Triangulation, Persistence, and Fukaya categories

Abstract

This paper introduces a new algebraic notion - triangulated persistence category (TPC) - that refines that of triangulated category in the same sense that a persistence module is a refinement of the notion of a vector space. The spaces of morphisms of such a TPC are persistence modules and this category is endowed with a class of weighted distinguished triangles. Under favourable conditions we show that the derived Fukaya category admits a TPC refinement and this is applied to deduce a global rigidity result for spaces of compact, exact Lagrangians in certain Liouville manifolds: we construct a metric on this space with intrinsic symplectic properties.

177 pages; 25 figures; the second chapter of this paper incorporates our earlier preprint, Triangulation and persistence: Algebra 101, arXiv:2104.12258; several imprecisions and minor errors are corrected in this version, without incidence on the main results