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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2012
    • (edited Sep 8th 2012)

    at principle of equivalence I have restructured the Examples-section: added new subsections in “In physics” on gauge transformations and on general covariance (just pointers so far, no text), and then I moved the section that used to be called “In quantum mechanics” to “Examples-In category theory” and renamed it to “In the definition of \dagger-categories” (for that is really what these paragraphs discuss, not any notion of equivalence in quantum mechanics, the application of \dagger-categories in that context notwithstanding)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2012
    • (edited Sep 8th 2012)

    And I move the “How to break equivalence-invariance” to the end. At least it should go after the “General definition”, I’d say.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 9th 2012

    So general covariance is an example of the (mathematical) principle of equivalence. Remind me, how should I think of the (physical) principle of equivalence? I think somewhere you just said it’s a result of using Riemannian spaces. So the physical principle of equivalence is part of a precondition for the mathematical principle of equivalence to have as special case general covariance?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2012
    • (edited Sep 9th 2012)

    Remind me, how should I think of the (physical) principle of equivalence?

    I am claiming that the mathematical formalization of the “principle ofequivalence” in the theory of gravity is (or was, in the time when people actually referred to it):

    1. the infinitesimal neighbourhood of every point in spacetime is a vector space (the tangent space);

    2. the Levi-Civita connection can be taken (up to its gauge freedom) to vanish on that neighbourhood (not it’s derivative, though, hence not the curvature). This is the existence of “Riemann normal coordinates”.

    In conclusion: in first order approximation around any point, gravitational dynamics looks like dynamics in flat space.

    So that “principle of equivalence” is not so very close to the mathematical “principle of equivalence” as we have on that page. What we have on the page principle of equivalence however directly subsumes what in gravity is the principle of general covariance.

    But of course the relation is via those Riemann normal coordinates (or can be understood that way): the principle of general covariance says that if you see a field of gravity gg on spacetime Σ\Sigma, you may regard it as being equivalent to the field of gravity g˜f *g\tilde g \coloneqq f^* g for any diffeomorphism f:ΣΣf : \Sigma \to \Sigma. The statement of Riemann normal coordinates is that you can always find such a diffeomorphism such that at a given point f *gf^* g induces vanishing Levi-Civita connection at that point, hence proving that up to gauge equivalence the dynamics in the infinitesimal neighbourhood of that point is that of flat space.

    That’s how these things relate. Schematically we have implications

    principle of equivalence in mathematics \Rightarrow principle of general covariance \Rightarrow principle of equivalence in physics

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2012

    I turned this into a quick note here. But need to quit now.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 7th 2018

    One day we should probably extract some text from that discussion at In the concept of †-categories. But for now, I added a link to this old MO discussion – Are dagger categories truly evil?.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2018

    Added a reference

    diff, v90, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    Touched the lead-in sentences, adding more hyperlinks (such as to higher structures).

    Removed two more pointless appearances of the word “evil”.

    diff, v93, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 24th 2021

    I wonder if that long discussion on dagger categories could be condensed and extracted now with the benefit of hindsight.

    • CommentRowNumber10.
    • CommentAuthorKeith Harbaugh
    • CommentTimeMar 28th 2021

    Updated inline link to: Makkai 1995 “Towards a Categorical Foundation of Mathematicics”

    diff, v94, current

    • CommentRowNumber11.
    • CommentAuthorGuest
    • CommentTimeMar 12th 2022
    This page says "It does not break equivalence-invariance to state that two morphisms are equal, given a common source and target; this is because a hom-set is a set, where equality is meaningful." But if I click on the "hom-set" link, it says that a hom-set is only a set for a locally small category. Does this matter?

    I think that maybe this claim and the corresponding claim in the next paragraph should be removed.
    • CommentRowNumber12.
    • CommentAuthorGuest
    • CommentTimeMar 12th 2022
    that's probably because hom-sets are large (i.e. classes) in a large category, and so aren't small (i.e. set).
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2022

    Yes, but size issues are not the point here, the point is the uniqueness of identities between elements, the choice of ambient universe does not change this. Of course it won’t hurt to add further clarifying remarks, if this seems confusing. But for that we first need to have the edit functionality back.

    • CommentRowNumber14.
    • CommentAuthorGuest
    • CommentTimeMar 12th 2022
    Is there a difference between "size issues," i.e. whether or not a collection of objects is too large to be a set, and the issue of whether the objects can be meaningfully compared for equality?
    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2022

    Yes, they’re entirely unrelated questions. A small collection can lack a strict equality, while a large collection can have one.

    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 14th 2022

    that’s probably because hom-sets are large (i.e. classes) in a large category, and so aren’t small (i.e. set).

    I’d like to push back against this: a large category can have small homs, eg the category of sets.

    The category of classes (of NBG classes, for example) is not locally small, and given two class functions between the same pair of classes, we can define their equaliser, and then ask if the equaliser is inhabited. This is a proposition of first-order logic.

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