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Atrium > Mathematics, Physics & Philosophy: How can physics intuition help (and hinder) a n-POV?

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    • CommentRowNumber1.
    • CommentAuthorfastlane69
    • CommentTimeJan 13th 2016
    • (edited Jan 13th 2016)
    To me, a 0-transfor is like (is?) position, a 1-trans like velocity, and a 2-trans like acceleration.

    So the best way I've been able to categorically describe (newtonian) physics is as a position 0-transfor embedded?mapped?adjoined? with a constant time.

    Is this correct? Or is this an example of my physics getting in the way of my CT? :)

    So how will my physics intuition help me understand CT and other categories and how will it hinder me?
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2016
    • (edited Jan 13th 2016)

    A useful match to physics goes like so:

    • an object (“0-morphism”) of an n-groupoid is like a field configuration;

    • a morphism (i.e. a 1-morphism) is like a gauge transformation turning one field configuration into another;

    • a 2-morphism is like a gauge-of-gauge transformations, turning one gauge transformation into another;

    • generally an nn-morphism is like a higher gauge transformation.

    See for instance the bachelor thesis Stacks in Gauge theory.

    • CommentRowNumber3.
    • CommentAuthorfastlane69
    • CommentTimeJan 13th 2016
    • (edited Jan 13th 2016)
    That's field theory though, not newtonian theory.... unless you are considering a velocity as a gauge transformation on positions which would be very cool!!!!

    I ask because several (all?) of CT concpets tickles my physics intuition... but I'm all too aware of trusting intuition blindly.

    So I'm trying to track down how many of the CT-like concepts that I learned for my Physics PhD are, to use the parlance, "universal" or "natural".
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2016
    • (edited Jan 13th 2016)

    Maybe your intuition will match the role that the path groupoid plays in physics. You may regard its objects as positions of a particle, and its morphisms as trajectories traced out by the particle.

    The infinitesimal version of this, where we keep only the derivatives of the trajectories at any position, is called the tangent Lie algebroid. But in a “synthetic differential geometry“-context this may indeed itself be regarded as a groupoid. If so, then this is a groupoid whose objects are points, hence positions, and whose morphisms are vectors and hence may indeed sensibly be thought of as velocities.

    • CommentRowNumber5.
    • CommentAuthorfastlane69
    • CommentTimeJan 13th 2016
    • (edited Jan 13th 2016)
    " path groupoid plays in physics. You may regard its objects as positions of a particle, and its morphisms as trajectories traced out by the particle.. "

    Sexy! Awesome! Thank you so much for that insight! That is exactly the kind of language I want to use!

    These link provided unanimously speak of smooth differential manifolds.
    My applications deal with discrete spaces where a differential is a difference
    Is the homotopy "smooth map" requirement violated if I use a discrete set of points (lattice and DeltaX?) versus a continuous number line (Manifold and dx?)?
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2016

    If you are happy with velocities not being vectors but being finite edges, then you may just as well consider the groupoid (or more general category) whose objects are the points in your discrete space, whose morphisms are the discrete paths, and whose composition of morphisms is the concatenation of paths. There is for instance the path category freely generated from a graph, maybe that’s just what you are looking for.

    • CommentRowNumber7.
    • CommentAuthorfastlane69
    • CommentTimeJan 13th 2016
    • (edited Jan 13th 2016)

    “There is for instance the path category freely generated from a graph, maybe that’s just what you are looking for.”

    You are making my day; yes, that may/is exactly what I need!

    I have a graph in front of me that I’m trying to decipher. It consists of frequency as objects/dots and discrete frequency intervals as morphisms/arrows.

    I’ve been using my “position/velocity” intuition on it and I was worried of just how far I could/should take it. Then I came across CT and it says (to me at least) that I can take that intuition this far and perhaps even further as long as I stay “categorical”, “natural”, or “universal”.

    So if keep the following transformations natural (or universal or categorical, not sure how to state this)

    Grf —> Grf:Path — >Grf:Path:Groupoid—–> my physics map

    Grf —> Grf:Path — >Grf:Path:Groupoid—-> my frequency map

    then I can use my intuition from one on the other because

    my physics map —> my frequency map.

    Yes/No?

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