May 31, 2007

Kock on Higher Connections

Posted by Urs Schreiber

Building on his synthetic description of parallel transport, which I mentioned a while ago in Kock on 1-Transport, Anders Kock has now worked out a notion of higher order connections using synthetic differential geometry:

Anders Kock
Infinitesimal cubical structure, and higher connections
arXiv:0705.4406v1

This is rooted in the world of strict $n$-fold groupoids: an $n$-connection here is an $n$-fold functor from cubical $n$-paths in a space to an $n$-fold groupoid $G_{(n)}$: $\nabla : P_n^{\mathrm{cub}}(X) \to G_{(n)} \,.$

Smoothness of this functor is described using synthetic differential reasoning (described in detail in his book).

In the strict cubical context, Anders Kock finds precisely the relation between parallel transport, curvature, and Bianchi identity which I describe in $n$-Curvature:

For any given parallel $n$-transport, the corresponding curvature is an $(n+1)$-transport. The latter is necessarily flat, meaning that its curvature $(n+2)$-transport is trivial. This flatness of the curvature $(n+1)$-transport is the (higher order) Bianchi identity.

Posted at May 31, 2007 12:25 PM UTC

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Re: Kock on Higher Connections

Kock’s paper looks nice.

I’m immediately struck by a comment in the Introduction: he thanks Ronnie Brown for persuading him to “think strictly and cubically”. Now, I understand what this means. But what does it mean?

It’s the “strict” rather than the “cubical” that I’m interested in. Most Café readers are presumably familiar with the way that we expect weak higher categorical structures to “model” homotopy types. But for years, Ronnie Brown and co-workers have been developing a way of doing higher algebraic topology with strict higher categorical structures. How is this possible?

The way they turn spaces into categorical structures is presumably different from the way we usually talk about. Indeed, the categorical structures that they end up with aren’t just strict $n$-categories. If I remember correctly, one of the most successful theorems (due to Loday?) is that “homotopy $n$-types are modelled by $cat^n$-groups”.

Can anyone explain what’s going on? I know I could go and read a bunch of papers, but… well, I haven’t got round to it at any point in the last ten or so years. If someone already gets the point and would like to explain it here, that would be wonderful!

Posted by: Tom Leinster on May 31, 2007 2:55 PM | Permalink | Reply to this

Re: Kock on Higher Connections

How is this possible?

So they claim that somehow one can circumvent the need for weak higher $n$-categories by instead using (strict) $n$-fold categories?

Say I have a 3-group (a 1-object trigroupoid) which is not strict and not equivalent to a strict 3-group. Would it be possible to conceive this equivalently as a strict 3-fold groupoid?

It’s not completely implausible that this can be done, since in an $n$-fold category we have more “freedom” than in an $n$-category, in general:

for instance we may pass from 2-fold categories to 2-categories when the horizontal and the vertical 1-morphism catgegories are the same. But they need not be the same.

Posted by: urs on May 31, 2007 3:06 PM | Permalink | Reply to this

Re: Kock on Higher Connections

Urs wrote:

So they claim that somehow one can circumvent the need for weak higher $n$-categories by instead using (strict) $n$-fold categories?

I don’t know. But if that is what’s going on, how do you turn a space into a strict $n$-groupoid?

Posted by: Tom Leinster on May 31, 2007 4:05 PM | Permalink | Reply to this

Re: Kock on Higher Connections

Sorry - when I wrote strict $n$-groupoid’, I meant strict $n$-fold ($=$ $n$-tuple) groupoid’.

Posted by: Tom Leinster on May 31, 2007 4:07 PM | Permalink | Reply to this

Re: Kock on Higher Connections

how do you turn a space into a strict n-groupoid?

Not sure. But isn’t there an obvious fundamental strict $\infty$-fold groupoid associated with every space $X$:

$k$-morphisms are maps $[0,1]^k \to X$ which are constant in a neighbourhood of the boundary.

Composition in all directions is the obvious gluing of these maps.

Am I overlooking a subtlety?

Posted by: urs on May 31, 2007 4:09 PM | Permalink | Reply to this

Re: Kock on Higher Connections

Am I overlooking a subtlety?

Okay, apart from the fact that this isn’t a groupoid… ;-)

Posted by: urs on May 31, 2007 4:10 PM | Permalink | Reply to this

Re: Kock on Higher Connections

In fact, I am not entirely sure I understand exactly how Anders Kock deals with this issue. Some synthetic magic might be going on which escapes me:

in definition 3 we have any $k$-fold/$k$-cubical/$k$-tuple groupoid, denoted $G_\bullet$, which is only required to have object set coinciding with base space.

Posted by: urs on May 31, 2007 4:19 PM | Permalink | Reply to this

Re: Kock on Higher Connections

Am I overlooking a subtlety?

Surely composition isn’t unital or associative?

Posted by: Tom Leinster on May 31, 2007 5:45 PM | Permalink | Reply to this

Re: Kock on Higher Connections

Right, I’d need to divide out by reparameterizations, certainly.

Posted by: urs on May 31, 2007 5:57 PM | Permalink | Reply to this

Re: Kock on Higher Connections

I don’t know if even that’s enough. Suppose we’re just trying to turn a space $X$ into a strict 1-groupoid, by taking the objects to be the points and the morphisms to be the set of continuous maps $[0, 1] \to X$ ‘divided out by reparametrizations’. What does the phrase in quotes have to mean in order to make this work?

My first idea was that a ‘reparametrization’ might be an endpoint-preserving self-homeomorphism of $[0, 1]$. So you’d identify two maps $[0, 1] \to X$ if one was the composite of a reparametrization with the other.

But this won’t do. A (strict 1-)groupoid has to have (strict) identities, and this doesn’t. In particular, constant paths aren’t identities: the composite $f \circ const_x$ of a path $f: x \to y$ with the constant path at $x$ is not a reparametrization of $f$, in the sense of the previous paragraph.

Posted by: Tom Leinster on May 31, 2007 8:33 PM | Permalink | Reply to this

Re: Kock on Higher Connections

To get a strict 1-groupoid of paths, namely the path groupoid $P_1(X)$, we let the space of objects be $X$ and let the space of morphisms be thin homotopy classes of paths which are constant near their boundary points.

Two paths are thinly homotopic if there is a homotopy between them whose differential has less than full rank everywhere.

(As described in John’s lecture Quantization and Cohomology, week 21.)

This applies to the smooth case. In the continuous case we can still revert to ordinary homotopy classes of paths.

In a similar manner we get (weak or strict) $n$-groupoids $P_n(X)$ of $n$-paths.

But actually for $n$-fold paths this might be more subtle. I have to think further about this.

Posted by: urs on May 31, 2007 8:49 PM | Permalink | Reply to this

Re: Kock on Higher Connections

In the continuous case, two paths are thinly homotopic if the homotopy connecting them

$H:I^2 \to X$

factors through a tree $T \subset I^2$.

Posted by: David Roberts on June 1, 2007 6:41 AM | Permalink | Reply to this

thin homotopy

In the continuous case, two paths are thinly homotopic if the homotopy connecting them […] factors through a tree […]

Hm, I think I see what you mean. Interesting. Would this give an equivalent definition of thin homotopy also for the smooth case? Do you have a reference?

Posted by: urs on June 1, 2007 12:14 PM | Permalink | Reply to this

n-fold versus n

By the way, we once talked about the issue whether to use 2-categories or double catgories for the description of parallel 2-transport here.

There we also mentioned the paper

R. Brown & G. Mosa, Double Categories, 2-Categories, Thin Structures and Connections, Theor. Appl. Cat. 5 7 (1999) 163-175 (pdf)

which nicely discusses how to pass from double categories to 2-categories when the horizontal and vertical morphism catgories coincide (and when some extra condition is met).

By the way, beware that, as Anders Kock also remarks in his introduction, the term “connection” as in the title of the above paper is completely unrelated to the one as appearing in “connection on a bundle”.

Posted by: urs on May 31, 2007 3:14 PM | Permalink | Reply to this
Read the post The second Edge of the Cube
Weblog: The n-Category Café
Excerpt: Differentiating parallel transport anafunctors to Cartan-Ehresmann connections.
Tracked: May 31, 2007 9:54 PM
Read the post Arrow-Theoretic Differential Theory
Weblog: The n-Category Café
Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 27, 2007 5:29 PM

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