15:30
Given a triangulated category A, equipped with a differential
Z/2-graded enhancement, and a triangulated oriented marked surface S, we
explain how to define a space X(S,A) which classifies systems of exact
triangles in A parametrized by the triangles of S. The space X(S,A) is
independent, up to essentially unique Morita equivalence, of the choice of
triangulation and is therefore acted upon by the mapping class group of the
surface. We can describe the space X(S,A) as a mapping space Map(F(S),A),
where F(S) is the universal differential Z/2-graded category of exact
triangles parametrized by S. It turns out that F(S) is a purely topological
variant of the Fukaya category of S. Our construction of F(S) can then be
regarded as implementing a 2-dimensional instance of Kontsevich's proposal
on localizing the Fukaya category along a singular Lagrangian spine. As we
will see, these results arise as applications of a general theory of cyclic
2-Segal spaces.
This talk is based on joint work with Mikhail Kapranov.