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The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial setof an $L_\infty$-algebra plays an important role in deformation theory and manyother areas of mathematics. Unfortunately, this construction only works over afield of characteristic $0$. The goal of this paper is to show that the notionsof Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for amuch 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 thecategory of unital associative algebras) with an arity $0$ operation admits atwisting procedure, a natural notion of Maurer-Cartan equation and under somemild additional assumptions can also be integrated to a Maurer-Cartansimplicial set. In particular, we show that the Koszul dual of theBarratt-Eccles operad and its $E_n$-suboperads admit Maurer-Cartan simplicialsets. In this paper, we will work over arbitrary rings.

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