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I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory.
To start with I produced a dictionary table, for inclusion in relevant entries:
Why the same final entry in these two rows?
| dependent sum | generalized homology spectrum | space of quantum states (“bra”) |
| dual of dependent sum | generalized cohomology spectrum | space of quantum states (“ket”) |
Elsewhere homology is related to observables.
Those “bra”-s, being “dual pure states” $\langle \psi \vert \,\colon\, \mathcal{H} \to \mathbb{C}$ are a kind of observable, namely reflecting the observation: “system is in state $\psi$”.
But without monoid structure on the base space (such as it being a loop space) there is no algebra structure induced on its homology, and hence no canonical structure of an “algebra of observables”.
Ah, Ok. Thanks!
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