What's the name of this flavor of n-category? - MathOverflow

What's the name of this flavor of n-category?

2010-11-12T03:34:16Z

I'm looking for the name of a certain n-category definition. (Someone explained it to me a couple of years ago. I remember the definition, but not the name. Without the name it's difficult to search for a citation. I want the citation in order to explain something we're not doing in a paper.)

For background, consider the Moore loop space $\Omega_r$ of loops of length $r$ (that is, parameterized by the interval $[0,r]$). We have a strictly associative composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. The main idea of an "xxxx" n-category is to imitate this idea in higher dimensions. The $k$-morphisms are parameterised by $k$-dimensional rectangles with sides of lengths $r_1,\ldots,r_k$. Gluing rectangles together gives $k$ different strictly associative ways to compose $k$-morphisms.

Question: What is "xxxx" above?

Bonus question: What's the best (or any) citation for this idea?

EDIT: It turns out the definition I was trying to remember is unpublished work of Ulrike Tillmann. But the version from Ronnie Brown linked to in David Roberts' answer is pretty similar (for my purposes, at least).

Answer by David Roberts for What's the name of this flavor of n-category?

2010-11-12T03:54:12Z

Ronnie Brown has a related idea, contained in this article:

Moore hyperrectangles on a space form a strict cubical omega-category
arXiv

discussed briefly here at the nLab.

If you are instead thinking of a globular $n$-category, the closest I know of is a Trimble n-category, but that doesn't use Moore paths, but paths of length 1 and the $A_\infty$-co-category structure on $[0,1]$.

Answer by Peter LeFanu Lumsdaine for What's the name of this flavor of n-category?

2010-11-12T08:10:17Z

Simpson-semistrict $n$-categories could be what you're after: $n$-categories where everything except the unit laws holds strictly, generalising one of the crucial properties of Moore path spaces? It's not a specific definition of $n$-category, but a strictness property which can be applied within various definitions.

Carlos Simpson has conjectured that these are enough to model homotopy types; Moore path space show this in dimension 1. I know very little about the details of this myself, I'm afraid, but what I have read about it is mostly from these sources plus their links and discussions:

Simpson, Homotopy types of strict 3-groupoids.
nlab: semi-strict $\infty$-category
nlab: Simpson’s conjecture (I can't figure out how to link this directly; the single-quote in the url seems to confuse markdown)
n-Category Café: Urs Schreiber, Semistrict Infinity-Categories and ω-Semi-Categories

I believe several people have been making some progress on it recently; eg Makkai mentioned some results along these lines at the latest Octoberfest.