Hi Jeff , sorry, I should have been more careful . What I mean on elementary terms is: i) a set of objects ii) horizontal and vertical arrows iii) horizontal 2-cells between pairs of parallel horizontal .. arrows

(iv) vertical 2-cells between paris of parallel vertical arrows (v) cells (squares) in squares of horizontal and vertical arrows (vi) cubes for a pair of squares connected at all four sides by 2-cells, horizontal at the horizontal sides, vertical at the vertical sides All of this composes in the expected way, i.e. objects, the horizontal arrows and 2-cells form a 2-category, likewise objets, vertical arrows, and vertical 2-cells. Moreover cubes compose in all three directions : horizontally along common vertical 2-cells, and vertically along common horizontal 2-cells, and in the front-to-back direction along common cells. I now believe, this is a case of what is called a 3-fold category (what I called a "triple category" before), where all arrows in one direction are identities making the double categories sharing this dimension into 2-categories. All the best, Ondrej

On 7 Dec 2010, at 18:35, Jeff Egger <jeffegger@yahoo.ca> wrote:

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Hi Ondrej,

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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 ?

Actually, it's not entirely clear to me what you mean by this (let alone whether there's a reference for it).

Heard out of context, I would have guessed that "double-2-category" should mean "2-category internal to 2-Cat".

This would entail, among other things:  a "2-category of objects" (whose cells I shall call "objects", "vertical arrows" and "vertical discs");  a "2-category of arrows" (whose cells I shall call "horizontal arrows", "squares" and "horizontal tubes"); and,  a "2-category of 2-cells" (whose cells I shall call "horizontal discs", "vertical tubes" and, um, "4-dimensional somethings").

[A horizontal tube is something whose boundary consists of two vertical discs glued to either end of a cylinder (which, in turn, consists of two squares glued together).]

But this is a special case of what I am trying very hard not to call a "double-double category"---i.e., a "quadruple category".  But that disagrees with what follows.

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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.

So perhaps you can give some more details?

Cheers, Jeff.

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