I’ll see — maybe if the name of the spammer is not too outrageous to start with (because I cannot change that short of deleting the whole message, I think).

]]>opening this thread took me to comment #31. If the past repeats I will be taken there even when more new comments are added.

yup. taken to #31 again.

I had to delete one spam comment posted shortly after #35.

Could you just edit the spam post and blank its content?

]]>I had to delete one spam comment posted shortly after #35.

Deleting spam seems to have these effects.

]]>Why did this entry appear near the top of the list of recent changes and then disappear? Did some nLab wizard delete it from the log? For what reason?

I’m posting this to see if that action screws up the database of what nForum posts I have seen (as has happened in the past) and and when I click to view this comment that history is corrupted and I get taken to some much prior post.

EDIT: opening this thread took me to comment #31. If the past repeats I will be taken there even when more new comments are added.

]]>also touched the wording of the Idea-section, for streamlining

]]>brought the list of references into chronological order, for it to be less odd (there is still much room left to improve on it)

also added pointer to:

- Sinan Yalin,
*Classifying spaces and moduli spaces of algebras over a prop*, Algebr. Geom. Topol.**14**(2014) 2561-2593 [arXiv:1207.2964v1, doi:10.2140/agt.2014.14.2561]

Updated references with DOIs and links to published versions.

]]>added note at the top regarding that PROP is different from Prop.

Anonymous

]]>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.

]]>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.

]]>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).

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 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 $\otimes = \invamp$?

]]>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”?

]]>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?

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.

]]>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.

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 #17: I don’t know the answer to this yet. I’d have to think some more on it.

]]>@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*.

@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?

]]>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.

]]>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?

]]>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.

]]>