Re: Composing modifications

[JeanBenabou <jean.benabou@wanadoo.fr>]

Dear Ronnie,
I usually enjoy your remarks and I would probably have enjoyed your  
last ones, except that I cannot read it. Has the sorry time finally  
arrived when one can no longer work in Mathematics unless he knows  
TeX, LaTeX, and/or other sophisticated word processings? Should I,  
and many others it would be too long to name, stop doing mathematics?

I'd appreciate answers from all colleagues


> Dear All,
>
> (this was sent earlier but had some html which may have got it  
> rejected).
> Thanks  very much for the info.
>
> After I sent off my email I did remember (just about) papers of Tom  
> Fiore and of Grandis/Pare which were relevant and was collecting  
> together references, but fortunately you beat me to it, and with  
> any interesting points.
>
> It may be convenient to have the definition of multiple composition  
> in KX, the cubical singular complex of X, on the record
> as follows:
> --------------------------------------------------------
> Let $(m) = (m_1, \ldots , m_n)$ be an $n$-tuple of positive integers
> and
> $$\phi_{(m)} : I^n \rightarrow [0, m_1] \times \cdots \times [0,
> m_n]$$ be the map $(x_1 , \ldots , x_n) \mapsto (m_1 x_1, \ldots ,
> m_n x_n).$ Then a {\it subdivision of type $(m)$} \index{singular
> $n$-cubes!subdivision of type $(m)$} of a map $\alpha : I^n
> \rightarrow X$ is a factorisation $\alpha = \alpha' \circ
> \phi_{(m)}$; its {\it parts} are the cubes $\alpha_{(r)}$ where $(r)
> = (r_1, \ldots , r_n)$ is an $n$-tuple of integers with $1 \leqslant
> r_i \leqslant m_i$, $i = 1, \ldots , n,$ and where $\alpha_{(r)} :
> I^n \rightarrow X$ is given by
> $$(x_1, \ldots , x_n) \mapsto
> \alpha'(x_1 + r_1 - 1, \ldots , x_n + r_n - 1).$$
>
> We then say that $\alpha$ is the {\it composite}of the
> cubes $\alpha_{(r)}$ and write $\alpha = [\alpha_{(r)}]$. The {\it
> domain} of $\alpha_{(r)}$ is then the set $\{(x_1,\ldots,x_n) \in
> I^n : r_i-1 \leqslant x_i \leqslant r_i, 1 \leqslant i \leqslant
> n\}$.
> ---------------------------------------------------------------------- 
> --------------
> (I have also put on  the arXiv an exposition of Moore  
> Hyperrectangles, which is the start of another approach. One  
> problem with this is that a homotopy is more general than a Moore  
> hyperrectangle as defined  there, since a homotopy is a path in a  
> space of such hyperrectangles.)
>
> These compositions satisfy an interchange law, but without  
> associativity (unlike the Moore approach) we do not get of course  
> any of what one might call `partitioning' results. One should get  
> these `up to homotopy' but this seems  difficult to formulate.
>
> In our work we also found the necessity of `connections', \Gamma^ 
> \pm _i, which have many advantages including bringing the cubical  
> theory a little nearer to the simplicial: a cube \Gamma^\pm_i  x  
> has two adjacent faces the same. In the strict theory, or at least  
> for groupoids, the compositions, at least of two  cubes,  are  
> recoverable from the connections, up to homotopy.
>
> Then of course one needs the more general n-fold objects. These  
> occur topologically from (n+1)-ads
> X_*= (X;A_1, ....,A_n) where the A_i are subspaces of  X. Let \Phi=  
> \Phi(X_*) consist of maps of an n-cube into X which take the faces  
> in direction i into A_i. Then \Phi has n compositions, making it a  
> `weak n-fold category' . (The amazing fact (Loday) is that \p_1 
> (\Phi, *), where * is the constant map, gets the structure of cat^n- 
> group = strict (!!) n-fold groupoid in groups, and these model weak  
> pointed homotopy (n+1)-types. )
>
> So I am just suggesting that cubical approaches have reached into  
> methods not easily approachable by simplicial or globular methods   
> and perhaps more can be made of this.
>
> Will more high powered approaches using operads help in all this?
>
> I should also mention Richard Steiner's paper
> Thin fillers in the cubical nerves of omega-categories <http://0- 
> ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/ 
> journaldoc.html?cn=Theory_Appl_Categ> <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=240201>, Theory App Categories (2006) 144--173 as  
> suggesting the possibility of using some kind of thin structure to  
> define weak cubical categories.
>
> My motivation all along was in terms of `algebraic inverses to  
> subdivision' .
>
> Ronnie
>

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