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One small question that has often occurred to me:
(Technical note: I chose the “Latest Changes” category, even though no change to monoidal bicategory was made yet, because monoidal bicategory appears to not have had a thread of its own yet, and it is not inconceivable that this page will evolve in the future and need a thread)
Off-hand I’m not sure which diagram you mean, but if you could point out the place in say Coherence for Tricategories where this appears, that might help.
I think it wouldn’t be that difficult drawing up the diagrams, and so maybe I can get started on that in the near future. (Much more difficult is tetracategories. (-: )
The diagram I was asking about in Gordon–Power–Street corresponds to the modification called $\mu$ and it is about (diagrams equivalent to) the diagram (which I have seen on several occasions apparently needlessly displayed like that, with the lower identity morphism displayed like that, as an edge)
$\array{ (o_0\otimes I)\otimes o_1 & \overset{a}{\rightarrow} & o_0\otimes(I\otimes o_1) \\ \uparrow & \Downarrow\mu & \downarrow \\ o_0 \otimes o_1 & \overset{\rightarrow}{\mathrm{id}_{o_0\otimes o_1}} & o_0\otimes o_1 }$Remarks.
and the $\rho$ight unitor
$\array{ & (o_0\otimes o_1)\otimes I & \overset{a}{\longrightarrow} & o_0\otimes (o_1\otimes I) & \\ & {\nwarrow} & \overset{\rho}{\Leftarrow} & {\nearrow } & \\ & & o_0\otimes o_1 & & \\ }$in view of which one could be inclined to give $\mu$ (incidentally, is there a usual word for $\mu$? pressing on with mixing English and Greek, one could of course call it the $\mu$iddle unitor…) in the form
$\array{ & (o_0\otimes I)\otimes o_1 & \overset{a}{\longrightarrow} & o_0\otimes (I\otimes o_1) & \\ & \nwarrow & \Downarrow\mu & \swarrow & \\ & & o_0\otimes o_1 & & \\ }$the only local (when it comes to taking the diagram as a mathematical object in its own right and using it for further purposes, well, then this is a different matter) problem I can find with is the aesthetic/psychological one that $\Downarrow$ looks like pointing from a 1-cell to a 0-cell.
So, do you agree that in the basic example of the modification $\mu$, there is no known reason except that otherwise the 2-cell arrow $\Downarrow$ would point to a 0-cell?
(Technical note: apologies for the “incremental posts”; there was a problem with the preview functionality, but I think I now know again how to use Preview and can avoid the incremental edits showing up too early)
I don’t know which texts you are looking at, but there is no harm in writing it just in the way you’d think is sensible:
$\array{ & (x \otimes I)\otimes y & \overset{a}{\longrightarrow} & x \otimes (I \otimes y) & \\ & \searrow & \overset{\mu}{\Rightarrow} & \swarrow & \\ & & x \otimes y & & \\ }$and I know of no reason why one should write it as the square. Who does it like that?
Oh sorry, did you say Garner-Gurski do that? If so, I would ask them why they do that. I wouldn’t do it, and indeed it’s not necessary most of the time, but there may be some reason others have in mind.
Who does it like that?
A surprisingly large (I mean, large in the large-relative-to-the-apparent-needlessness-of-that-convention sense of large) number of sources do this. One example is N. Gurski. An algebraic theory of tricategories, PhD thesis, University of Chicago, online-availabe version dated March 9, 2007, last diagram on page 158 (perhaps more stable reference: ‘antepenultimate diagram in Appendix B.1 of N. Gurski’s thesis’)
Perhaps needless to say, Gurski’s thesis is a wonderfully precise source to work with. Which is rather why I wonder why the diagram is drawn this way in it. I am not sure whether I would like to contact authors who use this draw-$\mu$-differently-than-$\lambda$-and-$\rho$-convention on account of this detail alone.
Concerning #5: A solution to this mystery is probably to be found in the two identities-between-2-cells appearing right before Definition 3.1.3 of “An algebraic theory of tricategories”, together with the somewhat-cryptic (to me) Remark 3.1.7 of the thesis:
” Remark 3.1.7. It should be noted that λ and ρ seem to have a different status than μ. In particular, the reader will note that the cells are not categorified versions of bicategory axioms, but instead categorified versions of useful results about constraint cells in a bicategory. See [22] for a proof of the one-object versions of these bicategorical results and to see how they assist in the proof of coherence for monoidal categories. Thus λ and ρ provide an interesting example of how new data arises in the categorification process. “
It seems the convention of drawing $\mu$ as a square is indeed explained by certain uses the diagram is put to, like guessed in #3 above.
[Technical note: like this post exemplifies, the current software of the nForum can render at least two different Greek fonts.]
What do you find cryptic about Remark 3.1.7?
It’s true that, as that remark says, $\mu$ has a bit of a different status from $\lambda$ and $\rho$, but I don’t think that explains drawing it in a different way. In particular, the coherence axioms just before Def 3.1.3 could equivalently be phrased in terms of a $\mu$ written in the more natural triangular order as Todd suggests, and might even look simpler that way.
I think Nick Gurski would be happy to hear from you and happy to answer this question. And once you do find an answer, I think it would be worth recording on the page tricategory for the benefit of future readers.
You are certainly right that Nick is wonderfully precise.
Yeah, I don’t know. Having played with some of the combinatorics of these things at one point of my life, I am somewhat familiar with the phenomenon of remark 3.1.7. You might be right that his decision is connected with that remark, but it’s opaque to me at the moment, and it’s hard for me to understand how that decision could be about anything essential. I’ll consider emailing him about this.
Thanks for the answers.
I think Nick Gurski would be happy to hear from you and happy to answer this question. And once you do find an answer, I think it would be worth recording on the page tricategory for the benefit of future readers.
I will consider doing so in due course, but it will still take some time and deliberation on my part.
By the way, on a literary note,
[removed by author, considered too distracting by himself]
I think a more appropriate heading in either case would be something simple such as “The shape of the middle unitor”.
In the meantime, before we hear back from Nick (if someone should write him), I’d like to point out that there are principled explanations behind the polyhedral shapes that appear in the traditional description of bicategories, tricategories as in Gordon-Power-Street, tetracategories as written up by me, etc. Some of the underlying formalism was partially written up by me a long time ago (made public on Baez’s website here), but only partially. I’m trying to write up something now which goes into more detail.
In particular, the Stasheff polytopes also known as associahedra famously make their appearance in the traditional descriptions of higher multiplicative data, which is to be expected of course since the associahedra form a universal contractible operad in a suitable sense (cf. the importance of universal contractible globular operads in Batanin’s work, etc.). To be sure, the Stasheff associahedra form a non-unital operad. What we are discussing now are the shapes for higher unital data. In those long-ago days when I was drawing up the infamous tetracategory axioms, I had a set of informal rules for how those unital shapes are generated and how they go in any number of dimensions; specializing to tricategories, the shape for $\mu$ is definitely a triangle. This is not to deny of course that there may be a good reason for having Nick’s $\mu$, but I would venture to say that in view of the principled explanations, that reason is nothing set in stone. My guess is that it’s an expedient convenient for a certain purpose.
In the interests of getting this wrapped up, I’ll write Nick myself.
Thanks for the anwers.
I had a set of informal rules for how those unital shapes are generated and how they go in any number of dimensions; specializing to tricategories, the shape for μ is definitely a triangle
This seems an important point. Then those rules of yours could have a bearing of the perennial problem of “describing the same thing by things of different shapes”. We should maybe not get into this here (what I was getting at with my comment about the new thread is to have an informed, precise and modern thread on what one might call higher model theory, which is more or less the tool of choice when it comes to handling the “when do descriptions have the same models” question, a thread aware of what is routinely handled by available technology and what is not—for example, the group in and around the Oxford computing lab nowadays seems to routinely speak model theoretic language, saying things like ” a signature is a list of allowed moves “, while in Tarskian times, signatures used to mean lists of essentially 0-dimensional marks on paper”, wirth regard to manipulations of string diagrams.)
How do your informal rules make the shape of $\mu$ definitely triangular?
Peter, I’m writing things up; once I’m done I can show everyone an answer to your last question.
Re # 7
What do you find cryptic about Remark 3.1.7?
Nothing in particular, but I’ll try to briefly describe something:
the reader will note that the cells are not categorified versions of bicategory axioms, but instead categorified versions of useful results about constraint cells in a bicategory.
I interpret as
the 3-cells-that-are-the-values-of-applying-the-evaluation-function-pasting-schemes—>3-cells are not -vertically-categorified bicategory-axioms, but vertically-categorified versions of equations which hold-not-by-fiat-but-by-necessity in every bicategory.
EDIT: caveat lector: while it is true that some axioms for higher categories are equations between cells, at this particular point, the author means “cell”, not “equations between cells”; see comment no. 16 below.
Minor quibble, to be gotten out of the way immediately: the author should have written “that the equations between the cells are”, not “that the cells are”, because in that sentence he puts “cells” on the same footing as “result”, but a cell, by itself, can never be a (axiom-derived-mathematical) “result”.
So, actually, I interpret this sentence as
the three equations-between-the-3-cells-that-are-the-values-of-applying-the-evaluation-function-pasting-schemes—>3-cells-to-the-pasting-schemes-which-appear-on-pages-25-and-26-of-the-thesis are not -vertically-categorified bicategory-axioms, but vertically-categorified versions of equations which hold-not-by-fiat-but-by-necessity in every bicategory.
And, well, this I believe (while, strictly speaking, to me, not seeing any patterns like those in the usual renditions of bicategory axioms is not a reason for making the strong claim that the equations “are not categorified versions of bicategory axioms”, but I cannot see $\mu$ playing any special role within this process-of-not-being-something. The author is explaining a special status of something by intimating that it somehow has a special status in some-kind-of-not-being-something, which to me seems rather cryptic.
Then the next sentence
“See [22] for a proof of the one-object versions of these bicategorical results” can be confusing to inexperienced readers (but of course: this is not the intended audience) by mentioning “one-object versions” which makes me think of horizontal categorification (and makes me thus doubt my exegesis given above).
This picking-apart of Remark 3.1.7 is not meant as a criticism of the, again, very useful text; one can find fault with just about any text.
EDIT: And, needless, but perhaps worthwhile to say: one should not expect too much, already for raw information-theoretic reasons, from squeezing out as much as possible from one single paragraph.
It is rather meant to answer to question cited at the beginning of this comment.
the author should have written “that the equations between the cells are”, not “that the cells are”, because in that sentence he puts “cells” on the same footing as “result”, but a cell, by itself, can never be a (axiom-derived-mathematical) “result”.
No, he shouldn’t; he meant what he said. The cells are not themselves “results”, but they are (as he said) categorified versions of results. What he means is that if you write down the three triangles in which $\lambda$, $\rho$, and $\mu$ live but interpret their boundaries as referring to data in a bicategory rather than a tricategory, then the commutativity of the triangle in which $\mu$ lived is one of the axioms of a bicategory, whereas the commutativity of the triangles in which $\lambda$ and $\rho$ lived are not axioms but theorems about a bicategory.
(There is the separate issue that those theorems could have been taken as axioms, at the cost of some slight theoretical redundancy. This would be a nicely uniform way to proceed though as we climb up the $n$-categorical ladder.)
added proper referencing for:
added proper referencing for these two items:
Mikhail Kapranov, Vladimir Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations, in: Proc. Symp. Pure Math. 56 Part 2 (1994), AMS, Providence, pp. 177–260 (doi:10.1090/pspum/056.2)
Mikhail Kapranov, Vladimir Voevodsky, Braided monoidal 2-categories and Manin-Schechtman higher braid groups, Journal of Pure and Applied Algebra Volume 92, Issue 3, 25 March 1994, Pages 241-267 (doi:10.1016/0022-4049(94)90097-3)
added proper referencing for:
added proper referencing for:
(curious how all the monoidal bicategory articles tend to go to Advances)
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