Abstract
Homological localizations of spaces and spectra have been a fundamental tool in algebraic topology since the 1970s, especially in the setting of chromatic homotopy. However, it is unknown whether the existence of cohomological localizations can be proved in ZFC or not. Although this is apparently a homotopy-theoretical problem, it turned out to be closely related to set-theoretical reflection principles and therefore to the existence of large cardinals. In this note we present the state of the art with enough background so that proofs of results are readable by both topologists and set theorists.
Supported by MCIN/AEI/10.13039/501100011033 under grant PID2020-117971GB-C22.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
J. Adámek and J. Rosický. Locally Presentable and Accessible Categories. London Math. Soc. Lecture Note Ser., vol. 189, Cambridge University Press, Cambridge (1994).
J.F. Adams. The sphere, considered as an H-space mod p. Quart. J. Math.12, 52–60 (1961).
J.F. Adams. Idempotent functors in homotopy theory. In: Manifolds – Tokyo 1973, pp. 247–253, Univ. of Tokyo Press, Tokyo (1975).
J.F. Adams. Localisation and completion, with an addendum on the use of Brown–Peterson homology in stable homotopy. Lecture notes by Z. Fiedorowicz on a course given at The University of Chicago in Spring 1973. Revised and supplemented by Z. Fiedorowicz, arXiv:1012.5020 (2010).
J. Bagaria. Large cardinals as principles of structural reflection. arXiv:2107.01580 (2021).
J. Bagaria, C. Casacuberta, A.R.D. Mathias and J. Rosický. Definable orthogonality classes in accessible categories are small. J. Eur. Math. Soc.17, 549–589 (2015).
A.K. Bousfield. The localization of spaces with respect to homology. Topology14, 133–150 (1975).
A.K. Bousfield. The localization of spectra with respect to homology. Topology18, 257–281 (1979).
A.K. Bousfield. Cohomological localizations of spaces and spectra. Unpublished (1979).
A.K. Bousfield and D.M. Kan. Homotopy Limits, Completions and Localizations. Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin-Heidelberg (1972).
E.H. Brown. Abstract homotopy theory. Trans. Amer. Math. Soc.119, 79–85 (1965).
E.H. Brown and F.P. Peterson. A spectrum whose \({\mathbb Z}_p\)-cohomology is the algebra of reduced p-th powers. Topology5, 149–154 (1966).
C. Casacuberta, D. Scevenels, and J.H. Smith. Implications of large-cardinal principles in homotopical localization. Adv. Math.197, 120–139 (2005).
A. Deleanu. Existence of the Adams completion for CW-complexes. J. Pure Appl. Algebra4, 299–308 (1974).
S. Eilenberg and N.E. Steenrod. Axiomatic approach to homology theory. Proc. Natl. Acad. Sci. USA31, 117–120 (1945).
P. Gabriel and M. Zisman. Calculus of Fractions and Homotopy Theory. Ergeb. Math. Grenzgeb., vol. 35, Springer-Verlag, Berlin-Heidelberg (1967).
P. Hilton, G. Mislin and J. Roitberg. Localization of Nilpotent Groups and Spaces. North-Holland Math. Studies, vol. 15, North-Holland, Amsterdam (1975).
M. Hovey. Cohomological Bousfield classes. J. Pure Appl. Algebra103, 45–59 (1995).
T. Jech. Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Math., Springer-Verlag, Berlin-Heidelberg (2003).
D.C. Johnson and W.S. Wilson. BP operations and Morava’s extraordinary K-theories. Math. Z.144, 55–75 (1975).
A. Kanamori. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Perspectives in Mathematical Logic, Springer-Verlag, Berlin-Heidelberg (1994).
A. Lévy. A Hierarchy of Formulas in Set Theory. Mem. Amer. Math. Soc., vol. 57, Amer. Math. Soc., Providence (1965).
J.P. May. Simplicial Objects in Algebraic Topology. The Univ. of Chicago Press, Chicago (1967).
M. Mimura, G. Nishida and H. Toda. Localization of CW-complexes. J. Math. Soc. Japan23, 593–624 (1971).
T. Ohkawa. The injective hull of homotopy types with respect to generalized homology functors. Hiroshima Math. J.19, 631–639 (1989).
A.J. Przeździecki. Homotopical localizations at a space. Topology Appl.126, 131–143 (2002).
D.G. Quillen. Homotopical Algebra. Lecture Notes in Math., vol. 43, Springer-Verlag, Berlin-Heidelberg (1967).
D.G. Quillen. Rational homotopy theory. Ann. of Math. (2)90, 205–295 (1969).
D. Ravenel. Localizations with respect to certain periodic homology theories. Amer. J. Math.106, 351–414 (1984).
J.-P. Serre. Groupes d’homotopie et classes de groupes abéliens. Ann. of Math. (2)58, 258–294 (1953).
G. Stevenson. Derived categories of absolutely flat rings. Homol. Homotop. Appl.16, 45–64 (2014).
D. Sullivan. Genetics of homotopy theory and the Adams conjecture. Ann. of Math. (2)100, 1–79 (1974).
D. Sullivan. Infinitesimal computations in topology. Publ. Math. IHÉS47, 269–331 (1977).
Acknowledgements
I thank Joan Bagaria for revising and correcting the set-theoretical content of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Casacuberta, C. (2023). Cohomological Localizations and Set-Theoretical Reflection. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-031-12244-6_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-12243-9
Online ISBN: 978-3-031-12244-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)