arXiv subjects

Samson Saneblidze

Publications and source records attributed to Samson Saneblidze.

22 records · Page 2Linked to original sources

arXiv2004

The twisted Cartesian model for the double path fibration

In the paper the notion of truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a permutocubical set that models in particular the path fibration on a loop space. The chain complex of this twisted Cartesian product in fact is a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras.

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arXiv2004

A cubical model for a fibration

In the paper the notion of truncating twisting function $τ:X\to Q$ from a simplicial set $X$ to a cubical set $Q$ and the corresponding notion of twisted Cartesian product of these sets $X\times_τQ$ are introduced. The latter becomes a cubical set whose chain complex coincides with the standard twisted tensor product $C_*(X)\otimes_{τ_*}C_*(Q)$. This construction together with the theory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations.

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arXiv2004

Diagonals on the Permutahedra, Multiplihedra and Associahedra

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

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arXiv2004

A Diagonal on the Associahedra

Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal Δ: C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We apply the diagonal Δto define the tensor product of A_\infty-(co)algebras in maximal generality.

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