What Does the Classifying Space of a 2-Category Classify?

December 4, 2006

What Does the Classifying Space of a 2-Category Classify?

Posted by Urs Schreiber

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My personal spy has just returned from the Nordic Conference in Topology that took place last week.

I hear that Tore A. Kro has new notes on his work with N. Baas and M. Bökstedt available online

N. Baas, M. Bökstedt, T. A. Kro
2-categorical K-theories.

They try to answer the question: What does the classifying space of a 2-category classify? Their answer is: for sufficiently well behaved topological 2-categories $C$ , the nerve of $C$ is the classifying space for charted $C$ -bundles.

Here a charted $C$ -bundle is essentially like what one would call the transition data for a 2-groupoid bundle #. The only difference is that no invertibility in $C$ is assumed. As a consequence, transition functions may go from patch $i$ to patch $j$ , but not the other way around.

The main application of this theory in these notes is a proof of the previously announced claim, that for $C$ the 2-category of Kapranov-Voevodsky 2-vector spaces the classifying space is the 2K-theory introduced by Baas, Dundas and Rognes. For $C$ the 2-category of Baez-Crans 2-vector spaces the classifying space is two copies of ordinary K-theory.

Posted at December 4, 2006 3:28 PM UTC

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36 Comments & 5 Trackbacks

Re: What does the Classifying Space of a 2-Category classify?

The Baas, Dundas, and Rognes paper is here.

Posted by: Allen Knutson on December 4, 2006 6:47 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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The Baas, Dundas, and Rognes paper is here.

Thanks. I should have included more links.

I did collect a couple of related links here.

A transcript of a summary talk by Birgit Richter on the BDR work is given here:

Seminar on 2-Vector Bundles and Elliptic Cohomology, II, III, IV, V.

Posted by: urs on December 4, 2006 8:42 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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Thank goodness that Urs has his own mathematical MI-6! 2-Categorical K-theories are certainly something very interesting to think about. I will try to understand better this idea, with the help of the useful `Seminar on 2-Vector Bundles and Elliptic Cohomology’.

Posted by: Bruce Bartlett on December 4, 2006 8:53 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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On p. 5, in example 2.4, the authors mention String bundles as those classified by a 2-category they call $2S$ .

Maybe I didn’t look at these notes closely enough, but I did not see the definition of $2S$ .

I would expect that the classifying space for $\mathrm{String}_G$ bundles, for fixed Lie group $G$ , is the realization of the nerve of the following sub-2-category of $\mathrm{Bim}(\mathrm{Hilb})$ :

objects are algebras Morita equivalent to the algebras generated by positive energy representations of $\L_k G$ , morphisms are bimodules for these algebras and 2-morphisms are bimodule homomorphisms.

On p. 4, example 2.2, a 2-category is introduced whose classifying space classifies line bundle gerbes. The category has a single object, has $\mathbb{CP}^\infty$ worth of 1-morphisms and a circle worth of 2-morphisms between any pair of 1-morphisms.

Noticing that $\mathbb{CP}^\infty \simeq BU(1)$ it seems to me that this category is essentially the one I described in a little essay called How many Circles are there in the World?.

Posted by: urs on December 5, 2006 12:10 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

The only difference is that no invertibility in C is assumed. As a consequence, transition functions may go from patch i to patch j, but not the other way around.The lack of inverses is familiar from the case
of fibrations in which instead of a group
one has only the topological monoid of self-homotopy equivalences of the fibre.

The patching should be replaced by a mapping cylinder.

Alternatively the transport along paths has no need to cancel.

Posted by: jim stasheff on December 5, 2006 4:57 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

What is a topological 2-category? It doesn’t seem to be defined in `2-vector bundles and forms of elliptic cohomology’ or `2-categorical K-theories’.

Posted by: Bruce Bartlett on December 5, 2006 3:11 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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What is a topological 2-category?

I’d guess they use it to mean a 2-category internal to topological spaces. I.e. a topological space of $n$ -morphisms for $0 \leq n \leq 2$ with all source-, target- and composition maps continuous.

Posted by: urs on December 5, 2006 3:22 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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Right - thanks. Its just that there is another notion of `topological 2-category’ inspired by John’s `HDA II : 2-Hilbert spaces’ paper. It works as follows.

Firstly, I assume everyone is familiar with the Grothendieck paradigm relating n-groupoids and topology:

(1)

\text{n-groupoids} \leftrightarrow \text{homotopy n-types}.

There is a `quantum cousin’ of this which relates topology, algebra, duality and higher categories:

(2)

\array{ & \text{Topology} & \text{Algebra} \\ & & \\ n=0 & \text{topological spaces} & \text{commutative H*-algebras} \\ n=1 & \text{topological groupoids} & \text{symmetric 2-H*-algebras} \\ n=2 & \text{topological 2-groupoids} & \text{symmetric 3-H*-algebras} }

and so on (By the way, all this weird stuff about $n-H^*$ -algebras can be found in the introduction of HDA II).

Lets see how it works. When $n=0$ , it is just the standard Gelfand-Naimark theorem relating topological spaces with commutative $H^*$ -algebras. (Of course, its really compact Haussdorff topological spaces and so on, but lets gloss over these details for the moment).

When $n=1$ , it’s the categorified Gelfand-Naimark theorem. This relates topological groupoids to symmetric 2- $H^*$ -algebras.

In this context, a topological groupoid is just a groupoid whose hom-sets are topological spaces. The objects don’t carry a topology! A symmetric 2- $H^*$ -algebra is just a nice linear, monoidal category with direct sums. The categorified Gelfand-Naimark theorem says these structures are the same. In the one direction you take `Rep’ while in the other you take `Spec’.

And the pattern continues. The important thing is that an $n$ -groupiod in this context is an $n$ -category, all of whose objects, morphisms, 2-morphisms etc. are weakly invertible, and whose $n$ -morphisms carry a topology. There is *no* topology on the lower morphisms.

The idea is that as $n \rightarrow \infty$ , we are `bumping’ topology out of the game! Thus a topological $\infty$ -groupoid will be the same thing as a discrete $\infty$ -groupoid. At this point we’ll have consummated Grothendieck’s dream - for we’ll have an equivalence between topological spaces, $\infty$ -groupoids and *quantum* $\infty$ -categories (linear $\infty$ -categories with duals).

Anyhow, the point is that there is a context in which a `topological 2-category’ could be conceived of differently.

Posted by: Bruce Bartlett on December 5, 2006 5:06 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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A good example of a topological groupoid, in the sense that the objects are discrete but the morphisms are topological spaces, is Cohen, Jones and Segal’s flow-line groupoid $C_f$ associated to a Morse function $f$ on a manifold $X$ . This comes from their paper on Morse theory and classifying spaces.

Recall that the objects of $C_f$ are the critical points of $f$ , and the hom-sets are the flow lines.

Posted by: Bruce Bartlett on December 5, 2006 5:48 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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Anyhow, the point is that there is a context in which a `topological 2-category’ could be conceived of differently.

Very worthwhile remark!

In fact, maybe one should have a closer look at what precisely Kro and collaborators call a topological 2-category.

Certainly, they want KV-2-vector spaces to form a topological 2-category, also in the semi-skeletal version where the collection of objects is the natural numbers and a morphism from $n$ to $m$ is an $m\times n$ matrix with entries being vector spaces.

This is rather similar to the example you mention:

A good example of a topological groupoid, in the sense that the objects are discrete but the morphisms are topological spaces

But I guess in the case of KV-2-vector space we do get a topological 2-category in the sense internal to $\mathrm{Top}$ , in that, for instance, source and target maps on 1-morphisms are indeed constant on connected components of the space of 1-morphisms.

Posted by: urs on December 5, 2006 7:06 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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‘Topological category’ is used to mean two different things:

categories internal to Top, which have a space of objects and a space of morphisms, and
categories enriched in Top, which have a set of objects and, for any two objects $x$ and $y$ , a space of morphisms $hom(x,y)$ .

The latter is the special case of the former where the space of objects has the discrete topology.

In general, whenever we have a category $K$ with finite limits, we can define both categories internal to $K$ and categories enriched in $K$ :

categories internal to $K$ , which have a $K$ -object of objects and a $K$ -object of morphisms, and
categories enriched in $K$ , which have a set of objects and, for any two objects $x$ and $y$ , a $K$ -object of morphisms from $x$ to $y$ , called $hom(x,y)$ .

When we have a functor

$F : Set \to K$

which preserves finite limits, we can use this to turn any category enriched in $K$ into a category internal to $K$ . That’s what we’re doing above, where

$F : Set \to Top$

sends any set to that space regarded as a set with the discrete topology.

In general, for $n$ -categories, we expect $n+2$ different levels of internalization. At one extreme we should have ‘complete internalization’, where there’s a $K$ -object of objects, a $K$ -object of morphisms, and so on up to a $K$ -object of $n$ -morphisms. At the other extreme we have plain old $n$ -categories, where there’s a set of objects, a set of morphisms and so on. Right next to that other extreme we have ‘enrichment’, where we have a set of objects, a set of morphisms, and so on — but for any two parallel $(n-1)$ -morphisms $f$ and $g$ we have an object in $K$ called $hom(f,g)$ .

When $n = 1$ we just have three choices: categories internal to $K$ , categories enriched in $K$ , and plain old categories.

I think this is cool.

Posted by: John Baez on December 6, 2006 1:56 AM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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Such groupoids enriched in $\mathbf{Top}$ are much like `simplicial groupoids’ - really groupoids enriched in $\mathbf{sSet}$ , and these model all unpointed homotopy types. There is a paper by Zhi-Ming Luo (math.AT/0301045) relating (presheaves of) simplicially enriched groupoids and 2-groupoids, but not simplicially-enriched 2-groupoids. Interesting …

Posted by: David Roberts on December 6, 2006 2:38 AM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

A topological 2-cat -
is it not a 2-cat in which all the structures are topological spaces or continuous maps?

jim

Posted by: jim stasheff on December 5, 2006 4:48 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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A topological 2-cat - is it not a 2-cat in which all the structures are topological spaces or continuous maps?

I think so #. Unless these authors have redefined this somewhere in the context of their work. But I am not aware of any such redefinition.

Posted by: urs on December 5, 2006 4:52 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

Jim wrote:

A topological 2-cat -
is it not a 2-cat in which all the structures are topological spaces or continuous maps?

This is one meaning, and probably by far the most common one. If you want to be painfully unambiguous, you can call this a 2-category internal to Top.

In the case of 1-categories, people often use K-category to mean a category enriched in K, and category in K to mean a category internal to K. I described the difference here.

In the case of 2-categories there are even more layers of distinction, but not many people seem to realize this yet.

Posted by: John Baez on December 6, 2006 2:05 AM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

Hi, Bruce! If you want to be disgustingly cool, you’ll write quotes like ‘this’ instead of like `this’. That’s one difference between this environment and TeX.

Posted by: John Baez on December 6, 2006 2:15 AM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

Great - thanks for the tip! The next cool thing I’d like to see is the ability to include .eps files into one’s comments… or perhaps an \xymatrix environment :-)

Posted by: Bruce Bartlett on December 6, 2006 12:03 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

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The next cool thing I’d like to see is the ability to include .eps files into one’s comments… or perhaps an \xymatrix environment :-)

You can include pictures, using the ordinary HTML img tag.

Somebody should write a script that reads in xypic code, runs it through a LaTeX compiler, and transforms the result into a .jpg or something.

Posted by: urs on December 6, 2006 12:14 PM | Permalink | Reply to this

Re: What does the Classifying Space of a 2-Category classify?

Bruce asked:

What is a topological 2-category?

Along with all of the answers above, it’s worth knowing that there is another (completely different) meaning of ‘topological category’ that I’ve soon used point-set topologists (who, like most working mathematicians, are using catgory theory as a language to point out useful features of specific categories). To them, a topological category is a category equipped with a faithful functor to Set that creates all limits and colimits. Examples include the category of topological spaces, the category of uniform spaces, and the category of convergence spaces. Examples do not include any category with homotopy-equivalent maps identified, nor the category of locales, nor any category of smooth spaces as far as I can tell. (These are point-set topologists, after all!)

Posted by: Toby Bartels on December 6, 2006 11:09 PM | Permalink | Reply to this

Concordance

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Behind the scenes we talked about this:

Some definitions in the above notes appear only after the terms defined appear in the theorems.

One such definition is concordance. This is defined in definition 7.1 on p. 28.

Two “2-bundles” (really: local transition data of 2-bundles) on $X$ are said to be concordant if there is a 2-bundle on $X \times \mathrm{interval}$ which restricts to the given ones on the boundary.

That’s an equivalence relation, and concordance classes are therefore denoted $\mathrm{Con}(X,\mathrm{something})$ .

That’s what one sees appear, for instance, in example 2.4 on p. 5.

(By the way: I greatly prefer anonymous comments over no comments at all. If you don’t feel like transmitting what you consider private communication over the entire web, with your name attached, but if you do feel like commenting on anything we talk about here, please consider dropping us an anonymous comment. )

Posted by: urs on December 5, 2006 3:48 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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Would it be worth checking what the associated charted $2 C$ -bundles for 2-categories, $2 C$ , of other 2-vector spaces? Baas et al. cover Kapranov-Voevodsky and Baez-Crans versions, which leaves Elgueta and other versions.

Posted by: David Corfield on December 7, 2006 1:47 AM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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Would it be worth checking what the associated charted $2C$ -bundles for 2-categories, $2C$ , of other 2-vector spaces?

I certainly think so, and I have ranted on that several times, for instance here.

The thing is this: while it is quite interesting that the “K-theory” of Baez-Crans 2-vector bundles is $K\times K$ and that of Kapranov-Voevodsky is $\mathrm{BDR}-2K$ , the original hope was that there is a kind of 2-vector bundle such that its K-theory is something more closely resembling elliptic cohomology, somehow.

This goal has not been achived yet, as far as I am aware.

And I argued that this is maybe no wonder: while the notions of 2-vector spaces used so far in these studies all have their raison d’être, they are all comparatively restricted, as compared with the most general notion of 2-vector space one would imagine.

Baez-Crans 2-vector spaces are $\mathrm{Disc}(k)$ -module categories. This is “relatively restricted” because $\mathrm{Disc}(k)$ (the discrete monoidal category over a field $k$ ) is so puny.

Kapranov-Voevodsky 2-vector spaces are module categories for something a little bigger, namely for $\mathrm{Vect}$ . But they are just a tiny subset of all $\mathrm{Vect}$ -module categories.

So, here is my first $n$ -Café Millenium Prize: One million Microeuros for the first one to compute the classifying space of charted $2C$ -bundles with

(1)

2C := \mathrm{Bim}(\mathrm{Vect}) \,.

Posted by: urs on December 7, 2006 9:27 AM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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The second $n$ -Café Millenium Prize is to categorify the Generalized Tangle Hypothesis. We have a classifying space for a 2-group. So now find a categorified Thom construction, using the best 2-vector spaces on offer.

What kind of thing has a normal 2-bundle with structure?

Posted by: David Corfield on December 7, 2006 4:20 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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So, Urs, what might correspond in the case of your favoured 2-vector spaces to the completion of a $k$ -dimensional real vector space as a $k$ -sphere? Presumably, the answer for a skeletal $(p, q)$ Baez-Crans 2-vector space would be a $q$ -sphere bundle over a $p$ -sphere.

Posted by: David Corfield on December 12, 2006 10:27 AM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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So, Urs, what might correspond in the case of your favoured 2-vector spaces to the completion of a $k$ -dimensional real vector space as a $k$ -sphere?

You, John and others have, in the meantime, thought much more about this than I have. I can try to say something, but the risk is that I throw the discussion back to a point you have long passed.

Anyway. If I understand you correctly, you are asking what the categorification of a projective space would be.

For an ordinary $k$ -vector space $V$ , and for $k^\times$ the multiplicative group inside $k$ (i.e. $k$ without 0), we have a $k^\times$ -action on $V$

(1)

k^\times \times V \to V

and passing to the projective space amounts to “dividing out” this action

(2)

P V \simeq V/{k^\times} \,.

My instinct would be to try to categorify this in the more or less obvious way.

I am considering 2-vector spaces to be suitable module categories

(3)

V

for an action by a (usually abelian and braided) monoidal category

(4)

C \,.

Inside $C$ , we find the Picard 2-group $P(C)$ . In terms of its suspension, this is simply the core of $\Sigma(C)$ . So $P(C)$ has all those objects of $C$ which have a dual object and all isomorphism between these.

The action

(5)

C\times V \to V

hence restricts to an action

(6)

P(C) \times V \to V

and we can think of this as the action of the (weak, in general) Picard 2-group on $V$ .

I guess it makes sense to try to divide out by this group action and address the result as a 2-projective space.

We should then also talk about how exactly to define the quotient of a category by the action of a 2-group. But let me put that aside for the moment.

The remaing question then is to identitfy the Picard 2-groups for various monoidal categories $C$ .

For Baez-Crans 2-vector spaces, which should be module categories for

(7)

C = \mathrm{Disc}(k)

the Picard 2-group is just

(8)

P(C) = \mathrm{Disc}(k^\times) \,.

For Kapranov-Voevodsky 2-vector spaces we have

(9)

C = \mathrm{Vect}_k

and hence

(10)

P(C) = 1d\mathrm{Vect}_k \,.

Same for general $\mathrm{Vect}$ -module categories.

Hm, I have the vague recollection that we were that far long before already.

Posted by: urs on December 12, 2006 10:55 AM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

Sorry, I wasn’t very clear. I was thinking about how to categorify the Thom Space construction, part of which involves forming a sphere bundle from a vector bundle. I was asking what the equivalent move might be for a 2-vector 2-bundle.

Posted by: David Corfield on December 12, 2006 1:30 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

how to categorify the Thom Space construction

Ah, I see.

Hm, we’d need an arrow-theoretic formulation of what it means to form a one-point compactification of a vector space.

Any ideas?

Posted by: urs on December 12, 2006 1:36 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

Well, here’s Toby telling us that:

The Stone Cech compactification functor from the category of topological spaces to the category of compact topological spaces is the left adjoint of the inclusion functor.

Posted by: David Corfield on December 12, 2006 4:30 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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The Stone Cech compactification functor from the category of topological spaces to the category of compact topological spaces is the left adjoint of the inclusion functor.

Ah, great. So what we need next, then, is a notion of topological 2-space and compact topological 2-space.

But that we essentially talked about above.

I’d say it looks like a save move to start by declaring that a compact topological 2-space is just a category enriched over - or internalized in - compact topological spaces.

If so, we have an obvious inclusion 2-functor from compact topological 2-spaces to topological 2-spaces.

This might likely have a weak adjoint.

And what you are looking for should be the action of this weak adjoint on a given kind of 2-vector bundle.

So now it looks as if all the abstract ingredients we need are there.

But apparently work is required for actually carrying through this procedure for a given case.

Posted by: urs on December 12, 2006 4:44 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

Now which varieties of vector 2-spaces are topological 2-spaces?

Posted by: David Corfield on December 12, 2006 5:07 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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Now which varieties of vector 2-spaces are topological 2-spaces?

Most of them are naturally enriched over something topological. Whether there is something more interesting than the discrete topology on the objects may depend.

Baez-Crans 2-vector spaces are, being categories internal to $\mathrm{Vect}$ automatically also categories internal to $\mathrm{Top}$ , as long as we have the standard topology on our (finite-dimensional) vector spaces.

Kapranov-Voevodsky 2-vector spaces are also “topological categories”, this is the example that got the discussion above started.

To think of an object in $\mathrm{Bim}(\mathrm{Vect})$ – an algebra $A$ – as a topological category we should think of it in terms of its image under $\mathrm{Bim}(\mathrm{Vect}) \stackrel{\subset}{\to} {}_{\mathrm{Vect}}\mathrm{Mod}$ , where it becomes the category of $A$ -modules. This is naturally enriched over $\mathrm{Top}$ , as morphisms here are linear spaces.

So I think about all flavors of 2-vector spaces that one would think of are topological categories.

Posted by: urs on December 12, 2006 5:22 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

There are other ways to think of Thom spaces that are more natural (i.e. that are already meaningful both in differential geometry and
in algebraic geometry): if E is a vector bundle on X, the Thom space of E is the quotient of E by the complement E-s(X) of the zero section s of E. This is the point of view adopted by Morel and Voevodsky in their homotopy theory of schemes (its naturality is due the strong link of this object with Grothendieck’s six operations). The interest of this is that we don’t need any metric. A very deep feature of Thom spaces is their link with projective spaces. Hence the questions: what is the 2-projective space of dimension n?
How to associate to a 2-vector bundle E a 2-projective space P(E)?

Posted by: Denis-Charles Cisinski on December 12, 2006 5:06 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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the quotient of $E$ by the complement $E-s(X)$

Hm, how is this quotient formed? I am not sure I understand which quotient is meant.

Posted by: urs on December 12, 2006 5:15 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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To form the Thom space of a vector bundle $E$ , you add a ‘point at infinity’ to it. In the simplest cases it amounts to this: you take each fiber $E_x$ and add a point at infinity to it to get a sphere; then you identify all these points at infinity.

This has the effect of making the complement of the zero section of $E$ contractible. That’s what Denis-Charles meant by ‘taking the quotient of $E$ by the complement of the zero section’.

Given a subspace $A \subseteq X$ , topologists write $X/A$ to mean the result of collapsing $A$ to a point — an honest quotient. But, you can also imagine a ‘homotopy quotient’ where you glue on just enough stuff to $X$ to make $A$ contractible, and that’s what’s going on here.

Posted by: John Baez on February 4, 2007 8:45 PM | Permalink | Reply to this

Re: What Does the Classifying Space of a 2-Category Classify?

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That’s what Denis-Charles meant by ‘taking the quotient of E by the complement of the zero section’.

Thanks! Cool that you rememebered this question after such a long while.

Posted by: urs on February 5, 2007 2:29 PM | Permalink | Reply to this

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Re: What Does the Classifying Space of a 2-Category Classify?

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2-categorical $K$ -theories is now out on the ArXiv. There are some differences from the version mentioned in the post. In particular, on page 6 we hear about foam bundles, a construction which

draws inspiration from work by J. C. Baez and S. Galatius.

Posted by: David Corfield on December 20, 2006 10:24 AM | Permalink | Reply to this

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