Symmetric Homotopy Theory for Operads and Weak Lie 3-Algebras

by Malte Dehling
Doctoral thesis
Date of Examination:2020-11-16
Date of issue:2021-01-14
Advisor:Prof. Dr. Chenchang Zhu
Referee:Prof. Dr. Chenchang Zhu
Referee:Prof. Dr. Bruno Vallette
Persistent Address: http://dx.doi.org/10.53846/goediss-8401

 

 

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Abstract

English

This thesis consists of two parts: In the first, we develop the homotopy theory of differential graded operads over any unital commutative ring. The main idea is to consider the symmetric group actions as part of the operadic structure and not of the underlying category. We introduce a new dual category of higher cooperads, a new higher cobar-bar adjunction with the category of operads, and a new notion of higher homotopy operads for which we establish the homotopy properties. In the second part, we introduce a category of weak Lie 3-algebras with suitable weak morphisms. The definition is based on the construction of a partial resolution of the Koszul dual cooperad of the Lie operad with free symmetric group actions. Weak Lie 3-algebras and their morphisms are then defined as solutions to Maurer-Cartan equations. We prove a version of the homotopy transfer theorem for weak Lie 3-algebras and provide a skewsymmetrization construction from weak Lie 3-algebras to 3-term L-infinity algebras. Finally, we provide some initial applications of weak Lie 3-algebras in higher differential geometry.
Keywords: Operads; Koszul Duality; Homotopy Algebras; Homotopy Operads; Homotopy Lie Algebras; Weak Lie 3-Algebras
 
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