标题： 范畴铺砌理论：通过边反转构造有向平面铺砌 显示英文标题

标题： Categorical Tiling Theory: Constructing Directed Planar Tilings via Edge Reversal

摘要： 平面镶嵌类似于代数拓扑中的单纯复形和其他复形，但从这个角度进行研究的却不多。 我们构建了与带有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镶嵌建模为该范畴上的预层的问题。 从组合学的角度来看，这相当于为镶嵌选择一个“对齐”方式：为每条边指定一个方向，并为每个多边形面片的边赋予一致的标签。 我们证明，对于固定的镶嵌，给定一个对齐方式，可以通过比较边的方向来表征该镶嵌的其他所有对齐方式。 然后，我们为每个顶点处有偶数个多边形的镶嵌构建了一个“反射性”对齐方式，并由此生成了一类具有优雅对称性质的对齐方式。 显示英文摘要

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.