double 2-categories

double 2-categories

[Ondrej Rypacek]

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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Re: double 2-categories

[Ronnie Brown]

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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Re: double 2-categories

[Ross Street]

I would also point to the papers by Andrée and Charles Ehresmann:

Multiple functors. II. The monoidal closed category of multiple  
categories.
Cahiers Topologie Géom. Différentielle 19 (1978), no. 3, 295–333.

Multiple functors. III. The Cartesian closed category ${\rm Cat}_{n}$.
Cahiers Topologie Géom. Différentielle 19 (1978), no. 4, 387–443.

==Ross

On 08/12/2010, at 1:13 AM, Ronnie Brown wrote:

> 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


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Re: double 2-categories

[Ondrej Rypacek]

Thanks! I think a special case of a 4-fold category is precisely what
I'm after, I just didn't know what are they called beyond the double
case.

Ondrej




On 7 December 2010 14:13, Ronnie Brown <ronnie.profbrown@btinternet.com> wrote:
> 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
>

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Re: double 2-categories

[Jeff Egger]

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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Re: double 2-categories

[Ondrej Rypacek]

> 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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