Collectanea Mathematica

, Volume 66, Issue 1, pp 149–160 | Cite as

Conditionally flat functors on spaces and groups

  • Emmanuel Dror Farjoun
  • Jérôme SchererEmail author


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.


Localization 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.


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Copyright information

© Universitat de Barcelona 2014

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsHebrew University of Jerusalem (Gig’at Ram)JerusalemIsrael
  2. 2.EPFL, SB MATHGEOMLausanneSwitzerland

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