C,(D2). Later Tamarkin and Tsygan [TTOO] found a rather simple algebraic construction of G,,, -> B,.
Operads and so no nice small set of higher homotopies for commutativity, though some explicit constructions of algebraic E00-operads were given [Smi94]. -algebras. In that case, what symmetry should we require of the homotopies for associativity? 13). For dg Lie algebras L, the standard construction is the graded symmetric coalgebra on the suspension of L and thus the n-ary brackets of an Lam-algebra have corresponding symmetry. 8). ' Thus an appropriate specification for a `strictly commutative' A,-algebra A, to be called a C,-algebra (also known as a balanced Ate-algebra), is via a coderivation on the graded Lie coalgebra on the suspension of A or equivalently in terms of a coherent set of n-ary products which vanish on shuffles.