Triangulated surfaces in triangulated categories

Abstract

For a triangulated category \mathcal A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S , we introduce a dg-category F(S,\mathcal A) parametrizing systems of exact triangles in \mathcal A labelled by triangles of S . Our main result is that \mathcal F(S,\mathcal A) is independent of 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 \mathcal A is the category of 2-periodic complexes of vector spaces, \mathcal F(S,\mathcal 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.