范畴平铺理论：通过边反转构造有向平面平铺
Categorical Tiling Theory: Constructing Directed Planar Tilings via Edge Reversal
AI总结：
本文提出范畴平铺理论，通过构造有限范畴和预层模型，将平铺对齐问题形式化，并利用边反转构造反射对齐，生成具有对称性的对齐族。
摘要：
平面的平铺类似于代数拓扑中的单纯复形及其他复形，但尚未从这一视角进行研究。我们构造了与带标记有向边的多边形相对应的有限范畴，并引入了将欧几里得或双曲平面的平铺建模为该范畴上的预层的问题。在组合上，这相当于为平铺选择一个“对齐”：为每条边指定一个方向，并为每个多边形平铺块的边指定一致的标记。我们证明，对于固定的平铺，给定一个对齐，我们可以通过比较边方向来刻画同一平铺的所有其他对齐。然后，我们为每个顶点处具有偶数个多边形的任何平铺构造一个“反射”对齐，并由此生成一大族具有优雅对称性质的对齐。
英文摘要：
Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce the problem of modeling tilings of the Euclidean or hyperbolic plane as presheaves over such a category. Combinatorially, this amounts to choosing an ``alignment'' for a tiling: a direction for every edge and consistent labels for the edges of each polygonal tile. We show that for a fixed tiling, given a single alignment we can characterize every other alignment of the same tiling by comparison of the edge directions. We then construct a ``reflective'' alignment for any tiling with an even number of polygons at each vertex, and from this generate a large family of alignments with elegant symmetry properties.
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。
