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Chetan Vuppulury (Oct 08 2021 at 19:28):I am trying to understand the properties of the 2-category of algebras and bimodules (defined here). What are some paces I can read more about this category? I can see that it has a symmetric monoidal structure given by the usual tensor product, but is it written up in detail anywhere?
And what else is known about this category? Does it have limits and colimits? Since it is considered one of the standard delooping of the usual category of modules, does it satisfy some sort of 2-abelian category axioms? Such as, does it maybe have biproducts?
Any references (other than Shulman's paper which is on the nLab) about this 2-category would be nice.
John Baez (Oct 08 2021 at 19:36):By the way, it's a bicategory rather than a 2-category. I guess the nLab has a policy of not distinguishing them, but if you're looking for good reference you might want to search using "bicategory".
John Baez (Oct 08 2021 at 19:38):You might find some good information by going to a more general case. An algebra is a Vect-enriched category with one object. A bimodule between algebras is a Vect-enriched profunctor between such Vect-enriched categories. So, the bicategory of which you speak is contained in the bicategory of "Vect-enriched profunctors".
John Baez (Oct 08 2021 at 19:39):And, painful as it might initially be, you can learn a lot of the basic properties of this bicategory by reading about "enriched profunctors".
Mike Shulman (Oct 08 2021 at 19:40):The symmetric monoidal structure is most naturally constructed by regarding it as the underlying bicategory of a double category, as in this paper and this one.
John Baez (Oct 08 2021 at 19:41):I didn't want to drop so many buckets of ice water on him at the same time, but yeah.
Mike Shulman (Oct 08 2021 at 19:43):Makes sense. But I don't know offhand of a detailed writeup of the monoidal structure on a bicategory of this sort, as he asked for, that doesn't (perhaps implicitly) use the double-categorical approach.
Mike Shulman (Oct 08 2021 at 19:43):The bicategory of Vect-enriched profunctors that John mentioned also has better formal properties, including biproducts.
John Baez (Oct 08 2021 at 19:45):Another thing, just to make him run away screaming: there's a really nice 3-group that captures useful information about commutative rings, algebras over these rings, and bimodules between these algebras: it's sometimes called the Brauer 3-group, and it contains within it the familiar ideas of Brauer group and Picard group.
Chetan Vuppulury (Oct 08 2021 at 19:48):Thanks a lot! I was trying to avoid double categories, but if that is the most natural way to do things, I better read up on that approach I guess.
There is also the bicategory of Vect enriched categories (and enriched functors) into which bicategory embeds (I think), does that not have good properties? (I've never seen profunctors,)
I really appreciate the buckets of cold water! And I am actually also interested in the fully dualisable subcategory which contains the Brauer 3-group. I am approaching this stuff from the perspective of families of TQFTs (with values in the bicategory of algebras and bimodules) and 2-vector bundles with dualisable fibres.
John Baez (Oct 08 2021 at 19:52):Chetan Vuppulury said:
Thanks a lot! I was trying to avoid double categories, but if that is the most natural way to do things, I better read up on that approach I guess.
There is also the bicategory of Vect enriched categories (and enriched functors) into which bicategory embeds (I think), does that not have good properties? (I've never seen profunctors,)
It does have good properties. The point of using a double category is that you can draw the enriched functors (e.g. algebra homomorphisms) as "vertical" arrows and the enriched profunctors (e.g. bimodules) as "horizontal" arrows.
John Baez (Oct 08 2021 at 19:52):It just helps to keep them straight in your mind.
John Baez (Oct 08 2021 at 19:53):We say this double category is "fibrant", or an "equipment", because we have the ability to turn vertical arrows into horizontal ones.
John Baez (Oct 08 2021 at 19:54):To conclude, I should say that Mike calls the functors "tight" morphisms and the profunctors "loose", perhaps reflecting the fact that the former compose associatively "on the nose" while the latter compose associatively only up to isomorphism.
John Baez (Oct 08 2021 at 19:55):He introduced this because some people use "horizontal" to mean what other people call "vertical", and vice versa. You can see us discussing that here.
John Baez (Oct 08 2021 at 19:57):If you don't like "tight" and "loose" morphisms, you can say "arrows" and "proarrows".
John Baez (Oct 08 2021 at 19:58):See, so little research is being done on category theory that we have to make up 5 terms for each concept to make it seem like there's more.
Mike Shulman (Oct 08 2021 at 20:00):Chetan Vuppulury said:
There is also the bicategory of Vect enriched categories (and enriched functors) into which bicategory embeds (I think), does that not have good properties? (I've never seen profunctors,)
Did you mean enriched functors or profunctors? John and I are talking about profunctors. The bicategory of algebras doesn't embed into that of categories and functors.
Mike Shulman (Oct 08 2021 at 20:01):The double category includes the functors as well as the profunctors, but it's the profunctor direction that the bimodules embed into.
Chetan Vuppulury (Oct 08 2021 at 20:02):Enriched functors, I was thinking of sending a ring to it's category of left modules and a bimodule to the enriched functor that tensors with . Is that not fully faithful?
Reid Barton (Oct 08 2021 at 20:02):It's fully faithful if you only allow colimit-preserving functors
Reid Barton (Oct 08 2021 at 20:03):Oh, enriched functors? Enriched over what though? Ab?
Reid Barton (Oct 08 2021 at 20:03):I think it doesn't change my answer.
Chetan Vuppulury (Oct 08 2021 at 20:04):Sorry, I meant faithful and reflects isos. And enriched over Vect, I meant to say algebra, not ring.
Mike Shulman (Oct 08 2021 at 20:08):That embedding is "faithful" in a bicategorical sense, meaning "locally fully faithful", i.e. acts fully-faithfully on hom-categories. It's also "conservative" in the analogous bicategorical sense of reflecting equivalences.
Reid Barton (Oct 08 2021 at 20:08):Oh also I should clarify that I meant that the whole assignment is fully faithful when viewed as a functor from Rings to (locally presentable Ab-enriched categories, colimit-preserving additive functors). You might have only meant that it was locally fully faithful... right, like Mike just said.
Mike Shulman (Oct 08 2021 at 20:09):As Reid says, it's bicategorically fully-faithful (an equivalence on hom-categories) if you restrict the codomain to contain only colimit-preserving functors. And when you do that, you've essentially reinvented profunctors!
Mike Shulman (Oct 08 2021 at 20:09):One definition of a profunctor from to is a colimit-preserving functor between their categories of (enriched) presheaves.
Reid Barton (Oct 08 2021 at 20:12):In plainer language, the statement I had in mind is that there is an equivalence on the level of Hom-categories, from
to
If you replace the latter category by the category of all additive functors from to , then you still get a fully faithful embedding (because by definition, the colimit-preserving functors form a full subcategory of all additive functors).
Chetan Vuppulury (Oct 08 2021 at 20:19):Mike Shulman said:
As Reid says, it's bicategorically fully-faithful (an equivalence on hom-categories) if you restrict the codomain to contain only colimit-preserving functors. And when you do that, you've essentially reinvented profunctors!
I see! That's wonderful! Where can I read about enriched profunctors?
John Baez (Oct 08 2021 at 20:28):Type "enriched profunctor" into the all-knowing Goog.
John Baez (Oct 08 2021 at 20:29):You'll probably get the nLab article on them.
John Baez (Oct 08 2021 at 20:30):You might want to learn what a profunctor is before learning about the enriched case. Even plain old profunctors are great!