Triangulated surfaces in triangulated categories

By Tobias Dyckerhoff, M. Kapranov. 2013

79 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bec8d40142c5b207e15082bbf95cf9500cfb5001

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Citations in the corpus

Perverse Schobers (2014, methodology): “For example, when X is a Riemann surface with marked points, the Fukaya category of X with coefficients in a constant (Z/2-graded) triangulated category A was defined in [DK].”
Flat surfaces and stability structures (2014, unknown): “to show that it agrees with other denitions found in the literature. Broadly speaking, these are either based on Floer intersection theory (see [17], [38], and many others), sheaves on ribbon graphs [39, 16, 31], or more direct approaches [1]. Our own approach to dene F(S) falls into the latter cate”
Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, methodology): “This result was first proved by Dyckerhoff and Kapranov and constitutes a cornerstone in their work [17], [18], [16], [15], to which we refer for the remarkable richness of this class of examples.”
Relative Calabi–Yau structures (2016, methodology): “We recall the context of [Dyc15] where the topological Fukaya category of a stable marked surface (S,M) is defined (based on the ideas of [DK13]).”
Geometry of canonical bases and mirror symmetry (2013, methodology): “It can be checked by using the Kontsevich combinatorial model [1,15,49,73] for the Fukaya–Seidel category as a category of locally constant sheaves on the Lagrangian skeleton for a surface with potential in the case of (C, et + e−t ), shown on Fig.”