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June 19, 2008

Fundamental 2-Groups and 2-Covering Spaces

Posted by John Baez

guest post by David Roberts

This is a talk prepared for the Categorical Groups workshop in Barcelona. With the technology at hand, why let funding issues stop me from presenting it? You can see the slides here:

Fundamental 2-groups and 2-covering spaces

Abstract: By our knowledge of fundamental groups we can discover the properties of covering spaces. We turn this on its head, and use 2-covering spaces, which have groupoids for fibres instead of sets, and consider fundamental 2-groups. In the process, it is necessary to define the fundamental 2-group of a topological groupoid, and investigate homotopy lifting properties. The induced map from the 2-covering space to the base leads us to a consistent notion of sub-2-group.

Thanks are due to everyone at the Café for comments and inspiration over the time this and related work was done.

Covering spaces are determined by what happens at the level of fundamental groupoids. If the base space is path-connected and pointed, we can be slack and just use fundamental groups. As a disclaimer, I’m going to sweep all finicky considerations like “semi-locally simply connected”, and its generalisations, under the carpet, and assume for now that all spaces are nice enough (all locally contractible spaces are nice enough). It is a well-known theorem that a pointed, path-connected covering space is uniquely determined by a subgroup of the fundamental group of the base, and vice-versa. If we forget the point, then it is determined by a conjugacy class of subgroups.

Instrumental in proving this result is the path lifting property of covering spaces. If we knew straight away that a covering space p:YX p:Y \to X is a fibration with fibre FF, then the answer leaps out at us using the long exact sequence in homotopy: 0=π 1(F)π 1(Y)π 1(X) 0 = \pi_1(F) \to \pi_1(Y) \to \pi_1(X) voila! π 1(p)\pi_1(p) is injective since FF is a discrete space, and hence has simply connected path components. The definition of a covering space is a long way from saying it is a fibration, so how is it we get the result? More importantly, what does this tell us about 2-covering spaces?

What we need to do is to bump the categorical dimension up a notch on everything we are working with. Thus we replace spaces with topological groupoids and groups with 2-groups. For me topological groupoids are groupoids internal to Top\mathbf{Top}, not groupoids enriched in Top\mathbf{Top}. The tricky part is getting a homotopy 2-group from a topological groupoid, since the category of topological groupoids one first thinks of does not have enough equivalences. There are ways to sort this out using intimidating words like ‘bicategory of fractions’ or ‘anafunctors’, but paths and homotopies are simple enough to sort out without such technology.

Recall that given a space MM with a cover WMW \to M we can form the (topological) groupoid W [2]W^{[2]} with object space WW and a unique morphism between any two points in the fibre over MM. The spaces MM we are interested in are the interval, square and cube, I kI^k for k=1,2,3k=1,2,3. Also, we will only be considering finite closed covers by intervals/polygons/polyhedrons.

I call a groupoid 𝔭\mathfrak{p} arising from a closed cover of II by intervals a partition groupoid, as it is determined by a partition of II. A path in a topological groupoid XX is just a functor 𝔭X. \mathfrak{p} \to X. Unwrapping the definition, this is precisely the sort of path for open covers that Urs has discussed, and that Moerdijk-Mrcun use to define the fundamental groupoid of a Lie groupoid. A path looks like a finite sequence of paths in the object space with the ‘jumps’ bridged by morphisms - these are the images of the morphisms of 𝔭\mathfrak{p}.

Given a pair of paths that agree at 0 and 1 (this is unambiguous, as the fibres of the cover over 0,1 are single points), we can try to define a homotopy between them. This requires defining the right sort of cover of the square. In my slides I give a list of conditions, including one that makes all vertices at most trivalent, but it is easier just to give a picture of an example:

polygonal cover of the square

Let 𝔥\mathfrak{h} be a groupoid arising from such a cover of the square. The two subgroupoids which arise from restricting this cover to the top and bottom edges of the square are called the final edge and initial edge respectively. A functor 𝔥X \mathfrak{h} \to X is a homotopy between the paths defined by the initial and final edges if the vertical edges of the square (or rather, the subgroupoids defined thereby) are mapped to XX through the trivial groupoid {*}\{*\}. Call the images of these two edges xx and yy. We can think of such a homotopy as a string diagram, since each edge is mapped to a path in the arrow space of XX, and each polygon is mapped to the object space of XX, the paths of arrows linking the edges of the polygons. At the trivalent vertices, a cocycle condition must be satisfied. “But this isn’t a string diagram,” I hear you cry, “it has lines escaping the edges!” But remember that each outside vertical edge is fixed at a point, so we can imagine that they are mapped to the identity arrows, and there are invisible borders on the square, mapped to xx and yy. If our topological groupoid is an ordinary groupoid, then this reduces to the usual notion of string diagram. Or if XX has one object, then this is a string diagram for a topological group. I often drop the adjective ‘modified’, which I use in the slides.

We can do the same thing for homotopies between string diagrams, but this time it is too hard for me to draw. Suffice it to say that homotopies between string diagrams use covers that look like covers of the cube by polyhedrons coming from something like a spin foam. In calculations these are generally rectilinear, but we allow more general such things.

Homotopy classes of string diagrams, where the homotopy is fixed around the edges of the square, are called 2-tracks. A 2-track between two constant paths at a point is what is generally known as an element of the second homotopy group, at least when we are dealing with good old spaces. To introduce the fundamental 2-group Π 2(X,x)\Pi_2(X,x) of a topological groupoid XX at the point xx, I will first describe the case when the topological groupoid is a space.

One description of the fundamental 2-group of a space, due to Cegarra and Garzon, is the fundamental groupoid of the loop space. This has based loops for objects, and (using the cartesian closed nature of your favourite version of Top\mathbf{Top}) 2-tracks for morphisms. The product in the 2-group is induced by the loop space product, which we know is homotopy associative and unital. The 2-tracks represented by the usual homotopies expressing this become the coherence morphisms in the 2-group. This description relies on the topology on the loop space, which is not something we have for the analogous object for topological groupoids. However, we can use the description using 2-tracks, which is due to Hardie-Kamps-Kieboom (used in defining the fundamental bigroupoid of a space). Simply replace paths in the space with paths in the topological groupoid, and 2-tracks in the space with 2-tracks in the topological groupoid. Again, this is not in general a strict 2-group.

A functor f:(X,x)(Y,y)f:(X,x) \to (Y,y) of pointed topological groupoids induces a functor Π 2(X,x)Π 2(Y,y)\Pi_2(X,x) \to \Pi_2(Y,y) of 2-groups. It is fairly immediate that the fundamental 2-group of a space, considered as a topological groupoid with only identity arrows, is equivalent to the usual one.

The point of all of this is that we want to relate the fundamental 2-group of a 2-covering space to that of the base. But I haven’t told you what a 2-covering space is, yet!

Definition: A 2-covering space is a functor of topological groupoids p:ZXp:Z \to X such that XX is a space, and there is an open cover U αX\coprod U_\alpha \to X such that there is a collection of topologically discrete groupoids D αD_\alpha, and a weak equivalence U×D αZ| U α U \times D_\alpha \to Z\big|_{U_\alpha} over U αU_\alpha.

Whoa! “Now what’s a weak equivalence?” I hear you grumble. Easy: it’s a fully faithful, essentially surjective functor such that the ‘surjective’ part admits local sections. This implies the fibres of pp are groupoids weakly equivalent to topologically discrete groupoids. This will help us in our task to categorify the old result about covering spaces above, since a topologically discrete groupoid is a 1-type, much as the fibres of a covering space are 0-types.

First we prove this theorem:

Theorem: If f:DYf:D \to Y is a weak equivalence and DD is topologically discrete, dDd \in D, the induced functor Π 2(D,d)Π 2(Y,f(d))\Pi_2(D,d) \to \Pi_2(Y,f(d)) is an equivalence.

Then we can calculate that Π 2(D,d)π 1(D,d)=D(d,d)\Pi_2(D,d) \simeq \pi_1(D,d) = D(d,d), so the fibres of a 2-covering space have trivial π 2\pi_2. Or, to put it another way, there is at most one 2-track between paths in a fibre.

It is possible to lift paths to a 2-covering space, but now we can only define the starting point up to an arrow (which are all ‘vertical’ - they project to identity arrows). This is reminiscent of Dold fibrations, which lifts of homotopies exist, where the starting point can only be specified up to a vertical homotopy. The following theorem corresponds to the uniqueness of path lifting for 1-covering spaces. There is a groupoids worth of paths over a given path in the base, due to the natural transformations available to us.

Theorem: The groupoid of lifts of a given path in XX, starting at xXx\in X, is weakly equivalent to the fibre of ZXZ \to X over xx.

It is also possible to lift homotopies: given a paracompact space YY a homotopy h:Y×IXh:Y \times I \to X and an anafunctor Y×{0}ZY \times \{0\} \to Z covering hh, we can lift the homotopy so that it agrees with the existing lift up to an ananatural transformation. If the 2-covering space trivialises over a numerable cover, it is possible to drop the paracompactness assumption on YY. That is a whole lot of jargon, but if you like just replace YY with a square, and the anafunctor with a string diagram.

We then use this to show

Theorem: Let p:ZXp:Z \to X be a 2-covering space and zZz \in Z. The induced map of 2-groups Π 2(Z,z)Π 2(X,p(z)) \Pi_2(Z,z) \to \Pi_2(X,p(z)) is faithful.

My thanks go to Mathieu Dupont for disabusing me of a misconception I had when initially trying to prove an early, incorrect, version of this theorem.

This relates to the notion of sub-2-group discussed by others present at the categorical groups workshop, e.g. Carrasco–Garzon–Vitale.

Also, the (homotopy) quotient G//HG//H of a map of 2-groups j:HGj:H \to G is (equivalent to) a groupoid if and only if the induced homomorphism π 2(j)\pi_2(j) is injective. This is the case if jj is faithful. We could consider the homotopy quotient Π 2(X,p(z))//Π 2(Z,z) \Pi_2(X,p(z)) // \Pi_2(Z,z) and compare it to the fibre, but that is for another time.

We therefore state this slogan:

The fundamental 2-group of a 2-covering space is a sub-2-group of the fundamental 2-group of the base.

Posted at June 19, 2008 9:13 AM UTC

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19 Comments & 1 Trackback

Re: Fundamental 2-Groups and 2-Covering Spaces

Why “2”?

Excuse my complete ignorance.

Posted by: Christine Dantas on June 19, 2008 12:29 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Why “2”?

A “2” indicates that something is one step up in categorical dimension compared to the ordinary.

An ordinary covering space is a projection PX P \to X where the fiber over each point is a (finite) Set.

When in some structure sets are consistently replaced by categories, one says one has moved up in categorical dimension by one and speaks about a “2-structure”.

So the idea is: think of an ordinary covering space whose fibers are sets as a “1-covering space”.

Then, when the fibers are allowed to be categories, we have a “2-covering space”.

For more details but still pedagogical, try for instance Baez-Dolan:Categorification.

Posted by: Urs Schreiber on June 19, 2008 1:11 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Christine wrote:

Why “2”?

Urs gave you a good answer. Let me give you an obnoxious answer:

Because we can’t do “3” yet!

Not very enlightening… but let me explain. Every concept in ordinary mathematics, based on set theory, is potentially the first of a sequence of richer and more interesting concepts based on categories, 2-categories, 3-categories and so on. For example: a set with multiplication and inverses is a group. A category with multiplication an inverses is a 2-group. A 2-category with multiplication and inverses is a 3-group. And so on.

But, we often get tired before reaching 3. So, you’ll see a lot of talk about 2-things on this blog.

“Categorification” is all about climbing this ladder and showing how it sheds light on math and physics. In geometry, categorification often goes along with an increase in dimension: for example, categorifying particles gives strings. So, Urs wisely calls the string a ‘2-particle’. Categorifying again we get 2-branes, which he wisely calls ‘3-particles’. It’s just clearer, in the long run, to say

1-particle, 2-particle, 3-particle, …

instead of

particle, string, 2-brane, 3-brane, …

How is this related to David Roberts’ talk? You’ll see that David has drawn a picture which looks like a string worldsheet chopped up into a polygons. The reason, very simply, is that nn-groups show up as symmetries of nn-particles, and he’s looking at n=2n = 2.

Posted by: John Baez on June 19, 2008 3:06 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Thank you for your answers, they help a lot!

Now I have a clear place to start.

Since you were often talking about “n” (hence of course the name of the blog), I was wondering about all the interest in “2”. Of course it makes sense that “2” is the first level in raising the complexity of the “categorification ladder”, so that you are evidently trying to understand “simpler” structures first. But I had to ask the “obvious” to be sure, and in return I receive very interesting and clarifying answers, given my complete ignorance of the subject. Thanks a lot.

Posted by: Christine Dantas on June 19, 2008 3:57 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

I’ll take the opportunity and ask another very basic question. I’m reading “Lie Groups, Lie Algebras, and Some of Their Applications” by Robert Gilmore, and it was very illuminating for me to read right in the first chapter the various structures from sets to groups, vector spaces, etc to algebras.

If I understand it correctly, categorification uses the group concept in an increasingly abstract sense. Is it possible to add more structure to categorification by the use of the algebra concept, for instance? I ask this because algebra is also a very useful concept in physics, so if one is willing to recast physics under the categorification concept, how does algebra fits in?

I’m sorry if I am making some confusion here. I still have to read Baez-Dolan’s paper, so you may spare your time from answering this one while I get the paper and read it…

Posted by: Christine Dantas on June 19, 2008 4:18 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

categorification uses the group concept in an increasingly abstract sense.

I’d put it more generally: it generalizes the concept of monoid, i.e. of a set with a product operation on it.

Is it possible to add more structure to categorification by the use of the algebra concept

Yes, certainly.

A group is a category with a single object all whose morphisms are invertible.

An algebra is a category with a single object such that the space of morphisms is a vector space.

Starting from this point of view and categorifying in various way leads to homological algebra, DG-categories, A A_\infty-categories, spaceoids and many other things.

Posted by: Urs Schreiber on June 19, 2008 5:56 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Thanks, that is very interesting and helpful.

Best regards,

Posted by: Christine Dantas on June 20, 2008 11:55 AM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Could you possibly comment on how your work is related to the work of Behrang Noohi and of Helen Colman ?
Posted by: Eugene Lerman on June 19, 2008 5:31 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Noohi’s treatment of covering stacks in the article you refer to is, despite a lot of very nice machinery, still a theory of 1-covering ‘spaces’. A lot of the results depend on the corresponding facts for ordinary covering spaces. It is similar to Grothendieck’s theory of discrete opfibrations: the total and base ‘spaces’ are categories, but the fibres are (equivalent to) sets.

The correspondence between topological groupoids and topological stacks is explored in Dorette Pronk’s 1996 article “Etendues and stacks as bicategories of fractions”. General principles tell us that since my 2-covering space is a groupoid over the space XX, it will give us, at the very least, a presheaf of groupoids over the (big or small) site defined by XX. I suspect less stackification will be needed to give a legit stack than a generic topological groupoid over XX.

The homotopy groups of stacks that Noohi defines would correspond to homotopy groups of topological groupoids calculated using anafunctors. Since my homotopy 2-group is defined using closed covers of small-dimensional cubes, it isn’t immediate that it gives the same result as if I calculate it using anafunctors. However, as I mentioned in the slides, loops given by partition groupoids are in a sense cofinal in loops given by anafunctors. This means we should get equivalent results using partition groupoids or anafunctors.

Colman’s paper is interesting, but I think the bicategory of fractions of Lie groupoids she defines is essentially a bicategory of 1-types, since the functors that are formally inverted are those that induce an equivalence of fundamental groupoids. I have written down somewhere an example of a functor between topological groupoids with distinct π k\pi_k for k2k\geq 2 that is an equivalence in her bicategory. In fact, one can use spaces, since they are examples of topological groupoids. The fundamental groupoid in that paper comes from the Moerdijk-Mrcun article

“Lie groupoids, sheaves and cohomology”, in Poisson Geometry, Deformation Quantization and Group Representations, Lond. Math. Soc. Lecture Notes Ser. 323 (2005) pp. 145–272.

which I refer in the slides, and implicitly above. The set of orbits of my fundamental 2-group is precisely the fundamental group as calculated by their fundamental groupoid.

Posted by: David Roberts on June 20, 2008 10:36 AM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Colman’s article is cool in that the bicategory she defines contains 1-types as represented by spaces, and 1-types as represented by ordinary groupoids. For a connected 1-type represented by a space, i.e. a BGBG for GG a discrete group, it is equivalent to a groupoid in her bicategory by this span: BGEGGBG BG \leftarrow EG \rtimes G \to \mathbf{B}G Clearly for a general 1-type one just takes a disjoint union of the above.

Posted by: David Roberts on June 22, 2008 2:59 AM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

I’ve got 5 minutes before I must rush off… I would like to know if by any chance David and Urs’s construction of the inner automorphism 3-group of a strict 2-group is related to the construction John and Danny used to prove their result about 2-bundles and classifying spaces (see pages 24 and 25 of the slides from John’s talk).

Posted by: Bruce Bartlett on June 19, 2008 6:27 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Hi Bruce,

the relation I could see was this.

Have to run now myself…

Posted by: Urs Schreiber on June 19, 2008 6:35 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Hi David,

I am very much enjoying this interpretation of transformations between anafunctors as “anahomotopies” which you give. That’s great. Very nice.

Here are some remarks:

slide 26:

maybe it would help the reader label the diagonal vertical arrows by γ i\gamma_i and maybe to give the functor hXh \to X a name.

slide 28: I am not sure I understand what you want to express on slide 28. What do you mean by “constant or rescaling”?

slide 30: am I right that a general natural transformation I am to think of as a 4-valent vertex here? But that we are supposed to be thinking of this always as decomposed into two 3-valent ones? If so, I understand what’s going on. If not, I might need help! :-)

Generally, I am a bit confused by your use of the word “string diagram”. I guess I can see what you mean, but I had thought “string diagram” has an established technical sense as “diagram Poincaré dual to pasting diagram in a 2-category”. That’s not exactly what you are talking about, is it?

Posted by: Urs Schreiber on June 20, 2008 6:12 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

slide 26: will get onto that. Expect an update next week.

slide 28: I think I’ll use colours to highlight the various features in the next edit. What is meant by “constant or rescaling” is that in the regions where the boundaries are vertical, the homotopies are constant at the given path. In the regions where the boundaries are diagonal, the homotopies are just scaling the path, to fit the new constant paths into the unit interval. Or, if you like, look at it from top-down, removing constant paths by shrinking their domains until they can be removed, expanding the neighbouring domains to take up the space.

slide 30: You are right - the obvious way to make a natural transformation into a homotopy is to have 4-valent vertices, and the naturality condition is taken care of by what the functor does on quadruple overlaps. It is possible though to break the natural transformation up into a ‘movie’, where each frame moves one segment of the path. To save space, I have chosen to move every second second segment of the path per frame. This is just to conform to my rules for ‘string diagrams’.

In the slides I’m a bit sloppy with the string diagram terminology, and I need to alter it a little. What it boils down to is that if the codomain topological groupoid is an ordinary groupoid, an anahomotopy (as you call it) is a string diagram. Toby talks about string diagrams for internal (2-)groups, using general, constant ‘elements’, but nothing stops us from making the strings in string diagrams actual paths.

Posted by: David Roberts on June 21, 2008 6:16 AM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

Hi David, okay I’ve finally been able to read through the slides and the above posts, nice stuff! It is good to see my eminent countryman Keith Hardie being mentioned in a talk. Actually the first time I heard about him was through Hellen Colman. In Cape Town he he gave a talk partially about “2-tracks”. Funnily enough, Dorette Pronck has worked on stuff together with Hellen Colman, so it’s all a big twisted circle.

Posted by: Bruce Bartlett on June 21, 2008 10:39 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

On Thursday some big geometric talks went down at the conference… and we at the n-cafe would do very well to assimilate into the matrix the material which was presented. I fear it will be down to Tim and I (Derek?) to report on these events, since they might not make it into a TWF… since due to a bad case of jetlag it might well be that the author of TWF was reporting back on the other talks while one or two of them were being presented :-)

Right now what is of most relevance is that in the beginning of his talk Pietro Polesello spoke about stuff very closely related to what David presented in his talk. Pietro and Ingo Washkies wrote a paper in 2004 about Higher monodromy, where they studied 2-covering spaces from the stacky angle… just as David was suggesting in his third bullet point at the end of the talk. I wasn’t aware of this paper - and Pietro told me indeed hardly anyone is aware of it. Did you know of it David?

The stuff about 2-covers was only the first half of his talk - the rest of his talk was about applying the 2-character to this construction, and how it might relate to Luztig’s character shaves.

Ironically Pietro gave his talk on exactly the same day that David posted his on this site, as well as John writing TWF :-) I informed him about David’s work when I saw him on Friday.

Posted by: Bruce Bartlett on June 21, 2008 11:11 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

I wasn’t aware of this paper

Tsk, pay attention, Bruce. And here.

Posted by: David Corfield on June 21, 2008 11:44 PM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

…so it’s all a big twisted circle.

We are in the business of twisting things! :)

I did have the higher monodromy paper on my hard drive, but since I downloaded it when it came out and I couldn’t really understand it, it mouldered in a forgotten corner, until it was mentioned recently here (links offered by David C above). Then on second, third and fourth readings I came to appreciate it. The ideas are quite big in it, especially things like “The constant stack with stalk a category CC is the stack of locally constant CC-sheaves”, and the analogous result for locally constant stacks, valued in a 2-category.

It was slightly remiss of me to not refer to Pietro’s paper - ‘twill be fixed in the next edit.

Posted by: David Roberts on June 22, 2008 2:58 AM | Permalink | Reply to this

Re: Fundamental 2-Groups and 2-Covering Spaces

I fear it will be down to Tim and I (Derek?) to report on these events

Please do.

And please do not eventually shy away from it by being too ambitious. Better a brief and sketchy summary posted than a beuatiful review exposition never done.

As you all know: the way to do this is to prepare a text just as you would for a comment to the blog, previewing it as a comment to ensure that it compiles, but then not posting it as a comment but sending it by email to one of the Café hosts so that they can post it as a guest entry.


Posted by: Urs Schreiber on June 22, 2008 6:03 PM | Permalink | Reply to this
Read the post Aldrovandi on Non-Abelian Gerbes and 2-Bundles
Weblog: The n-Category Café
Excerpt: A talk by Aldrovandi on different perspectives on gerbes and 2-bundles.
Tracked: June 23, 2008 12:43 PM

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