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Jonathan Beardsley (Jul 06 2023 at 05:47):Sorry, I dunno if I'm Zuliping correctly here. Does anyone know if there are references for promonoidal right Kan extensions? I was looking at older stuff by Day and Street and it seems like they do a lot of with promonoidal left Kan extensions. Does something go wrong with right Kan extensions?
Morgan Rogers (he/him) (Jul 06 2023 at 06:06):What does the "pro" in "promonoidal" refer to?
Nathanael Arkor (Jul 06 2023 at 06:16):Morgan Rogers (he/him) said:
What does the "pro" in "promonoidal" refer to?
It's the same "pro" as in "profunctor", because a [[promonoidal category]] is what you get when you replace the functors in a definition of a monoidal category with profunctors (though what that "pro" strands for is not entirely clear).
Mike Shulman (Jul 06 2023 at 06:30):Whatever it stands for, it's not the same as the "pro" in a [[pro-object]]. But I think both of them are just supposed to mean "not quite".
Morgan Rogers (he/him) (Jul 06 2023 at 06:56):Aha. @Jonathan Beardsley would you mind elaborating on what the left Kan extensions look like in this context? Sorry for the naive questions, but I suspect that with enough context I could answer the original question :sweat_smile:
John Baez (Jul 06 2023 at 10:57):Could it be that Day and Street are working at a high enough level of generality, with enough "op" symmetry around, that they discussed only promonoidal left Kan extensions because you can automatically deduce the corresponding results for promonoidal right Kan extensions just by formally taking the "op" of every category (or enriched category, or whatever) involved?
Just a thought.
Jonathan Beardsley (Jul 06 2023 at 16:46):Right. I think one could most concisely state it as saying a promonoid is a pseudomonoid in the bicategory of categories and profunctors between them
Jonathan Beardsley (Jul 06 2023 at 16:47):That's a good idea John. I was secretly hoping someone had already checked all the details and I could just find a statement like "Promonoidal right Kan extensions exist if ..." But oh well, maybe I'll actually have to go "read" something.
Jonathan Beardsley (Jul 06 2023 at 16:50):Wow the mobile interface is really bad
Jonathan Beardsley (Jul 06 2023 at 16:50):Or maybe it's just the interface plus my phone
fosco (Jul 06 2023 at 19:35):I am surprised left extension even exist, usually it's right extensions and lifts that exist in , can you provide more context for what's in the old stuff?
also, @Nathanael Arkor : https://en.wiktionary.org/wiki/πρό#Ancient_Greek
fosco (Jul 06 2023 at 19:36):the pro- in profunctor stands for "a 1-cell that acts on behalf of, or similarly to, a functor"
Jonathan Beardsley (Jul 06 2023 at 19:37):@fosco here's the paper about left promonoidal extensions http://www.tac.mta.ca/tac/volumes/1995/n4/v1n4.pdf
fosco (Jul 06 2023 at 19:39):I was not aware of that paper; "one always has to learn"
Jonathan Beardsley (Jul 06 2023 at 19:41):actually maybe i'm misunderstanding the basic idea of that paper.... like, what they mean by "left kan extension is promonoidal" or whatever. what i want to have is two promonoidal functors F:C→D and G:C→E and I want the right kan extension of F along G to give a promonoidal functor E→D
Jonathan Beardsley (Jul 06 2023 at 19:49):but i did have the same thought. Prof has all right kan extensions, but not necessarily left, I guess?
Jonathan Beardsley (Jul 06 2023 at 19:50):i wonder if the bicategory of promonoidal categories also has all right extensions and if they give the usual right kan extension on underlying categories
Jonathan Beardsley (Jul 06 2023 at 19:55):oh @fosco you have this question: https://mathoverflow.net/questions/352843/promonoidal-categories-as-s-algebras
Is this at all answered by remark 2.2 here
Day-promonoidal-functor-categories.pdf
fosco (Jul 06 2023 at 19:57):I'll look into that! I know it exists. Never really understood it. My bad. :grinning:
Kevin Arlin (Jul 06 2023 at 20:12):What goes wrong with left extensions in Prof? Does it only break for profunctors over some interesting enrichment base, or are you saying even over Set?
fosco (Jul 06 2023 at 20:15):Kevin Arlin said:
What goes wrong with left extensions in Prof? Does it only break for profunctors over some interesting enrichment base, or are you saying even over Set?
left Kan extensions tend to not exist globally, i.e. as an adjoint to precomposition, because precomposition tends to fail preserving limits (colimits, no problem): after all, composition is defined as a colimit!
Kevin Arlin (Jul 06 2023 at 20:17):Ah, right, interesting! Is it more or less only profunctors representable by functors you can left extend along?
Jonathan Beardsley (Jul 06 2023 at 20:19):I wrote this as an MO question, in case anyone comes up with an answer and wants it to be on the books: https://mathoverflow.net/q/450316/11546
Jonathan Beardsley (Jul 06 2023 at 20:20):One could try to answer the related question: if we have a monoidal bicategory C with all right Kan extensions is it true that the bicategory of pseudomonoids in C also has all right Kan extensions and, if so, are the computed in C?
Kevin Arlin (Jul 06 2023 at 20:32):Incidentally, it doesn't feel to me like the issue of extensions in Prof is relevant here, since we're talking about promonoidal functors, which are really a special case of monoidal functors, not about monoidal profunctors or something, no?
Jonathan Beardsley (Jul 06 2023 at 20:33):hmmmm, so in the specific case i'm interested in, i've got actual functors between promonoidal categories
Jonathan Beardsley (Jul 06 2023 at 20:33):but i don't think they're special cases of monoidal functors, since, again in my special case, the domain category is not in fact monoidal
Jonathan Beardsley (Jul 06 2023 at 20:33):but, yeah, all the profunctors are in fact functors
Kevin Arlin (Jul 06 2023 at 20:33):I mean via the Day convolution. But yeah, we're on the same page on the other point I think.
Jonathan Beardsley (Jul 06 2023 at 20:34):i don't know, maybe i should just write down what my actual question is in case there's some better way to do it
Kevin Arlin (Jul 06 2023 at 20:35):(ie ProMon is the same as the bicategory of monoidal presheaf categories where the monoidal product is cocontinuous in each variable, right?)
Jonathan Beardsley (Jul 06 2023 at 20:35):errrrrr, haha, i don't know. is this basically by taking the monoidal completion?
Kevin Arlin (Jul 06 2023 at 20:35):I'd be interested in the actual question. I've been thinking about the equipments of (co)multicategories lately, which seems closely related.
Jonathan Beardsley (Jul 06 2023 at 20:36):so my question is suuuuuper basic, so i suspect it will not be useful for difficult technical questions that actually arise in category theory, but i'll write it down
Kevin Arlin (Jul 06 2023 at 20:36):Oh, wait, I take it back. I was thinking about the Day convolution but not every cocontinuous monoidal functor between Day convolutions will come from promonoidal functors between the original promonoidal categories.
Jonathan Beardsley (Jul 06 2023 at 20:44):I was thinking recently about the construction of taking a commutative monoid A and producing from it the special Γ-space (really just a Γ-set), let's call it HA:Γ°→Set_∗, which takes <n> to Aⁿ, and the various maps are built from the multiplication on A, the unit of A, and the projections. Let me just write Γ instead of Γ° from now on, to cut down on the notation. It's known that if A is a commutative monoid then HA is lax monoidal, or in other words, is a monoid in the Day convolution monoidal structure of ΓSet. You can just check this explicitly on HA. But there seems to be something more general going on.
HA is basically determined by what it does on <0>, <1>, <2> and maps between them. That gives you the addition structure and the unit and tells you they play well together. So you might ask about restricting HA to the full subcategory of Γ spanned by <0>, <1> and <2>. This full subcategory, I'll call it Γ₂. Now, Γ₂ is not monoidal, since it's not closed under smash product, but it still has a partially defined monoidal structure that tells me how to construct <1>Λ<1>, <1>Λ<2> and <2>Λ<1>. I'm thinking of this as a promonoidal structure on Γ₂. With respect to this, the inclusion Γ₂→Γ is promonoidal as well. And of course HA:Γ→Set_∗ is promonoidal because it's actually monoidal, and I THINK that HA restricted to Γ₂ will also give a promonoidal functor Γ₂→Set_*.
Another fact is that HA can be constructed from HA restricted to Γ₂ by right Kan extension along Γ₂→Γ. So is the fact that HA is (pro)monoidal recoverable from the fact that HA is constructed by extending a promonoidal functor HA:Γ₂→Set_*, along another promonoidal functor Γ₂→Γ?
Jonathan Beardsley (Jul 06 2023 at 20:46):I think there's really a very general underlying question here which is something like: if I restrict a monoidal functor to a full subcategory of a monoidal category (which I think should be promonoidal) and then Kan extend back again, do I get a (pro)monoidal functor back?
Jonathan Beardsley (Jul 10 2023 at 16:47):alright i think this is probably easier than proving some generic thing about promonoidal right kan extensions. specifically, suppose I've got two categories, one symmetric monoidal and one symmetric promonoidal, say C and D, and a symmetric promonoidal functor (not a profunctor) functor F:C→D. now I can take the category of symmetric (pro)monoidal functors to some symmetric monoidal category E, SymMon(D,E), and ask about the restriction F':SymMon(D,E)→SymMon(C,E). I feel like F' must also be symmetric monoidal (w/r/t Day convolution). then one can ask (i) is it a left adjoint, say right adjoint R? and (ii) if so, for some symmetric promonoidal functor f:C→E is Rf equivalent to what I'd get if I computed the right Kan extension of f just as a functor?
Jonathan Beardsley (Jul 10 2023 at 16:48):And even better, in my case, the functor F is fully faithful