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I have come across lectures about Higher Topos Theory in mathematical physics and I am wondering about the explicit restriction to the category of smooth sets.

Why should the potentially possible laws of physics be restricted to live inside category of smooth sets? Is this category chosen because of experiments that indicate that we live in a "smooth" world? Could alternate possible laws of physics (by possible I mean internally consistent) that describe some other possible universe not constrained by relativity and QM "live inside" much less restricted set theoretical universe $V$?

Edit: By smooth set I mean a generalized smooth space that may be probed by smooth Cartesian spaces (https://ncatlab.org/nlab/show/smooth+set).

Lecture by Urs Schreiber: https://www.youtube.com/watch?v=Gr_BCr1rYVM&t=1791s

Qmechanic
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Pan Mrož
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1 Answers1

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There's no limit to what you are allowed to use in physics. The only requirement is that it be empirically adequate for modeling nature. Our current best models use vectors in Hilbert space (quantum mechanics) and fields on 4-dimensional Pseudo-Riemannian surfaces (general relativity), both of which are smooth sets. But that doesn't mean it's impossible for some deeper theory to approximate those theories in the relevant domains using something that is not a smooth set (Stephen Wolfram for example is currently attempting such a project).

Travis
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