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For a triangulated category with a 2-periodic dg-enhancement and a triangulated oriented marked surface , we introduce a dg-category parametrizing systems of exact triangles in labelled by triangles of . Our main result is that is independent of the choice of a triangulation of 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 is the category of 2-periodic complexes of vector spaces, turns out to be a purely topological model for the Fukaya category of the surface . 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.
Cite this article
Tobias Dyckerhoff, Mikhail Kapranov, Triangulated surfaces in triangulated categories. J. Eur. Math. Soc. 20 (2018), no. 6, pp. 1473–1524DOI 10.4171/JEMS/791