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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2014
    • (edited Feb 4th 2014)

    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:

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 27th 2023

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2023

    Those “bra”-s, being “dual pure states” ψ|:\langle \psi \vert \,\colon\, \mathcal{H} \to \mathbb{C} are a kind of observable, namely reflecting the observation: “system is in state ψ\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”.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 27th 2023

    Ah, Ok. Thanks!