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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