Ronnie Brown wrote; Has anyone formulated a cubical (square?) version of bicategory or higher?

Dominic Verity invented "double bicategories", which are the fully weakened version of double categories. Unfortunately, his thesis can only be obtained by special shipment from Australia. Someday someone will scan it and make it more widely available! Luckily, you can see the definition in this paper: Jeffrey Morton Double bicategories and double cospans http://arxiv.org/abs/math/0611930 For higher dimensions, try: Marco Grandis HIGHER COSPANS AND WEAK CUBICAL CATEGORIES (COSPANS IN ALGEBRAIC TOPOLOGY, I) http://tac.mta.ca/tac/volumes/18/12/18-12abs.html and also parts II and III of this series. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Ronnie, There are a number of "cubical" notions of bicategory or higher in the literature with which I am sure you are familiar; the earliest being Dominic Verity's double bicategories (which are really a particular kind of triple category), and later the weak (or pseudo) double categories studied in a series of papers by Bob Paré and Marco Grandis. However, none of these quite seem to fit the spirit of what you are asking for, and certainly do not describe the singular cubical complex of a space. One approach which is indicated in the literature is based on Michael Batanin's theory of higher operads. In particular, Tom Leinster, in the refined presentation of this theory given in his book, discusses the possibility of using it to give a notion of weak cubical omega-category, though he does not go into any details. The basic idea is straightforward. On the category of cubical sets (without diagonals or symmetries) one has the monad T for strict cubical omega-categories; and one aims to define the monad for weak cubical omega-categories as a suitable deformation T' of this monad. In the language of homotopy theory, one would like T' to be a cofibrant replacement for T; this then ensures both that T' looks sufficiently like T, and also that its algebras are homotopy-invariant in a suitable sense. To make this more formal, one defines a cubical operad to be a monad S on the category of cubical sets together with a cartesian monad morphism S => T. The existence of such a monad morphism---which I think will be unique if it exists---places some strong restrictions on the shapes of the operations out of which the monad S is built: a typical such operation will take an n_1 x n_2 x ... x n_k grid of k-dimensional hypercubes, and stick them together in some way to yield a single k-dimensional hypercube. This is precisely what one wants for a notion of (weak) cubical category. To ensure that the monad S looks sufficiently like the monad T, one needs the notion of a contraction (= acyclic fibration). In fact, we only need to know how to say that the monad morphism S => T has a contraction, and this is quite simple. Suppose we are given a n_1 x ... x n_k grid of k-dimensional hypercubes as before. Suppose we are also given operations s_1, ..., s_2k in S which indicate how to compose up the (k-1)-dimensional grids of hypercubes obtained as the faces of the original grid. Then we require that there should be given an operation of S which acts on n_1 x ... x n_k grids, and whose actions on each of the (k-1)-dimensional faces are given by s_1, ..., s_2k. There now arises a category whose objects are cubical operads S equipped with a contraction on S => T, and whose morphisms are monad maps over T which commute to the contractions in the obvious sense. This category is locally presentable, and hence has an initial object; which we declare to be the monad T' for weak cubical omega categories. One may in fact show that T' is a cofibrant replacement for T in a suitable model structure on cubical operads. One may of course specialise this construction to low dimensions, and I have just sat down to work it out in the case where n=2. Rather surprisingly, it seems that there might not be a finite axiomatisation of the resultant structure; at least, one does not immediately leap off the page which is the usual situation. The best I have been able to do is the following (which is not exactly the result of applying the above construction but is a perfectly good surrogate). Definition. A cubical bicategory is given by sets of objects, of vertical arrows, of horizontal arrows and of squares, satisfying the obvious source and target criteria, together with operations of identity and binary composition for vertical and horizontal arrows, satisfying no laws at all; and finally, for every n x m grid of squares (where possibly n or m are zero), and every way of composing up the horizontal and vertical boundaries using the nullary and binary compositions, a composite square with those boundaries. The coherence axioms which this structure must satisfy say that any two ways of composing up a diagram of squares must give the same answer. I would be very interested to know if anyone can extract from this definition a finite collection of composition operations on squares, and a finite collection of equations between them, which together generate all the others. The key obstable seems to be problem that identity 1-cells are not strict in either direction. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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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