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

University of Haifa

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	American 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

Keywords

Primary
Secondary

All Science Journal Classification (ASJC) codes

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.",

keywords = "Primary, Secondary",

author = "Baues, {Hans Joachim} and David Blanc",

note = "Funding Information: We thank the referee for his or her comments. The second author would like to thank the Max-Planck-Institut f{\"u}r Mathematik for hospitality while this research was carried out. He would also like to thank Bernard Badzioch, Wojtek Dorabia{\l}a, Mark Johnson, and Jim Turner for many useful discussions.",

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AU - Baues, Hans Joachim

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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N2 - 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.

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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DO - 10.1016/j.jpaa.2010.09.003

M3 - Article

SN - 0022-4049

VL - 215

SP - 1420

EP - 1439

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 6

ER -

Comparing cohomology obstructions

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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