JournalsjemsVol. 20, No. 6pp. 1473–1524

Triangulated surfaces in triangulated categories

  • Tobias Dyckerhoff

    University of Bonn, Germany
  • Mikhail Kapranov

    IPMU, Kashiwa, Japan
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For a triangulated category A\mathcal A with a 2-periodic dg-enhancement and a triangulated oriented marked surface SS, we introduce a dg-category F(S,A)F(S,\mathcal A) parametrizing systems of exact triangles in A\mathcal A labelled by triangles of SS. Our main result is that F(S,A)\mathcal F(S,\mathcal A) is independent of the choice of a triangulation of SS 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\mathcal A is the category of 2-periodic complexes of vector spaces, F(S,A)\mathcal F(S,\mathcal A) turns out to be a purely topological model for the Fukaya category of the surface SS. 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.

Cite this article

Tobias Dyckerhoff, Mikhail Kapranov, Triangulated surfaces in triangulated categories. J. Eur. Math. Soc. 20 (2018), no. 6, pp. 1473–1524

DOI 10.4171/JEMS/791