Triangulated surfaces in triangulated categories

TL;DR

This paper constructs a dg-category associated with a surface and a triangulated category, showing its independence from triangulation and linking it to the Fukaya category in a topological setting.

Contribution

It introduces a new dg-category F(S,A) parametrizing exact triangles labeled by a surface's triangulation, with invariance under triangulation changes and connections to Fukaya categories.

Findings

F(S,A) is Morita equivalent regardless of triangulation.

F(S,A) admits a canonical mapping class group action.

In the case of 2-periodic complexes, F(S,A) models the Fukaya category topologically.

Abstract

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.