Mathematics

Classical AQFT can be defined as a cosheaf A of C*-algebras on the
manifold of space-time M (or more generally on a suitable lorentzian
manifold playing such rôle), satisfying two axioms: locality, which
ensures that observables in an open region are a fortiori observables
in any superset of that region, and causality, ensuring that If U,V
are spacelike separated regions, then A(U) and A(V) pairwise commute
as subalgebras of the total algebra A(M).
Now what if we want to suitably categorify this notion, extending it
to the realm of tensor categories (that is, categories equipped with a
tensor functor subject to suitable axioms). Causality has to be
replaced by a higher-categorical analogue of commutativity,
generalizing (in fact, categorifying) the concept of commutators of a
subalgebra and Von Neumann algebras.