Inversion of elements in a group.
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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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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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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Exactly.
End of conversation
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