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Created universal algebra in a monoidal category
In the lab book metaphor, this page is some jottings of stuff that I'm pretty sure must be out there (as it's a fairly obvious thing to do) but have no idea of what it's called (hedgehogs, perhaps?). So I'd be grateful if someone strong in the ways of Lawvere theories could stop by and help me out.
(Plus I had to make up the notation and terminology as I went along so that's all horrible)
Hopefully the big box at the top of the page makes this clear!
Of course! A "PRO", now why didn't I think of that?
Seriously, thanks! I look forward to reading what you write.
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<p>A "PRO", now why didn't I think of that?</p>
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<p>Indeed. One of the good examples for bad terminology, I'd say. Unfortunately.</p>
Replied to Toby's question: In short, yes to the merge, but I don't feel that I have the expertise to do it.
Responded to the commentary with hopefully something a little clearer (be patient with me! Whilst I know quite well the little bit I'm trying to do, the general set-up is unfamiliar so I'll probably go on being unclear for a while). Forgot to say "thank you" for the answer so shall do so now: thank you.
somebody please add this reference to a relevant nLab entry
It's already at distributive law.
PS @ Zoran: Pick the ‘Markdown’ button below the comment field to make your link work as you intended.
But it should probably be also in an entry on PRO(P)s
OK, now it's in PROP.
Reply to Todd at universal algebra in a monoidal category
I'm also conversing there.
@ Urs: Todd added the reference there too!
More clarification at universal algebra in a monoidal category.
Zoran, I think that was a hang-over from when I didn't know what a PRO was. Now that I know, that page should be merged with PRO and the terminology corrected.
(Incidentally, what's the convention: do we keep mentioning that we've replied over there? Or do we assume that everyone who's interested is keeping an eye on that page? I guess an edit brings it to the top of the general n-lab RSS feed. I'm happy with either convention, but I have a mild preference for keeping on mentioning it here as I'm hoping that Gavin Wraith will join in)
it can't hurt to mention all edits here
A nonstrict monoidal category is indeed a pseudo-algebra for the PRO of monoids in Cat. Actually, I would regard PROs as unnecessary complication, since operads or monads suffice, but they have the same algebras and pseudo algebras.
There is one issue, though: a nonstrict monoidal category is usually defined in a "biased" way (having only binary and nullary tensor products), while a pseudo-algebra for the PRO of monoids in Cat is an "unbiased" monoidal category, having one n-ary tensor product for every n. The two are equivalent (and often in practice we speak as if our monoidal categories were unbiased anyway), but the equivalence of the two basically requires all of Mac Lane's coherence theorem for (biased) monoidal categories.
The only reference I have for this notion is a brief private conversation I had once with Martin Hyland; I don't recall if I asked or if he told me why. But I was hoping to construct a little story about it anyway, by calculating something like where is the ring of continuous functions on a space . I'll bet one can see some classical measures crop up there. In passing, I think I recall observing once that distributions supported at a point admit a coalgebra structure. All this stuff deserves to be recorded at the Lab.
Did you maybe accidentally switch the text filter (seen below the edit window)? At least for "Markdown" text filter it should work.
I didn't even know that we have text filters here, until I accidentally changed it by hitting some keys.
Ah, that was it -- thanks.
Hopefully now it's clear what I'm trying to do. Essentially, Toby and Todd's attempts to understand were correct and it was just my poor explanations getting in the way.
Yes, I believe it is more or less clear Andrew, and it's got me thinking about some things coalgebraic which I mean to set down soon. There's probably a nice story to be told.
Excellent! I look forward to reading it. If I'm allowed to make a request, can you put in references where you know them?
A lot of this stuff (or at least, the part relating to the bits I'm interested in) seems scattered across the literature with no particular thread to help find it. My coauthor and I would often argue something through only to find that some obscure paper from 50 years ago has the essential details, but we only found it almost by accident! Often, it would be some side remark that you made on the cafe!
Thanks!
@David (and potentially Andrew): apparently what I am calling "measure coalgebra" is better know as "universal measuring coalgebra", and is discussed in chapter 7 of Sweedler's book Hopf Algebras. I don't have access to this book however.
I plan to write a little more about that construction later. If someone has access to that book, then I might want to compare notes at some point.
On the other hand it does not look complicated.
Continuation of discussion in query box over at universal algebra in a monoidal category, just putting down some things that Andrew no doubt observed already to himself. Reference to paper by Barr.
Answer to Andrew's question 1 at universal algebra in a monoidal category, in the negative.
Returning to this, I now think that what I actually want is an operad but possibly without the symmetric group action (I need to check whether or not I truly have a symmetric monoidal structure). I hadn't realised the significance of the number of outputs: when the product is the cartesian product then, of course, a morphism into a product is a combination of morphisms into the single base object. I have a nice correspondence between the monoidal product in my category and the cartesian product on the underlying sets, but that works for morphisms out and I don't think it works for morphisms in. Therefore I need to ensure that I have only one output for my (generalised) operations.
Talk about going the long way round!
Incidentally, is there an operad version of a PRO? That is, without the symmetric group action?
D'oh. Follow the links, Luke: multicategory claims that what I'm after is a Stasheff operad.
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