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One difference between the absolute Galois groupoid of a field and the fundamental groupoid of a topological space is that for the former the set of objects you want to range over is not actually a set.

So I wonder if you could define the absolute Galois groupoid in von Neumann–Bernays–Gödel set theory (without using truncating cardinals)?

Are there papers on number theory written from this point of view?

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    $\begingroup$ Could you perhaps include the definition of the absolute Galois groupoid for the uninitiated? $\endgroup$
    – Emil Jeřábek
    Commented Dec 4, 2020 at 19:42
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    $\begingroup$ What does "without truncating cardinals" mean? Are you referring to Scott's trick? If so, why do you want to avoid it? It lets you formalize everything you want in $\mathsf{ZF}$ alone, with no need to talk about classes at all. $\endgroup$
    – Noah Schweber
    Commented Dec 5, 2020 at 2:00
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    $\begingroup$ As to the question, I suspect that the standard definition (whatever it is - I can't find a satisfying reference, but this MO question mentions it) already works as a definition in $\mathsf{NBG}$ - of a large groupoid, of course - so that really all or almost all texts in number theory essentially do this by default. Scott's trick can then be used to canonically produce an equivalent small groupoid, so that we can freely switch to $\mathsf{ZFC}$ anytime we want. $\endgroup$
    – Noah Schweber
    Commented Dec 5, 2020 at 2:07

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