Obstruction theory in model categories
J. Daniel Christensen, William G. Dwyer and Daniel C. Isaksen
MSC: 55S35, 55U35, 18G55 (primary); 18G30, 55P42 (secondary)
Department of Mathematics
University of Western Ontario
London, Ontario N6A 5B7
jdc@uwo.ca
Department of Mathematics
University of Notre Dame
South Bend, IN 46556
dwyer.1@nd.edu
Department of Mathematics
University of Notre Dame
South Bend, IN 46556
isaksen.1@nd.edu
Keywords: obstruction theory, closed model category, simplicial set,
spectrum
Many examples of obstruction theory can be formulated as the study
of when a lift exists in a commutative square. Typically, one of
the maps is a cofibration of some sort and the opposite map is a
fibration, and there is a functorial obstruction class that determines
whether a lift exists. Working in an arbitrary pointed proper model
category, we classify the cofibrations that have such an obstruction
theory with respect to all fibrations. Up to weak equivalence, retract,
and cobase change, they are the cofibrations with weakly contractible
target. Equivalently, they are the retracts of principal cofibrations.
Without properness, the same classification holds for cofibrations
with cofibrant source. Our results dualize to give a classification
of fibrations that have an obstruction theory.