We construct an explicit diagonal ∆ P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P → K [19] and its factorization through J. We introduce the notion of a permutahedral set Z and lift ∆ P to a diagonal on Z. We show that the double cobar construction Ω 2 C * (X) is a permutahedral set; consequently ∆ P lifts to a diagonal on Ω 2 C * (X). Finally, we apply the diagonal on K to define the tensor product of A ∞ -(co)algebras in maximal generality.

In the paper the notion of truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. Using this construction together with the theory of twisted tensor products for homotopy G-algebras a strictly associative multiplicative model for a fibration is obtained.

We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set X we associate a necklical set ΩX such that its geometric realization | ΩX|, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on |X| and the differential graded module of chains C * ( ΩX) is a differential graded associative algebra generalizing Adams' cobar construction.

We introduce the notion of a matrad M = {M n,m } whose submodules M * ,1 and M 1, * are non-Σ operads. We define the free matrad H ∞ generated by a singleton θ n m in each bidegree (m, n) and realize H ∞ as the cellular chains on a new family of polytopes {KK n,m = KK m,n }, called biassociahedra, of which KK n,1 = KK 1,n is the associahedron K n . We construct the universal enveloping functor from matrads to PROPs and define an A ∞ -bialgebra as an algebra over H ∞ .

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