Triangulated surfaces in triangulated categories

Understand

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.