The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of the European Mathematical Society


Full-Text PDF (385 KB) | Metadata | Table of Contents | JEMS summary
Online access to the full text of Journal of the European Mathematical Society is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
subscriptions@ems-ph.org
Volume 20, Issue 6, 2018, pp. 1473–1524
DOI: 10.4171/JEMS/791

Published online: 2018-04-30

Triangulated surfaces in triangulated categories

Tobias Dyckerhoff[1] and Mikhail Kapranov[2]

(1) University of Bonn, Germany
(2) IPMU, Kashiwa, Japan

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.

Keywords: Triangulated categories, ribbon graphs, topological Fukaya categories, mapping class groups

Dyckerhoff Tobias, Kapranov Mikhail: Triangulated surfaces in triangulated categories. J. Eur. Math. Soc. 20 (2018), 1473-1524. doi: 10.4171/JEMS/791