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.