  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.
