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Shea Levy (Oct 29 2020 at 14:01):In ordinary category theory, if I referred to a "point" or "global element" of an object X it would be understood I mean an arrow from . Thinking of multicategories as modelling the logic of, say, natural deduction, though, the relevant analogue seems more appropriate to be a multiarrow from the empty list, rather than from the singleton list containing the terminal object. Is this accepted terminology?
Reid Barton (Oct 29 2020 at 15:00):This terminology only really makes sense for vaguely Set-like or otherwise geometric categories. For example, in any category with a zero object, this notion is trivial, but an abelian group, say, has a perfectly good notion of element.
Reid Barton (Oct 29 2020 at 15:03):The more generally sensible notion is a map from the unit of a monoidal category, which in the case of a cartesian monoidal category is the same as a map from the terminal object. And in a multicategory the empty sequence indeed plays the role of the unit (more precisely, a unit object is an object which represents the functor ).