Obstruction Theory in Model Categories

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.Comment: 17 pages. v3 includes improved introduction and several other minor improvement

Phantom Maps and Homology Theories

We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X -> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X,HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor S -> A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.Comment: 25 pages, AMSLaTeX, to appear in Topolog