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I just thought I’d throw something out there. Would it not be great if we could come together and select something to work on as a kind of ’polymath’ project here at the nLab? I mean, there are a lot very knowledgeable people, with a broad common core knowledge. Would it not be great to select some topic, and work on it together, with the view of making a nice nLab page, or cluster of nLab pages, in the end?
In a sense, of course, that is what already happens. But I am thinking that it could be interesting to select something to really focus on, which everybody could try to offer some ideas on, in polymath-style, as time allows. I for one greatly enjoy collaborative work.
To make things more exciting, the topic should probably have some kind of novelty to it; that could be just novelty of exposition (giving a categorical treatment of something, say), or giving a proof of something that is folklore but not written down, or something with a new twist on it, or some really new thing.
Would anybody be interested in this, and have any suggestions for topics? I can make a few suggestions, but I don’t wish to put people off, or try to narrow things down too much; I would probably be more interested in suggestions of others. But, just to get the ball rolling, here are a few.
1) Gödel’s incompleteness theorem. An expositional project could be to carefully treat Gödel numbering, taking a categorical point of view. A more ambitious project would be to give a formulation and proof of Joyal’s folklore theorem. There was once a thread on this in the nForum where a few of us floated a few ideas.
2) Constructive topology. There are various approaches to this: formal topologies, locales, etc. But could one take a more synthetic approach?
3) Free co-completion. I think it should be possible to construct the free co-completion of a category internally to a 2-category with a reasonably weak amount of structure. This would be somewhat surprising: one would not need a Yoneda structure, an equipment, or any of that kind of thing. In polymath-style, I don’t know whether it will work, but I can throw out some ideas, and maybe somebody can take them further.
4) Semi-strict 4-groupoids. I have an idea for defining these as follows: enrich groupoids over the category of strict cubical 3-categories with its Gray-like monoidal structure. One obtains something different to any of the studied notions. They are weak in only one direction, but it is possible that this weakening is enough to model homotopy 4-types. I can throw out some ideas here, but mostly things are up in the air.
I can propose others, but that’s a start at least.
I forgot to say that probably the topic should aim to not require a huge amount of background. But let’s just see what comes up.
Likewise, I suggest that if done, it should be done on the blog, not here, for visibility and openness to driveby participants.
But Forum also has a blog form of posts, which can be chosen when posting a new ’discussion’. Category cafe is, well, not so technically structured for research, rather an opinion mill.
Unfortunately, I don’t think I’d have the time to actively participate in such a venture.
You seem to have a lot of interesting ideas, though, and I still think you ought to have your own personal ncatlab web(s). Such webs can be publicly visible but privately managed, or open to the public to make wiki revisions, or completely private (invisible to the public). I’d be particularly interested to see more details of what you have in mind for your (1).
Your (4) looks like something that Sjoerd Crans (who is no longer active in research mathematics AFAIK, unfortunately) would have put some serious thought into. If it were me, I’d contact Ross Street and Dominic Verity for some pointers to the literature.
the blog
(Sorry: what blog?)
Todd: on the left of the page you have two choices. One is start a new discussion. And another is start a new blog. The difference is mainly in appearance. Blog posts are threads with larger size fonts and so on and were a category created to replace the external end like the cafe.
Another, a bit unpleasant note. Being delighted with Richard’s enthusiasm (and breadth of vision), I still feel obligated to recommend to Richard as an older (though less successful) colleague to save more (but not all) of his immense intellectual energy for different format of projects which will do more to his personal career than our altruistic efforts in this community usually do (and to notice: Richard is trying to even raise the altruism and quality standard to even much higher level; with disclaimer that I am not wishing to diminish the publc value of altruistic and collective projects, as you can witness from immense time I spent contributing to Lab.); I mean to do a paper in collaboration with several people present here (in more traditional format) is unfortunately still more opportunistic than to do online project probably without a publication. And I do not like that the young bright people have to sacrifice to make the change (I am delighted that some of them are still enthusiastically willing to). Many of us sacrificed a lot to change the things, to go on for altruistic projects, do public discussions for common good or simply intellectual reasons, posting numerous posts and creating much public material and often ended in receiveing more of a critique than a praise and all that almost not counting when competing for real jobs. I am still an ethusiast (though a bit more cautious), and will further help, but when I see young talented people I would rather try to save them (at least by a warning like this, which I admit am uncomfortable but feel morally obligated to do) from many of troubles which come as a rule when you come to an age when good playing does not count any more (and then it is often late). I am sure Urs will resonate with my opinion, his immense contribution to modern mathematical public projects is immense, with only partial career recognition of his role (and invested time, thought, initiative and knowledge), though in his case I am sure (and he knows it) it will eventually be widely clear. Some other people try to go partly along similar paths, but many do not succeed even to stay in career path on the way. Our mutual support but also some wiseness in not being too wasteful from the conservative point of view should help to some important extent.
on the left of the page you have two choices.
Yes, right. I’ve never tried to use the blog option there.
I mean the n-cat cafe, purely because this is a format that Polymath has worked with so far.
Although all the projects so far basically had a Fields medalist sponsoring them, so people were happy to jump in because of a sort of halo effect (“I collaborated with Gowers/Tao/etc!”), and they didn’t need any career boost so could help others in this respect. If we had some big name jumping on board to ’run’ the project on the blog, I’m sure we’d get something pretty interesting happening. But I can’t think of someone suitable, sadly…
David, this may be the case, but my experience is that even an occasional witness-support of a top mathematician (even a Fields medalist), or order of thousand or more created entries in online mathematics exposition, often does less than accepted hard facts like a publication in traditional journal if that journal has solid impact factor (regardless the quality of the refereeing process there) or past grant/money award luck and mere length of official employment in a relevant institution (neither of these is a scientific achievement, but it counts!). Of course, people of the calibre of possibly getting to Princeton, Harvard or similar top institution have likely far much more sensitive committee audience in such institutions and there such a thing may really matter, but in the rest of the mundane world, just get those stupid impact factors, citations, and talk to the local people with influence to decision committees. Bad but real.
and they didn’t need any career boosts
I was behaving like that in about 2004-2008 when the professional world of mathematics was graceful to me (partly because I was not too old), but in practice almost everybody at a moderate or young age needs a career boost. I do credit a lot people who sacrifice consciously. There is enough money and workplace in mathematics for all considerably knowledgeable contributors, but somehow much is wasted in wars and pure waste (like let’s spent a grant carelessly because there is a deadline soon, or let’s apply for this grant because it will do good to our CV; I consider the latter and partly the first immoral. If there is no real need it is better to return the money to the agency than to waste it for nonsense (even if we are punished for that in future). But we have to fight also to change the policy to be able to extend the deadlines for spending in sufficiently justified scircumstances and that the agencies in science leave the unspent money within science at least and not return it to the same pool where the military funding is. The money spent by the agencies could be projected into future statistically in an ideal system, accounting for the average nonspent return and having a volatility regulations for deviations from expected behaviour; this may need a pressure on legislation).
Sorry, by ’they’ I meant the Fields medallists.
Thank you very much for the kind words, Zoran! It is a travesty that the immense creative work on the nLab has not been sufficiently recognised, especially, from what one picks up, the difficulties that Urs and yourself have had.
My personal situation is unusual; for a large number of reasons, I stopped working in academia at the end of summer last year, and do not have any current plans to return. But I am still first and foremost a mathematician, and will I think never stop being so! For a number of reasons, I struggle with making my work available; I have written hundreds, maybe thousands, of pages of mathematics that have not been made public because I am not completely satisfied with them, and I have numerous ideas which I have thought about and explored, and more which I would like to pursue. I would love to find a way to share this with people, and to be able to work with other people on these ideas, and I am hoping that the nLab, one way or another, can help with this.
Regarding #4: thanks Todd! Yes, I know Crans’ work, and I think most work on this, quite well. I have discussed these ideas a bit with Mark Weber.
Regarding the personal web, I would be open to this. I am conscious that the pages on cubical sets that I have been working on are not in a satisfactory state at present, but have very limited time to improve things; if people wish to move them over to a personal web and revert, that is absolutely fine with me. We could always move stuff back to the nLab later, or at any point.
Regarding the polymath ideas: maybe my use of the word ’polymath’ was not helpful in conveying what I had in mind. What I meant was just that there is a core of people working here who have broad and deep knowledge, with quite large overlap. It seems to me that extraordinary things could be achieved if the nLab had, at any given time, one or two ’feature projects’ which everybody tried especially hard to keep one eye on and keep up with, to keep pushing things forward. As I say, I myself have little time, but I have enough to be able to read a day’s discussion on a topic, and offer some ideas if I have them. But it’s just a suggestion; I quite understand people might be shy of making a commitment to this kind of thing.
It’s an interesting idea. I do feel kind of like category theory doesn’t really lend itself as well to that sort of project as other disciplines do — but I could be wrong. If there were such a project going forward that I was interested in, I would certainly make an effort to keep up and contribute, and I expect others would as well; indeed that’s what already happens on a smaller scale already with nForum discussions or nCafe posts. The problem would be in finding a topic that’s going to interest enough people, and has enough depth to it, to keep that going for longer.
Of your suggestions, I would probably be the most interested in semistrict 4-categories (with an eye towards semistrict n-categories). In addition to Crans’s work, some input ought to come from the “multitensors” approach of Weber and his collaborators (which you are probably aware of), as well as the authors of Globular and the thoughts of Makkai mentioned there. It’s something that a lot of people have thought about, but no one has really nailed down a satisfactory answer yet so far as I know.
Thanks for your thoughts, Mike! I agree that this is exactly what happens on a smaller scale at the nForum or nCafé.
I just sort of threw my suggestions out there, to have something to propose. I would prefer it if others came up with something! For instance, Mike, I think that you might well be able to make an excellent suggestion, if you find the opportunity to have a think about it.
Here is one more thrown out suggestion: something on understanding low-dimensional homotopy groups of spheres from a category theoretic, or combinatorial, point of view. I have explored at various times rigidifying Penon-like definitions of a weak n-groupoid as far as one can one in order to be able to make a calculation in the expected way (if the homotopy hypothesis holds); using some other specific low-dimensional notions of a higher category; and using finite topological spaces. One could tie this in with HoTT, and other things.
Here’s a wacky idea that I would probably enjoy, but I don’t know whether there would be enough other people that would get into it. Logicians and philosophers who study exotic logics and set theories are sometimes attracted to category theory as an application, because we seem to want to actually talk about things like “the category of all sets” that engender paradoxes in ordinary set theory. It might be interesting to gather a bunch of actual category theorists (or at least people who are familiar with and use categories in their work) to evaluate some of these proposals. Can we find a way to actually do category theory with a category of all sets in, say, NF? Linear set theory? Paraconsistent logic? Or can we find an argument that all such ideas are doomed to failure? Probably a lot of time would be spent just learning how to think in these exotic logics, but I would enjoy that too.
I would definitely join in on this!
From September on I will be working on computing the Adams-Novikov-type spectral sequence in equivariant cohomology for the equivariant cohomotopy groups , for a finite ADE-group and an ADE-orbifold, in terms of -equivariant complex cobordism of .
If anyone here wants to join in, that would be nice.
Mathematically, the equivariant ANSS is both a very natural next thing to consider as well as technically demanding (here is an MO comment by Tyler Lawson). Therefore it will be useful to have a very specific case in mind (for choice of equivariance, choice of degree, choice of domain) for which there is reason to expect something interesting as well as some intuition as for what may be going on. This I draw from physics (here).
It’s still some time until September. Meanwhile we are warming up with giving a detailed account of the classical (non-equivariant) situation (here).
An interesting further aspect would be to try to have HoTT methods be put to use for this undertaking. I have the feeling that Adams spectral sequences might turn out to be a killer application for HoTT: their computation on the one hand tends to demand sheer computational power, which a computer may help with, on the other hand their computation requires homotopy theoretic insight. Hence one would want both: a computer algebra system that natively knows about homotopy theory. (As in Kochmann, “Stable Homotopy Groups of Spheres – A Computer-Assisted Approach”, Lecture Notes in Mathematics, 1990 – but HoTT might be the kind of software that could take this to another level.)
I would be keen for that, Urs!
This might not be of sufficient interest to warrant a polymath-style investigation, but Mike’s #15 reminds me of a problem that I’ve not had success with and would love help with if anyone is interested. I might consider offering a cash prize (or some other reward, e.g., an expensive wine or single malt scotch) for a solution to the problem I’ll describe below.
At the time I entered graduate school, I wondered what a universe of enriched category theory would look like where one could freely take functor categories without concern with size issues. I even had pretensions that I would one day be writing my thesis on a notion that I called epistemologies, where I attempted to write down synthetic first-order axioms for such a situation. The axioms seem to skate on the edge of paradox, and I was never able to come up with a really satisfying model for them. That is what I want help with.
In one incarnation, an epistemology is more or less a cartesian bicategory (not necessarily locally posetal) in which every 0-cell of has a “free cocompletion” . Let me make that last part precise. For a bicategory , let denote the bicategory of objects, left adjoint 1-cells, and 2-cells between them: the standard notation for cartesian bicategories. (If for example is the bicategory of preorders = 2-enriched categories and 2-enriched profunctors between them, then will be the bicategory of preorders and preorder maps = 2-enriched functors.) We have an evident “inclusion” , and we say is potent (think: has powers) if has a right biadjoint , satisfying the Kock-Zöberlein or lax idempotence property. Recall that a bi-adjunction , with unit and counit and is lax idempotent if either of the following two conditions is satisfied:
The triangulator isomorphism (specified as part of the bi-adjunction data) is the counit of an adjunction ,
The triangulator isomorphism (again part of the bi-adjunction data) is the unit of an adjunction .
Normally I assume also in the definition of epistemology is that is, as a symmetric monoidal bicategory, compact in the bicategorical sense, meaning roughly that each 0-cell has a dual (inducing biadjunctions , etc.). Cf. the fact that classically, the bicategory of small -enriched categories and -profunctors, with cartesian closed say, has this property where the dual of a -category is .
So, to re-iterate, I define an epistemology to be a potent compact cartesian bicategory. Quite a lot of formal enriched category theory can be internalized in such , starting with the definition of the base where is the monoidal unit of . On the basis of the axioms, one may easily prove that the free cocompletion is an exponential , in the sense that maps (= left adjoints in ) of the form , are equivalent to maps of the form . Thus the unit has components of the form which by all rights should be thought of as yoneda embeddings. In other words, potency plus compactness yields an unrestricted -enriched presheaf construction, for any object , including for example itself.
This map corresponds to a map which plays the role of hom-functor. It turns out that maps behave like -enriched functors, and 2-cells as enriched transformations, and as I say quite a rich theory can be developed.
My basic question is: do epistemologies even exist?
In a literal sense, certainly they exist. If is any frame, then the bicategory of small -enriched categories and profunctors is an epistemology. But this is a somewhat boring example, because it (as a bicategory) is locally posetal.
So my question is: are there any epistemologies besides boring locally posetal ones? Or are epistemologies “inconsistent” in the sense that any two parallel 2-cells must coincide?
I don’t know. My intuition says epistemologies should exist, and maybe such things could be constructed syntactically; maybe one could develop a 2-categorical decision procedure for free epistemologies that is confluent and strongly normalizing, and one could check inequalities of 2-cells by reductions to normal forms. It would be even more exciting to me though if there were “natural models”. You could say that the situation is analogous to the early days of (untyped) lambda calculus, before Dana Scott produced real models of it in the 1960’s.
Presumably you’ve tried to reproduce version of the usual paradoxes like Freyd’s and Burali-Forti’s in an epistemology?
My memory is yes, I’ve tried to some extent, but I’ve felt stymied there as well. I’d need to get my head back in this game.
How about categories in the effective topos internally enriched over modest sets?
I’ve considered that as well, but wasn’t able to convince myself that the notion of “completeness” used for modest sets was sufficient for the purpose (I suspect it’s not).
I’m very much interested in that problem, Todd (for its own sake, no reward needed)! To ask one of the obvious questions, without having thought it through: in the one object case, so a cartesian monoidal category, what would an epistemology be, a little more explicitly? Is it clear that there are no interesting examples in this setting? I mean, on objects one cannot do anything, but is there anything interesting to observe on the level of 1-arrows and 2-arrows?
I think too that epistemologies should exist, for the following simple (possibly naive) reason: I think that an epistemology should be an ’algebraic’ notion in a higher categorical (3-categorical?) setting, and that for this kind of algebraic gadget there should be quite general construction theorems. Not that we are able to work effectively with this kind of algebraic structure at the moment, of course!
In response to Richard: so, the notion of epistemology has a number of moving parts, some of which get along with the condition of one object better than others. If you split up the notion into three parts, ’cartesian’, ’compact’, and ’potent’, it’s actually the ’cartesian’ that gets along least well. Let me get that out of the way first, so that I can get to the slightly more interesting stuff later.
One thing I should say is that a one-object cartesian bicategory in the Carboni-Walters (or Carboni-Kelly-Verity-Wood) sense is not the same as a cartesian monoidal category, which may mean that the naming isn’t very good. For the primary examples , the tensor product isn’t cartesian, although the restriction of this tensor to what I called is cartesian in the sense of being a cartesian product (in the sense appropriate for 2-categories, i.e., a 2-product). But that in itself spells trouble because it means that the one object of must be the 2-terminal object, and so is itself trivial. And then itself will be trivial as well once we add on the potency axiom, or even in more everyday situations such as if you have a tabularity condition (every arrow of is the composite of a right adjoint followed by a left adjoint).
However, in other variants of this notion of ’epistemology’, I’ve left off the cartesian bicategory condition and instead considered compact symmetric monoidal bicategories satisfying the potency condition. (In fact this variant is what I currently have on my nLab web page.) Now again I don’t think compact symmetric monoidal bicategories with one object are the same as compact symmetric monoidal categories (because the word ’compact’ is at the wrong level, being about duals of 0-cells in a 2-category instead of duals of 1-cells). In fact the unique 0-cell is the monoidal unit which is automatically dual to itself, and so I think we just get a symmetric monoidal category that satisfies the potency axiom.
These certainly exist by the way, and are typically non-posetal. In fact potent symmetric monoidal categories are the same as compact symmetric monoidal categories. Certainly every compact symmetric monoidal category is automatically potent as a bicategory. For, since every object has a right dual (i.e., is a left adjoint 1-cell if we think of it as a bicategory), it means the inclusion is just the identity which clearly has a (KZ) right (bi)adjoint given by the identity again.
(I think that can easily be converted to an argument that free epistemologies (in this variant sense) just about have to be locally non-posetal in general, since compact sm categories are. That’s partly the reason I’ve added on the cartesian bicategory condition: this makes for much more of an acid test. That and the fact that cartesianness makes epistemology structure property-like structure w.r.t. bicategories.)
In the other direction, given a potent symmetric monoidal category, to show it is compact it suffices to show the claim that is the identity . If denotes the unique 0-cell, we have and , and the equivalence sends to the left adjoint ; in the reverse direction it sends a map or left adjoint to the composite where by the KZ axiom. So ; if we show is a left adjoint (or a left dual in the symmetric monoidal category), then every is a left adjoint. But of course a right dual object such as in a symmetric monoidal category is also a left dual, so we are done.
Thanks for your interest in this! Let’s continue at the epistemologies thread that Mike started.
I was a bit away. For the moment just question about Urs 17. Urs is interested in equivariant cohomology for orbifolds. In my memory from Wisconsin days where for orbifold special orbifold cohomology is used, they considered the orbifold cohomology closer to Bredon cohomology then equivariant cohomology. Now why the extension of usual orbifold cohomology is to be considered instead in your project ?
Presently, to see the effect whose study the project is after, the tom Dieck splitting theorem is used at some point, and this holds for genuinely equivariant cohomology.
On the other hand, a while back I had started to suspect that possibly the effect is there even in naively equivariant cohomology. I need to come back to this and check.
A slight side note relating to Urs’ project: I saw recently the result that every orbifold is equivalent to a quotient stack for a Lie group action on a manifold: given that the target of Urs’ project here is ADE orbifolds, this gives me hope that one might conceivably write all such orbifolds as (or if you like), as the physics literature seems to assume. Then one is genuinely in the setting of equivariant stuff, rather than merely locally equivalent to an action groupoid.
I had seen that abstract, too. But lost the reference. Do you have it?
@Urs
Adem-Leida-Ruan 2007, Theorem 1.23 and Corollary 1.24, is where I saw it.
Looking back at the proof, it doesn’t seem like one can immediately get or action, since the construction goes via the frame bundle, and so one gets an -action for the dimension of the orbifold. hm…
Thanks! I have briefly recorded it here. But wasn’t there also a recent preprint on the arXiv on a statement like this? That’s what I seemed to remember having seen. But now I don’t find it.
Maybe you saw Global quotients among toric Deligne-Mumford stacks referring back to that book:
we restrict ourselves throughout this paper to orbifolds arising as quotient stacks [X/G], where G is a Lie group acting smoothly and properly on a smooth manifold X. It is worth noting that all reduced, or effective orbifolds – orbifolds whose local isotropy groups act effectively – are known to be of this type by a frame-bundle construction (see, for example, [1]),
Andre Henriques was studying this kind of questions on orbifolds as quotients by Lie groups (and also the case with a gerbe involved, needed to cover the non-effective case) in his thesis (among other questions on not-necessarily effective orbifolds), as I vaguely remember from old days.
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