Triangulated surfaces in triangulated categories

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.

55 pages, v2: references added and typos corrected, v3: expanded version, comments welcome

References in corpus (3)

• Matrix Factorizations and Representations of Quivers I
• Grothendieck ring of pretriangulated categories
• Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstrass cubics

Cited by in corpus (32)

• Flat surfaces and stability structures
• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Relative Calabi-Yau structures
• Wrapped microlocal sheaves on pairs of pants
• The nonequivariant coherent-constructible correspondence for toric stacks
• Non-characteristic expansions of Legendrian singularities
• On the combinatorics of exact Lagrangian surfaces
• Crossed simplicial groups and structured surfaces
• Topological Fukaya category and mirror symmetry for punctured surfaces
• Simplicial structures in higher Auslander-Reiten theory
• Mirror symmetry for the Landau-Ginzburg A-model $M={\mathbb{C}}^n,W=z_1\cdots z_n$
• The symplectic geometry of higher Auslander algebras: Symmetric products of disks
• Higher categorical aspects of Hall Algebras
• Cyclic structures and broken cycles
• 2-Segal spaces as invertible infinity-operads
• Perverse sheaves of categories and some applications
• Legendrian skein algebras and Hall algebras
• Harmonic Maps to Buildings and Singular Perturbation Theory
• Ginzburg algebras of triangulated surfaces and perverse schobers
• Geometry of canonical bases and mirror symmetry
• Perverse schobers and wall crossing
• Spherical adjunctions of stable $\infty$-categories and the relative S-construction
• Lagrangian Cobordisms in Liouville manifolds
• The Hall Algebras of Surfaces I
• Fukaya categories of higher-genus surfaces and pants decompositions
• Parabolic induction and perverse sheaves on h/W
• Line fields on punctured surfaces and twisted derived categories
• Formal Contact Categories
• Structured Topological Field Theories via Crossed Simplicial Groups
• Strebel Differentials and stable Matrix Factorizations
• Contraderived categories of CDG-modules
• A geometric realization of silting theory for gentle algebras