Groups arise naturally as automorphism groups. I tend to think of "group" as meaning "thing that is isomorphic to an automorphism group", and take the group axioms as a convenient way to characterize which things are groups.
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Replying to @ModelOfTheory
Yes, but what is an automorphism group? A collection of endomorphism tuples (m_1, m_2) such that m_1 * m_2 = m_2 * m_1 = 1, composing under the rule (m_1, m_2) * (n_1, n_2) = (m_1 * n_1, n_2 * m_2). But perhaps a 2 has been snuck in here implicitly…
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Replying to @RadishHarmers
Given a set with some sort of structure on it (which necessarily includes =), its automorphism group is the set of all functions from the set to itself preserving that structure, endowed with every operation that can be naturally assigned to such sets in a constructive manner.
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Replying to @ModelOfTheory @RadishHarmers
By preserving that structure, I mean in the iff sense, not just that structure must be carried over in one direction.
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Replying to @ModelOfTheory @RadishHarmers
Admittedly what I said doesn't apply to automorphism groups of objects in non-concrete categories. I think of these as "things that work in much the same way that sets with structure do".
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Replying to @ModelOfTheory
Alright, now the 2 comes from your focus on "functions"; that is, on particular 2-ary relations. But we might also look at 3-ary correspondence relations. E.g., R(x, y, z), such that for any choice of x, y, or z, there are designated corresponding values for the other arguments.
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Replying to @RadishHarmers @ModelOfTheory
Note that we can compose these 3-ary correspondences in the various ways that 3-simplices can be put together to make new 3-simplices, just as we could compose our familiar 2-ary correspondences in the various ways that 2-simplices can be put together to make new 2-simplicies.
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Replying to @RadishHarmers
So a triangulation of a triangle should correspond to a composition rule for these 3-ary correspondences? I don't see how.
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Replying to @ModelOfTheory
Yes, that's correct. Consider every directed edge of a triangle to be an ordinary automorphism (with the two opposite directions along an edge being ordinary inverses).
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Replying to @RadishHarmers @ModelOfTheory
Our correspondences are those triangles that commute. A triangulation of a large triangle into smaller commuting triangles also shows that the large triangle commutes.
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Oh I see. Composition rules subject to the condition that shared edges must agree, much like composition in groupoids is subject to the condition that shared vertices must agree.
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