多边形方程解的上同调
Cohomology for solutions of polygon equations
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中文总结 AI 辅助
本文针对多边形方程的集合论解构造允许着色的半单纯集以定义其（上）同调群，建立了其与满足特定条件的高Segal半单纯集的范畴等价性，特例恢复了Dyckerhoff-Kapranov的五边形方程解与2-Segal半单纯集的对应结果。
中文摘要 AI 辅助
多边形方程构成了一类推广五边形方程的方程族。本文中，我们构造了与多边形方程的集合论解相关的允许着色的半单纯集，并利用它们定义了对应的（上）同调群。我们研究了这些群的若干性质，建立了多边形方程的集合论解与满足特定条件的高Segal半单纯集之间的范畴等价性。作为特例，我们的结果恢复了Dyckerhoff-Kapranov证明的双射集合论五边形方程解与2-Segal半单纯集之间的对应关系。
英文摘要
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.
