Abstract:
The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of $A_\infty$-algebras and $D_\infty$-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.
Keywords:differential module with homotopy simplicial faces, $A_\infty$-algebra, $D_\infty$-differential module, realization of a simplicial topological space, SDR-data, category of $F_\infty$-modules, category of $D_\infty$-modules.
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