The n-Category Café

So I kept talking about how the sequence

$$X\hookrightarrow X//G\to BG$$

reminded me of the universal $G$-bundle in its groupoid incarnation

$$G\to G//G\to BG$$

but I had missed that you had already suggested the answer here.

The thing is that there is the universal $\mathrm{Set}$-bundle $$\begin{array}{c}{s}^{-1}\mathrm{pt}\\ \downarrow \\ T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow \\ \mathrm{Set}\end{array}$$

and that the action groupoid sequence is the pullback of that along the representation functor

$$\begin{array}{ccc}X& \to & s^{-1}\mathrm{pt}\\ \downarrow & & \downarrow \\ X//G& \to & T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Set}\end{array}.$$

Hm, that fits in beautifully with what I am trying to do:

In Section s of bundles … I pointed out that if you have a $G$ bundle given by a cocycle

$$g:Y^{•}\to BG$$

and if you choose a representation

$$\rho :BG\to \mathrm{Set}$$

of $G$, then the space of sections of the $\rho$-associated bundle is precisely the space of lifts

$$\begin{array}{ccc}& & X//G\\ & {}^{\sigma }\nearrow & \downarrow \\ Y^{•}& \stackrel{\rho }{\to }& BG\end{array}.$$

So the situation is

$$\begin{array}{ccccc}& & X& \to & s^{-1}\mathrm{pt}\\ & & \downarrow & & \downarrow \\ & & X//G& \to & T_{\mathrm{pt}}\mathrm{Set}\\ & {}^{\sigma }\nearrow & \downarrow & & \downarrow \\ Y^{•}& \stackrel{g}{\to }& BG& \stackrel{\rho }{\to }& \mathrm{Set}\end{array}.$$

Beautiful. It’s so obvious now that I see it, but I was blind to that until now.

David, thanks so much for making me see this.

Finally I am able to nicely unify the different points of view on sections of $n$-bundles which I have been talking about.

In case anyone remembers, I grew fond of the fact that a section is a morphism into the corresponding transport functor/cocycle (for instance here or here.)

In the attempt to put this into a larger perspective, I started characterizing them in terms of tangent categories (for instance here) and ever since bugged John about how there seems to be a relation to groupoidification.

Then, more recently, I started thinking here about that concept of section as a lift of the cocycle through the action groupoid. The reason is that in this formulation, in contrast to the other two above, all that ever appears are groupoids, and hence this formulation, if everything is Lie, can be differentiated and considered entirely in the world of $L_{\mathrm{\infty }}$-algebras (described here).

But it’s annoying to have $2\frac{1}{2}$ different cool abstract definitions of what a section of an $n$-bundle is. Of course in special examples one could see that all definitions agreed, but I had no clue what abstract mechanism related them.

Now I have. Thanks to your remark above. The solution is given in the following diagram (for 0-Cat, but the generalization is essentially obvious):

I have to point out further how very cool this is:

remember all that business about tangent categories: $T_{\mathrm{pt}}\mathrm{Set}$ is is defined to be a subcollection of maps of the standard interval $$I:=\{•\to \circ \}$$ into $\mathrm{Set}$. Now we see how it all comes together with the theory of sections:

Our transformation into the cocycle is an element of

$$\mathrm{Hom}(Y^{•}\otimes I,\mathrm{Set})\simeq \mathrm{Hom}(Y^{•},\mathrm{Hom}(I,\mathrm{Set})).$$

That’s how the $\mathrm{Hom}(I,\mathrm{Set})$ appears. Then the constraint that this is actually a transformation starting at the terminal cocycle yields the constraint appearing in the definition of tangent categories: the $\{•\}\hookrightarrow \{•\to \circ \}$ has to be fixed.

It’s pretty remarkable what is gained by realizing universal bundles not as spaces, but as $n$-groupoids. Not only is $EG$ then the right home for the curvature (when shifted further with one $B$), but also for the sections (when not shifted.) Sounds weird on first sight. But works rather neatly.

I am wondering about the pushout of $$\begin{array}{ccc}BG& \stackrel{\rho }{\to }& \mathrm{Set}\\ \downarrow \\ BEG\end{array}$$

in 2Cat. Here the top left item is the group $G$ regarded as a one-object groupoid. The bottom item is the one-object 2-groupoid with $G$ as its 1-morphisms and a unique 2-morphism between any two 1-morphisms.

Does a strict pushout exist? Or a slightly weak version?