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arXiv · 2608.17263

Cohomology for solutions of polygon equations

Serban Matei Mihalache·Tomoro Mochida
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Abstract

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff--Kapranov.

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Serban Matei Mihalache, Tomoro Mochida. 2026-08-18. Cohomology for solutions of polygon equations. https://arxiv.org/abs/2608.17263

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