熵、上闭链及其图论表示
Entropy, cocycles, and their diagrammatics
AI总结：
该研究将群上闭链的熵相关构造转化为平面网络图演算，得到的幺半范畴补充了现有熵的范畴与算子研究方法。
摘要：
论文第一部分阐释了如何用平面三价图网络对群$G$上取值于其表示的一上闭链和二上闭链进行编码：图的边由$G$的元素标记，表示的元素置于区域内，并配有相应的图操作规则。当该群为半直积时，可通过涉及的两个子群的交叠网络得到类似的表示形式。\nM. Kontsevich与J.-L. Cathelineau已证明，如何通过直线仿射对称群上的2-上闭链，来解释有限随机变量的熵和无穷小二重对数，包括它们的四项函数关系。\n我们将他们的构造转化为一种图演算，用于评估描述合适幺半范畴中态射的平面网络。具体而言，四项关系变为类似于结合律方程的网络等式。由此得到的幺半范畴补充了现有的熵的范畴论和算子论研究方法。
英文摘要：
The first part of the paper explains how to encode a one-cocycle and a two-cocycle on a group $G$ with values in its representation by networks of planar trivalent graphs with edges labelled by elements of $G$, elements of the representation floating in the regions, and suitable rules for manipulation of these diagrams. When the group is a semidirect product, there is a similar presentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy of a finite random variable and infinitesimal dilogarithms, including their four-term functional relations, via 2-cocycles on the group of affine symmetries of a line. We convert their construction into a diagrammatical calculus evaluating planar networks that describe morphisms in suitable monoidal categories. In particular, the four-term relations become equalities of networks analogous to associativity equations. The resulting monoidal categories complement existing categorical and operadic approaches to entropy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
