Dear Colleagues, I have found the `very good' properties of cubical sets essential for many aspects of my work, and are used in our new book `Nonabelian algebraic topology', (downloadable pdf from my web pages) which puts in one place the work with Higgins et al; with regard to the monoidal closed structures this is especially seen in Chapter 15, whose Introduction p. 513 mentions a crucial natural transformation \eta: \rho (X_*) \otimes \rho(Y_*) \to \rho(X_* \otimes Y_*) where X_*, Y_* are filtered spaces, and \rho gives the cubical higher homotopy groupoid, which is almost `obvious' in this cubical context, and so can be translated to other contexts. Whether these `very good properties' are `excellent' in a formal sense I do not know, but it would be very surprising if that failed! I do not find analogous properties simplicially. I got the impression from John Moore that although Dan Kan's initial work on combinatorial homotopy was cubical (see his first PNAS Note) the problems of the homotopy type of the realisation of the cartesian product discouraged any further cubical work, as the simplicial method went smoothly, although often technical. So nobody tried to fix the cubical theory. Our `fix', i.e. the connections, was forced on us for other reasons, initially to relate crossed modules and double groupoids; they were crucial for the 2-d van Kampen theorem (all done in the 1970s) and then for higher analogues. I doubt if much of this local-to-global stuff could have been conjectured, let alone proved, simplicially. On the other hand, there are two papers on classifying spaces for equivariant crossed complexes with Golasinski, Porter and Tonks where simplicial methods are used, because they link nicely with work on coherence of Cordier and Porter. A recent paper using cubical methods which could be relevant for Dmitry is Faria~Martins, J. and Picken, R. Surface {H}olonomy for {N}on-{A}belian 2-{B}undles via {D}ouble {G}roupoids}.{Adv. Math.} 226(2011) 3309–3366. Ronnie On 15/09/2011 12:56, Dmitry Roytenberg wrote:
Dear colleagues,
This raises the following question. In homotopy theory one almost always considers categories enriched in simplicial sets, endowed with the classical Kan-Quillen model structure, but are cubical sets (with or without connections) equally good for that purpose? In his "Higher Topos Theory" Lurie considers monoidal model categories S enjoying certain additional properties and proves that for each such "excellent model category" S the category Cat(S) of small S-enriched categories admits a model structure with a particularly nice description of fibrations and weak equivalences (that S = simplicial sets with the classical model structure and the Cartesian product is excellent is a result of Dwyer and Kan; the corresponding model structure on Cat(S) is due to Bergner).
So my question is, is it known whether the category of cubical sets, with or without connections, admits an excellent model structure? The reason I am asking is because in geometric applications I have in mind cubes are often easier to work with than simplices (e.g. it is easier to compute integrals of differential forms over cubes).
Thank you in advance.
Dmitry
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