The n-Category Café

So how many of our questions got answered?

I had long and very interesting discussions with Nils Baas, that I am very grateful for. I tried to attract Nils Baas to our first $n$-Café Millenium Prize Question. But I am not sure if I did succeed.

So how many of our questions got answered?

As it goes with good conferences, there is always more to talk about than time permits.

And that even holds if one is crazy enough to visit Toronto and be content with College Street being the only part of the city one visits.

Among the things that I was looking forward to talk about was Toby’s reconstruction of 2-bundles from local data.

We did find time for that, if maybe not as much as the topic deserves.

Interestingly, it turns out that, as far as I could figure out, Toby’s reconstruction does produce a $G_2$-2-bundle that is trivial as a fiber bundle of categories $B\to X$, i.e. where the category $B$ is of the form $B=X\times F$. All the nontriviality is pushed into the $G_2$-action $B\times G_2\to B$.

In fact, something roughly similar happens for the reconstruction of global transport 2-functors that I am working on with Konrad Waldorf.

So how many of our questions got answered?

One nice thing was that I received a couple of answers to questions that I did not even know I should ask! :-)

One nice insight was this:

André Joyal, in his talk, mentioned various important concepts in category theory. One of the them was related to factorization systems.

In that business, one considers collections $E$ and $M$ of morphisms that (are supposed to) behave like epi- and monomorphisms.

In particular, one considers squares $$\begin{array}{ccc}A& \stackrel{\mathrm{epi}}{\to }& B\\ \downarrow & & \downarrow \\ C& \stackrel{\mathrm{mono}}{\to }& D\end{array}$$ where the top morphism is an epimorphism and the bottom morphisms is a monomorphism.

Of special interest, then, is the situation where a square of the above kind admits a diagonal $$\begin{array}{ccc}A& \stackrel{e}{\to }& B\\ \downarrow & \swarrow & \downarrow \\ C& \stackrel{m}{\to }& D\end{array},$$ for every choice of vertical arrows, such that everything still commutes

If that is the case, one says that the epimorphism $e$ is orthogonal to the monomorphism $m$.

There are more or less obvious generalizations of this concept for the case where everything lives in $n$-categories.

(I am grateful to Igor Bakovic for helpful discussion of this and of the related literature. In as far as I have forgotten the details again, that’s completely my own fault.)

What made all this interesting for me was that I recognized my own pet structures in this general structure that people are considering:

In my talk in Toronto I described how we can put local structure on an $n$-functor $$\begin{array}{c}{P}_n(X)\\ \downarrow \mathrm{tra}\\ T\end{array}$$ by putting it into a square $$\begin{array}{ccc}{P}_n(U)& \stackrel{p}{\to }& P_n(X)\\ {\mathrm{tra}}_U\downarrow & \Downarrow \sim & \downarrow \mathrm{tra}\\ T\prime & \stackrel{i}{\to }& T\end{array},$$ where $p$ is epi and $i$ is mono.

If $p$ happens to be “orthogonal” to $i$ in the sense of factorization system theory, i.e. if we can fill this diagram $$\begin{array}{ccc}{P}_n(U)& \stackrel{p}{\to }& P_n(X)\\ {\mathrm{tra}}_U\downarrow & \swarrow & \downarrow \mathrm{tra}\\ T\prime & \stackrel{i}{\to }& T\end{array},$$ then this means, in the context I consider, that $\mathrm{tra}$ has the given structure not just locally, but already globally.

By itself that’s not deep or anything. But I enjoyed seeing this contact to a larger context.

I took most of the photos; Dan Christensen took a few.