MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Return to Question

2 added answer

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EDIT: It turns out the definition I was trying to remember is unpublished work of Ulrike Tillmann.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

1

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

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?