Re: double 2-categories

[Ondrej Rypacek <ondrej.rypacek@gmail.com>]

> 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:
>
>> 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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