### On n-Transport, Part I

#### Posted by Urs Schreiber

Ever since John planted this idea into my brain, I have been thinking about (“parallel”) transport along $n$-paths.

Essentially, $n$-transport is nothing but some $n$-functor. But regarding an $n$-functor as describing $n$-transport makes us want to do certain things with it.

Most notably, we tend to have the desire to

$\;\;\;\;$ $\bullet$ locally trivialize

a given globally defined $n$-transport.

Moreover, given a local trivialization, all we often really want to know about it are its

$\;\;\;\;$ $\bullet$ local transitions.

The story of $n$-transport begins with making these two concepts precise. This is what I talk about in this *part I*.

While the above may appear rather unspecific, it is noteworthy that the character of a particular $n$-transport with local trivialization depends in its essence only on a single datum, namely a morphism

of codomains. I’ll say in detail what this means in the pdf linked to below.

What is nice about this is that all kinds of well-known and not-so-well-known structures under the sun turn out to be local transition data for $n$-transport with respect to some such morphism $i$.

You stick one choice of $i$ into the $n$-transport formalism, turn the crank (and it’s a canonical crank), and out drops some concept that people have written pages about to define. Or that people have not defined yet - but should have.

Here are some examples.

The motivating toy example is obtained by
letting $i$ be the natural embedding $\Sigma(\mathrm{End}(\mathbb{C}^n)) \stackrel{i}{\to} \mathrm{Vect}_\mathbb{C}$. Then $i$-transport is a **vector bundle** with connection and $i$-trivialization is a local trivialization.

The following example is very similar, but more subtle: let $\Sigma(\Sigma(\mathbb{C})) \stackrel{i}{\to} \Sigma(1\mathrm{DVect})$. Then $i$-transitions are precisely **line bundle gerbes** with connection.

Or take $\Sigma(H\to G) \overset{i}{\to} \Sigma(\mathrm{Tor}_G(H))$. Then $i$-transitions are precisely **nonabelian principal bundle gerbes** with (fake-flat) connection.

There is an iterative notion of locally trivializing an $i$-transition. This leads to interesting trivializations coming from identity morphisms:

Let $i$ be the identity morphism on the $n$-group $\Sigma^n(U(1))$. Then $i$-transitions are

Deligne cocycles.

Let $i$ be the identity morphism on an arbitrary $n$-group. Then $i$-transitions are

nonabelian differential cocycles, characterizing nonabelian gerbes with connection.

There are more details and more examples, but this should give the idea.

Actually, as you may have guessed, there is not only a morphism of codomains involved, but also a morphism, $p$, of domains. Varying this leads to interesting examples of another aspect of transport.

Let $X$ be a space with a $G$ action, and let $p$ be the projection of $n$-paths on $X$ onto $n$-paths in $X/G$. Then $p$-local transitions describe **$G$-equivariant** versions of the above concepts.

For $n \gt 1$ equivariance has new qualitative aspects that are unknown for $n=1$. Physicists know that parallel transport of strings knows not just *orbi*folding, but also ** orientifolding**. It turns out that orientifold transport is nothing but a special case of $\mathbb{Z}_2$-local $(1\mathrm{DVect} \overset{i}{\to} \mathrm{Bim}_\mathbb{C})$-transition of 2-transport.

This had been understood originally by a detour through 2D CFT reasoning. For suitable choices of $i$, 2-transport also seems to know about 2D QFT. This is in fact one of my main motivations at the moment, but here I’ll spare that maybe for later.

In forthcoming contributions, I would like to discuss these examples in more detail. They are all closely related to several topics that have already appeared here ($\to$).

But for that to be fruitful, everybody first has to absorb the basic notion of $i$-trivialization, from which everything follows by crank-turning.

Since this stuff cannot sensibly be reasoned about without diagrams, this entry continues with the following pdf.

## Re: On n-Transport, Part I

Urs, TraTriTra was not at all what I expected, so in addition to commenting on what you do say, I’ll inlcude comments on what I expected.

line 2 of body in THE base category of fibres would be clearer if you said category of morphisms of fibres

in classical bundle theory, $i$-trivial would be reducible (from $T$ as structure group to $T'$) and in fact $i$ there need not be an injection cf. $\mathrm{Spin}(n) \to O(n)$

I’m not up on ‘adjoint equivalence’ - why not just equivalence?

how about commenting on $i$-trivial for $T'$ the one object, one morph cat? as in ordinary bundle speak?

$P_U$ –>> $P$ ?? the very notation suggests the restriction of $P$ to e.g. an open set $U$

what’s proper all about? I don’t recognize a bundle analog

$P^{[2]}$ denotes composable pairs?

p.6 where did $\bar t$ come from? and why should there be one?

p.7 the 3 bar sign denotes ?some kind of equivalence? without invoking a 3-morphism?

the right hand pictures up top are very reminiscent of the picture of homtopy associativity

let $p^*_{23}g = a$,

$p^*_{13}g = \bar b$ that is, reverse the arrow

so $p^*_{12}g = \bar b \bar a$

$p^*_1 t = c$,

so $p^*_3 \bar t =bc$

then $p^*_2 \bar t =a(bc)$

and $p^*_2 t =(ab)c$

leaving room for an associating homtopy!! though I think the double arrow would then be drawn horizontally

****************************************************

now back to basics

for 1-transport, $P$ is the path space traditionally or thin homotopy classes thereof (though again, why use inverses?)

for 2-tranport, $P$ is the space of maps of $I^2$ into $T$ ?? or a 2-simplex (cf.Serre fibration) into $T$ with the 2-simplex regarded as a 2-morphism??

where is this studied in detail - before trying for $n$?

this is fun