Hi Ondrej,
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.
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]