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?




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EDIT: It turns out the definition I was trying to remember is unpublished work of Ulrike Tillmann.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



What's the name of this flavor of n-category?
Ronnie Brown has a related idea, contained in this article:



> 
> Moore hyperrectangles on a space form a strict cubical omega-category  
> 
> [arXiv](http://arxiv.org/abs/0909.2212)
> 
> 
> 


discussed briefly [here at the nLab](http://ncatlab.org/nlab/show/Moore+path+category).


If you are instead thinking of a globular $n$-category, the closest I know of is a [Trimble n-category](http://ncatlab.org/nlab/show/Trimble+n-category), but that doesn't use Moore paths, but paths of length 1 and the $A\_\infty$-co-category structure on $[0,1]$.


