Triangulated surfaces in triangulated categories
For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S...
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| Format: | Journal article |
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2013
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| _version_ | 1826258396961046528 |
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| author | Dyckerhoff, T Kapranov, M |
| author_facet | Dyckerhoff, T Kapranov, M |
| author_sort | Dyckerhoff, T |
| collection | OXFORD |
| description | For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces. In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine. |
| first_indexed | 2024-03-06T18:33:21Z |
| format | Journal article |
| id | oxford-uuid:0a6911eb-3185-44a6-a406-881fbf6f11f7 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:33:21Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | oxford-uuid:0a6911eb-3185-44a6-a406-881fbf6f11f72022-03-26T09:23:44ZTriangulated surfaces in triangulated categoriesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:0a6911eb-3185-44a6-a406-881fbf6f11f7Symplectic Elements at Oxford2013Dyckerhoff, TKapranov, MFor a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces. In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine. |
| spellingShingle | Dyckerhoff, T Kapranov, M Triangulated surfaces in triangulated categories |
| title | Triangulated surfaces in triangulated categories |
| title_full | Triangulated surfaces in triangulated categories |
| title_fullStr | Triangulated surfaces in triangulated categories |
| title_full_unstemmed | Triangulated surfaces in triangulated categories |
| title_short | Triangulated surfaces in triangulated categories |
| title_sort | triangulated surfaces in triangulated categories |
| work_keys_str_mv | AT dyckerhofft triangulatedsurfacesintriangulatedcategories AT kapranovm triangulatedsurfacesintriangulatedcategories |