The n-Category Café

Regarding “Grothendieck topology” – those are all good points, of course (and not the first time such points have been made). I wouldn’t call it particularly “the author’s choice of terminology” in this instance, since that might suggest that the author was doing something idiosyncratic instead of just following a tradition (however irreflectively :-) ).

It reminds me of another “controversy” that came up at the Café some time ago: use of the word “schizophrenic” as in “schizophrenic object” (something used to induce a concrete duality, such as ‘2’ playing a double role in Stone duality). Tom Leinster complained about the propriety of this word, quite rightly in my opinion, and there was some discussion about alternatives. Anyway, as the Café and the nLab build strength and momentum and become forces to be reckoned with, I’m all for taking advantage of the gathering strength and coming to a consensus on what we regard as improved terminology, with a view toward a brighter future. :-)

On the specific matter of “Grothendieck topology”, I like something along the lines of “local operator”. Recalling something Lawvere once wrote, I’d prefer “local modal operator”, since inserting $j$ in front of a predicate $\phi$ is trying to express a modality of the form “it is locally the case that $\phi$”.

We should certainly alert the reader of the existence of differing terminologies, but it seems that it might be desirable for the nLab as a whole to use consistent terminology. Any thoughts?

Yes indeed. A glossary page with nlab terminology in the first column would help.
Recall my problem with lax versus pseudo versus sh = infty

There is an interesting site

http://jeff560.tripod.com/mathword.html

which is an ongoing attempt to catch the earliest occurrance of mathematical words

Searching the n-lab for pseudo,
I am told of lots of entries containing the word
but search for pseudo-tensor or pseudo-functor
or without the - I find nothing

and I can’t create an entry becasue I don’t know the definitions

might also be good to have a page

pseudo versus lax

I think concrete technical questions, at whatever level of understading, well deserve to be placed in the corresponding $n$Lab entry. Please continue asking such quiestions there.

I think if and when a questions asked at an $n$Lab entry turns out not to have a straightforward answer, or if the answer is controversial or the like, i.e. whenever the question requires discussion, this discussion should be moved here to the blog.

What about adding a comment to the HowTo page like:

“Once you edit a page, your name will appear at the bottom, grayed out with a question mark since you don’t have a user page yet. You may take this as an invitation to create your user page and tell us about yourself. But if you don’t want to or don’t have the time right now, you should also feel free to just make a blank user page for yourself, containing only ‘category: people’ (so that you show up in the list of contributors).”

The point for adding “category: people” is that they will show up when you click “people”, so you can see a list of names of people who have contributed.

The comments here are quite sensible, except that most of them are based on this misconception.

Here is the problem: category: people already contains one person that is not a contributor (James Dolan), and presumably it will eventually have further articles on people unlikely to contribute (Bill Lawvere, maybe Ross Street), and even people who couldn't possibly contribute (Max Kelly, Saunders Mac Lane).

On the other hand, if you want a list of contributors, you click Authors at the top of the page.

Of course, we could restrict category: people to contributors; that is more convenient to look at than Authors for a simple list. And we could add, say, category: biography if we want to pick out the articles on Jim Dolan and Saunders Mac Lane.

But the main point is this: Nothing has to be done to get people's names on a list of contributors. In fact, they couldn't take it off the list if they tried!

I felt “pressured” to create my own page just to cosmetically get rid of the greyed out background.

Ah, see, I think that such pressure is a good thing. But if people are intimidated (as you suggest), then your removing that intimidation would be good. A note on HowTo would also be good (but I don't think that we should encourage them to only write category: people, since that's not necessary to join a list of contributors and therefore really adds nothing).

The right place to report typos on a page is through the “Edit” link near the bottom of the page.

It's a highly automated process; instead of giving you a form to write to a human about the typo, this link simply presents you with the full contents of the page in the original markup. Correct the typo there, hit the “Submit” button, and the software will automatically correct the typo on the page itself in real time.

(You need Javascript and cookies, for some security or anti-spam reason that Jacques knows. But you can delete the cookies afterwards if you want.)

The general plea from Urs is a good one, but may I ask that there is some morediscussion of some pedagogical issues in this discussion section as well. This particularly relates to enrichment and may help clarify things with regard to the mathematical aspects as well.

When going towards enriched settings it is easy for those in the know to do something that is correct but not completely transparent. The task of dragging concepts from the dark is hard to do, but sometimes especially with enriched concepts to get the idea to work one has to rewrite the original unenriched concept slightly. If the nLab is to be useful not only for us who are already reasonably skilled at this process, but also for the beginner in the area, then we need something more like:

Original unenriched concept:

Adapted description with reasons why, intuition etc. for the slight change in viewpoint, then

Enriched concept.

I have been trying to use something like this approach in working with weighted limits and colimits, with just partial success. (It is not that easy to do!) If someone wants to see my efforts in the longer menagerie notes just ask.

The same thing goes for categorification. I am trying to explain categorifcation to some analysts and mathematical physicists for some seminars and the problem is to find the right approach direction.

I would greatly appreciate discussion of these slightly pedagogic issues on the blog.

I edited $(-1)$-groupoid to clarify the poset situation; you are right that $\Omega$ is always a partially ordered object in the topos.

Enriched topos theory is actually a deep and (at least, to me) unclear subject. So much of topos theory seems to depend on cartesianness. I’ve occasionally thought about what an enriched topos might be, but never really come up with anything really satisfactory. If the enrichment isn’t cartesian monoidal, then the internal logic of an enriched topos would probably be linear logic, but how to interpret linear logic internally in some category is also not an obvious question. One expects that perhaps “quantales” (closed monoidal suplattices) will play a role.

On the other hand, one can certainly define sheaves of anything one likes. It may be a bit harder if the site is also enriched. But I don’t know a whole lot of examples. In one approach, spectra are defined to be a certain sort of sheaf of topological spaces, although the word “sheaf” isn’t usually used.

If we ever did anything like this, should we settle on a naming convention? For example: [[Paper: Groupoidification Made Easy]] or [[Article: Groupoidification Made Easy]]

That shouldn't be necessary, as long as the titles don't conflict with any normal page titles. Most of the time, capitalisation should be enough to avoid that, and we can always make a rule that you have to add something to your paper title (for example, [[On Day convolution]] to avoid clashing with [[Day convolution]]). Actually recognising and searching for papers (from among the other articles) is what the categories are for. Probably one category will be enough, but we can always add more later if we want.

Otherwise, I agree with everything that you say.

NQ-supermanifolds are equivalent to Lie ∞-algebroids, this is not obvious but is an important fact that must be noted

I doubt it’s even true - super implies Z/2 graded

even if true, would it mean the defintions are equivalent or just that the categories are equivalent, e.g. every foo is equivalent to a bar and every bar is equivalent to a foo??

For what its worth, I like the direction the nLab is going. It makes sense to me to have as many pages as you think are necessary. It is easy enough to link things together. Plus, we can always consolidate things later once a clear picture emerges.

But on the topic of “Duplication”, I’ve seen an example or two where content was copied from one page to another. I’m not sure that is a good idea. Then it becomes difficult to keep track. If someone modifies the content on one page, but not the other where the same content appears, then things can become messy. I suggest that we try to avoid copying and pasting content from one page to another and instead provide links. I have been tempted to undue cases where I’ve seen this, but have been hesitant to “delete” stuff even if it was just copied from another page.

Toby wrote:

However, there’s a much more important difference between us and Wikipedia, which is that Wikipedia is a compendium of established knowledge, while we are pushing the frontiers. This means that many of our definitions will be tentative, and even some that we are sure about will need justification for the uninitiated. With that understanding, it’s very useful to have a page on each term, so we can explain on each page the points relevant to that term. To the extent that we’re sure of an identification of one term with another, we can justify the more obscure term on its page and send people to the more common term’s page to read all the facts. To the extent that our identifications are uncertain or in flux, we can clearly distinguish on each term’s page the known properties of that term’s referent from the conjectured properties, a distinction that won’t always be the same from term to term.

Yes, I agree.

One place where this issue is currently occupying my attention is the circle of entries which revolve around cohomology and $\infty$-stacks.

It was after reading (again) Mike Shulman’s work on enriched homotopy theory and re-reading in the light of this the work by Toën on $\infty$-stacks that I came to think that there is a nice big(ger) and more nicely more coherent picture to be described here. I started indicating this picture at $\infty$-stack homotopically and am further developing this currently more privately over at Nonabelian homotopical cohomology and fiber bundles. I am thinking that this should eventually make me go back to rearrange all the existing entries on higher cohomology and tell the story from this perspective.

But since this is more work with all the links in there, for the time being I think I will develop this at Nonabelian homotopical cohomology and fiber bundles and then impose the changes to the $n$Lab in one strike later.

In general I agree with what you say. For pairs of terms that are equivalent, but apparently different, I think can be useful to have a discussion from different points of view on the page of each name. But for pairs that are really just synonyms, it makes more sense to me for one page to just say “see blah.”

In the last-mentioned case, I think it is good to keep “truth value” separate from all the negative thinking pages, since many people may care about truth values but not want to wade through anything talking about (-1)-categories. However, since currently the content and points of view of “(-1)-category,” “(-1)-groupoid,” and “0-poset” are very very similar, I would be in favor of having two of those just say “see the third,” and my choice would probably be for the third to be (-1)-category.

I agree with what you said about copying being useful for starting pages, but only when we expect the new page to soon go in a different direction. I personally don’t have any expectations that (-1)-groupoid and 0-poset will contain any notably information or points of view from (-1)-category in the forseeable future; do you? If I want to change or add something to one of those pages, I don’t want to have to do it on all three, but if they all duplicate each other the only alternative is for them to become randomly out of sync (each containing only those changes made by people who happened to pick that page to edit, rather than one of the other two).

I wonder if there is a way to automate this. Is there a way to distinguish between a minor polish and the addition of content? For example, if less than 20 characters have changes, it is a minor polish. Then have the “Recently Revised” page only display modification that involve more than 20 characters?

How difficult is it to modify the source code? I’m not a programmer, but code give it a shot.

A couple questions for the community on this:

• Does creating a page ever count as “minor polishing?” Or should creation of new pages always be noted? Consider, for instance, bijection.

• Should every continuation of a discussion be noted, as in “continued discussion with so-and-so at X page”?

One caveat, maybe:

I am getting the impression that it is hard to plan something like “our activity on the $n$Lab concerning this and that point will be precisely like such” without actually trying to do it first.

Maybe a quicker road to a satisfactory result is:

- just start doing something. First something which is not overly sensitive. See how it develops, see how it can be steered and how not. Then after some practical experience has been gained, go back here and see if one can find an informed consensus on how to proceed on similar, then maybe more ambitious, projects in the future.

But I wonder whether we shouldn’t establish some naming convention for such pages, perhaps akin to the “alpha” naming convention in BibTeX.

Good point. Yes, I agree. So which convention should we use?

What’s the alpha-convention?

A compromise might be to have a

category: redirect

on a page with the abbreviated name that points to the full name.

For example, [[Bro73]] being a page the contains:

See [[Brown – Abstract Homotopy Theory and Generalized Sheaf Cohomology]]

category: redirect

Do we need reference pages for every reference? Maybe we do; it could be useful, if we get the names sorted out. But I initially populated it with the Elephant and Categories Work (books so well known that they have nicknames!) because I actually wanted to write something about them, not because I thought that the bibliographic data needed to be written down on a page. (In fact, the bibliographic data was pretty sparse; I suppose that I expected people to search the Internet if they really wanted more.)

Regarding a citation convention, why not use the MRNumber, at least where that is available? And the arXiv number where that is available (with MR beating arXiv if both are available)? The only objection to that that I can think of is that neither is particularly memorable, but to be honest I don’t think that any convention is going to eliminate the need to look up the exact reference each time. By my estimation, [Bro73] could refer to about 60 different papers.

I’ve yet to contribute to the nLab so I don’t know how easy it is to implement a new type of reference. Something like the [[word]] code but maybe [[mr:mrnumber]] or [[arxiv:arxivref]] automatically puts in all the rest of the bumph. That also helps Jim’s request since MathSciNet can export to BibTeX. The Front used to do this for the arXiv but I don’t think that it does anymore.

Actually, I think that we should not pick one. I think that we should write […]

Yes, I agree. I am in favor of following your suggestion.

I prefer not being forced to write anything at all between the factors, provided that the context already forces which order is meant. For example, if someone refers to an adjunction $F \dashv U$ and then refers to the monad $U F$, I’ll know what is meant (unless, of course, the adjunction is given as ambidextrous!).

So, someone correct me if I am wrong, but my experience is that apart from some category theorists and some computer scientists, the convention that $g f = g\circ f$ is fairly universal among mathematicians. Since composition is such a fundamental operation, I think that using more than one convention, or a convention which is unfamiliar to one’s reader, is a significant strike against the reader’s sympathy and comprehension. And given that we are trying to change the world in other ways already, I think we should take every opportunity to make the non-category-theorist reader comfortable. So I would prefer we adopt the convention that $g f = g\circ f$, with the option to write $f; g$ if we prefer (with explanation, or perhaps a link to [[order of composition]]).

I have the solution!

Why don’t we convince category theorists to write morphisms like

$$g:y\leftarrow x$$

and

$$f:z\leftarrow y$$

so that

$$f\circ g:z\leftarrow x$$

and EVERYONE would be happy :)

The whole issue comes about because someone unwisely chose to use $\to$ long ago :)

Will Trimble-weak-$n$-categories be “not weak enough”? How does one tell?

Well, I think the only way to tell “for sure” is to prove that they’re equivalent, or inequivalent, to some notion which is accepted to be “weak enough.” A way to get a good hint as to whether they’re weak enough would be to ask whether Trimble $n$-groupoids model all homotopy $n$-types.

Evidence seems to suggest that in order to be “weak enough” you have to have either weak interchange or weak units, but one or the other could be strict. Certainly they can’t both be strict, since in that case the Eckmann-Hilton argument happens strictly and collapses braided and symmetric things. See Simpson’s conjecture.

Now, I see that Eugenia Cheng mentions on p. 19 that this is the definition Peter May gave in his talk Operadic categories, $A_\infty$-categories and $n$-categories (which I haven’t really read yet, to be frank).

If you do read it, don’t put a lot of effort into the details. Peter May now admits that a lot of his claims in that paper are rubbish. But the general idea of iterated enrichment, in a setting more general than Trimble’s, is interesting and worth pursuing.

I have only looked at very little of the literature on Trimble-weak-n-categories

AFAIK there is only very little of that; in fact the three or four references you found, Urs, authored by Tom Leinster, Eugenia Cheng, and Nick Gurski, are all that I know of. I’m grateful to them for all they’ve done: they’ve generalized and brought this sort of definition into more of the mainstream (comparing with e.g., Batanin’s approach), and have suggested some new potential applications.

Andrew wrote:

This page is getting tricky to follow. Is there any possibility of hosting a forum on the same site as the $n$Lab? That’s the right place for discussions.

We already have a general discussion page on the $n$Lab, but it became very hard to follow, so we decided that here is the right place for discussions.

Perhaps any discussion where lots of people are talking about lots of different subjects is going to get hard to follow, unless someone keeps reorganizing it to make it follow some semblance of a logical pattern.

I think the motto on the $n$Lab homepage isn’t bad: you go to the $n$Lab to work, and go to the $n$Café to chat. Meaning: things that we want to be visible in some permanent, well-organized way should be on the $n$Lab, while free-roaming conversation belongs here.

But perhaps we should have more blog entries about more different aspects of the $n$Lab?

Actually, it’s interesting that you bring this up, Eric, because I’ve been vaguely thinking about passing from $Top$ to some more combinatorial notion of space, to which we could apply some of the ideas Urs and I have been kicking around recently. But I don’t want to attempt saying anything about this just yet.

It’s quite true that most functors from a topological context to an algebraic one lose lots of information – you’d never be able to retrieve the space from the algebraic model up to homeomorphism, for example. But the homotopy hypothesis says that in cases of interest, you should at least be able to recover the space up to homotopy equivalence, which is a lot coarser as equivalence relations go, but neverthless interesting.

Why would we believe it? Because there are lots of partial results that point in this direction. One particular incarnation of the homotopy hypothesis is about “$n$-types”, where we say that a continuous map $f: X \to Y$ is an $n$-equivalence if it induces an isomorphism on just the homotopy groups up to $\pi_n$. One of the very earliest “success stories” was for the case $n=1$, where the fundamental group(oid) functor does the trick. The functor going back the other way is

$$B: Groupoid \to Top$$

which maps a groupoid to its classifying space, and the crucial result is that for any space $X$ there is a map

$$X \to B \Pi_1(X)$$

which induces an isomorphism on $\pi_0$ and $\pi_1$. Already this isn’t a trivial result.

Things quickly get more complicated as we move up in dimension, with a fairly substantial literature surrounding the case $n = 2$. An early success story here is the recognition that “crossed modules”, which we’ve been discussing a bit in nLab as you know, can be used to classify connected 2-types. At some point is was recognized (thinking especially of mathematicians like Ronnie Brown) that these structures were connected with categorical groups, or monoidal groupoids, and by now this theory has been pretty well worked over.

Of course I won’t go into all the history (not that I know it myself), but there’s also this famous 600-page letter “Pursuing Stacks” by Grothendieck, which is now archived at Bangor, which outlines what Grothendieck called the fundamental $n$-groupoid, and which is conjectured to model $n$-types. This to me is an awfully tantalizing conjecture, and it’s something I’d like to spend more time with in the company of Urs and others. It was this idea which motivated me to develop these “Trimble $n$-categories” which we’ve been discussing.

In a more combinatorial direction, there is an old idea that simplicial sets are, for present homotopical purposes, basically as good as general spaces: in technical terms, any space is weak homotopy equivalent to the realization of a certain simplicial set called its simplicial singular complex. Anyway, for various technical reasons, the experts tend to like using simplicial sets, and here we have an actual theorem instead of just a homotopy hypothesis, if we’re sufficiently sneaky about it. For, one notion of $\infty$-groupoid is the notion of Kan complex, and it’s a theorem that any simplicial set is weakly equivalent to a Kan complex! This is actually an useful and important result which today is embedded in the general theory of model categories.

So summarizing, there’s plenty of reason I think to have faith in the homotopy hypothesis, and there’s no doubt that combinatorial or discrete models will play an important role in the game.

If you see something missing from the nLab wiki that you think should be there, why don’t you add it yourself? That is the whole point of the wiki *confused*

FireFox decides it is of type “text/html” instead of “application/xhtml+xml” and any MathML is wrongly displayed.

That’s a generic problem for which there is no easy fix.

When served over the Web, the file extension is irrelevant. The MIME-type of the page is determined by the Content-Type header sent by the webserver. Firefox (and other XHTML-capable browsers) get application/xhtml+xml; Internet Explorer (in particular) gets text/html.

When viewing files on your local disk, the browser has only the file extension to rely upon. If we used .xhtml, IE would barf. If we use .html, then nobody barfs, but Firefox won’t render the MathML.

I guess the question comes down to: what do you intend to do with the exported files?

One answer is: import them into another Instiki installation. In that case, the filename extensions are irrelevant.

That applies whether the Instiki installation is on a remote webserver, or running on your local machine.

Another answer is: build a static website out of these pages. In that case, too, the webserver takes care of setting the Content-Type header.

A third answer is: view these as static files on my local machine, without importing them into Instiki, or serving them with a locally-running webserver.

In that case, you have a problem. Whatever file extension you choose, some browsers won’t handle it correctly.

Un unzipping the file there are several duplicates with different sizes and timestamps.

That’s simply a consequence of the decision, by the owners, to avoid cleaning up the site by removing obsolete pages.

There’s a “Philosophy” page and a “philosophy” page. The former consists, in its entirety, of a pointer to the latter. Similarly, there’s a “jim stasheff” and a “Jim Stasheff” page, etc.

Well, I don’t claim to be at all competent in software-related matters, so as a general rule, don’t place your hopes too high.

That said, I do pay attention when people come by to improve on things I write in nLab, and this was no exception. The point has been noted.

We agreed that “$n$-category” is as weak as possible, and everything else has to be explicitly qualified.

Yes, there are probably inconsistencies. Your help is appreciated. The general rule is: if you come across anything you find annoying or even psychological harmful: drop everything else, hit “edit” and improve it! :-)

… or, I should add, if you don’t want to or cannot improve on it, drop lots of these green query boxes saying “this is unclear”, “what is it really that is meant here?”, “this conflicts with the defintion above”, etc.

Hi Eric,

here is the quick idea:

in a cubical set, you are guaranteed for every $n$-cell (which I draw as a 1-cell) $$a \stackrel{f}{\to} b$$ that there is the identity $(n+1)$-cell (which I’ll draw as a 2-cell) of the form $$\array{ a &\stackrel{f}{\to} & b \\ \downarrow^{Id} &\Downarrow^{Id}& \downarrow^{Id} \\ a &\stackrel{f}{\to} & b } \,.$$

A cubical set is said to have a connection (no relation to the parallel transport meaning of “connection”!) if in addition it has for every $n$-cell $a \stackrel{f}{\to} b$ also $(n+1)$-cells of the form

$$\array{ a &\stackrel{f}{\to} & b \\ \downarrow^{f} &\Downarrow^{Id}& \downarrow^{Id} \\ b &\stackrel{Id}{\to} & b } \,.$$