Title: Higher-Categorical Associahedra Show Chinese title

Title: 高阶范畴结合多面体

Abstract: The second author introduced 2-associahedra as a tool for investigating functoriality properties of Fukaya categories, and he conjectured that they could be realized as face posets of convex polytopes. We introduce a family of posets called categorical $n$-associahedra, which naturally extend the second author's 2-associahedra and the classical associahedra. Categorical $n$-associahedra give a combinatorial model for the poset of strata of a compactified real moduli space of a tree arrangement of affine coordinate subspaces. We construct a family of complete polyhedral fans, called velocity fans, whose coordinates encode the relative velocities of pairs of colliding coordinate subspaces, and whose face posets are the categorical $n$-associahedra. In particular, this gives the first fan realization of 2-associahedra. In the case of the classical associahedron, the velocity fan specializes to the normal fan of Loday's realization of the associahedron. For proving that the velocity fan is a fan, we first construct a cone complex of metric $n$-bracketings and then exhibit a piecewise-linear isomorphism from this complex to the velocity fan. We demonstrate that the velocity fan, which is not simplicial, admits a canonical smooth flag triangulation on the same set of rays, and we describe a second, finer triangulation which provides a new extension of the braid arrangement. We describe piecewise-unimodular maps on the velocity fan such that the image of each cone is a union of cones in the braid arrangement, and we highlight a connection to the theory of building sets and nestohedra. We explore the local iterated fiber product structure of categorical $n$-associahedra and the extent to which this structure is realized by the velocity fan. For the class of concentrated $n$-associahedra we exhibit generalized permutahedra having velocity fans as their normal fans. Show Chinese abstract

Abstract: 第二作者引入了2-关联多面体，作为研究Fukaya范畴函子性性质的工具，并猜想它们可以作为凸多面体的面偏序集来实现。 我们引入了一类称为范畴$n$-关联多面体的偏序集，它们自然地扩展了第二作者的2-关联多面体和经典的关联多面体。 范畴$n$-关联多面体为仿射坐标子空间树排列的紧化实模空间的层偏序集提供了一个组合模型。 我们构造了一族称为速度扇的完全多面体扇，其坐标编码了碰撞坐标子空间对的相对速度，且其面偏序集是范畴$n$-关联多面体。 特别是，这给出了2-关联多面体的第一个扇实现。 在经典关联多面体的情况下，速度扇退化为Loday对关联多面体的实现的正常扇。 为了证明速度扇是一个扇，我们首先构造了一个度量$n$-括号的锥复形，然后展示从该复形到速度扇的分段线性同构。 我们证明了速度扇（不是单纯性的）在相同的射线集上具有规范的光滑旗三角剖分，并描述了第二个更细的三角剖分，该三角剖分提供了一个新的辫排列扩展。 我们描述了速度扇上的分段单模映射，使得每个锥的像都是辫排列中锥的并集，并强调了与建筑集和嵌套多面体理论的联系。 我们探讨了范畴$n$-关联多面体的局部迭代纤维积结构以及这种结构在速度扇中实现的程度。 对于集中$n$-关联多面体类，我们展示了具有速度扇作为其正常扇的广义排列多面体。