* double 2-categories @ 2010-12-07 12:59 Ondrej Rypacek 2010-12-07 14:13 ` Ronnie Brown ` (3 more replies) 0 siblings, 4 replies; 6+ messages in thread From: Ondrej Rypacek @ 2010-12-07 12:59 UTC (permalink / raw) To: 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/ ] ^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: double 2-categories 2010-12-07 12:59 double 2-categories Ondrej Rypacek @ 2010-12-07 14:13 ` Ronnie Brown 2010-12-09 9:36 ` Ross Street [not found] ` <4CFE4105.5020902@btinternet.com> ` (2 subsequent siblings) 3 siblings, 1 reply; 6+ messages in thread From: Ronnie Brown @ 2010-12-07 14:13 UTC (permalink / raw) To: Ondrej Rypacek; +Cc: categories 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/ ] ^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: double 2-categories 2010-12-07 14:13 ` Ronnie Brown @ 2010-12-09 9:36 ` Ross Street 0 siblings, 0 replies; 6+ messages in thread From: Ross Street @ 2010-12-09 9:36 UTC (permalink / raw) To: Ronnie Brown; +Cc: Ondrej Rypacek, categories 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/ ] ^ permalink raw reply [flat|nested] 6+ messages in thread
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* Re: double 2-categories [not found] ` <4CFE4105.5020902@btinternet.com> @ 2010-12-07 16:08 ` Ondrej Rypacek 0 siblings, 0 replies; 6+ messages in thread From: Ondrej Rypacek @ 2010-12-07 16:08 UTC (permalink / raw) To: Ronnie Brown; +Cc: categories 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: double 2-categories 2010-12-07 12:59 double 2-categories Ondrej Rypacek 2010-12-07 14:13 ` Ronnie Brown [not found] ` <4CFE4105.5020902@btinternet.com> @ 2010-12-07 18:35 ` Jeff Egger [not found] ` <456981.42294.qm@web110605.mail.gq1.yahoo.com> 3 siblings, 0 replies; 6+ messages in thread From: Jeff Egger @ 2010-12-07 18:35 UTC (permalink / raw) To: Ondrej Rypacek, categories 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/ ] ^ permalink raw reply [flat|nested] 6+ messages in thread
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* Re: double 2-categories [not found] ` <16A709CA-BA0D-4E71-8994-F700356973D3@gmail.com> @ 2010-12-07 19:31 ` Ondrej Rypacek 0 siblings, 0 replies; 6+ messages in thread From: Ondrej Rypacek @ 2010-12-07 19:31 UTC (permalink / raw) To: Jeff Egger; +Cc: categories > 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 6+ messages in thread
end of thread, other threads:[~2010-12-09 9:36 UTC | newest] Thread overview: 6+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2010-12-07 12:59 double 2-categories Ondrej Rypacek 2010-12-07 14:13 ` Ronnie Brown 2010-12-09 9:36 ` Ross Street [not found] ` <4CFE4105.5020902@btinternet.com> 2010-12-07 16:08 ` Ondrej Rypacek 2010-12-07 18:35 ` Jeff Egger [not found] ` <456981.42294.qm@web110605.mail.gq1.yahoo.com> [not found] ` <16A709CA-BA0D-4E71-8994-F700356973D3@gmail.com> 2010-12-07 19:31 ` Ondrej Rypacek
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