The n-Category Café

an entry higher parallel transport that has a section on flat ∞-parallel transport in Top

Another remark along these lines is now at $Top$ as a cohesive $\infty$-topos.

I’m having trouble getting my .pdf reader to read this.

Not sure what I can do. The new version has a bunch of colorful graphics of simplices. On my system this does display, but when I point my pdf reader to these graphics pages it suffers from something that makes it becomes slow to the point that it might seem it has frozen.

Hi Todd,

you write:

I recently asked a question on Math Overflow which seems to be related to this.

I am not sure in which way you think of Jim’s notes here as related to your question there. Methinks what Jim is describing here is rather orthogonal to the discussion there. (But of course all these things are related in a web of relations.)

What I do see is a relation between the MO-answer by Bill Thurston that you have checked as correct (or whatever that check mark means): the simplicial model for $B G$ involving connections that Thurston sketches is what in Jim’s latest article Čech cocycles for differential characteristic classes is denoted $\mathbf{B}G_{diff}$.

I think in the notes posted in this entry here, Jim is doing something else. To see if we can agree on it, let me try to summarize the situation in my words:

Jim considers topological fibrations $E \to B$ with fiber $F$. Write $Aut(F)$ for the “automorphism $\infty$-group” of $F$, which we may think of as realized as a topological group. In any case we have a delooping $B Aut(F)$ and the fibration $E \to B$ is classified by a map $B \to B Aut(F)$. It is the $F$-$\infty$-bundle associated to the $Aut(F)$-principal $\infty$-bundle that is classified by $B \to B Aut(F)$.

Now we want to think of this situation in terms of higher parallel transport and (flat!) topological $\infty$-connections. We may simply apply the fundamental $\infty$-groupoid functor

$$Sing : Top \to sSet$$

to the situation to obtain for every $F$-fibration a “flat $\infty$-parallel transport” (what Jim in his note calls a “representation up tp homotopy”)

$$Sing B \to Sing B Aut(F) \,.$$

Here since $Sing$ is right adjoint and in fact even an equivalence in the correct sense, we may pass it through the delooping and the automorphisms and think of this as

$$\theta : Sing B \to B Aut F \,,$$

where now the symbols on the right are implicitly with their meaning in $sSet$.

This is, I think, what unwinds in components to Jim’s definition 3.1.

So what happens is that we re-think the topological classifying map in terms of parallel transport, by invoking the homotopy-hypothesis-equivalence.

And then theorem 8 and the statement of section 4 follow from the above statement of classification of fibrations in $Top$, using that $Sing$ is an equivalence: every fibration $E \to B$ admits a flat $\infty$-parallel transport $Sing B \to B Aut(F)$ and may be reconstructed from it.

Or rather, conversely, what Jim does in his note is to provide explicit combinatorial formulas to present this general abstract construction concretely.

The situation to compare this to is the canonical connection on a $G$-principal bundle for $G$ a discrete group, this is essentially unique and flat, and its holonomy entirely classifies that bundle. Here we have the $\infty$-version of this situation:

we are to think of a topological space/simplicial set analogously as a discrete $\infty$-groupoid, hence of $Aut(F)$ as a discrete $\infty$-group. Accordingly, we find that every $Aut(F)$-principal $\infty$-bundle has an essentially unique flat $\infty$-connection and may be reconstructed from the parallel transport of the flat $\infty$-connection.

In conclusion, I think if we apply this to your MO question, we find ourselves in a trivial setup: we’d need to assume the group $G$ in your setup to be discrete. Then $Sing B G$ is canonically modeled as the groupoid $(G \stackrel{\to}{\to} \bullet)$ which we may think of as $\{\bullet \stackrel{g}{\to} \bullet\}$ where now indeed every arrow is labeled by the holonomy of the unique and flat connection evaluated around it (since $G$ is discrete). Then the intuition that the map $\Omega B G \to G$ is given by evaluating parallel transport becomes true in a precise but essentially tautological sense.

To go beyond that tautological sense, one needs to consider smooth structure and non-dicrete $\infty$-groups.

If you have trouble seeing the graphics,
just e-mail me for a copy (how primitive!)

I will respond to Todd’s question on mathoverflow