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Matteo Capucci (he/him) (Jan 07 2025 at 22:01):What do you call a commutative monoid in quantales with their tensor product? Is anyone aware of def in the literature?
Matteo Capucci (he/him) (Jan 07 2025 at 22:02):(IMO quantale is such a lame name btw, they should be called monoidal lattices, making these rigoidal lattices)
Mike Shulman (Jan 07 2025 at 22:44):"monoidal suplattices", right?
Mike Shulman (Jan 07 2025 at 22:45):Normally I would think of the suplattice structure as analogous to addition, so that the monoidal operation is analogous to multiplication and a quantale is already a ring-ish thing.
Mike Shulman (Jan 07 2025 at 22:46):What do you call a commutative monoid in the category of rings with its tensor product?
Matteo Capucci (he/him) (Jan 07 2025 at 23:05):Mike Shulman said:
"monoidal suplattices", right?
indeed
Matteo Capucci (he/him) (Jan 07 2025 at 23:06):Mike Shulman said:
What do you call a commutative monoid in the category of rings with its tensor product?
uhm, a commutative algebra?
Matteo Capucci (he/him) (Jan 07 2025 at 23:06):Mike Shulman said:
Normally I would think of the suplattice structure as analogous to addition, so that the monoidal operation is analogous to multiplication and a quantale is already a ring-ish thing.
yeah that's a fair point :thinking:
Mike Shulman (Jan 08 2025 at 01:35):A commutative algebra generally means a commutative monoid in the monoidal category of -modules for some commutative ring (or field) , which is equivalent to a commutative ring together with a ring homomorphism .