### 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.

## 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

weakhigher categorical structures to “model” homotopy types. But for years, Ronnie Brown and co-workers have been developing a way of doing higher algebraic topology withstricthigher 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!