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Hi Frédéric,
interesting questions. I wish I had more time for this right now (once again…). Hopefully somebody else has. Zoran will know more, once he comes back online.
Added some stuff to this page, including a (very basic) definition of colored PROPs, after work of Donald Yau. Also included one example and talked a bit about the possibility of having enriched PROPs. Linked to dioperad which doesn’t exist yet, but maybe should just redirect to polycategory anyway.
I fixed two things: (1) in a colored prop it shouldn’t just be that the set of objects is the set of lists, but that the monoidal structure is given by concatenation of lists, and (2) every prop defines a polycategory, not vice versa.
Thanks Mike! I guess… haha now I’m actually confused about what “X defines a Y” even means. I was thinking “given a polycategory, there is an underlying PROP.”
Wait, I think I’m just genuinely confused about the mathematics here. What is meant by “every PROP defines a polycategory?” Given an arbitrary PROP we can produce a polycategory, is what you’re saying? Considering that PROPs can compose along lists with multiple objects in them, this would seem to indicate that many PROPs cannot be represented by a polycategory, right?
Well, the problem is that a (representable) polycategory has two tensor products, whereas a prop is with respect to just one. The two tensor products can coincide, so a prop can give rise to a polycategory, but not the other way.
I added the example of bialgebras to prop, which is perhaps a paradigmatic case.
Hm, ok. This is not clear from the entry on polycategories (i.e. what the two tensor products are). Certainly PROPs have two composition products (though one of these becomes the monoidal product in the representation at PROP). I mean, I’m trying to find a good reference on any of this stuff, and not having a ton of success, but are PROPs more general than polycategories, less general, or neither?
So a representable polycategory is essentially the same as a linearly distributive category, which for categorically-minded readers may give the clearest impression of what polycategories really are. (Polycategories were invented I believe by Manfred Szabo, coming from more of a proof theory tradition, where logical sequents $A_1, \ldots, A_m \to B_1, \ldots, B_n$ are classically interpreted as entailments from a conjunction $A_1 \wedge \ldots \wedge A_m$ to a disjunction $B_1 \vee \ldots \vee B_n$.) Linearly distributive categories (q.v.), which have two tensor products related by a strength, are to polycategories as monoidal categories are to multicategories.
Added: perhaps what makes it even clearer for the categorically-minded are the genuine examples of linearly distributive categories based on star-autonomous categories, where the two tensor products are really De Morgan dual to each other. (Linearly distributive categories aren’t much more general than that, because every linearly distributive category embeds fully and faithfully and linearly-distributively into a $\ast$-autonomous category.)
Props are less general than monoidal categories, and monoidal categories are examples of (and considerably less general than) linearly distributive categories, which are representable polycategories.
Added model structure on simplicial PROPs, morphisms of PROPs, and the endomorphism PROP example.
Thanks for all your additions, Jon!
Here is one hint on the coding: it turns out that for anchoring references, it is better to put the anchor name at the beginning as in
* {#HR1} [[Phillip Hackney]] and [[Marcy Robertson]], _On the Category of PROPs_, [arXiv:1207.2773v2](http://arxiv.org/pdf/1207.2773v2.pdf).
instead of at the end, as in
* [[Phillip Hackney]] and [[Marcy Robertson]], _On the Category of PROPs_, [arXiv:1207.2773v2](http://arxiv.org/pdf/1207.2773v2.pdf). {#HR1}
because sometimes with the latter software gets mixed up and mis-identifies the intended link.
Perhaps what was confusing is that when a prop gives rise to a polycategory – that is, when you take a prop and consider its underlying polycategory – there is a lot of the structure that you don’t see any more. As you said, a prop has operations allowing you to compose along any number of objects (including zero, which is the “tensor product”), while when you make a polycategory out of it, you forget all those operations except the ones for composing along exactly one object. And that’s why you can’t go the other way: given a polycategory, you can’t make a prop because you don’t know how to compose along more (or less) than one object.
The definition of prop we have on the page right now is the original “Adams-Mac Lane” one. I think I’ve read that there is another “graphical” one (maybe from here?) that is almost the same, but differs somehow in that the 0-ary-0-coary operations don’t commute as strictly. Does anyone know more about that than me?
Todd, you might know the answer to this: are props conservative over polycategories? That is, do the prop axioms imply any properties of the operations on a polycategory that the polycategory axioms don’t? One way to say that precisely, I guess, is to generate the free prop $F_prop G$ and the free polycategory $F_poly G$ on a “polygraph” $G$ and ask whether the unique functor of polycategories $F_poly G \to F_prop G$ is faithful.
@Urs: thanks for pointing that out! I’m very very new at doing anything more complicated than writing LaTeX on here, so all the tips are appreciated. I also basically just copied and pasted the +-- {: .num_defn }
stuff, so I hope that’s not too messed up. If the nlab didn’t automatically collapse all my revisions you’d see that I made MANY MANY horrible mistakes!
@Mike: okay yeah that makes sense. It seems like it’s sort of when you take a PROP to a category. You just lose a bunch of information?
@Jon, no problem at all, I am glad to see you that trying your hand on it. But if you want to experiment with stuff before changing a given page, you may want to use the Sandbox.
Re #17: I don’t know the answer to this yet. I’d have to think some more on it.
Up in #4 Jon said
Linked to dioperad which doesn’t exist yet, but maybe should just redirect to polycategory anyway.
Is a dioperad literally the same as a one-object (enriched) polycategory? Glancing at the definition I don’t see any obvious difference.
Re #16:
The definition of prop we have on the page right now is the original “Adams-Mac Lane” one. I think I’ve read that there is another “graphical” one (maybe from here?) that is almost the same, but differs somehow in that the 0-ary-0-coary operations don’t commute as strictly. Does anyone know more about that than me?
what they say in the linked paper is
In classical PROP’s there are two a priori different compositions, horizontal and vertical… In particular $A(0,0)$ carries two multiplications which satisfy the middle interchange relation and thus make $A(0,0)$ a commutative monoid by the classical Eckmann-Hilton argument….In graphical PROP’s however… there is no such graph representing vertical composition [$A(n,0) \times A(0,m)\to A(n,m)$]; rather there is only a graph with two connected components which represents horizontal composition. Therefore, in the case of graphical PROP’s $A(0,0)$ carries only one composition (the horizontal) and is thus not necessarily a commutative monoid.
However, at the moment I don’t buy their definition of graphical PROP at all. They don’t actually give a precise definition, but the best I can extract from what they write is that graphical PROPs are the algebras for a polynomial monad of the form
$\mathbb{N}\times \mathbb{N} \leftarrow DirectedLoopFreeGraphs_* \to DirectedLoopFreeGraphs\to \mathbb{N}\times \mathbb{N}.$where $DirectedLoopFreeGraphs$ and $DirectedLoopFreeGraphs_*$ denote respectively the set of isomorphism classes of directed loop-free “graphs” (in their sense) and the set of such isomorphism classes of graphs with one vertex marked. But this doesn’t really make sense to me because directed loop-free graphs can have automorphisms that interchange vertices, so that the operation of “inserting a graph at a vertex into another graph”, which is supposed to define the monad multiplication, doesn’t seem to make sense as an operation on isomorphism classes.
On MO, Donald Yau has pointed out a citation for the statement “dioperads are one-object symmetric polycategories” (attributed there to Tom Leinster), so I’ve added a remark and redirect to polycategory.
It looks like the discussion died out here, but isn’t the page as written currently wrong when it says
Note that PROPs are strictly more general than polycategories since in a PROP we can compose along many objects at once
since this actually means PROPs are less general in that they require a more powerful operation?
Yes, I think you’re right. At least, insofar as it makes sense to say that either one is “more general” than the other, which isn’t clear to me. There’s a right adjoint forgetful functor from props to polycategories, but it’s not full. Would we say that “sets are more general than groups”?
I just got rid of any use of “general” in comparing them. I think the properad article makes the same mistake, but I’m not familiar enough to know off-hand and not interested enough to make sure that I’m right, so I’ll leave it alone.
Incidentally, is there a simple way to characterize the PROPs among the symmetric polycategories? Are they “just” symmetric polycategories that are representable on both sides where $\otimes = \invamp$?
I suppose the way you are comparing coloured PROPs and symmetric polycategories is as structures whose morphisms have sequences of objects (or “elements of a freely generated monoid”) as inputs and outputs. Another way is to see a coloured PROP as a symmetric monoidal category (which “happens” to be strict and have a freely generated monoid of objects), and use the equivalence between generic SMCs and two-sided representable symmetric polycategories with a choice of universal morphisms, such that $\otimes = \parr$ and a bunch of other compatibilities. But the two are very different: going “from PROPs to polycategories”, in the first case you would take only the generating set as the set of objects of the polycategories, where in the second case you would take the entire generated monoid.
In both cases, I don’t think it makes sense to “characterise PROPs among polycategories”: in the first case, as Mike points out, you don’t really have a way of composing morphisms as you would in a PROP; in the second case, “corresponding to a PROP” is the property of a choice of universal morphisms, more than a property of the polycategory itself.
I don’t know what “a property of a choice of universal morphisms means — universal morphisms being unique up to unique isomorphism, any property possessed by one choice of them should also be possessed by any other choice. I would say that “being a prop” is not a property of a polycategory, instead it’s extra structure on a polycategory (additional composition operations).
To clarify, the point I was making is that there are two different functors from PROPs to polycategories:
the one that forgets some composition operations;
the inclusion of PROPs into symmetric monoidal categories, followed by the inclusion of SMCs into linearly distributive categories (as the “degenerate” ones with $\otimes = \parr$), followed by the equivalence between linearly distributive categories and two-sided representable polycategories with a choice of universal morphisms.
In #26 Max seemed to be thinking of the first one, but then mentioned representable polycategories, which are only produced by the second one.
As Mike says, the first functor is not full, so “being a PROP” is best seen as extra structure on a polycategory, in this sense. For the second one, if we restrict to categories of strict monoidal functors, then restrict to functors of representable polycategories that preserve a choice of universal morphisms on the nose (and a specific choice really matters here), we do get a full functor and can speak of the “property” of being a PROP, but at the cost of introducing this extra structure.
Ah, I see. I’m still not sure that second functor is full either, though: when props are considered as (strict) smcs, a prop morphism isn’t just a (strict) sm functor but has to send generating objects to generating objects.
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