Re: double 2-categories

[Jeff Egger <jeffegger@yahoo.ca>]

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.





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