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Stream: learning: questions

Topic: Morphisms from the monoidal unit

Paolo Perrone (Aug 16 2023 at 08:19):

Is there a standard name for morphisms from the monoidal unit in "algebraic" contexts?
In Markov categories and other similar theories we call them "states". I'm not sure though that I would use that term for, say, a map $\mathbb{Z}\to G$ of Abelian groups, corresponding to a unique element of $G$. What's the accepted terminology?

Dylan Braithwaite (Aug 16 2023 at 08:26):

You could just call them 'points'. That's at least consistent with the terminology in [[pointed abelian groups]]/ [[pointed objects in a monoidal category]]

Mike Shulman (Aug 16 2023 at 14:55):

Or "elements"

Reid Barton (Aug 16 2023 at 15:00):

Maybe "global elements", depending on context.

Mike Shulman (Aug 16 2023 at 15:03):

"Global elements" makes me think specifically of morphisms out of the terminal object. I'm not sure what I would think if I read it in a context of a non-cartesian monoidal category, but I would be a bit confused.

Mike Shulman (Aug 16 2023 at 15:04):

"monoidal elements"?

Todd Trimble (Aug 16 2023 at 15:10):

Monoidal points, perhaps.

Reid Barton (Aug 16 2023 at 15:29):

I'm thinking about sheaves of abelian groups, for example.
A "(generalized) element" might also be a homomorphism from any object.

Mike Shulman (Aug 16 2023 at 16:18):

Ok, yes, in sheaves of abelian groups I could accept "global element" to mean a morphism out of the unit object, since (1) the terminal object is also initial, so maps out of it are uninteresting, and (2) maps out of the unit object are the same as global elements, in the ordinary sense, of the underlying sheaf of sets.