proc. Natl. Acad. Sci. USA., 41 (1955), No. 12, 1092– 1096.
- Kan D.M., “Abstract Homotopy. II”, Proc. Natl. Acad. Sci. USA., 42 (1956), No. 5, 225– 228. doi:10.1073/pnas.42.5.255.
- Leinster T., Basic Category Theory, Cambridge University Press, 1st ed., vol. 143, 2014.
- MacLane S., Homology, Classics in mathematics, Springer-Verlag, Berlin, 1995.
- May P.J., Simplicial Objects in Algebraic Topology, University of Chicago Press, vol. 114, Chicago, 1992.
- Porter T. and Kamps K.H., Abstract Homotopy and Simple Homotopy Theory, World Scientific, 1st ed., Singapore, 1997.
- Ruiz R., “Overview on models in Homotopical Algebra”, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur., 28 (2004), No. 106, 100–121.
- Ruiz R., “Change of Models in Algebraic Topology”, Thesis (Ph.D), Temple University, Pennsylvania, 1977, 311 p.
- Ruíz R., “Introducción a la teoría de homología general”, https://n9.cl/hdk6, [citado 4 de agosto 2021].
- Vick J.W., Homology Theory: an introduction to algebraic topology, Springer-Verlag New York, 2nd ed., vol. 145, New York, 1994.