A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting of disjoint unions of disks in $M$ with maps inclusions that are surjective on connected components (see e.g. Proposition 2.19 here). This means that factorization algebras (as well as there non-unital version) induce a "monodromy functor" from exit paths to spaces, and one can show even more:
Theorem 5.5.4.10 in Lurie's "Higher Algebra" shows that the category of non-unital locally constant factorization algebras is equivalent to the category of factorizable constructible sheaves on $\operatorname{Ran}(M)$. Further, Remark 5.5.4.12 shows there is a fully faithful embedding of the category of (unital!) locally constant factorization algebras into the category of factorizable cosheaves on the extended Ran space $\operatorname{Ran}^+(M)$, which is formed by adding a generic point $\emptyset$ to the usual Ran space.
While I think that I have some intuition about why the first statement should be true (by trying to imagine how exit paths in the Ran space could be considered as paths joining together in $M$), the second claim puzzles me a lot. $\operatorname{Ran}^+(M)$ is not Hausdorff so special care must be taken when applying the usual technology for stratified spaces, and even if the exodromy theorem works I don't really have an intuition of how exit paths here should look like (although it doesn't seem unreasonable that adding this extra point gives us a unit in out algebraic structure).
Is there a more extensive treatment of this idea somewhere in the literature? Maybe answering Lurie's open question about the essential image of the above-mentioned fully faithful embedding? And as I am already asking, is my intuition right that exit-paths in the Ran space, corresponding to joining paths in the manifold, are in direct relation to Feynman-diagrams and many-point-functions in physics?
