double 2-categories
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
participants (4)
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Jeff Egger -
Ondrej Rypacek -
Ronnie Brown -
Ross Street