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Cole Comfort (Jan 19 2023 at 13:34):Does anyone have a reference for the notion of concategory mentioned in the following slides:
https://www.cl.cam.ac.uk/events/syco/2/slides/levy.pdf
The basic idea is that a concategory is like a polycategory where instead of cutting along a single object; one cuts along the whole list of objects at once or cuts them together by tensoring the domain and codomain lists together. The notion of morphism act pointwise like in the case of polycategories, which distinguishes them from strict monoidal categories and functors.
I suspect that this can be formalized in terms of a generalized polycategory arizing from something like the doubly iterated finite list monad, but like I said the literature is rather sparse.
Nathanael Arkor (Jan 19 2023 at 14:05):The reason the literature appears sparse is because what Levy calls "concategories" are usually called (coloured/multisorted) [[PROs]] in the literature. However, Levy is making a conceptual distinction between the traditional notion of PRO (as a kind of theory / freely generated monoidal category), and a polycategory-like structure (analogous to the conceptual distinction between an operad and a multicategory).
Cole Comfort (Jan 19 2023 at 14:24):The problem I am running into is trying to define a lax normal functor from a prop to a "2-prop" (displayed props). But this functor doesn't strictly preserve identities on tensors of generating objects, only on the generating objects themselves.
Maybe I am just missing a subtlety in the definition of a morphism of props. It is not just a strict monoidal functor, but moreover it has to send n fold tensors of generators in the first category to n fold tensors of corresponding generators in the second category. So a morphism of props can be defined as a function between colours as function between homs in the obvious way.
So perhaps the appropriate notion of lax normal functor between 2-props should be more like the corresponding notion for polycategories, as opposed to that for monoidal categories, where one doesn't require that the tensors of identities are sent to tensors of identities (because it is not even a well formed thing to ask for in the polycategory case), only that the identies of generating colours are sent to identities of generating colours.
Cole Comfort (Jan 19 2023 at 14:32):I guess I should just try to work out what the Grothendieck-Benabou correspondence is for PRO(P)s.
Nathanael Arkor (Jan 19 2023 at 15:04):By "2-PROP", do you mean a 2-categorical analogue of a PROP? (I haven't come across the terminology "displayed PROP" before.)
Cole Comfort (Jan 19 2023 at 15:17):Yes, by a (multicoloured) 2-PROP, I mean what a symmetric monoidal bicategory is to a symmetric monoidal category. And a displayed PROP would be a lax normal functor from a prop into the strictification of categories and profunctors seen as a 2-PROP with respect to the cartesian product. And by the Grothendieck-Benabou correspondence I mean the equivalence between the lax functors category out of an indexing PROP into (considered as a PROP) and the slice category of PROPs over (all of this stuff preserving the PROP structure of course)
Cole Comfort (Jan 19 2023 at 15:20):I think that this is the correct setting to make this correspondence give you the stricitification theorems for monoidal and symmetric monoidal categories. Trying to write a paper on this, but these details are insanely tricky.
Mike Shulman (Jan 19 2023 at 18:14):Nathanael Arkor said:
However, Levy is making a conceptual distinction between the traditional notion of PRO (as a kind of theory / freely generated monoidal category), and a polycategory-like structure (analogous to the conceptual distinction between an operad and a multicategory).
There's a conceptual distinction between an operad and a multicategory?
Nathanael Arkor (Jan 19 2023 at 18:40):I suppose it depends on whom you're asking. The terminology "coloured operad" and "multicategory" refer to the same definition, but have generally been traditionally used in somewhat different contexts, and, as I see it, the "PRO" terminology fits more within the same context as "operad", and Levy's "concategory" terminology fits more within the same context as "multicategory". Perhaps I should have said "historical conceptual distinction": one could very well use the same terminology with each of the different perspectives.
Mike Shulman (Jan 19 2023 at 19:07):I wish that were true, but nowadays it seems to me that lots of people use "operad" in contexts where "multicategory" would make more sense.
Nicolas Blanco (Jan 19 2023 at 19:10):I haven't look in details at these, but afair concategories also appeared in Ross Duncan's PhD thesis under the terminology compact polycategories.
Cole Comfort (Jan 19 2023 at 20:13):Nicolas Blanco said:
I haven't look in details at these, but afair concategories also appeared in Ross Duncan's PhD thesis under the terminology compact polycategories.
Thank you! I have read his thesis before, but reading it now, I realize he only requires that the identity be preserved on every object, not on lists of objects, this is exactly the definition I need.
Nicolas Blanco (Jan 19 2023 at 20:47):I haven't checked the details so just some rough ideas based on intuition: a representable concategory should be a compact closed category, similarly a representable 2-concategory should be a compact closed bicategory (the notion has been worked out by Mike Stay). Since Prof is a compact closed bicategory (see Mike Stay's paper) you have a 2-concat Prof. Now the only difference between a concat and a regular polycat is the way you compose polymaps. So the Bénabou-Grothendieck correspondence should work in this setting too.
Nathanael Arkor (Jan 19 2023 at 20:55):Nicolas Blanco said:
I haven't look in details at these, but afair concategories also appeared in Ross Duncan's PhD thesis under the terminology compact polycategories.
My understanding was that Duncan's "compact polycategories" were the same as properads (called "pluricategories" by Kavanagh)?
Cole Comfort (Jan 19 2023 at 21:22):Nicolas Blanco said:
I haven't checked the details so just some rough ideas based on intuition: a representable concategory should be a compact closed category, similarly a representable 2-concategory should be a compact closed bicategory (the notion has been worked out by Mike Stay). Since Prof is a compact closed bicategory (see Mike Stay's paper) you have a 2-concat Prof. Now the only difference between a concat and a regular polycat is the way you compose polymaps. So the Bénabou-Grothendieck correspondence should work in this setting too.
Yes this seems to be the case. Can you think of any non representable concategories? The distinction of morphisms between concategories and the similarity to polycategories also begs the question if concategories arise as the generalized polycategories for a distributive law of monads
Cole Comfort (Jan 19 2023 at 21:33):Nathanael Arkor said:
Nicolas Blanco said:
I haven't look in details at these, but afair concategories also appeared in Ross Duncan's PhD thesis under the terminology compact polycategories.
My understanding was that Duncan's "compact polycategories" were the same as properads (called "pluricategories" by Kavanagh)?
The difference appears to be that the data of pluricategories includes an identity morphism for each list of objects, however Duncan only asks for an identity for every object. Although you can get an identity for each list by tensoring things together, it need be preserved by the appropriate morphisms between pluricategories.
Cole Comfort (Jan 19 2023 at 21:36):And this is a crucial difference if you are concerned with displayed categories because the normal morphisms are totally different.
Nathanael Arkor (Jan 19 2023 at 21:43):I hadn't noticed that Kavanagh's pluricategories require an identity for each list. That seems an unusual requirement.
Amar Hadzihasanovic (Jan 20 2023 at 16:55):Concategories also have cut along the empty list of objects, aka parallel composition (they are seemingly what I'd call coloured PROs).
Properads, on the other hand, have cut along non-empty lists (only connected diagrams compose).
Presumably pluricategories are like that too, if the nlab page is to be trusted.
Which one are you after?
Amar Hadzihasanovic (Jan 20 2023 at 16:57):The distinction between units on lists and units on items is, I think, only relevant in the second case, because you cannot parallel-compose single-item units in a properad.
But you can in a concategory.
Amar Hadzihasanovic (Jan 20 2023 at 17:02):Btw my paper "Weak units, universal cells, and coherence via universality for bicategories" is, mainly, about how to get units via representability in a non-unital (and many-object) version of properads where only non-empty lists are allowed as sources and targets.
Amar Hadzihasanovic (Jan 20 2023 at 17:02):Perhaps there is something there that you can use.
Cole Comfort (Jan 20 2023 at 17:45):Amar Hadzihasanovic said:
Perhaps there is something there that you can use.
Yes this is a lot of nice stuff to work with, and these distinctions are very nuanced. But although you can get units for lists by cutting together atomic units along the empty list, the natural notion of structure preserving morphism wouldn't be required to preserve them. And this is precisely the structure which I need.
Cole Comfort (Jan 20 2023 at 17:47):unless this somehow is a consequence of the other axioms
Amar Hadzihasanovic (Jan 20 2023 at 18:45):Don't morphisms preserve parallel composition/cuts along the empty list? The cut-together units will "act" like units, so they are the unique units of that type; so if morphisms preserve both single-item units and parallel composition, they will also send the cut-together units to cut-together units, no?
Cole Comfort (Jan 21 2023 at 20:58):Amar Hadzihasanovic said:
Don't morphisms preserve parallel composition/cuts along the empty list? The cut-together units will "act" like units, so they are the unique units of that type; so if morphisms preserve both single-item units and parallel composition, they will also send the cut-together units to cut-together units, no?
I think I see what is going on in my attempt to work out the Grothendieck-Benabou correspondence in this setting. Although (the aforementioned structure preserving) functors between concategories preserve the induced identity on lists by the argument you sketched, if you switch to thinking about lax normal functors between 2-concategories, then normality on atomic objects does not seem to imply normality on lists of objects. Because cutting along the empty list is only preserved up to natural transformation, and not natural isomorphism.
This is a very strange thing that given two structures, the categories generated by their structure preserving morphisms are the same, but if you categorify them then they diverge.
James Deikun (Jan 22 2023 at 02:18):I feel like lax normal functors are not actually the direction to look in here. They occur in the classic displayed category correspondence because Prof is a virtual equipment and lax normal functors are the natural morphisms of virtual equipments (at least, when they happen to have composition). Because concategories are not "many-to-one" like virtual equipments you'll probably get a more natural correspondence if you look for a type of morphism that is natural in a many-to-many setting, with a good look at the data you need on the other side of the correspondence as your guide.
James Deikun (Jan 22 2023 at 02:40):In particular, look at the preimage of a particular object in a Hom under a functor of concategories and what algebraic structure connects it with the preimages of its inputs and outputs. These will be your "profunctors", more or less. But also look at the fact that you need data for both serial and parallel composition, the parallel composition will significantly complicate things compared to "lax normal functors".