Thanks! I think a special case of a 4-fold category is precisely what I'm after, I just didn't know what are they called beyond the double case. Ondrej On 7 December 2010 14:13, Ronnie Brown <ronnie.profbrown@btinternet.com> wrote:

I am not sure why there is the restriction to having 2-categories as edge arrows. They could be double categories, perhaps.  Would this then be any more general than a 4-fold category?

A definition of n-fold category is given in

34.  (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981) 371-386.

and this also contains a definition of what was later called a globular set, giving a notion of what we now call a strict globular n-category, though the emphasis in the paper is on the groupoid case.

Ronnie

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