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Todd,
when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?
Thanks!
I’ll get to it when I can, probably sometime later today.
I have removed the first query box and inserted a proof of one of Max Kelly’s lemmas. I’ll get to the other in a bit, the one that says $\lambda_1 = \rho_1$.
That’s great, thank you, Todd!
I’ll have a look as soon as the Lab wakes up again…
Yes, it’s slow, isn’t it? But I managed to stick in the other lemma as well. I’ll finish up by describing what Joyal and Street do (will have to be later today).
Thanks, Todd.
Looking at what you have now, I wonder if the section Definition – Other coherence conditions should not be moved to the Properties-section, where already a stub section “Properties - Coherence” is waiting with a link to coherence theorem for monoidal categories, which in turn linke to Mac Lane’s proof of the coherence theorem for monoidal categories.
Somehow all this would deserve to be put coherently in one place. What do you think? Do you have any plans with this material?
all this would deserve to be put coherently in one place
Heh. Good one.
Anyway, yes, I agree with you. I have to be doing other things now, but if you would like to rearrange the material, please go right ahead. I was mainly trying to take care of Adam’s queries (that have now been removed).
Looking at the two nLab articles you linked to – they could use some more work. “Mac Lane’s proof” is really long and might look scarier to the reader than it actually is. Hopefully I’ll get some time soon to give them a crack.
Okay, if I may, might play with rearranging the material in some way a little later. Thanks.
Added to the References-section at monoidal category right at the beginning a pointer to the pretty comprehensive set of lecture notes:
some super cryptic Anonymous added the following reference which I have rolled back
Anonymous put back again this reference which I have again reverted. The two edits come from Bell Canada in Montreal but they are different IP which means we can’t use IP blocking.
Should we put a note in monoidal category#references telling him to desist and directing him to this thread in the nForum?
Thanks for dealing with this Rod. That might be a reasonable strategy; put it in a query box probably.
He did it a gain so I put in a query box.
Edit. I checked wikipedia monoidal category and he did the same thing there which I also removed.
Looks like he also did the same on August 31 at coherence theorem for monoidal categories and Mac Lane’s proof of the coherence theorem for monoidal categories. I’ve removed the links.
There is something strange now in Mac Lane’s proof of the coherence theorem for monoidal categories. There are many new general sections (apparently not belonging in this entry, but rather in monoidal category) even before the Contents, and the first actual section (“Introduction and statement”) is labeled section 18.
There was a ’>’ right at the start that was mucking things up! I removed it and it looks much better!
Great, thanks!
He back, reinserting his reference into every thing its been removed from. I haven’t reverted them yet.
Do we need to contact Bell Canada and see if they can get him to stop or just block all the Bell Canada Montreal IP addresses?
Maybe we should just precede his references with a query box that says “1337777.OOO is a deranged crackpot that insists on including this reference and every time we remove it he adds it back “
Bah. Can we tell whether anyone legitimate is using those IP addresses?
Mike:
Do you mean, any legitimate nLab user?
At any rate, trying to block this user by IP address is unlikely to be fruitful; I note that their latest re-addition is from yet another IP address, 70.29.194.190. Blocking Bell Canada’s entire IP block (or at least their entire Montreal block) seems quite the sledgehammer.
Is the Instiki config/spam_patterns.txt file used by the nLab wiki? If so, adding things like:
1337777\.OOO
Maclane pentagon is some recursive square
github\.com/1337777
to that list might be another way forward. To get around that, the user would need to change the name they’re using, change the name of their article, and change the name of their GitHub account (or create a new one).
(Also, this user appears to have started ’contributing’ to the Coq-club list: https://sympa.inria.fr/sympa/arc/coq-club/2017-08/msg00048.html.)
Gah, just realised the user also has a GitLab account, so
gitlab\.com/1337777
would be another thing to add to that list.
20: Good idea, I’ve just done that .
Ok I’ve removed 1337777”s references from monoidal category, coherence theorem for monoidal categories, and Mac Lane’s proof of the coherence theorem for monoidal categories.
and also the query box
+-- {: .query} __1337777.OOO__. Stop trying to insert your reference until you have explained and discussed it in the [nForum: monoidal-category](https://nforum.ncatlab.org/discussion/4226/monoidal-category). It is annoying to keep having to remove it. =--
Let’s see if the blocking works and see if 1337777 is determined enough to work around it.
EDIT: I’ve also removed his reference from Wikipedia Monoidal_category, Coherence_theorem, and Coherence_condition.
Thanks everyone!
He’s back on all three pages. DId the spam list changes not propagate to the running code?
He bypassed the spam filter by subtly modifying the offending keywords (underscores, extra slashes, etc.). I think maybe I should just block the keyword 1337777
for a while.
Could the ’Block or report user’ at https://github.com/1337777 be used? There’s an option “Contact Support about this user’s behavior”.
It seems unlikely that there will ever be a legitimate use of 1337777. Reporting him to github seems like a good plan too.
Hm, looking at the user’s change to the ’monoidal category’ page, the spam filter should have still blocked the edit via the patterns for “Maclane pentagon is some recursive square” and “1337777.OOO”. Did the restart of Instiki, so that the new patterns get included, maybe not complete properly?
I just removed a new insertion of the link. The second part of that link makes more sense than the first part, which is just two diagrams, but is also very difficult to read and does not fit in that part of the reference list.
29: if you look at the source,
1337777\.OOO , _Maclane pentagon is some recursive_ _square ...
The backslash makes it a different string than 1337777.OOO, and similarly the _ _
“escapes” the second string. Anyway, let’s see how he gets around this…
Ah, good point, I’d not looked at the page source ….
Yes, will indeed be interested to see if this user is able to work around your latest change. :-)
He’s certainly persistent!
Indeed, rudely so.
The latest readdition gets around the spam filter by using HTML character entities:
1337777.OOO
I’ve taken a look at the Instiki source, and the patterns used in spam_patterns.txt
are actually Ruby regexes. So maybe an entry like this could be used:
(?:1|&#\d{2,4};|&#x\d{2,4};)(?:3|&#\d{2,4};|&#x\d{2,4};)(?:3|&#\d{2,4};|&#x\d{2,4};)(?:7|&#\d{2,4};|&#x\d{2,4};)(?:7|&#\d{2,4};|&#x\d{2,4};)(?:7|&#\d{2,4};|&#x\d{2,4};)(?:7|&#\d{2,4};|&#x\d{2,4};)
Also note that the user seems to be following this thread.
How does wikipedia deal with people like this?
@Mike:
The wiki software used by Wikipedia, MediaWiki, has functionality to deal with this sort of situation. Instiki doesn’t seem to have functionality to e.g. allow the site admin to lock a page against edits until further notice.
I’m surprised if simply locking a few pages temporarily is usually sufficient. But it would certainly be a nice option to have.
Reporting the user to his ISP seems like a reasonable decision to me.
Instiki doesn’t seem to have functionality to e.g. allow the site admin to lock a page against edits until further notice.
It is pretty easy to manually hardcode this though, as I’ve just done.
Oh, you’re right, of course - temporarily locking pages is not necessarily going to be sufficient. It’s a game of Chicken; who’s going to give up first? Still, MediaWiki has finer-grained functionality available:
MediaWiki offers flexibility in creating and defining user groups. For instance, it would be possible to create an arbitrary "ninja" group that can block users and delete pages, and whose edits are hidden by default in the recent changes log. It is also possible to set up a group of "autoconfirmed" users that one becomes a member of after making a certain number of edits and waiting a certain number of days. Some groups that are enabled by default are bureaucrats and sysops. Bureaucrats have power to change other users' rights. Sysops have power over page protection and deletion and the blocking of users from editing. MediaWiki's available controls on editing rights have been deemed sufficient for publishing and maintaining important documents such as a manual of standard operating procedures in a hospital.
Whereas Instiki seems to only provides access control at the ’web’ level, not the ’page’ level.
From experience, I don’t actually have much confidence in large ISPs actively addressing this sort of situation adequately, but I guess it’s worth a go at this point (as might be contacting GitHub and GitLab about this user as well).
@Adeel:
Nice!
1337777.OOO is back, adding links to at least
On it.
I have tightened the spam filter now. I will not say exactly how, as, as mentioned above, the user may be watching this thread. I have not committed the change anywhere, so there is nowhere to find it (unless one has access to the server).
I will now see if I can clear up in the database (update: actually will have to postpone until later; I will also attempt to strengthen the spam filter further, what I’ve done now will probably only stop the posts for a while).
Have now cleared up everything with
1337777.OOO
with the HTML character in the middle, including some older pages. There exist some historical ones (only in revision history I think) without the HTML character in the middle, I’ll try to remove those tomorrow.
I’ve also tightened the spam filter a little further.
Thanks very much for the alert, Rod!
The page Mac+Lane’s+proof+of+the+coherence+theorem+for+monoidal+categories still contains links of similar type at the bottom.
Thanks for raising this, deleted these revisions from September now, and blocked the IP address as well in nginx, because it has been used consistently.
I actually don’t know how the author managed to get those edits through; my attempts to reproduce the spam result in the spam filter blocking the edits correctly. It is possible that I misread the date to be from an earlier year than this year (i.e. it was just something that I forgot to clear up).
Mention earlier in the definition section that a monoidal category is just a category. Also add paragraph about how the definition of the monoidal structure of a category relies on the monoidal structure of the parent 2-category, in accordance with the microcosm principle. See discussion at https://nforum.ncatlab.org/discussion/11003/if-defining-monoidal-category-as-monoid-is-circular-then-sos-our-definition-of-monoidal-category/
This looks good to me, thanks.
Yes, I think you’re right. Why don’t you fix it?
I added a few words; does that help?
Not according to mine.
Added to the Idea section the fact that monoidal categories can be considered as one-object bicategories (and added relevant material to bicategory, under the Examples section).
Given a commuting diagram with an isomorphism, whiskering it with the inverse of that isomorphism gives a commuting diagram of the same shape as before but with that one arrow now pointing in the opposite direction.
Now once the perimeter of that big diagram commutes apply this reversal to the top right morphism. The resulting top right triangle is then seen to commute, and hence so does the original triangle in question.
[ duplicate removed ]
I mean whiskering. But just convince yourself that given a commuting triangle of isomorphisms, there is a corresponding commuting triangle with the direction of any one of the arrows reversed and labeled by the inverse of the original morphism. It’s immediate, by the definition of inverse morphisms.
Added related concepts:
Removed several duplicates and rearranged the list.
Re #72:
It looks correct to me as stated. A monoid internal to other monoidal 2-categories would, in general, no longer be a monoidal category, so the suggested change in the 3rd paragraph of #72 wouldn’t really work. The intention is as picked up in the 4th paragraph, and that is what the entry is saying.
Which is not to say that the wording in the entry could not be improved on.
Hereby moving the following old query box discussion out of the entry to here:
—- begin forwarded discussion —
+–{.query}
Ronnie Brown I entirely understand that most monoidal categories in nature are not strict, and CWM gives an example to show that you cannot even get strictness for the cartesian product. On the other hand, for the cartesian product we get coherence properties directly from the universal property.
Now the tensor product in many monoidal categories in nature comes from the cartesian product, but with more elaborate morphisms. Thus the tensor product of vector spaces comes from bilinear maps. The associativity of this tensor product comes from looking at trilinear maps, and so derives from the associativity of the cartesian product. In a sense, this tensor product is as coherently associative as the cartesian product, which could means that in a rough and ready way we do not need to worry.
My query is whether there is a study of this kind of argument in categorical generality?
Peter LeFanu Lumsdaine: The setting for a statement like this would presumably be the connections between monoidal categories and multicategories, which are discussed very nicely in Chapters 2 and 3 of Tom Leinster’s book. As far as I remember he doesn’t give anything that would quite make this argument, and I don’t know the literature of these well enough to say whether it’s been done elsewhere, but I’d guess it has, or at least that it would be fairly straightforward to give in that terminology. The statement would look something like:
“If $\mathbf{C}$ is a multicategory generated by its nullary, unary and binary arrows, $C$ its underlying category, and $\otimes$, $1$ are functors on $C$ representing the nullary and binary arrows of $C$, then $\otimes$ and $1$ form the tensor and unit of a monoidal structure on $C$.”
The ugly part of this is the generation condition, which will be needed since we only start with $\otimes$ and $1$ (indeed, some stronger presentation condition might be needed, actually). The unbiased version, where we have not just $\otimes$ and $1$ but an $n$-ary tensor product for every $n$, is essentially given in Leinster’s book, iirc, and doesn’t require such a condition.
=–
— end forwarded discussion —
Re #72, #73, I can see why that sounds odd to Sam’s ear.
Note that, in accordance with the microcosm principle, just as defining a monoid in a 1-category requires that the 1-category carry its own monoidal category structure, defining a monoidal category in the 2-category of categories requires that the 2-category carry a monoidal structure as well.
Since just before it says “a monoidal category is a pseudomonoid in the cartesian monoidal 2-category Cat”, how about:
Note that, in accordance with the microcosm principle, just as defining a monoid in a 1-category requires that the 1-category carry its own monoidal category structure, defining a monoidal category as a pseudomonoid in the 2-category of categories requires that this 2-category carry a pseudomonoidal structure as well.
If I were to express this thought I would erase the existing paragraph and start again from scratch, more directly to the point:
Notice how the very definition of monoidal categories above invokes the Cartesian product of categories, namely in the definition of the tensor product in categories. But the operation of forming product categories is itself a (Cartesian) monoidal structure one level higher up in the higher category theory ladder, namely on the ambient 2-category of categories. This state of affairs, where the definition of (higher) algebraic structures uses and requires analogous algebraic structure present on the ambient higher category is a simple instance of the general microcosm principle.
Okay, thanks.
Since all this discussion was sitting inside one humongous Definition-environment, I have now taken it apart into several numbered Definitions and Remarks. Also added more cross-links between such items where they referred to each other, fixed a bunch of links (somebody once did a lot of work on this entry without knowing how to code links in Instiki…).
Also added missing subsection headers. (Previously, the discussion of the 2-category $MonCat$ was sitting in the subcategory for “Strict monoidal categories”…)
In the definition of strict monoidal categories I fixed the wording: Now the ambient $Cat$ is of course regarded as a 1-category, not as a 2-category, unless we are trying to defeat the point laboriously made further above.
Yes! But if we have examples of (pseudo-)monoids in other monoidal 2-categories, then this would be a good point to mention them/link to them.
I’ve fiddled with the wording a little
The ability to define pseudomonoids in any monoidal 2-category is an example of the so-called microcosm principle, where the definition of (higher) algebraic structures uses and requires analogous algebraic structure present on the ambient higher category.
Maybe “necessity” instead of or in addition to “ability”: I think the point being made is that in defining monoidal categories one (secretly, maybe) needs to appeal to monoidal 2-category structure.
I guess the “ability” is implicitly about “sufficiency”, so that with the later “needs” both are covered, but yes there should be better wording. Why “secretly”?
But then we don’t even say this at microcosm principle, which just mentions the sufficiency part:
In higher algebra/higher category theory one can define (generalized) algebraic structures internal to categories which themselves are equipped with certain algebraic structure, in fact with the same kind of algebraic structure. In (Baez-Dolan 97) this has been called the microcosm principle.
So what in fact is the case? Is it both necessary and sufficient that higher structure be in place?
I said “secretly” because the point of this discussion is (as far as I see) that when one looks at the standard definition of monoidal categories, it is typically not made explicit that an ambient 2-categorical monoidal structure is being used, this happens tacitly or secretly in the background. We are adding a remark highlighting this pedantic subtlety.
Regarding sufficiency or necessity: In Lurie’s actual realization of the microcosm principle (here) it is both: algebras over an $\infty$-operad $\mathcal{O}$ are defined internal to $\mathcal{O}$-monoidal $\infty$-categories.
Incidentally, the $(\infty,1)$-categorical formulation resolves what in the original formulation of the principle looks like an infinite regression: To define monoidal categories we don’t actually need the monoidal 2-category $Cat$ but just its $(2,1)$-category core (since the coherence 2-morphisms that it is to supply are all invertible).
Re #85, #86:
It was a wide-spread mistake of old-school higher category theorists to think that to obtain a good theory of $n$-categories one needs to first define $(n+1)$-categories, because, so the logic went, the collection of all $n$-categories is bound to form an $(n+1)$-category which is needed to provide the ambient context for dealing with $n$-categories, notably to discuss their coherence laws.
This perceived infinite regression was arguably one of the reasons why the field of higher category theory was, by and large, stuck and fairly empty, before the revolution.
The error in the above thinking was to miss the fact that coherences only ever take value in invertible higher morphisms, so that a decent theory of $n$-categories is available already inside the $(\infty,1)$-category of $n$-categories.
This insight breaks the impasse: First define $(\infty,1)$-categories all at once, and then find the tower of $(\infty,n)$-categories on that homotopy-theoretic foundation.
The microcosm principle is an archetypical example of the need for this perspective: The coherences (unitor, associator, triangle, pentagon) on a monoidal category are all invertible, hence can be made sense of already inside the $(2,1)$-category of categories, functors, and natural iso-morphisms between them.
Thank you for that chunk of wisdom! I was definitely on track to falling into that way of thinking. In response to #80, I wonder if certain combinatorial species (those closed under product, so not trees, but forests, for example) are monoidal category objects in the monoidal 2-category of combinatorial species, with product given by the “star product” of combinatorial species. I’ll have to think about it a bit more in detail.
On the history lesson, when did the idea of $n$-categories get refined into the idea of $(n,m)$-categories? When I was first casually reading about higher categories, it took a long time before I really encountered the latter being given any serious attention, but that could very well just be an artifact of what I was reading.
Certainly by Lectures on n-Categories and Cohomology, but I think it was much earlier.
Regarding serious attention: This began with the use of $(\infty,n)$-categories by Lurie in the classification of TQFTs and the article on Goodwillie calculus.
I remember the revelation when opening this, having been brought up with the old-school ideas forever “towards an $n$-category of cobordisms” (tac:18-10). Suddenly there was a definition that worked.
The drama of the eventual lifting of the impasse of old-school higher category is also reflected in Voevodsky’s “breakthrough” through his “greatest roadblock” by realizing that (my slight paraphrase): “categories are not higher sets but higher posets; the actual higher sets are groupoids” (here).
This is referring to old-school higher category theory folklore being fond of the fact that “groupoids are just certain categories”. While true, it mislead people into not recognizing that homotopy theory is the foundation of higher category theory, not the other way around. Only when this was turned around and put on its feet did higher category theory start to run.
We’re wondering about such matters in a conversation from 2012 beginning here:
When I was learning about the higher dimensional program from John Baez all those years ago, I took it that n-categories were to be the basic entity. Then n-groupoids were to be thought of a special case of n-categories, particularly useful because homotopy theorists had worked out very powerful theories to deal with the former. The trick was to extend what they’d done, but to an environment with no inverses.
Do you think that what you’re finding here about the difficulty of directed homotopy type theory suggests that in some sense n-groupoids shouldn’t be thought of as a variant of something more basic?
I wonder if we have the points made in #85 and #87 on the nLab anywhere.
Interesting that old quote. Yes, that’s the point.
I have a vague memory of digging out, in a similar conversation years ago, quotes that explicitly make the error mentioned in #87. I am pretty sure where to look for them, but would have to search again. Maybe it’s not worthwhile.
I suppose if A. Joyal had been more into publishing his insights, the drama could have been shortcut by about two decades.
I felt this was all well-understood by now, but it wouldn’t hurt to have an $n$Lab entry on it. I might try to start something later on the weekend.
I was intrigued by the above and for the historical record, I looked back at my letters to Grothendieck from 1983. I pointed out there that Kan complexes were a good model for infinity groupoids and that there were several good candidates for infinity categories. (I do not seem to have explicitly mentioned weak Kan complexes / quasi-categories, but about that time Cordier and I started working on both fibrant SSet-categories and on quasicategories. We did not seem to appreciated the importance of the (infty,1)-idea however.) We had a sketch of the theory of weak Kan complexes to include the analogues of limits and colimits, ends and coends, but never wrote that up, as Jean-Marc felt that the SSet-categories would be more acceptable to both homotopy theorists and category theorists. Our write up of the ends and coends stuff in that latter setting took a lot longer that we had expected due to health issues and excessive teaching loads. We put that SSet-category view forward in the paper Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, but that paper, which had been essentially finished several years earlier, was initally rejected by another journal on the basis that ‘homotopy theorists did not need such a categorical way of looking at homotopy coherence’, or some such wording. It received a good report from the referee for TAMS however.
There were thus people who were looking at what eventually became quasi-category theory at about the same time as Joyal’s lovely approach was being developed, and with the Bangor approach to strict omega categories etc. the idea of doing all dimensions at once was pushed quite firmly. It should be also mentioned that, of course, Ross Street, Dominic Verity , Michael Batanin, and others in Sydney were putting forward a parallel vision at that time; (Edit) see for instance here for the Australian view in 2004. In the category theory conferences of the time there were talks which were more top-down, doing all dimensions at one by concentrating on the coherence questions, as well as those which were approaching the definition from the bottom-up.
I also remember, I think it was Maxim Kontsevich. giving a talk (probably 1992), which used A_infty categories and this was clearly linked in his mind and for many of the category theorists in the audience, to that of ’doing infinity category theory in all dimensions’ albeit for him it was based on a more algebraic dg-cat like structure.
I think the idea that one could do all dimensions at once was therefore well represented in talks during the 1980s and 90s, but some people preferred to be cautious and to try to understand the low dimensional weak categories (bicategories, tricategories, etc) which were combinatorially very tricky, and were therefore avoided by some (I would say that if one uses homotopy coherence and in particular higher operads (which we missed completely in our approach in the 1980s) , the combinatorics becomes more manageable, but can be hard work!)
By the way, the Grothendieck correspondence is due to be published some time next year I think.
Joyal did not just have an “approach” (nor just a “pursuit” “towards” a goal) as many had. He had seen and then worked out the theory, essentially what is now called $(\infty,1)$-category theory.
It wasn’t as widely known as it should have. I remember him opening a talk on quasi-categories in 2007 at the Fields Institute with the words “In this talk I want to convince you that higher category theory exists.” An innocent sounding statement, but somewhat damning to a room full of people supposedly all working on higher categories.
Nowhere in what I wrote was I suggesting that André had not put in a lot of hard work in developing the theory, and I was agreeing with you, Urs, that there were some in the 1980s and 90s who were still trying to do the inductive process. You are remembering 2007, I am remembering 15 to 20 years earlier, so there is no inconsistency between what you are saying and what I wrote. What is disappointing is that after that 24 year period, André still felt he had to justify that higher category theory existed, especially after the Minnesota conference of 2004, where a large number of people had met to discuss the state of the theory, and there were many talks about the various approaches. It was not 100% certain at that time which of the many versions were going to survive the race, nor if they were all equivalent.
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