Abstract:
On the basis of the colored version of Koszul duality, the notion of a differential module with $\infty$-simplicial faces is introduced. By using the homotopy technique of differential Lie modules over colored coalgebras, the homotopy invariance of the structure of a differential module with $\infty$-simplicial faces is proved. A relationship between differential modules with $\infty$-simplicial faces and $A_\infty$-algebras is described. The notions of the chain realization of a differential module with $\infty$-simplicial faces and the tensor product of differential modules with $\infty$-simplicial faces are introduced. It is shown that the chain realization of a tensor differential module with $\infty$-simplicial faces constructed from an $A_\infty$-algebra and the $B$-construction over this $A_\infty$-algebra are isomorphic differential coalgebras.
Keywords:differential module with $\infty$-simplicial faces, $A_\infty$-algebra, colored differential module, colored differential algebra, Koszul duality, chain realization of differential modules, $B$-construction, category of differential Lie $C$-modules, SDR-data, differential $R_\infty$-module.
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