  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.
