Higher homotopy associativity

Iwase, Norio; Mimura, Mamoru

doi:10.1007/bfb0085229

Cited by 46 publications

(87 citation statements)

References 7 publications

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“…The multiplihedra {J n+1 }, which serve as parameter spaces for homotopy multiplicative morphisms of A ∞ -algebras, lie between the associahedra and permutahedra (see [18], [6] The multiplihedron J n+1 can also be realized as a subdivision of the cube I n . For n = 0, 1, 2, set J n+1 = P n+1 .…”

Section: Faces and Edges Represented By Elements Of The First Five CLmentioning

confidence: 99%

Diagonals on the permutahedra, multiplihedra and associahedra

Saneblidze

Umble

2004

Homology, Homotopy and Applications

115

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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.

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Section: Faces and Edges Represented By Elements Of The First Five CLmentioning

confidence: 99%

Diagonals on the permutahedra, multiplihedra and associahedra

Saneblidze

Umble

2004

Homology, Homotopy and Applications

115

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“…Boardman and Vogt [1] fleshed out the definition in terms of painted trees; a detailed combinatorial description was then given by Iwase and Mimura [10]. Saneblidze and Umble relate the multiplihedron to co-bar constructions of category theory and the notion of permutohedral sets.…”

Section: Introductionmentioning

confidence: 99%

Marked tubes and the graph multiplihedron

Devadoss

Forcey

2008

Algebr. Geom. Topol.

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Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron.

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“…The fact that ϑ n,m = Id when 1 m, n 2 implies K n+1,m+1 = P m+n ; also, K n,2 ∼ = K 2,n is the multiplihedron J n for all n (see [14], [3], [10], [9]). The faces 24|13 and 1|24|3 of P 3,1 are degenerate in K 4,2 since ϑ 3,1 (24|13) = 24|1|3 and ϑ 3,1 (1|24|3) = 1|2|4|3; and dually, the faces 24|13 and 2|13|4 of P 1,3 are degenerate in KK 2,4 since ϑ 1,3 (24|13) = 2|4|13 and ϑ 1,3 (2|13|4) = 2|1|3|4.…”

Section: P P -Factorization Of Proper Cellsmentioning

confidence: 99%

Matrads, biassociahedra, and $A_{\infty}$-bialgebras

Saneblidze¹,

Umble

2011

Homology, Homotopy and Applications

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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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Higher homotopy associativity

Cited by 46 publications

References 7 publications

Diagonals on the permutahedra, multiplihedra and associahedra

Diagonals on the permutahedra, multiplihedra and associahedra

Marked tubes and the graph multiplihedron

Matrads, biassociahedra, and $A_{\infty}$-bialgebras

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