I will introduce the notion of the 'Poincaré' or 'Koszul' dual of a stratified space with tangential structure (TS), whose construction in general is as yet an open problem. Then I will outline (the finished part of) ongoing work on defining a functorial field theory, given, as input, a disk-algebra with TS. This recovers the framed case, which was proposed by Lurie (later picked up by Calaque and Scheimbauer): duals of stably-framed bordisms are euclidean spaces with flag-like stratifications. In particular, this notion explains the 'shape' of the higher Morita category of En-algebras when expressed in terms of factorization algebras, and gives a natural definition of Morita categories of disk-algebras with any TS. If time permits, I will propose a simple Poisson-structured version of this procedure which should construct, using Poisson additivity, extended classical gauge theories given only the 1-shifted Poisson algebra of bulk observables.