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Jon Sterling (May 28 2020 at 23:07):This is a naive question, but I am curious. Is there something analogous to classifying topoi for symmetric monoidal theories, in the same sense that I can find a classifying topos for an algebraic theory by considering presheaves on the lex envelope of the theory (or something like that)?
What I am looking for is some kind of a (perhaps generalized) "space" whose (generalized) points are models of a given PROP. If it's a topos, that's even better but I don't know if it is possible to hope for that.
Peter Arndt (May 29 2020 at 02:12):It depends on where you want to consider the models of your PROP: The most general kind of place where it makes sense to talk of a model of a PROP is a symmetric monoidal category. A model is then simply a symmetric monoidal functor from your PROP to that category, and you can not improve anything about your PROP if you want to cover all these examples.
If you are only interested in models living in a better kind of place, like in category whose symmetric monoidal structure is given by finite products, then you can boost up your PROP to become a finite product category (an algebraic theory): There is a forgetful functor from finite product categories to symm. monoidal cats and it has a left 2-adjoint - let's call it FP. This 2-adjunction, for a finite product category C, gives you the Hom-isomorphism
FinProdCat( FP(YourPROP), C) = SymMonCat( YourPROP , C) = Models of your PROP in C.
So you could say that FP(YourPROP) is the classifying finite-product-category of your PROP. (you can imagine that FP(YourPROP) arises by freely adjoining diagonals and projections to YourPROP -- those two things are missing in a general symmetric monoidal category)
If you are only interested in models of your PROP living in a topos with its product monoidal structure, then you can boost this up further, as you wrote yourself, by taking presheaves on the lex envelope of FP(YourPROP). Actually this construction works for a more general kind of model: Taking Lex completion and then presheaves is a left 2-adjoint from finite product cats to the 2-category of cocomplete and finitely complete categories, and colimit and finite limit preserving functors. Toposes also happen to live there and geometric morphisms have such functors as their left adjoint.
The chain of adjunction isomorphisms now looks longer:
CocompleteFinLimCat(Psh(Lex(FP(YourPROP))), C)=FinLimCat(Lex(FP(YourPROP)), C) = FinProdCat( FP(YourPROP), C) = SymMonCat( YourPROP , C) = Models of your PROP in C.
If you want to allow models not only in categories with finite products but with more general symm. monoidal structures, but which are cocomplete, then you can pass to the symmetric monoidal cocompletion - this is the category of presheaves with the Day convolution symm. monoidal structure. So it happens to be a topos, but now you don't place it in the world of toposes and geometric morphisms (and if you place it there it would classify something else, not models of your PROP).
Nice picture, isn't it? So I guess to see your PROPs as something geometric you have to consider models in a world where you know how to see things geometrically (after passage to the opposite category, usually)
Jon Sterling (May 29 2020 at 03:25):Peter, this is a beautiful picture you have painted, thank you so much! I will ponder it some more before following up.
sarahzrf (Jun 03 2020 at 20:06):Peter Arndt said:
If you are only interested in models living in a better kind of place, like in category whose symmetric monoidal structure is given by finite products, then you can boost up your PROP to become a finite product category (an algebraic theory): There is a forgetful functor from finite product categories to symm. monoidal cats and it has a left 2-adjoint - let's call it FP. This 2-adjunction, for a finite product category C, gives you the Hom-isomorphism
FinProdCat( FP(YourPROP), C) = SymMonCat( YourPROP , C) = Models of your PROP in C.
So you could say that FP(YourPROP) is the classifying finite-product-category of your PROP. (you can imagine that FP(YourPROP) arises by freely adjoining diagonals and projections to YourPROP -- those two things are missing in a general symmetric monoidal category)
hey... a little while ago, @Chloe was showing me how to get the spectrum of a ring as a locale directly by forming a quantale of ideals and then turning it into a frame. is there something similar to what you posted for, say, benabou cosmoi?
Morgan Rogers (he/him) (Jun 04 2020 at 09:45):Oh hey what's the multiplication in the quantale of ideals? elementwise multiplication? I need to see what that does for monoids! Can you dm me a reference?
Peter Arndt (Jun 04 2020 at 14:36):sarahzrf said:
hey... a little while ago, Chloe was showing me how to get the spectrum of a ring as a locale directly by forming a quantale of ideals and then turning it into a frame. is there something similar to what you posted for, say, benabou cosmoi?
Are you asking whether there is a 2-left adjoint associating a cosmos to a symmetric monoidal category? (I mean that would be something similar to what I posted) ... I have no idea. I don't even know what are the 'right' morphisms between cosmoi. I also have no idea whether the sym. mon. cocompletion I mentioned at the end gives you a cosmos (it's complete, but is it monoidally closed?).
Peter Arndt (Jun 04 2020 at 14:42):Or were you asking for something like a spectrum construction for those things that the 2-left adjoints spit out?
The one thing I know in this direction are Awodey/Forssell/Breiner's spectra of topoi - here is a nice account of this. To a given topos they associate a sheaf of local toposes on a space, much like you do it for rings. No need for locales there, though, it really is a space.
Peter Arndt (Jun 04 2020 at 14:52):Ah, another thing popping into my mind now is Balmer's spectrum of a tensor-triangulated category. If you have a tt-category, you can form a space whose points are its "thick subcategories" (closed under sums and summands) that are tensor-ideals, with a kind of Zariski topology. You can endow it with a sheaf of rings (basically: to a thick subcategory you associate the endomorphism ring of the tensor unit of the original category localized at that thick subcategory). The amazing theorem is this: If you start with a scheme, take the derived category of coherent O_X-modules and do this process you get back the scheme! Here is Balmer's ICM talk on this. So you can treat tt-categories as if they would come from a scheme.
Peter Arndt (Jun 04 2020 at 15:14):---------------TLDR: ---------------
Here a list of ways of seeing PROPs geometrically:
-- form a topos by first making it into a Lawvere theory and then taking the classifying topos for that.
-- form a cocomplete monoidal category, by passing to the presheaves with Day convolution product. Then see it as a geometric object like Martin Brandenburg does here
-- take the derived category of abelian groups in one of the last two categories; then you have a tt-category and can take its Balmer spectrum
-- if you somehow manage to make your PROP into a symmetric closed monoidal cocomplete category, then you can also see it as an affine 2-scheme like Chirvasitu and Johnson-Freyd do
sarahzrf (Jun 04 2020 at 15:53):@Morgan Rogers yeah, elementwise!
sarahzrf (Jun 04 2020 at 15:54):i don't know of a reference—in fact, i don't know if she knows a reference; i think she was kinda pulling together some stuff that she hadnt herself seen all in one place, cant remember
sarahzrf (Jun 04 2020 at 15:56):but i can tell you that in general, elementwise [binary] operations on sets are (0, 1)-day convolution
sarahzrf (Jun 04 2020 at 16:07):so if you're thinking of the quantale of ideals in a monoid
Chloe (Jun 04 2020 at 16:08):i don't know a reference for it
Chloe (Jun 04 2020 at 16:08):umm
Chloe (Jun 04 2020 at 16:09):i don't wanna claim that it's original per se, I wouldn't be surprised if it's written down somewhere
Chloe (Jun 04 2020 at 16:09):but i came up with the construction independently
sarahzrf (Jun 04 2020 at 16:09):hmm... considering the monoid as a monoidal discrete (0, 1)-category, presheaves w/ day convolution gives the power set of the monoid under elementwise multiplication
sarahzrf (Jun 04 2020 at 16:10):but i dont think it restricts you to ideals—why do ideals pop up in the topos of (1, 1)-presheaves & not here?
sarahzrf (Jun 04 2020 at 16:12):wait, hold on... truth values in PSh(M) for M a monoid are the ideals in M right
sarahzrf (Jun 04 2020 at 16:12):or did i imagine that
sarahzrf (Jun 04 2020 at 16:13):/me sighs and works it out
sarahzrf (Jun 04 2020 at 16:44):ugh i think i might have made a mistake somewhere because what im looking at looks quite odd
sarahzrf (Jun 04 2020 at 16:44):ill just let morgan answer later :upside_down:
sarahzrf (Jun 04 2020 at 16:47):Peter Arndt said:
sarahzrf said:
hey... a little while ago, Chloe was showing me how to get the spectrum of a ring as a locale directly by forming a quantale of ideals and then turning it into a frame. is there something similar to what you posted for, say, benabou cosmoi?
Are you asking whether there is a 2-left adjoint associating a cosmos to a symmetric monoidal category? (I mean that would be something similar to what I posted) ... I have no idea. I don't even know what are the 'right' morphisms between cosmoi. I also have no idea whether the sym. mon. cocompletion I mentioned at the end gives you a cosmos (it's complete, but is it monoidally closed?).
augh, sorry, i started reading your post, was too tired to read the whole thing, skimmed the rest, somehow missed that it actually does look quite relevant to what i meant to ask about, and posted my question
Morgan Rogers (he/him) (Jun 04 2020 at 16:48):sarahzrf said:
ill just let morgan answer later :upside_down:
Or now! Yes, the truth values in presheaves on a monoid are the right ideals in the monoid (equivalently, the sieves on the monoid when it is viewed as a category)
sarahzrf (Jun 04 2020 at 16:49):i was thinking roughly of something like "an adjunction between cosmoi and grothendieck topoi" to match the construction of a locale from a quantale i'd seen, but that does look like it might be in the rest of your post, or something close enough for me to figure out the rest or why i want something different after all
Morgan Rogers (he/him) (Jun 04 2020 at 16:51):But I hadn't noticed that the pointwise operation makes this into a quantale, this is really interesting, especially since the quantale is symmetric iff the monoid is commutative, and I've been looking for criteria in these presheaf toposes which correspond to the underlying monoids being commutative! (@Jens Hemelaer)
Morgan Rogers (he/him) (Jun 04 2020 at 16:52):(or at least, that's what it seems at first glance, maybe the structure collapses somehow)
John Baez (Jun 04 2020 at 16:57):Peter Arndt said:
-- take the derived category of abelian groups in one of the last two categories; then you have a tt-category and can take its Balmer spectrum
"Balmer spectrum"? What a great pun! Balmer was the German high school teacher who found a formula for the frequencies of some kinds of light emitted by hydrogen, which led to a good formula for the spectrum of hydrogen.... and then the Bohr atom!
John Baez (Jun 04 2020 at 16:57):• Balmer series, Wikipedia.
Joe Moeller (Jun 04 2020 at 16:58):I keep forgetting and re-remembering that I met Morgan at TICT, and somehow I'm always reminded when I see him talking about presheaves on monoids :smile:
Peter Arndt (Jun 04 2020 at 17:00):John Baez said:
• Balmer series, Wikipedia.
I know - I love this pun :joy:
Peter Arndt (Jun 04 2020 at 17:01):sarahzrf said:
i was thinking roughly of something like "an adjunction between cosmoi and grothendieck topoi"
Ah, ok, I don't know whether that exists. And I didn't address it anywhere
sarahzrf (Jun 04 2020 at 17:02):o
sarahzrf (Jun 04 2020 at 17:03):well, chloe has now pointed out to me elsewhere that thinking about "cosmoi" instead of "cocontinuous tensor categories" might be wrong since whats a morphism of cosmoi
sarahzrf (Jun 04 2020 at 17:03):and rly i probably do only want cocontinuity here
Peter Arndt (Jun 04 2020 at 17:05):For cocontinuous tensor cats I would guess there is such an adjunction, but I'm not sure. All these 2-adjunctions, usually fall out of a scary paper by Blackwell/Kelly/Power
sarahzrf (Jun 04 2020 at 17:05):sorry for not reading your posts very carefully :sweat_smile: im tired
Peter Arndt (Jun 04 2020 at 17:06):No no! My posts are too long! I have to work on this!
sarahzrf (Jun 04 2020 at 17:06):oh, of course it's from australia
Peter Arndt (Jun 04 2020 at 17:13)::koala: :sos:
Joe Moeller (Jun 04 2020 at 17:18):Some details of 2-monads are given in Johnson and Yau's book. They point to Lack's A 2-categories companion for an overview of some more details as well.
John Baez (Jun 04 2020 at 18:46):Joe is working a lot on cocontinuous tensor categories these days.
John Baez (Jun 04 2020 at 18:47):Well, I'm assuming those are cocomplete symmetric monoidal categories where the tensor product distributes over colimits - correct me if I'm wrong!
Chloe (Jun 04 2020 at 18:55):I like that you say "cocontinuous tensor categories" instead of "cocomplete tensor categories"
John Baez (Jun 04 2020 at 18:57):I said it just because @sarahzrf said it. I actually call these things "symmetric 2-rigs". To me a "2-rig" is cocomplete monoidal category where the tensor product distributes over colimits.
Peter Arndt (Jun 04 2020 at 19:08):Cool, thanks for the pointer to Lack's text! And what book Johnson and Yau is that?
John Baez (Jun 04 2020 at 19:10):Probably this:
Morgan Rogers (he/him) (Jun 05 2020 at 11:17):[Mod] Morgan Rogers said:
But I hadn't noticed that the pointwise operation makes this into a quantale, this is really interesting, especially since the quantale is symmetric iff the monoid is commutative, and I've been looking for criteria in these presheaf toposes which correspond to the underlying monoids being commutative! (Jens Hemelaer)
I've just been playing with this. Annoyingly, the pointwise product doesn't interact well with the canonical action of a monoid on its set of right ideals (by "inverse images"), so it seems like we don't get a nice morphism in the topos of actions in full generality. Oh well..!
sarahzrf (Jun 05 2020 at 23:49):ooh hold on, i think i know why this is...
sarahzrf (Jun 05 2020 at 23:49):in the (0, 1) case i was describing, the monoid becomes a monoidal structure
sarahzrf (Jun 05 2020 at 23:49):but what you're working with is instead turning them into an endomorphism monoid