DESCRIBING A POSSIBILITY OF NON-TRIVIAL CONTINUATION OF SIMPLICIAL SETS IN TERMS
OF THE HOMOLOGY OF THE HOCHSCHILD COMPLEX 

Ladoshkin Mikhail Vladimirovich, Candidate of physical and mathematical sciences, associate professor, head of sub-department of mathematics and methods of mathematics teaching, Mordovia State Pedagogical Institute named after M. E. Evsevev (11a Studencheskaya street, Saransk, Russia), m01051977@mail.ru

512.662.1

10.21685/2072-3040-2017-3-6

Background. The process of creating analogues of algebraic structures that persist during the transition to homotopy is currently an urgent problem of algebraic topology. Previously, the author built a higher simplicial set, which was a stable homotopy analogue of the simplicial object described by the objects on which this structure exists. The obtained results were compared with the results of V. A. Smirnov for simplicial sets and turned out to be significantly different. Following the general method of homotopically stable analogues studying this article considers the structure of many kinds of simplicially objects’ continuations up to homotopically stable analogues. With this in mind, the work reveals the link of the Hochschild homology with a possibility of non-trivial continuation of simplicial sets.
Materials and methods. In the work all the claims and constructions are shown in the field of characteristisc two. Such a method allows to avoid defining characters and keeping track of them when they change. In algebraic topology, all the main results obtained in the field of characteristic two, can be generalized for the case of arbitrary fields, which is a technical problem.
Results. The article describes a condition under which nontrivial continuation of a simplicial set to its homotopically stable analogue – higher simplicial sets. The wording of this condition is given in terms of the homology of the Hochschild complex.
Conclusions. The results of the article allow to draw a conclusion about the possibility of continuing a non-trivial simplicial structure, and explore the question of the number of non-trivial structures with accuracy up to isomorphism.

Simplicial set, Hochschild homology, crossing cochains, higher simplicial sets, equivalence of crossing cochains, isomorphism of simplicial sets

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