The n-Category Café

You have all the push-forwards you want

This I didn’t say. The recent discussion on Limits and push-forward was supposed to educate me further on the question of push-forward of fiber-assigning functors (or transport functors). I wish I understood this better.

I was hoping to be able to reduce the question to some nice abstract machinery, making full use of the fact that all my $n$-bundles with connection are $n$-functors ($\omega$-functors).

Push-forward of these is Kan-extension. In this comment, following lots of help from Robin Houston, I recall how the formula for the Kan extension of a functor, applied to a very simple toy case, does precisely encode the idea of push-forward of vector bundles, in that the coend formula sums up all the contributions of fibers that are being sent to the same point downstairs.

So if we consider an ordinary vector bundle as a fiber-assigning functor $$F : X \to Vect \,,$$ where $Vect$ is the category of not-neccesarily finite vector spaces, then the Kan-extension of that functor along some map $X \to Y$ produces the fiber assigning functor on $Y$ which represents the pushed-forward vector bundle. (One needs to be careful with the extra structure around, though, like smoothness for instance).

I am trying to use the fact that Kan extensions work in the arbitrary enriched context and that I think I am able to realize $n$-vector spaces entirely in terms of strict globular $n$-categories to define the push-forward of my fiber-assigning (parallel transport) $V$-functor $$X \to T$$ for $V = \omega\mathrm{Cat}$ to be the (left) Kan extension along the corresponding morphism. I was thinking that general existence theorems on the Kan extension when the domain $V$-category is small (as it is for the cases of interest) would guarantee that this exists, and I’d just need to work it out.

But I am not there yet.

I should say what this $\omega$-categorical formulation of $\omega$-vector spaces is that I am referring to:

given an $L_\infty$-algebra $g$ and a cochain complex $v$ (all assumed to be finite dimensional here) I am defining a representation of $g$ on $v$ to be an extension of the DGCA $CE(g)$ by the DGCA $\wedge^\bullet v^*$: $$\wedge^\bullet v^* \leftarrow CE(g,v) \leftarrow CE(g) \,.$$ This is supposed to be the dual differential version of the action groupoid $$V \to V//G \to \mathbf{B} G \,.$$ Ordinary reps of Lie algebras are of this form and every $L_\infty$-algebra has an adjoint rep on itself with this definition. The BRST-complex is also a special case, as we discussed.

So that makes me think that this definition is “good” and “right”. But this then means that I can deduce what the good and right $\omega$-vector spaces are which we need to $\omega$-vector bundles, since they should come from reps of $L_\infty$-algebras.

We integrate an $L_\infty$-algebra $g$ by applying first the functor $$S : DGCAs \to SmoothSpaces$$ which is part of the adjunction induced by the ambimorphic sheaf of forms, and then form path $\omega$-groupoids, $\Pi_n$ or $\Pi_\omega$.

We can apply this to the entire sequence defining the $L_\infty$-rep $$\Pi_\omega \circ S (\wedge^\bullet v^* \leftarrow CE(g,v)\leftarrow CE(g)) \,.$$

The right item will integrate to $\mathbf{B}(-)$ of the “simply connected” Lie $\omega$-group integrating $g$, so this wants to become $$V \to V//G \to \mathbf{B}G$$ with the left $\omega$-groupoid $$V = \Pi_\omega(S(\wedge^\bullet v^*))$$ being the $\omega$-vector space that $G$ is represented on.

Such a $V$ is a bit of a mixture of pure chain complexes (“Baez-Crans $\infty$-vector spaces”) and something richer. Its space of $k$-morphisms is the collection of $v$-valued differential forms on $D^k$ modulo this and that and being closed or satisfying some closure relations depending on the differential structure of $v$.

Then an associated $\omega$-vector bundle would be a transport functor $$\Pi_\omega(x) \to \omega\mathrm{Cat}$$ with local $V$-structure following our general prescription, possibly with some quotients applied on the right $\omega$-groupoid if we don’t want to talk about the “simply connected” structure group $G$ obtained from the integration process, but some quotient by it, usually by $\mathbf{B}^k \mathbb{Z}$s.

I was thinking that by just using abstract nonsense on $\omega Cat$-enriched functors I get a guarantee of push-forwards of such $\omega$-vector bundles. But I still need to think much more about this.

Yes. Only that $Bim(Vect)$ will probably be too simple minded and we will eventually need to look at the classifying space of the bicategory of $X$-algebras and their bimodules, where $X$ is something like “$C^*$” or “von Neumann” or “algebras of local nets of free fermions on the circle” or some other flavor of algebras.

In fact, it should be a parameterized notion of algebra, where the parameter is an elliptic curve. For each elliptic curve there would be a corresponding kind of algebra (“of observables of the CFT on that curve”) and hence a corresponding classifying space for the corresponding elliptic cohomology.

I did talk to 2.5 topologists about this who eventually agreed that computing $|Bimod(something)|$ should be done. I haven’t heard back of further progress along this line for a while, though.

Meanwhile, I am wondering if we need to transfer the problem from $\infty$-categories to topological spaces.

What I’d maybe rather want to do is understand elliptic cohomology not in terms of spectra, but in term of $\infty$-categorical cohomology.