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Brendan Murphy (Feb 08 2024 at 07:18):Given a monoidal category and the structure of a concrete category on it, ie a forgetful functor to the category of sets, what compatibility should we demand in order for this to deserve the name "concrete monoidal category"? For many "sets with structure" categories like topological spaces, posets, varieties over an algebraically closed k, manifolds, the forgetful functor is strong monoidal with respect to the cartesian monoidal structure. But also very interesting is modules over a commutative ring, where the functor isn't strong but it is lax and the internal homs are still ordinary functions satisfying a predicate. In both cases the forgetful functor is corepresented by the monoidal unit (because the unit is a monoid this implies the forgetful functor are lax). But we also might think of k-algebras as "concrete monoidal", maybe, and here the forgetful functor is corepresented by a different object, namely k[X]
Simon Burton (Feb 08 2024 at 19:50):This is like asking for a concrete bicategory... or a 2-Yoneda lemma, if there is such a thing ..
Mike Shulman (Feb 08 2024 at 19:54):Are you thinking that a monoidal category is a 1-object bicategory? That's true of course, but it doesn't seem to me to go in the direction Brendan is thinking about.
Mike Shulman (Feb 08 2024 at 19:55):If you want to emphasize the "internal homs are ordinary functions satisfying a predicate" point of view, you could ask that the forgetful functor is a (lax) [[closed functor]] whose comparison map is a monomorphism.
Mike Shulman (Feb 08 2024 at 19:56):(Between monoidal categories, a lax closed functor is equivalent to a lax monoidal functor, but emphasizing the closed structure is relevant for "internalizing" concreteness with internal-homs.)
John Baez (Feb 08 2024 at 19:57):@Simon Burton wrote:
This is like asking for a concrete bicategory... or a 2-Yoneda lemma, if there is such a thing ..
The Yoneda lemma for bicategories is a conceptually nice way to show every bicategory is equivalent to a 2-category, since it embeds any bicategory into the bicategory , which is actually a 2-category.
Simon Burton (Feb 08 2024 at 21:04):So yeah, are these pieces fitting together yet? A "concrete monoidal category" would be a one object bicategory with a (forgetful?) bifunctor to Cat.
Mike Shulman (Feb 08 2024 at 21:15):Usually a [[concrete category]] is one with a faithful functor to Set. Brendan is asking about when such a concrete category has a monoidal structure, but its forgetful functor still lands in Set -- not Cat. That's one reason I said I don't think viewing a monoidal category as a 1-object bicategory is helpful for this question.
Brendan Murphy (Feb 08 2024 at 21:53):Simon Burton said:
So yeah, are these pieces fitting together yet? A "concrete monoidal category" would be a one object bicategory with a (forgetful?) bifunctor to Cat.
I think this would turn out to be a monoidal category equipped with a faithful action on some other category (although I'm not 100% sure what faithful means here). This is concrete in the same sense that cayley's theorem shows every group is a concrete permutation group, but I think that sense of concrete is different than what we mean by a concrete category
Martti Karvonen (Feb 09 2024 at 09:42):As a (small) monoidal category is a pseudomonoid in Cat, a natural guess would be that a concrete monoidal category is a pseudomonoid in the 2-category of concrete categories. Defined this way, the resulting notion is sensitive to how one defines the latter - for instance, in Joy of Cats concrete functors are functors making the relevant triangle commute strictly, which would imply that the forgetful functor should be strict monoidal. If instead one requires the relevant triangle to commute up to a specified iso, you'd get monoidal categories whose forgetful functor is strong monoidal. I guess one could also work in a 2-category of "(op)lax concrete functors", in which case the resulting pseudomonoids have a forgetful functor that is (op)lax monoidal.
I haven't really thought what the conditions on 2-cells being "concrete" imply about the monoidal structure, but I'd guess that they won't rule out examples of interest.
John Baez (Feb 09 2024 at 19:33):I guess you want to allow lax monoidal forgetful functors in your definition of "concrete monoidal category" if you want Vect with its usual tensor product to count as a concrete monoidal category.
John Baez (Feb 09 2024 at 19:36):(Brendan said essentially this at the start.)
Oscar Cunningham (Feb 16 2024 at 16:20):Often in monoidal categories the underlying set of an object is given by the functor , where is the monoidal unit. So one very strong condition on a concrete monoidal category would be to demand precisely this.