Comparing cohomology obstructions

TL;DR

This paper demonstrates the isomorphism among three types of cohomology theories and uses this to unify different approaches to realizability problems in algebraic topology.

Contribution

It establishes the equivalence of Baues-Wirsching, Dwyer-Kan, and Andre-Quillen cohomologies under certain conditions, linking various realizability obstructions.

Findings

Proves isomorphism of three cohomology theories.

Unifies different approaches to realizability problems.

Introduces the notion of a mapping algebra as a key tool.

Abstract

We show that three different kinds of cohomology - Baues-Wirsching cohomology, the (S,O)-cohomology of Dwyer-Kan, and the Andre-Quillen cohomology of a Pi-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-Wirsching, the diagram rectifications of Dwyer-Kan-Smith, and the Pi-Algebra realization of Dwyer-Kan-Stover. Our main tool in this identification is the notion of a mapping algebra: a simplicially enriched version of an algebra over a theory.