Comparing cohomology obstructions

We show that three different kinds of cohomologies – Baues–Wirsching cohomology, the (S∗,O)-cohomology of Dwyer and Kan, and the André–Quillen cohomology of aΠ-algebra  – are isomorphic, under certain assumptions. This is then used to identify the cohomological obstructions in three general approaches to realizability problems: the track category version of Baues and Wirsching, the diagram rectifications of Dwyer, Kan, and Smith, and the Π-algebra realization of Dwyer, Kan, and Stover. Our main tool in this identification is the notion of a mapping algebra: a simplicially enriched version of an algebra over a theory.