Na minha lista:
| Main Authors: | , , |
|---|---|
| Formato: | Preprint |
| Publicado em: |
2024
|
| Assuntos: | |
| Acesso em linha: | https://arxiv.org/abs/2409.03633 |
| Tags: |
Adicionar Tag
Sem tags, seja o primeiro a adicionar uma tag!
|
| _version_ | 1866916383508398080 |
|---|---|
| author | Backman, Spencer Bottman, Nathaniel Poliakova, Daria |
| author_facet | Backman, Spencer Bottman, Nathaniel Poliakova, Daria |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher-Categorical Associahedra Backman, Spencer Bottman, Nathaniel Poliakova, Daria Combinatorics Algebraic Geometry Metric Geometry Symplectic Geometry 53D37, 51M20, 52B05, 14T90, 14A21 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. |
| title | Higher-Categorical Associahedra |
| topic | Combinatorics Algebraic Geometry Metric Geometry Symplectic Geometry 53D37, 51M20, 52B05, 14T90, 14A21 |
| url | https://arxiv.org/abs/2409.03633 |