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I got a question at the email address from someone who read the arXiv copy of my 2006 dissertation.
I was wondering why category physicists prefer 2-groupoids to double groupoids in applications to string-like theories. Isn’t a D-brane more like a section of a “double fibre bundle” than a 2-bundle?
I have wondered this myself occasionally. But not having a good physical intuition for branes one way or another, and preferring -categories to double categories as mathematical objects, I’ve been happy to stick to -bundles. So it’s a good question, but I don’t know the answer.
After chatting about cat^n groups on MO, I was wondering if it is feasible to do a bundle theory with cat^2 groups as the ’groups’ which are acting. It may be related to the double category point at the next level.
Could this have to do with shapes? It seems (I could be wrong) 2-groupoids are typically associated with simplices or bigons while double categories seem to be associated with squares. Squares seem to be less studied than simplices and bigons.
Just thinking out loud…
@Eric Just as I was. :-)
@Toby In the double category associated with a 2-groupoid the two directions look pretty much the same sine there will be ’connection’ between them. Perhaps the brane viewpoint has a distinguishable direction so two bundles with different information in them. Ronnie used to mention something that Kirill MacKenzie used to look at based on even more ancient ideas of Pradines, but as my wording suggests this was not yesterday!
I was wondering why category physicists prefer 2-groupoids to double groupoids in applications to string-like theories.
A double groupoid is just a particular way to present a 2-groupoid.
Isn’t a D-brane more like a section of a “double fibre bundle” than a 2-bundle?
In full beauty the field on the D-brane is a cocycle in twisted K-theory. Saying that properly requires talking -groupoids.
Hi Urs,
Once in a while (like after reading your comment #5) I get the sense that learning -categories should actually be simpler than learning -categories for some finite . You’ve probably made such an argument before, which would explain my sense of deja vu as I write this. Is that a fair statement?
The important bit is to distinguish between the abstract concept and its model. There is one abstract concept “-groupoid” and its abstract special cases “-groupoid” and then there are lots of models and incarnations of this. Trouble arises when mistaking the models for the concept.
That’s the problem of evil, by the way ;-)
To save Mike from saying it:
in my above comment I am implicitly assuming that by “double groupoid” we mean one where horizontal and vertical morphisms come from the same groupoid. There are more general double groupoids.
For a better comment I would need to know what exactly the original anonymous question was referring to, precisely.
The important bit is to distinguish between the abstract concept and its model.
Gah :o
I hope to understand statements like this some day. You say it like it should be so obvious, but there is a “considerable” effort required before the words will even parse correctly. I’m definitely nowhere near being able to parse the words.
Anyway, we’ve moved to a new flat that does not require a train ride (I walk to work), but this is a temporary move while we shop around for a flat to buy. We just put a deposit on a flat that would require an hour train ride each way. If we go through with it, I’ll have plenty of time to study this stuff :)
Urs wrote:
A double groupoid is just a particular way to present a 2-groupoid.
What ?? Double category is more general than a 2-category, so why the invertibility of arrows would force that double groupoid reduces to the 2-groupoid case in which ??
Zoran, see #8.
Why would I call it double groupoid then if I mean the special case when it reduces to a simpler thing ? If somebody asks about the difference he wants the case when the generalities differ. But then the question is just at the right place ? Why would the brane business lead to such a degeneration from (general) double groupoid to 2-groupoid (in say, vertical direction).
Zoran,
I read the question as being about the equivalent situation. That’s where i can make sense of the words “prefer” and “double fibre bundle”. Also, twisted bundles on branes are sections of 2-bundles. So if the other structure is equivalent to that…
But if you know what precisely the question was referring to, I’d be interested.
I did not spot the word “prefer” indeed as something very formal here. My understanding still is that the question is about how much D-brane geometry suggests the combinatorics of arrows in double category as most appropriate. You see, Segal had first definition of rational CFT without laxness in 2-categorical sense built in (what was inconsistent). For him this was essentially OK. But truly consistent mathematics approach has to deal appropriately with the 2-categorical aspect (if I understood Igor Križ and others right) of gluing of the analytic surfaces. Segal still prefers his version without the complication. Not that it is equivalent, or even literally correct to the lax version, but there are so many other aspects he is concerned with. It is like when somebody prefers doing category theory without reference to size/set theoretic issues. As long as one is far from tricky issues truly depending on size it is OK; but the two approaches are, in the intended generality, not equivalent.
(As far as 11, I skipped reading 8 originally because in the first line you referred to Mike. If you want me not to skip your clarifications, please do not hint in line 1 about addressing the text to another specific person.)
Of course, we should not dwell on the question any more if you do not see in which aspect the person could see D-brane geometry more akin to combinatorics of double categories. I know to little to see it.
I also interpreted the question as Zoran did: not why people prefer one representation to another for a thing, but why people prefer to study one thing rather than another thing.
Perhaps the answer to the original question in Urs’s
the field on the D-brane is a cocycle in twisted K-theory
but then more details might also be nice.
I have replied to the person who originally sent me the question and pointed them to this discussion. So perhaps they will chime in and clarify the question.
So perhaps they will chime in and clarify the question.
Please do chime in. I’d love to hear the original person’s thoughts.
but then more details might also be nice.
Some day I’ll write more on D-branes and twisted differential K-theory into the nLab. A good reference is Freed’s Dirac charge quantization and generalized differential cohomology.
The general statement is that gauge fields in physics, and higher gauge fields in supergravity and string theory, are cocycles in flavors of differential cohomology.
But let’s see, now I have a question:
double-groupoids, triple groupoids, -fold groupoids, in the weakest sense of the word, what do they form? Is their homotopy theory different from that of 2-types, 3-types, -types?
If it is different, what is it? If it is not different, then using these yields the same notion of principal infinity-bundle and connection on an infinity-bundle as any other model.
At cat-n-group it says that “n-fold groups” model the homotopy types of connected (n+1)-types.
That concerns the strict version (all the more remarkable).
If -fold groups in full generality are just another model for groupal homotopy -types, then -fold principal bundles in full generality will just be another model for -bundles.
@Urs Cat-groups are (n+1)-fold groupoids and model (n+1)-types, extending (and I always have to do the check here) cat-groups are double groupoids (with special properties of course) and model 2-types.
My question is do (n+1)-fold bundles exist that are somehow torsors for a Cat^n group action and would that be useful. Cat^n groups in their crossed n-cube form are neat to work with as they are almost just groups with n normal subgroups picked out analoguous to crossed modules being almost (group,normal subgroup) pairs.
Cat-groups are (n+1)-fold groupoids and model (n+1)-types, extending (and I always have to do the check here) cat-groups are double groupoids (with special properties of course) and model 2-types.
Yes, I know. That’s why I said: it looks like -principal -bundles in the world of -fold groupoids are principal -bundles over -types hence should be equivalent to any other definition of -bundles.
It would be easy to decide once we’d have a model structure on -fold groupoids for all . Did anyone consider this?
You had written : ’ “n-fold groups” model the homotopy types of connected (n+1)-types,” hence my reaction. I think that someone at Granada did this, but looking at a copy of the thesis in which I thought it was I cannot find it. Julia Cabello and Antonio Garcon did something for n-hypergroupoids but not as far as I can tell for cat^n groupoids.
Here in nlab we talk mainly about -principal bundles, not for higher . The model structure Urs points will present the -categorical structure. Thus, while the -groupoids for various can be possibly quite comparable, it is possible that in -world higher categorical bundles can be quite dependent on weather we take iterated or different kind of higher categories, and this is not detectable only by the model structure. All theh comparison theorems like Eilenberg-Zilber and so on are in homotopy world, so in more precise world of I would expect that there are possibly substantial differences. Of course, it is only a speculation.
I believe that Eilenberg-Zilber can be made considerably more explicit than it is usually made out to be. In that case with explicit constructions at the homotopy level we could be looking at doing calculations explicitly instead of working module WEAK equivalences. This has always seemed to me to be a good thing to try for.
I am not sure that the n-type formulation is that much related to the one. I am confused on this but crossed complexes are ’2-types with tails’ so are linearised above level 2. It is not that the k-cells are equivalences up there above level 2, it is that in the simplicial nerve of the thing there are unique thin fillers above a certain level. This means Whitehead products and stuff like that are trivial up there. As I said this question of linearising above a dimension does not feel to be that related to the indexation.
Crossed complexes are linearised above dimension 2 and have a beautiful homotopy structure, and correspond to -groupoids, but 2-crossed complexes correspond to …. what? As I said, I am not sure.
You had written : “n-fold groups” model the homotopy types of connected (n+1)-types
Oh, I see. I had put the quotation marks to indicate that this is not the standard terminmology. But it seemed to me to be the suggestive terminology.
Here in nlab we talk mainly about (∞,1)-principal bundles, not (∞,n) for higher n. The model structure Urs points will present the (∞,1)-categorical structure
I think for principal -bundles the -setup is by definition sufficient. Because these are things which should have an -groupoid of cocycles, precisely what an -category provides.
This is different for -vector bundles. They naturally live in an -topos, due to the fact that there are interesting non-invertible morphisms between vector bundles ( or not).
Urs, you see I wrote in 25
while the -groupoids for various can be possibly quite comparable, it is possible that in -world higher categorical bundles can be quite dependent on weather we take iterated or different kind of higher categories
and you are answering about higher groupoidal bundles. You see, I meant the principal bundles with structure -category rather than -groupoid.
Of course you may object to talk about principal bundles with more general coefficients than n-groupoids (I know that you do not like the idea when people consider e.g. categorical bundles, and I see that you would like that every morphism between generalized torsors is an equivalence what seems not to be in some meaningful generalizations). Let me throw few sources of justification, instead of an argument. The various modern ramifications of Galois theory goes beyond the groupoidal case, and principal bundles are geometric way of looking at such questions, so I do consider it relevant. Also many descent problems give higher categories of descent data, rather than descent groupoids and it is nice to talk via principal bundles about those descent problems. For a bit inappropriate example in similar direction, the community I am coming from considers various things like noncommutative Hopf algebroids as coefficients for noncommutative principal bundles, and we have also coalgebra-Galois extensions and so on. Of course in discussion above I did not mean so general descent and bundle theories but rather just n-categorical bundles. For example, Igor in ongoing work is doing bicategorical bundles (not bigroupoidal) and has some nice Galois type theorems there and connects some of those to questions of Galois and homotopy theories. Nothing noncommutative there.
Can there be a principality condition
for non-groupal ?
As we all know, the groupoidal case is the replacement of simultaneous satisfaction of freeness and transitivity of the action. Given a non-groupoidal 1-category , Street defines an -torsor (trivialized on a cover) by a bit different definition (page 25 in the file)
Street’s notion has its role in the categorical Galois theory of Janelidze as advanced also in their joint work.
Urs’s Utrecht colleague Ieke Moerdijk has instead in his book (Classifying spaces and classifying topoi, LNM 1616), page 2, a different notion/definition of an -torsor by listing separately adapted condition of freeness, transitiveness (and nonemptiness). I do not know if his 3 conditions can be replaced by a single condition similar like in groupoidal case. His definition looks very natural and it plays role in the torsor reinterpretation of a deep Diaconescu’s theorem.
Yes, and it always seemed to me that these torsors over a category are principal with respect to the oo-groupoid presented by i under the Thomason structure.
But we had this discussion before.
I vote for not changing the meaning of the term “principal action”. But of course it’s just terminology. Not so important.
But this is just a digression from my point. My point was the plausible speculation that there is a homotopy-relavant higher categorical aspect in which higher categories and n-tuple categories could have a non-equivalent role at level .
But we had this discussion before.
I did not get this - I mean if one looks at the category of all “torsors” over a 1-category , say in sense of Moerdijk, how is this category equivalent as -category to -category of torsors over whatever -groupoid ? Or is it equivalent to some intrinsically defined subcategory of it ? (what would be strange but interesting, I mean defining a cohomology via a distinguished subcategory of cocycles).
For me, one likes to talk about spaces in terms of categories of sheaves. Principality implies the descent along torsors for any stack over the site of spaces; (category of) equivariant objects on the total space of the torsor correspond to the category of usual objects on the quotient. This is related to Galois theory, and this extends well an d straightfoward even to many noncommutative situations like Hopf algebroids, noncommutative stacks and so on. The descent along to Hopf algebra version of principal bundles is a very active field since 1990 article of Hans-Juergen Schneider titled Principal homogeneous spaces for Hopf algebras in Israel J. Math (see Hopf-Galois extension). Urs said:
I vote for not changing the meaning of the term “principal action”. But of course it’s just terminology.
Well, for principal action I do not know, but under torsor, one should list things like Street’s definition over categories. There is a famous theorem that relates the stack property with the behaviour of limits weighted by such torsors (am I right ? I mean the same theorem does not hold if one takes just limits weighted by groupoidal torsors ? Mike ? ). On the other hand terminology “torsor” for Hopf algebroid and similar situations is well established, though one does not have that the category of all such is a groupoid: in some cases, I think, there are morphisms which are not invertible among torsors.
I hope you understand that I disagree that it is just a terminology. It is a view about what is a natural generality for studying phenomena related to principality, Galois theory and so on and I think these issues are extremely interesting to discuss and work on.
The principality condition that Urs mentions in #29 is just the statement that is a saturated anafunctor. Makkai calls an anafunctor with a category as codomain saturated when the underlying anafunctor between cores is saturated. He does this for logical reasons, namely that saturated anafunctors’ values are determined precisely up to isomorphism (and with a level of uniqueness). But this must be very different from the sort of things that Zoran has in mind. This is the sort of thing I was trying to address with my discussion on flat functors, because somewhere on the Lab I read that the generalisation of a -torsor is a flat functor . I’ll think a little about Street’s definition, and see if it is something I already know. Is the Moerdijk definition short and simple? If so, could someone write it in here (or better, and nLab page)?
David, entry torsor with structure category with Moerdijk’s definition exists from March 28, 2009. I just updated it recording the Street’s reference. I’ll try to discuss the flatness aspect after I remind myself a bit. Now I have to do some errants first.
how is this category equivalent as -category to -category of torsors
No, there won’t be such an equivalence. What I mean is that if you replace morphisms by spans as happens in these constructions, then you are really working with the groupoidification of the category.
I do not expect that all the nice (say Galois type) information which works nice for torsors over categories would survive and will be extractable from groupoidified version. For example consider a cartesian category and the 2-category of pseudofunctors . Street proves that an object of is a stack (edit: where covers are elements of a left calculus of fractions on as in Street’s paper) iff admits all colimits weighted by -torsors from to (spans already involved, Street’s definition; is an internal category in ). I do not see how such nice theorems could be preserved under complications with additional spans in the picture. If you have a nice theorem like some category equivalence what you seem not to claim than OK, but otherwise I see no way to replace non-groupoidal situation with groupoidal for the purposes like Galois theory and classification of locally constant objects which is measured by cocycles corresponding to principal actions.
Anyway, I updated the torsor with structure category with a section with the Street’s definition. The diagram is via codecogs.
By the way, I rolled back to codecogs version at split equalizer. The SVG replaced by anonymous user had arrows at geometrically displaced points and the sourcecode was not useful for human reading and reusing in other diagrams. If I have a diagram involving a split equalizer I want to take a sourcecode there and reuse it in another wiki page.
David, about flatness, Street wrote his definition in the preprint
and in some later papers. He recently extended and revised his 1981 preprint with the help of Dominic Verity:
There they say
Section 3 brings us to the notion of a -torsor where is a category in a finitely complete category . In topos theory it has been suggested that (in view of Section 8.3 of Johnstone 1977, Topos theory) a -torsor in a topos should perhaps be a flat discrete fibration over . However, we know of no results concerning such torsors (a la Johnstone, underlined by Z. Š.) with not a groupoid except that the topos would then be the -torsor classifier. Instead, the definition taken here is that a -torsor is a discrete fibration over which is locally representable (that is, on passing to a cover, looks like some . This leads to a category of -torsors. We contend that one-dimensional cohomology is the study of the categories .
Next paragraph after that is relevant to my discussion with Urs,
It will be clear (after Theorem 4.9) that a -torsor amounts precisely to a -torsor where is the groupoid obtained from by restricting to invertible morphisms in . So one may ask: is it not sufficient to consider -torsors where is a groupoid? No! For what we in fact have is an equivalence of categories and it is not possible to glean all the important information about a category from the associated groupoid . For example, in Section 6 we give a technique for deriving results about finiteness in a topos, vector bundles, and other local structures, which could not be obtained by restricting to groupoids.
Urs, this is kind of things I was talking about: at the level of objects one may have enough, but there are noninvertible arrows among torsors with structure category ! Maybe you meant the same in one of your comments above. Among applications he talks in the last sentence about torsor method to classify locally constant structures. You pointed above that for associated bundles one also has non-invertibility. Well, it seems that Street says he knows how to treat this via torsors in categorical sense.
Well, it seems that Street says he knows how to treat this via torsors in categorical sense.
I mean, it is not a problem to do associating, but the classification of associated things is not the same as the classification of original torsors. However it is my impression that one of the applications alluded to is that if we consider the associated bundles as locally constant structures then for the noninvertible morphisms among those one can look at nonabelian cohomology in the sense of classification of certain torsors with structure category. This is a side comment which may be wrong impression.
P.S. I updated entry 39.
I’m not surprised that Street ends up with -torsors being just -torsors, as in the first paper you referenced, he said that torsors trivialised over are pretty much just internal functors (i.e. a Cech cocycle) with some local triviality. Since is a groupoid, it only sees . Apart from the way he defines local triviality, which I’ll look at today, it looks like Street’s notion lines up with Makkai’s (implicit) definition - via saturated anafunctors. Thanks for tracking these references down, Zoran!
The OP (Rachel Martins) responds!
Hi, thanks for posting my question Toby!
The physics scenario that seemed to ask for a double category is…..imagine if you wanted to give a categorical model for the Higgs interacting with a cloud of ups and downs, themselves interacting strong and weak (but to simplify I’m only considering weak for now and restricting to internal space). If wanted to model the gauge bosons and the Higgs with 1 and 2-morphisms respectively then I suppose what would be needed in this description would be a double groupoid in the general sense so not a 2-groupoid (nor a 2-group) because you want the ups and downs to begin as left and right and end up with right or left chirality, which requires 4 objects not 2…..I hope I have begun to construct something related to this in the contexts of spectral triples and Fell bundles (sort of algebra bundles over groupoids) but this physics picture also seems intuitively like a D-brane doesn’t it? - Especially as the gauge bosons and Higgs are differential forms in Connes and Chamseddine’s picture.
On the other hand, very naive I know, don’t processes in space-time also look more like paths over (general) double groupoids too?
I started looking at the paper Urs suggested.
Rachel: If you’re still having technical problems joining and posting to this forum, send me details by email and I may be able to help. Or email Andrew Stacey.
Hi Rachel,
John wrote some step-by-step instruction to help people join his Azimuth forum. I would imagine the process should be pretty much the same for the nForum (with proper substitutions Azimuth Forum -> nForum of course)
Hi Rachel,
At one point I thought (not deeply, though) that double groupoids could be used to model for spacetime so as to make a difference between causal paths (i.e. worldlines) and spacelike paths, which are used to capture the topology of (local) spacelike slices. I don’t know if it would work but there is certainly a difference between the sort of paths one wants to look at parallel transport along (due to fermion interactions with gauge fields for example) and the sort of paths that tell you there is a hole in your space at the present time (in your frame of reference etc etc).
Zoran, thanks for that reference. So it does sound as if they consider category-bundles associated to groupoid-principal bundes. But I should have a look. (Not likely to be soon, though.)
Toby, thanks for forwarding that attempt at a clarification.
Rachel, to be frank, though, I am not following the motivation yet. The paragraph that I do follow is this question:
don’t processes in space-time also look more like paths over (general) double groupoids too?
So this is getting to an important point: maybe a double groupoid of 2-paths in a space looks more natural, but it is just a different but equivalent incarnation of the 2-groupoid of paths.
David, same comment to you: certainly it is a plausible idea that causal structure on spacetime can be encoded in terms of non-invertible morphisms, see for instance the discussion at The path (oo,1)-category of a Lorentzian space, but I don’t see that it matters for this whether we talk double categories or 2-categories.
To further the discussion, I think it would be really useful if somebody promiting n-fold categories in applications here would present us with some concrete example that he or she is thinking of, which is supposedly interesting for some application and which supposedly cannot be modeled by 2-categories.
I suspect that it is not that things cannot be modelled by 2-categories but that some double category model may be more transparently ’physical’.
I suspect that it is not that things cannot be modelled by 2-categories but that some double category model may be more transparently ’physical’.
So that’s what I have been suggesting all along in this thread, that using double-stuff here is just a certain choice of model for using 2-stuff.
My answer to the original question: “Why not define double bundles instead of 2-bundles?” would be “It doesn’t make a difference, up to equivalence.” But then others seemed to argue that it can make a difference. The details of that will depend on precisely what 3-category of double-things we speak about. I can certainly imagine that you can dream up a definition of that which would make for an inequivalent theory, but I am yet to see some motivation for why that would be good or expected.
I know this kind of discussion from the old days when I started giving talks about 2-transport. The first thing people would ask me when I drew a bigon always was: why don’t you draw a square. The answer was: you can always think of your square as a bigon. It doesn’t matter.
(Of course there is some technical fine print to “it doesn’t matter”. )
I think the reaction may be related to non-reversible phenomena in some directions. I tried in a paper to suggest that simplicially enriched categories were useful in modelling evolving situations, where a ’situation’ is ’spatial’ for instance and the change is not naturally reversible. This does make the physically silly assumption of a preferred ’direction’, thought of as ’time’, but is quite related to directed homotopy as put forward by Grandis. In those cases as one direction is not a groupoid, you cannot safely change (?) the double category to a 2-category, can you?
I tried in a paper to suggest that simplicially enriched categories were useful in modelling evolving situations, where a ’situation’ is ’spatial’ for instance and the change is not naturally reversible.
Sure. And simplicially enriched categories are a model for -categories. And Just as we may think of an -category = -groupoid as a space in homotopy theory, we may think of an -category as a space in directed homotopy theory. (Personally, i would take that as the definition of “directed homotopy theory”, but that’s another discussion).
Specifically, let be a topological poset, hence a causal topological space, for instance that underlying a causal Lorentzian space. Define its fundamental -category to be given by the quasi-category whose -norphisms are maps
such that the spine of is mapped to a future directed path (meaning that for in the spine we have in the poset ).
This clealry is a quasi-category, since the retracts of inner horns
are order-preserving on the spines (right?).
Next you write:
In those cases as one direction is not a groupoid, you cannot safely change (?) the double category to a 2-category, can you?
How did you jump from your discussion of directed homotopy theory in terms of simplicially enriched categories to double categories? I am not sure I see the argument that you intend to make.
In that model, one direction is not reversible, so if you try to use ’double’ techniques you are likely to think in terms of a double category with one direction being groupoidal, whilst the other is not. (NB. I am trying here to see (and to model) why someone may prefer double categories to 2-categories, not to argue that the 2-cat version is not equivalent.) Of course, perhaps a more natural way is to pass to the fundamental groupoid of each hom so as to get what Baues et al call a track 2-category, i.e., 2-cells are vertically invertible, so is just a groupoid enriched category.
I wonder about looking at directed homotopy from that angle as the discussion in Marco’s book makes it sort of clear that loops in the basic category can cause some problems in directed homotopy if you take that line, but I have a disagreement with some of the points that Marco makes. (Perhaps I do not understand his view well enough.)
I put the above definition of the fundamental (infinity,1)-category of a directed space into the nLab. It’s supposed to be the obvious and evident definition, but please have a critical look.
41 David
Apart from the way he defines local triviality, which I’ll look at today, it looks like Street’s notion lines up with Makkai’s (implicit) definition - via saturated anafunctors. Thanks for tracking these references down, Zoran!
Look, in the groupoid case, according to Street-Verity article this definition is not due Street but earlier due Joyal. Joyal gave this definition in a talk around 1980 which they quote. Then Street went on into repairing category case in Johnstone with the input from Joyal’s idea. That is my understanding of their footnotes. The point is that while the objects (cocycles) correspond to the gropupoid cocycles, the morphisms among those do not. This is what they emphasis in the quoted text above. Now are you claiming that one can enhance in category case the morphisms among the anafunctors to include more of them so that one has noninvertible ones in the nongroupoidal case ?
You have four corners of a square labelled, u_L, d_L, u_R, d_R.
So we are talking quarks now, I suppose. Okay, so you have a square. Where is the double category? What do you want to compose with what?
With only two nodes, the up and downs can shuffle into one another but the change as left goes to right disappears.
I gather this line is meant to be the argument why something has to be a double category instead of a 2-category?
No, let me be frank, all this sounds confused. Or at least way too vague to be either right or wrong. If we want to continue this discussion, we first need to get our feet on the ground and say something that – if maybe not precise – is a tad more tangible.
Now are you claiming that one can enhance in category case the morphisms among the anafunctors to include more of them so that one has noninvertible ones in the nongroupoidal case ?
Yes - there are non-invertible transformations between anafunctors where the codomain is a non-groupoidal category, simply because transformations between anafunctors , are just ordinary (internal) natural transformations between and .
if you wanted to use higher groupoids to describe the Higgs […]
How? Why?
[…] taking a left-handed quark undergoing the weak interaction (path from up to down say), into a right-handed one undergoing the weak interaction, then wouldn’t you need (some kind of) bundle over a double groupoid rather than over a 2-groupoid?
I can’t answer the question because I don’t follow its assumptions. In which sense do you imagine to “describe the Higgs” in terms of any groupoids at all?
You need to tell us what you have in mind. What are the objects that you have in mind? What are the morphisms that you have in mind? What is the composition operation on these morphisms supposed to be?
Here is what I can guess:
you are thinking of objects as forming the set of species of fundamental particles in some model
then you think of 1-morphisms as being certain reaction processes that these can undergo. So it seems you want to think of an arrow
and of an arrow
Are these arrows now supposed to be morphisms? What are the other morphisms, then? What are their labels? What is the composition operation on them?
(Do you by any chance maybe imagine some aspect of a monoidal category whose string diagram calculus would be Feynman diagrams?)
Next it seem to me that you are envisioning an arrow between arrows
Is that right? What would be other such 2-arrows? What do you envision the composition law on these 2-arrows to be? What horizonally? And what vertically?
Let me ask you bluntly: what do you actually know about double categories? Do you know how double categories “with connection” are equivalent to 2-categories?
Yeah I knew about the results by Brown and Spencer that double groupoids with folding or “connection” are equivalent to several things including 2-groups and that “holonomy” can be shown to correspond to holonomy in a fibre bundle
Notice the term “connection” on a double category in the sense of Brown and Spencer has nothing to do with connections on fiber bundles and their holonomy.
When I mentioned that I was using C*-categories and Fell bundles I meant to answer your question as to what the morphisms would be and how they would compose.
But did you? So your groupoid is supposed to have the extra structure of a -catgeory. But what is the underlying groupoid itself? What are the objects, what are the morphisms, how should they compose?
You don’t need to answer that. But that’s what i was trying to find out from you.
Notice the term “connection” on a double category in the sense of Brown and Spencer has nothing to do with connections on fiber bundles and their holonomy.
I know that is true in principle and in general but I thought that I had seen an example where the two terminologies coincided and I guessed that must have been the motivation for Brown and Spencer having used that word.
It’s a Fell bundle C-category *over a finite groupoid. If the space were (compact) “space-time” instead of internal space, it would be a compact groupoid of paths on a (Hausdorff) space. I am being cautious about saying double Fell bundle over double groupoid because I haven’t finished defining that properly!
…why use (higher) categories including groupoids to model the Higgs mechanism?
…because some view the Higgs as the internal space counterpart of the graviton and if people want to quantise gravity on the whole lot of space-time plus internal space, then I guess categorifying spectral triples might be one first step. In my naïve stuff, it seems that 1-categories only are needed but there’s so much people are doing with higher categories and space-time it would seem odd if internal space didn’t need them as well.
(…I registered and see that a “space-time” double groupoid (i.e. a double category of paths on a (locally) compact Hausdorff space where all morphisms are iso) is equivalent to a 2-groupoid.)
@Rachel I seem to remember that Ronnie chose the terminology because for the imagery of linking horizontal and vertical and(?) also because of some work of a differential geometer (Wirsic? or similar) who had adapted Ehresmann’s treatment of connections to something that looked feasible.
…because some view the Higgs as the internal space counterpart of the graviton
I like that point of view! Except for the extra subtlety that we are passing to non-commutative spaces, this means identifying the Higgs as one component of a connection on spacetime.
and if people want to quantise gravity on the whole lot of space-time plus internal space, then I guess categorifying spectral triples might be one first step.
You need to categorify spectral triple if you want to describe not the QFT of point quanta, but that of string quanta. This is in work of Wendlandt, Kontsevich and Soibelman. Yan Soibelman is currently writing up something on this for our book.
Urs is co-writing a book?! With Soibelman?
Googling and looking them up on Arxiv, I found notes by 2 of those authors on algebras, is that specifically what Urs is referring to regarding categorification of spectral triples? Glancing though, it looks very sophisticated mathematically and they seem to have a similar point of view to that of Bertozinni et al who consider spectral triples as objects and morphisms going between triples, whereas in my (naïve as I keep saying) picture a spectral triple is in and of itself a category.
Ta to the other people who also directed messages to me.
Urs is coediting the AMS volume Mathematical Foundations of Quantum Field and Perturbative String Theory where Soibelman is one of the main contributors.
The ideas of Kontsevich on relation between CFT, Riemann geometry and some categorification via spectral triples and so on is not available online; Soibelman collaborated with Kontsevich and wrote a long sketch of those ideas. Urs had that, now few years old, manuscript which I saw at some point. We will see a better version in the volume.
I look forward to seeing it when it’s finished.
I look forward to seeing it when it’s finished.
We are closing in on it. Now we have reached the stage that the official deadline for authors is in a few weeks and we have heard from most of them requests that they need more time. So everything is going according to plan. ;-)
Wow, great, I thought it will take another couple of years to see it!
another couple of years
No, no, official deadline for submission of manuscripts was originally some time this autumn, now it is something like new year. I have little idea and control how exactly things will proceed, that’s now much in the AMS hands, but I guess next summer we should be able to hold the thing in our hands.
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