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 On 07/12/2010 12:59, Ondrej Rypacek wrote:
Dear all,
Is there a standard reference for what could be called a double-2-category, by which I mean a double category where the categories of horizontal and vertical arrows are 2-categories ? It would be a special case of a "triple category", I guess, where there are objects, arrows in three directions, cells for each distinct pair of the directions, and cubes surrounded by cells.
Many thanks, Ondrej
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