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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2012
    • (edited Jan 16th 2013)

    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!

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 11th 2012

    I’ll get to it when I can, probably sometime later today.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 11th 2012

    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 λ 1=ρ 1\lambda_1 = \rho_1.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2012

    That’s great, thank you, Todd!

    I’ll have a look as soon as the Lab wakes up again…

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 11th 2012

    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).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2012
    • (edited Sep 11th 2012)

    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?

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 11th 2012

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2012

    Okay, if I may, might play with rearranging the material in some way a little later. Thanks.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2013

    Added to the References-section at monoidal category right at the beginning a pointer to the pretty comprehensive set of lecture notes:

    • CommentRowNumber10.
    • CommentAuthorRodMcGuire
    • CommentTimeAug 31st 2017

    some super cryptic Anonymous added the following reference which I have rolled back

    • CommentRowNumber11.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 1st 2017

    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?

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeSep 1st 2017

    Thanks for dealing with this Rod. That might be a reasonable strategy; put it in a query box probably.

    • CommentRowNumber13.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 1st 2017
    • (edited Sep 1st 2017)

    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.

  1. 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.

    • CommentRowNumber15.
    • CommentAuthorYaron
    • CommentTimeSep 2nd 2017

    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.

    • CommentRowNumber16.
    • CommentAuthorTim_Porter
    • CommentTimeSep 2nd 2017
    • (edited Sep 2nd 2017)

    There was a ’>’ right at the start that was mucking things up! I removed it and it looks much better!

    • CommentRowNumber17.
    • CommentAuthorYaron
    • CommentTimeSep 2nd 2017

    Great, thanks!

    • CommentRowNumber18.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 2nd 2017
    • (edited Sep 2nd 2017)

    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 “

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeSep 3rd 2017

    Bah. Can we tell whether anyone legitimate is using those IP addresses?

    • CommentRowNumber20.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 3rd 2017

    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.)

    • CommentRowNumber21.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 3rd 2017

    Gah, just realised the user also has a GitLab account, so

    gitlab\.com/1337777
    

    would be another thing to add to that list.

    • CommentRowNumber22.
    • CommentAuthoradeelkh
    • CommentTimeSep 3rd 2017
    • (edited Sep 3rd 2017)

    20: Good idea, I’ve just done that .

    • CommentRowNumber23.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 3rd 2017
    • (edited Sep 3rd 2017)

    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.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeSep 3rd 2017

    Thanks everyone!

    • CommentRowNumber25.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 4th 2017

    He’s back on all three pages. DId the spam list changes not propagate to the running code?

    • CommentRowNumber26.
    • CommentAuthoradeelkh
    • CommentTimeSep 4th 2017

    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.

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 4th 2017

    Could the ’Block or report user’ at https://github.com/1337777 be used? There’s an option “Contact Support about this user’s behavior”.

    • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeSep 4th 2017

    It seems unlikely that there will ever be a legitimate use of 1337777. Reporting him to github seems like a good plan too.

    • CommentRowNumber29.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 5th 2017

    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?

    • CommentRowNumber30.
    • CommentAuthorTim_Porter
    • CommentTimeSep 5th 2017
    • (edited Sep 5th 2017)

    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.

    • CommentRowNumber31.
    • CommentAuthoradeelkh
    • CommentTimeSep 5th 2017

    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

    • CommentRowNumber32.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 5th 2017

    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. :-)

    • CommentRowNumber33.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2017

    He’s certainly persistent!

    • CommentRowNumber34.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 7th 2017

    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.

    • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2017

    How does wikipedia deal with people like this?

    • CommentRowNumber36.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 7th 2017
    Why not simply report this user to Bell Canada and allow them to deal with it?

    Spam is certainly covered by their policies.

    Their whois information states:

    Comment: For abuse cases please use abuse@sympatico.ca
    • CommentRowNumber37.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 7th 2017

    @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.

    • CommentRowNumber38.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2017

    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.

    • CommentRowNumber39.
    • CommentAuthoradeelkh
    • CommentTimeSep 8th 2017

    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.

    • CommentRowNumber40.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 8th 2017

    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).

    • CommentRowNumber41.
    • CommentAuthorAlexisHazell
    • CommentTimeSep 8th 2017

    @Adeel:

    Nice!

    • CommentRowNumber42.
    • CommentAuthorRodMcGuire
    • CommentTimeMar 9th 2018

    1337777.OOO is back, adding links to at least

  2. On it.

    • CommentRowNumber44.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 9th 2018
    • (edited Mar 9th 2018)

    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).

    • CommentRowNumber45.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 9th 2018
    • (edited Mar 9th 2018)

    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).

    • CommentRowNumber46.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 9th 2018
    • (edited Mar 9th 2018)

    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!

    • CommentRowNumber47.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 23rd 2018
    • (edited Dec 23rd 2018)

    The page Mac+Lane’s+proof+of+the+coherence+theorem+for+monoidal+categories still contains links of similar type at the bottom.

  3. 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).

  4. Number the last natural isomorphism in this definition.

    relrod

    diff, v114, current

  5. It seems to me that the facets of the 4-simplex oriental are the edges, not the vertices, of the pentagon identity.

    DavidMJC

    diff, v115, current

    • CommentRowNumber51.
    • CommentAuthorziggurism
    • CommentTimeMar 13th 2020

    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/

    diff, v119, current

    • CommentRowNumber52.
    • CommentAuthorziggurism
    • CommentTimeMar 13th 2020

    latex fix

    diff, v119, current

    • CommentRowNumber53.
    • CommentAuthorMike Shulman
    • CommentTimeMar 13th 2020

    This looks good to me, thanks.

    • CommentRowNumber54.
    • CommentAuthorGuest
    • CommentTimeMar 28th 2020
    Is there a typo in the expression just before the text "is the cartesian associator"?
    It seems to me like the parens are misplaced on one of the terms.
    • CommentRowNumber55.
    • CommentAuthorMike Shulman
    • CommentTimeMar 28th 2020

    Yes, I think you’re right. Why don’t you fix it?

  6. Hi,
    In enriched category, section "2 -Definition", it is written that "one may think of a monoidal category as a bicategory with a single object".
    Is it possible to add a sentence about this in this page ?
    • CommentRowNumber57.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 24th 2020

    I added a few words; does that help?

    • CommentRowNumber58.
    • CommentAuthorluidnel.maignan
    • CommentTimeApr 25th 2020
    • (edited Apr 25th 2020)
    Your revision did not make it apparently.
    The last revision was March 30, 2020 according to the history page.
    • CommentRowNumber59.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 25th 2020
    • (edited Apr 25th 2020)

    Not according to mine.

  7. Sorry, I was expecting a change in the monoidal category page.
    • CommentRowNumber61.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 26th 2020

    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).

    diff, v122, current

  8. I checked both edits, they are great! Thank you :)
    • CommentRowNumber63.
    • CommentAuthorGuest
    • CommentTimeSep 8th 2020
    Can someone please help me understand why this holds "Since all the arrows are isomorphisms, it suffices to show that the diagram formed by the perimeter commutes" .As a newcomer to category theory this is not at all obvious to me.
    • CommentRowNumber64.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2020

    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.

    • CommentRowNumber65.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2020
    • (edited Sep 8th 2020)

    [ duplicate removed ]

    • CommentRowNumber66.
    • CommentAuthorGuest
    • CommentTimeSep 8th 2020
    What do you mean by whiskering?
    • CommentRowNumber67.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2020

    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.

    • CommentRowNumber68.
    • CommentAuthorJohn Baez
    • CommentTimeNov 16th 2020

    Added: every monoidal category is equivalent to a skeletal strict monoidal one. (FinSet, ×\times) is, but (Set, ×\times) is not.

    diff, v126, current

    • CommentRowNumber69.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 18th 2021

    Added related concepts:

    Removed several duplicates and rearranged the list.

    diff, v129, current

  9. add link to page dedicated to coherence theorem

    Antonin Delpeuch

    diff, v133, current

    • CommentRowNumber71.
    • CommentAuthorvarkor
    • CommentTimeNov 7th 2021

    Corrected reference to Mac Lane’s terminology.

    diff, v135, current

    • CommentRowNumber72.
    • CommentAuthorsamwinnick
    • CommentTimeNov 7th 2021
    I think there is a mistake in the wording here:

    "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 structure, defining a monoidal category in the 2-category of categories requires that the 2-category carry a monoidal structure as well. In this case we are implicitly employing the cartesian monoidal structure on Cat, so"...

    It should read ..."defining a monoidal category in a 2-category requires"... instead of ..."defining a monoidal category in the 2-category of categories requires"...

    The rest I think makes sense since in this article one is defining monoidal categories not internal to some exotic 2-category but in the 2-category Cat, which uses the cartesian monoidal structure of Cat.

    I don't want to change it myself because I am just learning about this stuff for the first time.

    -Sam Winnick
    • CommentRowNumber73.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2021
    • (edited Nov 8th 2021)

    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.

    • CommentRowNumber74.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2021

    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 C\mathbf{C} is a multicategory generated by its nullary, unary and binary arrows, CC its underlying category, and \otimes, 11 are functors on CC representing the nullary and binary arrows of CC, then \otimes and 11 form the tensor and unit of a monoidal structure on CC.”

    The ugly part of this is the generation condition, which will be needed since we only start with \otimes and 11 (indeed, some stronger presentation condition might be needed, actually). The unbiased version, where we have not just \otimes and 11 but an nn-ary tensor product for every nn, is essentially given in Leinster’s book, iirc, and doesn’t require such a condition.

    =–

    — end forwarded discussion —

    diff, v136, current

    • CommentRowNumber75.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 8th 2021

    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.

    • CommentRowNumber76.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2021

    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.

    • CommentRowNumber77.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 8th 2021

    That sounds good to me. I’ll make the change.

    diff, v137, current

    • CommentRowNumber78.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2021
    • (edited Nov 8th 2021)

    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 MonCatMonCat was sitting in the subcategory for “Strict monoidal categories”…)

    In the definition of strict monoidal categories I fixed the wording: Now the ambient CatCat 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.

    diff, v138, current

    • CommentRowNumber79.
    • CommentAuthorsamwinnick
    • CommentTimeNov 9th 2021
    Got it. My confusion was: I thought the intended meaning of the remark in question was that it's possible to define a monoidal category in any monoidal 2-category, not just in the Cartesian monoidal 2-category Cat. But now I see that no such claim is being made, rather, our attention is just being drawn to the fact that we use the monoidal product (cartesian product) in the 2-category Cat in our definition of 'monoidal category', an instance of the microcosm principle.
    • CommentRowNumber80.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2021

    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.

    • CommentRowNumber81.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 10th 2021

    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.

    diff, v139, current

    • CommentRowNumber82.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2021

    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.

    • CommentRowNumber83.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 10th 2021

    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?

    • CommentRowNumber84.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2021
    • (edited Nov 10th 2021)

    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.

    • CommentRowNumber85.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2021
    • (edited Nov 10th 2021)

    Regarding sufficiency or necessity: In Lurie’s actual realization of the microcosm principle (here) it is both: algebras over an \infty-operad 𝒪\mathcal{O} are defined internal to 𝒪\mathcal{O}-monoidal \infty-categories.

    Incidentally, the (,1)(\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 CatCat but just its (2,1)(2,1)-category core (since the coherence 2-morphisms that it is to supply are all invertible).

    • CommentRowNumber86.
    • CommentAuthorsamwinnick
    • CommentTimeNov 11th 2021
    Isn't this discussion of necessity vs sufficiency sort of analogous to the difference of structure and property?:

    sufficiency : property :: necessity : structure

    By that I mean, there exist a non-monoidal category C and a sense in which one can define a monoid object M in C, by specifying a functor \otimes, associator, unitors, etc for M. In this sense, having C monoidal guarantees that this _can_ be done. So we are referring to "properties" of this object M.

    On the other hand, we _need_ C to be a monoidal category in order to be able to define monoid objects whose monoidal structure is canonical. Here M is equipped with structure inherited from C.

    If what I wrote makes sense, I wonder if it would be relevant to talk about this nuance between necessity and requirement in the stuff, structure, property page, say with a hyperlink on the word "ability" or "necessity" on this article.

    I wonder if the first notion (sufficiency/property) is not so good from the perspective of category theory. In its defense, there are certainly categories where only certain objects have something special about them. For example, elliptic curves, among curves, have an addition law. But there is no biproduct for algebraic curves is there? Nevertheless, the notion of addition on elliptic curves isn't completely arbitrary; we still "can" define group objects here in a meaningful sense.

    Thank you both for your replies. This has been really helpful for me (as is the entire website). Also I hope to understand the last remark (#85) by Urs in the not-so-distant future. In the meantime, I am content with the following non-circular recipe: (1) Cat is a category; (2) Cat has products (thinking of pairs of sets as (x,y)={{x},{x,y}} to prove existence, but rarely ever again thinking of pairs like this); (3) monoidal categories are defined in terms of Cat and its finite Cartesian product operation; (4) categories with finite products are monoidal categories with respect to these; (5) define bicategory again using (Cat,\times); (6) in addition to being monoidal under the Cartesian product, Cat is a strict bicategory when using natural transformations for its 2-morphisms; (7) the 2-category Cat is a monoidal 2-category with respect to Cartesian products. Out of curiosity, do people not like this way of thinking about things because of step (2)? What is the reason for preferring a different recipe--using the "(2,1)-category core", as you say? Is there an nlab page that has the answer to this?
    • CommentRowNumber87.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2021
    • (edited Nov 11th 2021)

    Re #85, #86:

    It was a wide-spread mistake of old-school higher category theorists to think that to obtain a good theory of nn-categories one needs to first define (n+1)(n+1)-categories, because, so the logic went, the collection of all nn-categories is bound to form an (n+1)(n+1)-category which is needed to provide the ambient context for dealing with nn-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 nn-categories is available already inside the (,1)(\infty,1)-category of nn-categories.

    This insight breaks the impasse: First define (,1)(\infty,1)-categories all at once, and then find the tower of (,n)(\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)(2,1)-category of categories, functors, and natural iso-morphisms between them.

    • CommentRowNumber88.
    • CommentAuthorsamwinnick
    • CommentTimeNov 11th 2021

    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.

    • CommentRowNumber89.
    • CommentAuthorHurkyl
    • CommentTimeNov 11th 2021

    On the history lesson, when did the idea of nn-categories get refined into the idea of (n,m)(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.

    • CommentRowNumber90.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 11th 2021

    Certainly by Lectures on n-Categories and Cohomology, but I think it was much earlier.

    • CommentRowNumber91.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2021

    Regarding serious attention: This began with the use of (,n)(\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 nn-category of cobordisms” (tac:18-10). Suddenly there was a definition that worked.

    • CommentRowNumber92.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2021

    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.

    • CommentRowNumber93.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 12th 2021
    • (edited Nov 12th 2021)

    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.

    • CommentRowNumber94.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2021

    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 nnLab entry on it. I might try to start something later on the weekend.

    • CommentRowNumber95.
    • CommentAuthorTim_Porter
    • CommentTimeNov 12th 2021
    • (edited Nov 12th 2021)

    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.

    • CommentRowNumber96.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2021

    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 (,1)(\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.

    • CommentRowNumber97.
    • CommentAuthorTim_Porter
    • CommentTimeNov 12th 2021

    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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