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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EDIT: It turns out the definition I was trying to remember is unpublished work of Ulrike Tillmann.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).

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are you familiar with arxiv.org/abs/math.CT/0107188? maybe there is an answer in his "chatty" bibliography –  Sean Tilson Nov 12 '10 at 13:19
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2 Answers 2

up vote 7 down vote accepted

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 $[0,1]$.

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Thanks, that's helpful. The paper by Brown matches what I remember pretty well, but I thought there was some other name for this. I'll wait and see if any other answers are forthcoming. –  Kevin Walker Nov 12 '10 at 4:27
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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:

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

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Not quite what I'm after, but interesting nevertheless. Thanks. –  Kevin Walker Nov 12 '10 at 15:22
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