### 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!

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

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:

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.

## 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 $C$ 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

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.