Baković and Jurčo on Classifying Topoi for Topological Bicategories

February 14, 2009

Baković and Jurčo on Classifying Topoi for Topological Bicategories

Posted by John Baez

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Igor Baković is an energetic young mathematician from Croatia… I bet we’ll be hearing a lot from him as time goes on. He’s very excited about topos theory, nonabelian cohomology, 2-bundles and the like. I met him in Göttingen last week, and he said he and Branislav Jurčo were almost done with a paper on this subject. Now it’s out!

Igor Baković and Branislav Jurčo, The classifying topos of a topological bicategory.

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If you know about the classifying space for a topological group, you should be almost ready to understand the classifying space for a topological 2-group… but this is just a special case of the classifying space for a topological bicategory:

Nils Baas, Marcel Bökstedt and Tore Kro, Two-categorical bundles and their classifying spaces.

Let me sketch how this works. Jack Duskin described a way to turn a bicategory $C$ into a simplicial set $N C$ with one 0-simplex per object, one 1-simplex per morphism, one 2-simplex per 2-morphism of the form $\alpha : f g \Rightarrow h$ …

…where the 3-simplices describe composition in the bicategory, the 4-simplices record the fact that a bicategory satisfies the Mac Lane’s pentagon identity…

…and from then on up it’s sort of dull. This simplicial set is called the ‘nerve’ of the bicategory:

John Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories.

The same trick applies to a topological bicategory $C$ , and coughs up a simplicial space $N C$ . There’s a way to ‘geometrically realize’ any simplicial space and get a space, and if we do this to $N C$ we get the classifying space of the topological bicategory. Under some mild conditions this space classifies ‘principal $C$ -2-bundles’ — where of course the quoted phrase needs to be interpreted carefully to make sense!

But now Baković and Jurčo have constructed a classifying topos for a topological bicategory. Quoting their abstract, with a few changes in notation:

For any topological bicategory $C$ , the Duskin nerve $N C$ of $C$ is a simplicial space. We introduce the classifying topos $B C$ as the Deligne topos of sheaves $Sh(N C)$ on the simplicial space $2 C$ . It is shown that the category of topos morphisms from the topos of sheaves $Sh(X)$ on a topological space $X$ to the Deligne classifying topos $Sh(N C)$ is naturally equivalent to the category of principal $C$ -bundles. As a simple consequence, the geometric realization of the nerve $N C$ of a locally contractible topological bicategory $C$ is the classifying space of principal $C$ -bundles (on CW complexes), giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical $K$ -theory. We also define classifying topoi of a topological bicategory $C$ using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.

Of course there should really be some sort of classifying 2-topos for a topological bicategory. I believe Igor Baković is thinking about this now.

Posted at February 14, 2009 11:54 PM UTC

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Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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Let me engage in a bit of speculation about classifying 2-toposes, since I’ve been thinking a bunch about 2-toposes recently.

Fortunately, the classifying 2-topos of anything is almost certainly going to be a Grothendieck 2-topos, so we don’t need to worry about the right definition of “elementary 2-topos.” The classifying 2-topos of an ordinary 2-category $C$ should probably just be the functor 2-category $[C^{op},Cat]$ . I would guess that for any other 2-topos $K$ , 2-geometric morphisms $K\to [C^{op},Cat]$ could be identified with flat functors $C\to K$ , and that when $K=St(X)$ is the 2-topos of stacks on some space $X$ , these can be identified with $C$ -torsors over $X$ and thereby be related to somebody’s notion of “2-bundle.”

If $C$ is a topological 2-category, then by analogy with the 1-categorical case, it would be natural to construct its classifying 2-topos via “2-codescent” in the 3-category of 2-toposes, using the 2-toposes of stacks on the spaces $N C_0$ , $N C_1$ , and $N C_2$ of objects, morphisms, and 2-cells in $C$ . I would guess that an object of this 2-topos could thereby be identified with

A stack $X$ on $N C_0$ , together with
A morphism $f:d_1^{\star} X\to d_0^{\star} X$ of stacks on $N C_1$ , and
A 2-cell $\alpha:d_0^{\star}(f) \circ d_2^{\star}(f) \to d_1^{\star}(f)$ of stacks on $N C_2$ , such that
appropriate cocycle and degeneracy conditions are satisfied, and
$\alpha$ is an isomorphism when restricted to the subspace of $N C_2$ consisting of 2-cell isomorphisms in $C$ .

The last condition means that I’m really thinking of the nerve of a 2-category as a stratified simplicial set. If the topology of $C$ is discrete, this condition is needed to recover $[C^{op},Cat]$ ; otherwise we’re going to get only (normal) lax functors.

Posted by: Mike Shulman on February 16, 2009 9:10 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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The classifying 2-topos of a (discrete) bicategory

\mathcal{B}

is indeed the 2-category [

\mathcal{B}^{op}

,Cat] of homomorphisms of bicategories, which we might call the 2-category of 2-presheaves on

\mathcal{B}

. Equivalently, one can take for the classifying 2-topos of

\mathcal{B}

the 2-category

B\mathcal{B}

of (right) actions of

\mathcal{B}

from my thesis “Bigroupoid 2-torsors”. The latter point of view is internal, and allows us to define the classifying 2-topos

B\mathcal{B}

of a topological bicategory

\mathcal{B}

as the 2-category of

\mathcal{B}

-stacks, where a

\mathcal{B}

-stack

\mathcal{P}

is a continuous functor

\pi : \mathcal{P} \to B_0

from the stack

\mathcal{P}

to the discrete category

\mathcal{B}_0

of objects of the bicategory

\mathcal{B}

, together with a continuous action of

\mathcal{B}

which satisfy categorified action axioms. I believe that this classifying 2-topos can be constructed as a (tricategorical) colimit from the 2-toposes of stacks

St(N\mathcal{B}_0)

St(N\mathcal{B}_1)

St(N\mathcal{B}_2),\ldots,

on spaces in the Duskin nerve

N\mathcal{B}

\mathcal{B}

, precisely as you described. The right notion of a 2-bundle over a topological space

X

is given by principal actions of the bicategory

\mathcal{B}

: the action of

\mathcal{B}

\pi : \mathcal{P} \to B_0

is principal if an “action bicategory”, which I introduced in my thesis, is 2-filtered (in the sense of Dubuc and Street) or biflitered (in the sense of Kennison), since these two notions are equivalent. Then the Diaconescu’s theorem for 2-toposes gives a natural biequivalence

(1)

2-Top(St(X),B\mathcal{B}) \sim_{bi} 2-Bun(X,\mathcal{B})

between the 2-category

2-Top(St(X),B\mathcal{B})

of 2-geometric morphisms from the Grothendieck 2-topos

St(X)

of stacks over the topological space

X

to the classifying 2-topos

B\mathcal{B}

, on the one side, and the 2-category

2-Bun(X,\mathcal{B})

of principal 2-bundles under

\mathcal{B}

over

X

, on the other side.

Posted by: Igor Bakovic on February 17, 2009 11:24 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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Very cool! All this 2-topos stuff seems to be in the air everywhere. I look forward to reading your paper on the 2-Diaconescu theorem. Is your definition of (Grothendieck) 2-topos the same as the one I’m using (taken from Street)?

Posted by: Mike Shulman on February 18, 2009 3:43 AM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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Somehow, I still don’t need to worry about “the right” definition of the Grothendieck 2-topos (and not even to mention elementary 2-topos) because everything we did so far is over (the fixed) topological space $X$ , so the only possible definition is the corresponding Grothendieck 2-topos $St(X)$ of stacks over $X$ . But Zoran mentioned some intriguing subtleties in his comments, which are relevant even in this case, and they are connected to two possible ways in which we can view stacks as internally or externally complete objects.

To make this more precise, the more intuitive definition of a stack, adopted by Grothendieck and Giraud, uses an external point of view which consider stacks as a sheaf theoretical objects, using the respective notion of a fibered category. Thus, a stack $P$ over the topological space $X$ is defined as a pseudofunctor $P : Ouv(X)^{op} \to Cat$ (where $Ouv(X)$ denotes the category whose objects are open subsets of $X$ and morphisms are their inclusions), which satisfy effective descent conditions for both objects and morphisms.

On the other side, from an internal point of view, stacks are certain internal groupoids in the Grothendieck topos $Sh(X)$ of sheaves on the topological space $X$ . This latter point of view was adopted by Joyal and Tierney in their seminal paper

A,Joyal, M. Tierney, Strong Stacks and Classifying Spaces, Proceedings of the International Conference on Category Theory, Como 1990, Lecture Notes in Mathematics 1488, 213 - 236

in which they introduced closed model structure on the category $Cat(Sh(X))$ of internal categories in the Grothendieck topos $Sh(X)$ . Then stacks can be seen as a sort of injective objects in $Cat(Sh(X))$ and the fibrant objects for Joyal-Tierney model strucuture on $Cat(Sh(X))$ are called strong stacks, since they represent a strengthening of the notion of stack.

The dichotomy between internal and external point of view is encoded in a well known adjuction

(1)

\Gamma : Bun(X) \leftrightarrow Set^{Ouv(X)^{op}} : \Lambda

where $\Gamma : Bun(X) \to Set^{Ouv(X)^{op}}$ is the cross-section functor, which is a right adjoint to the bundle of germs functor $\Lambda : Set^{Ouv(X)^{op}} \to Bun(X)$ , which assigns to each presheaf $P$ in $Set^{Ouv(X)^{op}}$ its bundle of germes, which are defined as filtered colimits over neighborhoods of points in $X$ . It is also well know that the above adjunction (1) restricts to an equivalence

(2)

\Gamma : Etale(X) \leftrightarrow Sh(X) : \Lambda

between the category $Etale(X)$ of etale spaces over $X$ and the category $Sh(X)$ of sheaves over $X$ .

My main point is that a categorified version of an adjuction (1) should gives a (bi)adjunction

(3)

\Gamma_2 : 2-Bun(X) \leftrightarrow Cat^{Ouv(X)^{op}} : \Lambda_2

between the 2-category $2-Bun(X)$ of 2-bundles over $X$ and the 2-category $Cat^{Ouv(X)^{op}}$ of fibered categories over $X$ , whose objects are pseudofunctors $P : Ouv(X)^{op} \to Cat$ . Then the 2-functor $\Lambda_2 : Cat^{Ouv(X)^{op}} \to 2-Bun(X)$ , defined as the filtered (bi)colimit over neighborhoods of points in $X$ , would be the left (bi)adjoint to the 2-functor $\Gamma_2 : Cat^{Ouv(X)^{op}}) \to 2-Bun(X)$ of cross-sections, and this (bi)adjunction would similarly restricts to a (bi)equivalence

(4)

\Gamma_2 : 2-Etale(X) \leftrightarrow St(X) : \Lambda_2

between the 2-category $2-Etale(X)$ of etale categories over $X$ (which may be equivalently seen as the 2-category of internal categories in $Etale(X)$ ) and the 2-category $St(X)$ of stacks over $X$ .

These are the two equivalent ways (internal and external, respectively) in which we can define the Grothendieck 2-topos corresponding to the topological space $X$ . Unfortunately, I never saw that someone completely described a (bi)adjunction (3) or (bi)equivalence (4) in the existing literature!

There’s a Grothendieck construction of a filtered colimit of any functor $F :A \to Cat$ in Exposé VI of

M. Artin, A. Grothendieck, J. L. Verdier, SGA4, 1963/64 Springer, Lecture Notes Vol 270 (1972)

which was recently extended by Dubuc and Street to the bicolimit of the (strict) 2-functor $P : A \to Cat$ from the 2-filtered (strict) 2-category $A$ in

E. Dubuc, R. Street, A construction of $2$ -filtered bicolimits of categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 47 no. 2 (2006), p. 83-106

but for the construction of (bi)adjuction (3) we really do need a 1-filtered bicolimit of the pseudofunctor, and not a strict 2-functor!

I believe that these can be achieved by a modification of Dubuc and Street’s construction… but maybe somebody is already more familiar with this, and can save me of lot of (maybe unnecessary) work!?

Posted by: Igor Bakovic on February 18, 2009 9:17 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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That’s very interesting; I had not thought of the comparison “sheaves = etale spaces” as analogous to the comparison “stacks = internal categories in sheaves,” but now I do see the resemblance.

What is the advantage of the internal viewpoint? The fact that you can “strictify” a stack to make it into an internal category in sheaves has always seemed to me like a curiosity, perhaps sometimes technically useful, but sort of a doing-violence technique similar to replacing a bicategory by a strict 2-category. One thing that makes me feel this way is that so many naturally occurring stacks (such as the self-indexing of a topos) are only pseudofunctors (= fibrations).

Also, you can only expect to do that for stacks over 1-sites, since only then do you have a meaningful topos of sheaves in which to look at internal categories. Just like you can only make sheaves into etale spaces when you have sheaves on a posite (aka (0,1)-site), or equivalently on a locale/space, rather than a general 1-site.

Regarding the necessity of a general definition of (Grothendieck) 2-topos, what is the statement of your “Diaconescu’s theorem?” What I think of as Diaconescu’s theorem is about geometric morphisms from a general topos into a presheaf topos; but perhaps the point is that hypothesis on the domain is unnecessarily strong?

Posted by: Mike Shulman on February 19, 2009 2:51 AM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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Actually, I think a more precise analogy to the comparison between

2-sheaves (= stacks) on a 1-site $C$ and
internal 1-categories in the category of 1-sheaves on $C$

would be the comparison between

1-sheaves on a (0,1)-site $C$ , such as $O(X)$ and
internal 0-categories (many-object equivalence relations) in the category of (0,1)-sheaves on $C$ , such as $O(X)$ itself.

These latter are generally called $O(X)$ -valued sets. And in both cases, in order to get the correct morphisms in the second approach, you need to either restrict to “good” objects (stacks, or complete $O(X)$ -valued sets) or use some sort of generalized morphisms (anafunctors, or $O(X)$ -valued functional relations).

Posted by: Mike Shulman on February 19, 2009 4:32 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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Mike wrote:

The fact that you can “strictify” a stack [make it a true functor instead of a pseudofunctor -urs] to make it into an internal category in sheaves has always seemed to me like a curiosity, perhaps sometimes technically useful, but sort of a doing-violence technique similar to replacing a bicategory by a strict 2-category. One thing that makes me feel this way is that so many naturally occurring stacks (such as the self-indexing of a topos) are only pseudofunctors (= fibrations).

One way to make it look more natural is to invoke a Yoneda argument: for $X$ an $n$ -stack which is given by a pseudofunctor that is not necessarily a true functor, the functor $\bar X := Hom_{n-Stacks}(Y(-),X)$ with $Y$ the $n$ -Yoneda embedding of the site into $n$ -stacks on it is a strict functor, and ought to be equivalent as an $n$ -stack to $X$ .

In this context it may be noteworthy that there is a well-developed theory of $\infty$ -stacks which relies entirely on the concept of representing $\infty$ -stacks by ordinary contravariant functors from the site to a 1-category of $\infty$ -groupoids: this has been developed by Jardine, Joyal, Lurie, Toën and others (in alphabetical order here…) using a suitable model structure on simplicial presheaves. A good survey is

B. Toën, Higher and derived stacks: a global overview.

I’d suppose that this can in particular be taken as generalizing the Joyal-Tierny work that Igor mentioned.

Also, you can only expect to do that for stacks over 1-sites,

That’s of course true. On the other hand, one way to realize stacks on non-1-categorical sites is to pass to derived stacks, which amounts to taking the site to be enriched over $\infty$ -groupoids in a suitable way. This is again well modeled by the above methods.

Posted by: Urs Schreiber on February 19, 2009 9:07 AM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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for $X$ an $n$ -stack which is given by a pseudofunctor that is not necessarily a true functor, the functor $\overline{X}:=Hom_{n-Stacks}(Y(-),X)$ with $Y$ the $n$ -Yoneda embedding of the site into $n$ -stacks on it is a strict functor, and ought to be equivalent as an $n$ -stack to $X$ .

$\overline{X}$ is a strict functor for $n=1$ , but I don’t see why it would be for arbitrary $n$ . For $n=1$ it is strict because the 2-category of 1-categories is strict (at least, assuming choice so that we can use functors instead of anafunctors), hence so is the 2-category of 1-stacks, and thus so is its hom-functor. Likewise, if you choose Kan complexes as your model for $\infty$ -groupoids, then the $(\infty,1)$ -category of $\infty$ -groupoids can be made strict (the simplicially enriched category of Kan complexes), and it works. But in general, I don’t see why the $(n+1)$ -category of $n$ -categories would be strict. At the least, it will depend on carefully choosing your model for $n$ -categories.

Also, even when $\overline{X}$ is a strict functor, is it necessarily a sheaf?

On the other hand, one way to realize stacks on non-1-categorical sites is to pass to derived stacks, which amounts to taking the site to be enriched over $\infty$ -groupoids in a suitable way. This is again well modeled by the above methods.

But if your 2-site has non-invertible 2-cells, then it’s not enriched over $\infty$ -groupoids. I’m interested in actual $\infty$ -stacks, not just $(\infty,0)$ -stacks.

Posted by: Mike Shulman on February 19, 2009 4:21 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

I do not know how relevant is the following remark. Hirschowitz and Simpson wrote in their famous work on n-stacks another sort of strictification result for their version of n-stacks in the language of Segal categories. Link (242 pages, in French, but look for section 18, from page 182):
Descente pour les n-champs (pdf)

Posted by: Zoran Skoda on February 19, 2009 6:14 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

Marta Bunge reported at the category conference in Calais last june (2008) about her joint work with Claudio Hermida which is about some other aspects of Diaconescu-type results in categorical dimension 2. Her slides from Calais can be found here There are also some conjectural statements about higher n, with an interesting definition when an n-functor between strict n-categories should be called a fibered n-category . His definition is recursive. Hermida was telling me in an email in 2005 that he had such a definition and much of related picture, but that he was too busy with another job, preventing him from writing a complete theorem about Grothendieck n-fibrations.

The internalization aspects on Bunge’s slides seem to generalize the approach from her earlier treatment in 1-categorical case. Namely in a paper in Cahier she gave an alternative proof of a 1-dimensional Diaconescu’s theorem which gives much new insight. Maybe now when the Cahiers is free online, the paper will be more widely read by non-super-specialists. Actually she has 3 papers in Cahiers which I see related to the issue in 1-dimensional case. Here are the links:

Internal presheaves toposes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 18 no. 3 (1977), p. 291-330.

Stack completions and Morita equivalence for categories in a topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no. 4 (1979), p. 401-436.

Bunge, Marta; Pare, Robert. Stacks and equivalence of indexed categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no. 4 (1979), p. 373-399.

In their current work, Bunge and Hermida use the results of Mauri and Tierney:

Two-descent, two-torsors and local equivalence Journal of Pure and Applied Algebra 143, n.1-3, pp. 313-327 (1999).

Mauri’s thesis used to be available online; now it can be found using the wayback machine. Mauri and Tierney have defined the 2-torsors internally to a topos, and they have classification result used by Bunge and Hermida in that context. One should think of their 2-torsors as internal principal bundles for 2-groups in a topos; by the usual nonsense one replaces internal groupoids by stacks.

Igor Baković is working in his thesis more generally in the sense that he allows bigroupoids in place of structure 2-group; on the other hand he considers set theoretical and smooth contexts rather than internal to topos, though I know from the conversations with him that he thought much through how to do it in Barr-exact categories in spirit of a remarkable article

John Duskin, An outline of non-abelian cohomology in a topos : (I) The theory of bouquets and gerbes, Cahiers…, 23 no. 2 (1982), p. 165-191.

Igor worked on 2-bouquets, stimulated by Branislav as well. Let us hope these further works by Hermida (n-fibrations), Hermida-Bunge (above), Baković and Baković-Jurčo will soon be out as well.

Posted by: Zoran Skoda on February 18, 2009 4:08 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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What’s a ‘bouquet’? The general idea, rather than a precise definition, would be most helpful.

Posted by: John Baez on February 18, 2009 7:00 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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To give you the general idea what’s a bouquet, it would be enough to say that it is just an internal version of a gerbe!

Even the formal definition is enlightening: a bouquet $\mathcal{B}$ is an internal groupoid in a Grothendieck topos $\mathcal{E}$ which satisfies following conditions:

(i) $\mathcal{B}$ is locally nonempty, i.e. the canonical terminal morphism $B_0 \to 1$ is an epimorphism

(ii) $\mathcal{B}$ is locally connected, i.e. the canonical morphism $B_1 \to B_0 \times B_0$ is an epimorphism.

In the paper which Zoran mentioned, Duskin showed that for any bouquet $\mathcal{B}$ the fibered category $Tors(\mathcal{B})$ of $\mathcal{B}$ -torsors is a gerbe over $\mathcal{E}$ , and conversely that any gerbe $\mathcal{G}$ over $\mathcal{E}$ gives rise to a bouquet $\mathcal{B}$ such that we have an equivalence $\mathcal{G} \simeq Tors(\mathcal{B})$ of gerbes over $\mathcal{E}$ . Moreover, equivalence classes of gerbes correspond to Morita equivalence classes of bouquets.

Duskin proved this important result in nonabelian cohomology by showing that a canonical functor

(1)

\mathcal{A} : hom(-,\mathcal{B}) \to Tors(\mathcal{B})

which takes a morphism $f : X \to B_0$ , as an object in the fiber over $X$ of a small fibration $hom(-,\mathcal{B})$ , to the pullback $\mathcal{B}$ -torsor $f^{*}\mathcal{B}$ over $X$ , is an associated stack!

Since this was a genuine example of internal vs. external point of view, in the special case of gerbes rather the stacks (which I described in my reply to Mike above), I was motivated to think what would be an internal version of a 2-gerbe, which were defined externally by Larry Breen in

L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque 225, 1994

as locally non empty and locally connected 2-stacks (in bigroupoids) over the topological space $X$ .

Then I defined a 2-bouquet 2 $\mathcal{G}$ as an internal bigroupoid (in the sense of Bénabou) in the Grothendieck topos $\mathcal{E}$ which satisfy exactly the same axioms (i) and (ii) as in the definition of a bouquet above. For such 2-bouquet 2 $\mathcal{G}$ , I used a so called small 2-fibration $hom(-,\mathcal{B})$ , which I explicitelly described in my thesis , in order to prove the categorification of the associated stack functor (1). The main result of one of my forthcoming papers

I. Bakovic, The small 2-fibration and the associated 2-stack, in preparation

(which I hope to finish soon and make it available to all of you) is that a canonical 2-functor

(2)

\mathcal{A} : hom(-,2\mathcal{G}) \to 2-Tors(2\mathcal{G})

which takes a morphism $f : X \to 2G_0$ , as an object in the fiber over $X$ of a small 2-fibration $hom(-,2\mathcal{G})$ , to the pullback 2 $\mathcal{G}$ -torsor $f^{*}2\mathcal{G}$ over $X$ , is an associated 2-stack!

Posted by: Igor Bakovic on February 18, 2009 11:01 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

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To get back to your question John, and to give you one more hint about bouquets:

if you take for a Grothendieck topos $\mathcal{E}$ the category $Etale(X)$ of étale spaces over the topological space $X$ , then a bouquet in $Etale(X)$ is precisely a bundle gerbe in the traditional sense.

Posted by: Igor Bakovic on February 18, 2009 11:09 PM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

Just a test to bump this onto the front page.

Posted by: Jacques Distler on February 25, 2009 5:44 PM | Permalink | PGP Sig | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

Safari seems happy now. Thanks!

Posted by: Greg Egan on February 26, 2009 7:53 AM | Permalink | Reply to this

class. spaces for bicategories

Today this preprint appeared:

P.CARRASCO, A. M. CEGARRA, A. R. GARZON,

Nerves and classifying spaces for bicategories , arXiv:0903.5058

Posted by: Zoran Skoda on March 31, 2009 3:34 AM | Permalink | Reply to this

Re: class. spaces for bicategories

would welcome comments before I read it

Posted by: jim stasheff on March 31, 2009 4:16 PM | Permalink | Reply to this

Re: class. spaces for bicategories

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In brief:

The authors consider ten different constructions (simplicial, bisimplicial, even pseudosimplicial) which could be considered the nerve of a bicategory, take the geometric realisation, and show they are all homotopy equivalent. They then consider a Thomason-type homotopy colimit result for diagrams of bicategories and the associated Grothendieck construction (suitably generalised).

I think the main application that would interest me is in the case when one has a diagram of 2-groups $F:I^{op} \to 2Grp.$ The resulting bicategory $\int_I F$ over $I$ really is a bicategory.

How about this example: recall that for a connected pointed CW-complex $(X,x)$ , there is a crossed module $\pi_2(X,X_1,x) \to \pi_1(X_1,x)$ which captures the 2-type of $X$ . Call this the fundamental crossed module of $X$ .

Given a small category $I$ , the classifying space $BI =|NI| \simeq BI^{op}$ is a CW complex, and so for each object $i\in I$ , we can define the fundamental crossed module of $BI^{op}$ at $i$ and hence a 2-group. An arrow in $I^{op}$ induces change of basepoints for the fundamental crossed module by conjugating with the corresponding 1-cell in $BI^{op}$ , considered as a path. That this process is functorial is clear. Denote this functor by $\Pi:I^{op} \to 2Grp.$

Thus we have a diagram of strict 2-groups and so a diagram of bicategories. The Grothendieck construction $\int_I \Pi$ given in the paper is a bicategory over $I$ , and Theorem 7.4 in loc. cit. gives a long exact sequence relating the homotopy groups of $I$ , $\int_I \Pi$ and the fibre $\Pi(i)$ over a given basepoint $i\in I$ . These are calculated as the homotopy groups of the respective classifying spaces (by the results of the paper, any one of the models therein can be used for the classifying space of $\int_I\Pi$ ). Since the diagram of bicategories factors through $2Grp$ in this case, the fibres are connected 2-types, and so the long exact sequence tells us that $I$ and $\int_I\Pi$ have isomorphic homotopy groups for dimensions 4 and above, $pi_3(p)$ is an injection (for $p:\int_I\Pi \to I$ ), $pi_1(p)$ is a surjection, and there is an 8-term exact sequence relating the rest of the homotopy groups.

We could also replace the fundamental crossed module with the fundamental crossed complex on a single point, truncated so the associated globular object was a one-object 3-groupoid, then consider the associated diagram of 2-groupoids, forgetting the monoidal structure thereon.

This latter has a better theoretical motivation, because then the exact sequence described above tells us that $\int_I \Pi$ is 3-connected.

Posted by: David Roberts on April 2, 2009 5:00 AM | Permalink | Reply to this

Re: Baković and Jurčo on Classifying Topoi for Topological Bicategories

There is now a final published version available:

Igor Baković, Branislav Jurčo, The classifying topos of a topological bicategory, Homology, homotopy and applications 12 (1), 279-300 (2010), (published files).

Posted by: Zoran Skoda on April 23, 2010 3:57 PM | Permalink | Reply to this

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February 14, 2009