I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in refs 1, 2, and 3. The formulation was originally crafted for topological QFTs (which don't have local observables) and conformal QFTs, but it can also be used for traditional QFTs on pseudo-Riemannian manifolds, as explained in this excerpt from ref 1:
A category of QFT exists for each fixed spacetime dimension $d$ and a structure $S$ on manifolds. Here, the structure $S$ can be e.g. smooth structure, Riemannian metric, conformal structure, spin structure, etc. ... They are supposed to satisfy the standard axioms of Atiyah and Segal..., properly modified for the structure $S$.
...and from ref 3:
In a convenient axiomatization of quantum field theory..., the structure of a $d +1$ dimensional quantum field theory includes... a functor $\Phi$ from the category of closed $d$-manifolds [more precisely: *-manifolds] into the category of Hilbert spaces. ... Here ∗ can be any extra structure: for example an orientation, a spin structure, a complex structure, or a metric. ... Depending on the structure we obtain different types of quantum field theories: topological, `spin,' conformal, etc.
Question: How are local observables encoded in this formulation of QFT?
Section 2.2 in ref 1 answers the question for conformal QFTs, using the state-operator correspondence, but I don't know how to generalize this to not-necessarily-conformal QFTs. Ref 4 seems to partly address question, but only indirectly (and I'm having trouble parsing the higher-category-theoretic math).
Tachikawa (2017), "On 'categories' of quantum field theories" (https://arxiv.org/abs/1712.09456)
Monnier (2019), "A modern point of view on anomalies" (https://arxiv.org/abs/1903.02828)
Dijkgraaf and Witten (1990), "Topological gauge theories and group cohomology" (https://projecteuclid.org/euclid.cmp/1104180750)
Schreiber (2008), "AQFT from n-functorial QFT" (https://arxiv.org/abs/0806.1079)
Monnier (2014), "A modern point of view on anomalies" (http://www.maths.dur.ac.uk/lms/109/talks/1866monnier.pdf)
Freed (2014), "Anomalies and Invertible Field Theories" (https://arxiv.org/abs/1404.7224)
Sati and Schreiber (2011), "Survey of mathematical foundations of QFT and perturbative string theory" (https://arxiv.org/abs/1109.0955)
TQFT associates a category to a manifold (on Physics SE)