Comparing cohomology obstructions

Hans Joachim Baues; David Blanc

doi:10.1016/j.jpaa.2010.09.003

Comparing cohomology obstructions

Hans Joachim Baues, David Blanc

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 a. Π-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.

Original language	English
Pages (from-to)	1420-1439
Number of pages	20
Journal	Journal of Pure and Applied Algebra
Volume	215
Issue number	6
DOIs	https://doi.org/10.1016/j.jpaa.2010.09.003
State	Published - Jun 2011

Bibliographical note

Funding Information:
We thank the referee for his or her comments. The second author would like to thank the Max-Planck-Institut für Mathematik for hospitality while this research was carried out. He would also like to thank Bernard Badzioch, Wojtek Dorabiała, Mark Johnson, and Jim Turner for many useful discussions.

Keywords

Primary
Secondary

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jpaa.2010.09.003

Cite this

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title = "Comparing cohomology obstructions",

abstract = "We show that three different kinds of cohomologies - Baues-Wirsching cohomology, the (S*,O)-cohomology of Dwyer and Kan, and the Andr{\'e}-Quillen cohomology of a. Π-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.",

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AU - Blanc, David

N1 - Funding Information: We thank the referee for his or her comments. The second author would like to thank the Max-Planck-Institut für Mathematik for hospitality while this research was carried out. He would also like to thank Bernard Badzioch, Wojtek Dorabiała, Mark Johnson, and Jim Turner for many useful discussions.

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AB - 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 a. Π-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.

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Comparing cohomology obstructions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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