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Stream: theory: category theory

Topic: Polycategories and PROPs; common generalization ?

Alexander Gietelink Oldenziel (Apr 15 2024 at 16:04):

Both PROPs and Poly-categories are gadgets formalizing multi-input multi-output processes.
In PROPs there is extra structure than in poly-categories in that you can compose along multiple objects at the same time.

So... which one (if any) is the more general concept? Is there some common generalization?

One way I could imagine thinking about it is that for a given PROP one can consider the subPolyCategory of morphisms that are composed out of morphisms that are only composed along a single object.

I'm not sure how natural, canonical this is.
How should I think about common generalization of multi-input multi-output processes?

Nathanael Arkor (Apr 15 2024 at 16:35):

PROPs generalise properads which generalise polycategories.

Nathanael Arkor (Apr 15 2024 at 16:35):

In properads you may also compose along multiple objects at once. The difference in a PROP is that you can also compose along zero objects.

Mike Shulman (Apr 15 2024 at 18:20):

I would say that polycategories are more general than PROPs, since every PROP has an underlying polycategory in which you remember only the composition along single objects, but not conversely.

Mike Shulman (Apr 15 2024 at 18:20):

In the same way that, say, abelian groups are more general than rings.

Nathanael Arkor (Apr 15 2024 at 18:58):

Yes, I suppose I was thinking that one has a wider range of operations at one's disposal, but I think "generalise" is probably misleading in this context.

Chris Barrett (Apr 16 2024 at 07:02):

I think of PROPs and polycategories as very roughly analogous to monoidal categories and linearly distributive categories, respectively (the latter being, iirc, the representable polycategories). In this case, linearly distributive categories are more general, because they assume less structure and you can recover monoidal categories by identifying the two "tensor" and "parr" monoidal structures.

Cole Comfort (Apr 16 2024 at 07:46):

Linearly distributive categories actually generalize monoidal categories: by setting both monoidal structures to be equal, then all of the operations of monoidal categories live within linearly distributive categories.

This can not be said for PROPs and polycategories because there is no tensor.

Amar Hadzihasanovic (Apr 16 2024 at 07:55):

I think what Chris said is fine, in the sense that there is a commutative (up to natural isomorphism) diagram of functors
3223f01d-0e85-451d-8019-477fb4c2dd61.jpg

Amar Hadzihasanovic (Apr 16 2024 at 07:58):

(Actually should be symmetric monoidal categories and l.d. categories on top, or pros and planar polycategories on the bottom, sorry for the confusion)

Amar Hadzihasanovic (Apr 16 2024 at 07:59):

The top functor identifies a (symmetric) monoidal category with a "degenerate" (symmetric) linearly distributive category with tensor = par

The bottom functor forgets the composition operations that are not allowed in a (non-planar) polycategory

The left one embeds (symmetric) monoidal categories as the representable pro(p)s

The right one embeds (symmetric) l.d. categories as the representable (non-planar) polycategories

Amar Hadzihasanovic (Apr 16 2024 at 08:03):

I didn't say what the morphisms are in the top row; it should be strong (symmetric) monoidal functors on the top left, and functors that are lax wrt tensor and oplax wrt par on the top right, I think.

Nathanael Arkor (Apr 16 2024 at 08:10):

In what sense are monoidal categories representable PROs? This seems suspicious, because PROs encode more general composition than polycategories, so I would expect that a representable PRO would encode more general tensors than a representable polycategory.

Nathanael Arkor (Apr 16 2024 at 08:11):

A PRO has multiary domain and multiary codomain of morphisms, so a representable PRO should have two tensor products, no?

Nathanael Arkor (Apr 16 2024 at 08:12):

If there is a sense in which representable PROs are monoidal categories, it seems in a different sense to that in which linearly distributive categories are representable polycategories.

Amar Hadzihasanovic (Apr 16 2024 at 08:20):

A universal morphism in a PRO is the same as an invertible morphism, so when a PRO is representable the "two" tensor products coincide up to isomorphism (the inverse of a universal morphism for "tensor" is a universal morphism for "par").

But it is true that we need to adjust the definitions on the two sides if we want to end up with "a single monoidal structure" instead of "a pair of naturally isomorphic monoidal structures".

Amar Hadzihasanovic (Apr 16 2024 at 08:21):

The coincidence of universal with invertible in a PRO follows from the same argument as the proof that an isomorphism in a category is the same as a morphism that is split epi and mono.

Nathanael Arkor (Apr 16 2024 at 08:51):

By "universal morphism", you mean a morphism $A, B \to A \otimes B$ such that $\mathcal C(X_1, \ldots, A \otimes B, \ldots, X_m; Y_1, \ldots Y_m) \to \mathcal C(X_1, \ldots, A, B, \ldots, X_m; Y_1, \ldots Y_m)$ is invertible, or something else?

Amar Hadzihasanovic (Apr 16 2024 at 09:08):

Yes, that's what I mean. In a pro it also makes no difference if you include the "context" $X_1, \ldots, X_m$ or not.

(In a polycategory your definition would correspond to what is called "strongly universal", which is the notion of universality that is stable under composition).

Amar Hadzihasanovic (Apr 16 2024 at 09:12):

In a pro you have "binary" and "n-ary" identities, so the inverse image of $\mathrm{id}_{A,B} \in \mathcal{C}(A, B; A, B)$, which is a morphism $A \otimes B \to A, B$, turns out to be an inverse for your universal morphism $A, B \to A \otimes B$.

Nathanael Arkor (Apr 16 2024 at 09:25):

Right, I see. The way I should have been thinking about it is that, by having more general composition operations, the constraints on the tensor products are stronger, and so the more general the composition, the closer one gets to trivialising the induced monoidal structure(s).

Nathanael Arkor (Apr 16 2024 at 09:26):

Amar Hadzihasanovic said:

Yes, that's what I mean. In a pro it also makes no difference if you include the "context" $X_1, \ldots, X_m$ or not.

Are these observations written out anywhere?

Nathanael Arkor (Apr 16 2024 at 09:26):

It seems like it would be useful to have a reference for representability of these different structures somewhere. (Probably this has even been mentioned on Zulip before.)

Amar Hadzihasanovic (Apr 16 2024 at 09:27):

@Aaron David Fairbanks is working on it.

Mike Shulman (Apr 16 2024 at 16:53):

Representability of PROPs was mentioned on Zulip recently and someone cited my paper A practical type theory for symmetric monoidal categories. I claim no originality, but at least it's written out there with a sketch of a proof that representable PROPs are equivalent to symmetric monoidal categories.

Nathanael Arkor (Apr 16 2024 at 19:54):

One thing that it would be nice to have a reference for in the literature is the fact that representability in the sense of Definition 2.2 of "A practical type theory for symmetric monoidal categories" does coincide with representability in the polycategorical sense, i.e. a notion of strong universality.

Mike Shulman (Apr 16 2024 at 20:15):

Yes. Although it's a 1-line proof by the Yoneda lemma.