Re: Composing modifications

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

Dear All,

Has anyone formulated  a cubical (square?) version of bicategory or higher?

The model is surely the singular cubical complex KX of a topological
space. This has a great deal of  structure of multiple compositions and
tensor products which we have  fairly easily written down and exploited
in papers with Philip Higgins. By working with filtered spaces and
taking certain homotopy classes we also get (non trivially!) associated
strict structures.

There is also a notion of fibrant (=Kan) cubical set, and of fibration,
which seems not available globularly (?). So one formulation of a weak
cubical omega-category is to say it comes with a cubical fibration to a
strict cubical omega-category. (Such exists for RX_*, the cubical
singular complex of a filtered space.

(with P.J. HIGGINS), ``Colimit theorems for relative homotopy groups'',
{\em J. Pure Appl. Algebra} 22 (1981) 11-41.
)  This is a definition with one example and no theorems (as yet)!

By contrast, there is a singular globular complex GX of a space, see for
example

`A new higher homotopy groupoid: the fundamental  globular
$\omega$-groupoid of a filtered space', Homotopy, Homology and
Applications, 10 (2008), No. 1, pp.327-343.

but I think nobody has written down an axiomatisation.

Multiple compositions are difficult (for me, at any rate) in the
globular (and simplicial!) situation, so I tend to prefer the simple
minded approach. I have spent many happy hours subdividing squares by
horizontal and vertical lines into lots and lots of little squares!

Ronnie

David Leduc wrote:
> Dear Nick and Tom,
>
> Thank you very much for your replies. It is very helpful.
>
> I had in mind to form a tricategory of bicategories, therefore I guess
> I was talking of what Tom calls strong transformations. They are also
> called weak??? I am a bit confused by the lax, weak, pseudo, strict
> and so on terminology in higher category theory. Nick, could you
> confirm that you were talking of strong transformations in your mail?
> Now I am not sure anymore what are natural transformations in category
> theory. They are strict transformations, right?
>
> Unfortunately, I do not have a copy of the paper "Coherence for
> tricategories" by Gordon, Power and Street. I guess such reference
> would help me a lot with such questions.
>
> I have another question. For strong and strict (and maybe lax?)
> transformations, we have the interchange law relating vertical and
> horizontal composition.  What is the equivalent of interchange law for
> compositions of modifications?
>
> Thank you,
>
> David

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