Triangulation, Persistence, and Fukaya Categories

This memoir introduces a new algebraic notion: that of a triangulated persistence category (TPC), which refines the notion of a triangulated category in the same sense that a persistence module refines that of a vector space.The spaces of morphisms of such a TPC are persistence modules, and the 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 we apply this 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.