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arXiv · 2303.16706

A Lie theoretic approach to the twisting procedure and Maurer-Cartan simplicial sets over arbitrary rings

Niek de Kleijn·Felix Wierstra
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Abstract

The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial set of an $L_\infty$-algebra plays an important role in deformation theory and many other areas of mathematics. Unfortunately, this construction only works over a field of characteristic $0$. The goal of this paper is to show that the notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for a much larger number of operads than just the $L_\infty$-operad. More precisely, we show that the Koszul dual of every unital Hopf cooperad (a cooperad in the category of unital associative algebras) with an arity $0$ operation admits a twisting procedure, a natural notion of Maurer-Cartan equation and under some mild additional assumptions can also be integrated to a Maurer-Cartan simplicial set. In particular, we show that the Koszul dual of the Barratt-Eccles operad and its $E_n$-suboperads admit Maurer-Cartan simplicial sets. In this paper, we will work over arbitrary rings.

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Niek de Kleijn, Felix Wierstra. 2023-03-29. A Lie theoretic approach to the twisting procedure and Maurer-Cartan simplicial sets over arbitrary rings. https://arxiv.org/abs/2303.16706

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