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February 14, 2009

Baković and Jurčo on Classifying Topoi for Topological Bicategories

Posted by John Baez

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!

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:

Let me sketch how this works. Jack Duskin described a way to turn a bicategory CC into a simplicial set NCN C with one 0-simplex per object, one 1-simplex per morphism, one 2-simplex per 2-morphism of the form α:fgh\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:

The same trick applies to a topological bicategory CC, and coughs up a simplicial space NCN C. There’s a way to ‘geometrically realize’ any simplicial space and get a space, and if we do this to NCN C we get the classifying space of the topological bicategory. Under some mild conditions this space classifies ‘principal CC-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 CC, the Duskin nerve NCN C of CC is a simplicial space. We introduce the classifying topos BCB C as the Deligne topos of sheaves Sh(NC)Sh(N C) on the simplicial space 2C2 C. It is shown that the category of topos morphisms from the topos of sheaves Sh(X)Sh(X) on a topological space XX to the Deligne classifying topos Sh(NC)Sh(N C) is naturally equivalent to the category of principal CC-bundles. As a simple consequence, the geometric realization of the nerve NCN C of a locally contractible topological bicategory CC is the classifying space of principal CC-bundles (on CW complexes), giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical KK-theory. We also define classifying topoi of a topological bicategory CC 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

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 CC should probably just be the functor 2-category [C op,Cat][C^{op},Cat]. I would guess that for any other 2-topos KK, 2-geometric morphisms K[C op,Cat]K\to [C^{op},Cat] could be identified with flat functors CKC\to K, and that when K=St(X)K=St(X) is the 2-topos of stacks on some space XX, these can be identified with CC-torsors over XX and thereby be related to somebody’s notion of “2-bundle.”

If CC 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 NC 0N C_0, NC 1N C_1, and NC 2N C_2 of objects, morphisms, and 2-cells in CC. I would guess that an object of this 2-topos could thereby be identified with

  • A stack XX on NC 0N C_0, together with
  • A morphism f:d 1 Xd 0 Xf:d_1^{\star} X\to d_0^{\star} X of stacks on NC 1N C_1, and
  • A 2-cell α:d 0 (f)d 2 (f)d 1 (f)\alpha:d_0^{\star}(f) \circ d_2^{\star}(f) \to d_1^{\star}(f) of stacks on NC 2N C_2, such that
  • appropriate cocycle and degeneracy conditions are satisfied, and
  • α\alpha is an isomorphism when restricted to the subspace of NC 2N C_2 consisting of 2-cell isomorphisms in CC.

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 CC is discrete, this condition is needed to recover [C op,Cat][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

The classifying 2-topos of a (discrete) bicategory \mathcal{B} is indeed the 2-category [ op\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 BB\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 BB\mathcal{B} of a topological bicategory \mathcal{B} as the 2-category of \mathcal{B}-stacks, where a \mathcal{B}-stack 𝒫\mathcal{P} is a continuous functor π:𝒫B 0\pi : \mathcal{P} \to B_0 from the stack 𝒫\mathcal{P} to the discrete category 0\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 0)St(N\mathcal{B}_0),St(N 1)St(N\mathcal{B}_1),St(N 2),,St(N\mathcal{B}_2),\ldots, on spaces in the Duskin nerve NN\mathcal{B} of \mathcal{B}, precisely as you described. The right notion of a 2-bundle over a topological space XX is given by principal actions of the bicategory \mathcal{B}: the action of \mathcal{B} on π:𝒫B 0\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)2Top(St(X),B) bi2Bun(X,)2-Top(St(X),B\mathcal{B}) \sim_{bi} 2-Bun(X,\mathcal{B})
between the 2-category 2Top(St(X),B)2-Top(St(X),B\mathcal{B}) of 2-geometric morphisms from the Grothendieck 2-topos St(X)St(X) of stacks over the topological space XX to the classifying 2-topos BB\mathcal{B}, on the one side, and the 2-category 2Bun(X,)2-Bun(X,\mathcal{B}) of principal 2-bundles under \mathcal{B} over XX, 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

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

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 XX, so the only possible definition is the corresponding Grothendieck 2-topos St(X)St(X) of stacks over XX. 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 PP over the topological space XX is defined as a pseudofunctor P:Ouv(X) opCatP : Ouv(X)^{op} \to Cat (where Ouv(X)Ouv(X) denotes the category whose objects are open subsets of XX 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)Sh(X) of sheaves on the topological space XX. 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))Cat(Sh(X)) of internal categories in the Grothendieck topos Sh(X)Sh(X). Then stacks can be seen as a sort of injective objects in Cat(Sh(X))Cat(Sh(X)) and the fibrant objects for Joyal-Tierney model strucuture on Cat(Sh(X))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)Γ:Bun(X)Set Ouv(X) op:Λ\Gamma : Bun(X) \leftrightarrow Set^{Ouv(X)^{op}} : \Lambda

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

(2)Γ:Etale(X)Sh(X):Λ\Gamma : Etale(X) \leftrightarrow Sh(X) : \Lambda

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

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

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

between the 2-category 2Bun(X)2-Bun(X) of 2-bundles over XX and the 2-category Cat Ouv(X) opCat^{Ouv(X)^{op}} of fibered categories over XX, whose objects are pseudofunctors P:Ouv(X) opCatP : Ouv(X)^{op} \to Cat. Then the 2-functor Λ 2:Cat Ouv(X) op2Bun(X)\Lambda_2 : Cat^{Ouv(X)^{op}} \to 2-Bun(X), defined as the filtered (bi)colimit over neighborhoods of points in XX, would be the left (bi)adjoint to the 2-functor Γ 2:Cat Ouv(X) op)2Bun(X)\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)Γ 2:2Etale(X)St(X):Λ 2\Gamma_2 : 2-Etale(X) \leftrightarrow St(X) : \Lambda_2

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

These are the two equivalent ways (internal and external, respectively) in which we can define the Grothendieck 2-topos corresponding to the topological space XX. 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:ACatF :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:ACatP : A \to Cat from the 2-filtered (strict) 2-category AA in

E. Dubuc, R. Street, A construction of 22-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

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

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

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

would be the comparison between

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

These latter are generally called O(X)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)O(X)-valued sets) or use some sort of generalized morphisms (anafunctors, or O(X)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

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

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

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

X¯\overline{X} is a strict functor for n=1n=1, but I don’t see why it would be for arbitrary nn. For n=1n=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 (,1)(\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)(n+1)-category of nn-categories would be strict. At the least, it will depend on carefully choosing your model for nn-categories.

Also, even when X¯\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 (,0)(\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

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

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 01B_0 \to 1 is an epimorphism

(ii) \mathcal{B} is locally connected, i.e. the canonical morphism B 1B 0×B 0B_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()Tors(\mathcal{B}) of \mathcal{B}-torsors is a gerbe over \mathcal{E}, and conversely that any gerbe 𝒢\mathcal{G} over \mathcal{E} gives rise to a bouquet \mathcal{B} such that we have an equivalence 𝒢Tors()\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)𝒜:hom(,)Tors()\mathcal{A} : hom(-,\mathcal{B}) \to Tors(\mathcal{B})

which takes a morphism f:XB 0f : X \to B_0, as an object in the fiber over XX of a small fibration hom(,)hom(-,\mathcal{B}), to the pullback \mathcal{B}-torsor f *f^{*}\mathcal{B} over XX, 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 XX.

Then I defined a 2-bouquet 2𝒢\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 2𝒢\mathcal{G}, I used a so called small 2-fibration hom(,)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)𝒜:hom(,2𝒢)2Tors(2𝒢)\mathcal{A} : hom(-,2\mathcal{G}) \to 2-Tors(2\mathcal{G})

which takes a morphism f:X2G 0f : X \to 2G_0, as an object in the fiber over XX of a small 2-fibration hom(,2𝒢)hom(-,2\mathcal{G}), to the pullback 2𝒢\mathcal{G}-torsor f *2𝒢f^{*}2\mathcal{G} over XX, 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

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)Etale(X) of étale spaces over the topological space XX, then a bouquet in Etale(X)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

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 op2Grp. F:I^{op} \to 2Grp. The resulting bicategory IF\int_I F over II really is a bicategory.

How about this example: recall that for a connected pointed CW-complex (X,x)(X,x), there is a crossed module π 2(X,X 1,x)π 1(X 1,x)\pi_2(X,X_1,x) \to \pi_1(X_1,x) which captures the 2-type of XX. Call this the fundamental crossed module of XX.

Given a small category II, the classifying space BI=|NI|BI opBI =|NI| \simeq BI^{op} is a CW complex, and so for each object iIi\in I, we can define the fundamental crossed module of BI opBI^{op} at ii and hence a 2-group. An arrow in I opI^{op} induces change of basepoints for the fundamental crossed module by conjugating with the corresponding 1-cell in BI opBI^{op}, considered as a path. That this process is functorial is clear. Denote this functor by Π:I op2Grp. \Pi:I^{op} \to 2Grp.

Thus we have a diagram of strict 2-groups and so a diagram of bicategories. The Grothendieck construction IΠ\int_I \Pi given in the paper is a bicategory over II, and Theorem 7.4 in loc. cit. gives a long exact sequence relating the homotopy groups of II, IΠ\int_I \Pi and the fibre Π(i)\Pi(i) over a given basepoint iIi\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 IΠ\int_I\Pi). Since the diagram of bicategories factors through 2Grp2Grp in this case, the fibres are connected 2-types, and so the long exact sequence tells us that II and IΠ\int_I\Pi have isomorphic homotopy groups for dimensions 4 and above, pi 3(p)pi_3(p) is an injection (for p: IΠIp:\int_I\Pi \to I), pi 1(p)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 IΠ\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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