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• CommentRowNumber1.
• CommentAuthorvarkor
• CommentTimeMar 25th 2020

Clarified relationship between polycategories, properads and PROPs.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMar 25th 2020

Hmm… I preferred the original text. It might be worth adding a remark that PROPs also allow composition along zero objects at once, giving a “tensoring” operator, but I think that the PROP level of generality is the more natural one and ought to be emphasized over properads.

• CommentRowNumber3.
• CommentAuthorvarkor
• CommentTimeMar 26th 2020
• (edited Mar 26th 2020)

Could you explain why you think PROPs are the more natural notion than properads? To my eyes, polycategories and properads seem the most tightly related concepts, because they’re both natural generalisations of multicategories, depending on whether you generalise the partial or multiple notion of composition. While PROPs also seem like a natural notion, they seem further removed from polycategories. I’m not attached to the wording, and it could be worth having a note about how PROPs are “better” (in whatever sense), but if you think properads should be de-emphasised, it would be really good to have an explanation why (on the page). Thanks :)

(Edit: I agree mentioning the composition over zero objects is a good idea.)

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeMar 26th 2020

Well, as I said, PROPs also generalize the multiple notion of composition in a polycategory; it’s just a question of whether you generalize the set $\{1\}$ to the set $\{1,2,3,\dots\}$ or the set $\{0,1,2,3,\dots\}$. I think the latter is nearly always the more natural generalization, e.g. monoids are more important than semigroups, categories are more important than semicategories, categories with all finite products are more important than categories with only nonempty finite products, connected spaces should generally be considered to be nonempty, etc.

In fact, though, I think PROPs are more natural even than polycategories, so it’s kind of backwards to motivate PROPs by talking about whether they’re closer or further from polycategories anyway. (-:

• CommentRowNumber5.
• CommentAuthorvarkor
• CommentTimeMar 26th 2020
• (edited Mar 26th 2020)

Well, as I said, PROPs also generalize the multiple notion of composition in a polycategory; it’s just a question of whether you generalize the set $\{1\}$ to the set $\{1,2,3,\dots\}$ or the set $\{0,1,2,3,\dots\}$.

Stated in this way, it does seem natural to allow $0$. On the other hand, if you view the composition geometrically, it also seems natural to allow contractions of connected graphs, but not disconnected ones. So, I can see an argument for naturalness of both concepts independently. (I admit to having had little experience working practically with either properads or PROPs, however, so your intuition for what is natural is better than mine! My intuition stems mainly from my understanding of multicategories, in which this isn’t a question that even makes sense.)

In fact, though, I think PROPs are more natural even than polycategories, so it’s kind of backwards to motivate PROPs by talking about whether they’re closer or further from polycategories anyway. (-:

In any other context, starting with PROPs could well be the best place to start. However, in the context of a page on polycategories, the closeness of different concepts to polycategories seems like a relevant metric to use :P I’ve revised the text to make PROPs seem less like a second-class citizen on the page: do you think this is any better? My motivation for the ordering was solely based on the inclusion of concepts $\text{polycategories} \subset \text{properads} \subset \text{PROPs}$. Perhaps it would be better to have a specific page on comparisons between related operadic notions, though: then we avoid giving any one concept precedence.