Conditionally flat functors on spaces and groups
- 59 Downloads
Consider a fibration sequence \(F\rightarrow E\rightarrow B\) of topological spaces which is preserved as such by some functor \(L\), so that \(LF \rightarrow LE \rightarrow LB\) is again a fibration sequence. Pull the fibration back along an arbitrary map \(X\rightarrow B\) into the base space. Does the pullback fibration enjoy the same property? For most functors this is not to be expected, and we concentrate mostly on homotopical localization functors. We prove that the only homotopical localization functors which behave well under pull-backs are nullifications. The same question makes sense in other categories. We are interested in groups and how localization functors behave with respect to group extensions. We prove that group theoretical nullification functors behave nicely, and so do all epireflections arising from a variety of groups.
KeywordsLocalization Flatness Fiberwise localization Variety of groups
Mathematics Subject Classification (2000)55R05 20E22 55P60 55P65 55R70 20E10 20F14
This work started when the first author visited the EPFL in Lausanne and the facilitation of this working visit was greatly appreciated. We would like to thank Boris Chorny and Marino Gran for enlightening discussions, putting this work in perspective respectively with properness and reflective subcategories. We would like to thank also the referee for his careful reading and the improvements he suggested.
- 3.Bousfield, A. K.: Constructions of factorization systems in categories, J. Pure Appl. Algebra 9(2), 207–220 (1976/77)Google Scholar
- 8.Casacuberta, C., Rodríguez J. L., Scevenels D.: Singly generated radicals associated with varieties of groups. Groups St. Andrews: in Bath. I, pp 202–210 (1997)Google Scholar
- 9.Chachólski, W.: On the functors CWA and PA. Duke Math. J. 84(3), 599–631 (1996)Google Scholar
- 11.Dwyer, W. G., Farjoun, E. D.: Localization and Cellularization of Principal Fibrations. Alpine Perspectives on Algebraic Topology, pp. 117–124 (2009)Google Scholar
- 12.Everaert, T., Gran, M.: Protoadditive functors, derived torsion theories and homology, Preprint 2011, http://arxiv.org/abs/1111.5448
- 13.Dror Farjoun, E.: Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics, vol. 1622. Springer, Berlin (1996)Google Scholar
- 15.Mal’cev, A.I.: Algebraic systems, Posthumous edition. In: Smirnov, D., Taĭclin, M. (eds.) Translated from the Russian by B. D. Seckler and A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 192, pp xii+317. Springer, New York (1973)Google Scholar