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.
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 ∞ .
Let I = (Z 3 , 26, 6, B) be a 3D digital image, let Q(I) be the associated cubical complex and let ∂Q(I) be the subcomplex of Q(I) whose maximal cells are the quadrangles of Q(I) shared by a voxel of B in the foreground -the object under study -and by a voxel of Z 3 B in the background -the ambient space. We show how to simplify the combinatorial structure of ∂Q(I) and obtain a 3D polyhedral complex P (I) homeomorphic to ∂Q(I) but with fewer cells. We introduce an algorithm that computes cup products on H * (P (I); Z 2 ) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R 3 .
An A ∞ -bialgebra is a DGM H equipped with structurally compatible operations ω j,i :is an A ∞ -algebra and H, ω j,1 is an A ∞ -coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End (T H).To Jim Stasheff on the occasion of his 68th birthday.
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