Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
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.)
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 to the set or the set . 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. (-:
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 to the set or the set .
Stated in this way, it does seem natural to allow . 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 . 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.
Clarify that poly cats are for classical linear logic
Why classical? Isn’t it the case that you need duals (i.e. *-polycategories) for classical linear logic?
1 to 9 of 9