# Journal of the European Mathematical Society

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**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