Compare that with “strict $n$-fold groupoids model weak homotopy $n$-types” (Loday). Grothendieck’s reaction was to exclaim: “That is absolutely beautiful!” Loday and I showed how you can do calculations of homotopy types using this model, and Ellis and Steiner developed a more concrete model called “crossed $n$-cubes of groups”. This relates quite clearly to classical work on $n$-ad homotopy groups. I.e. given an $n$-ad of spaces, one looks at all the sun $r$-ads for $r \laqslant n$ and all the generalised Whitehead products.

Now the homotopy groups exist somehow in the interior of these nonabelian models, so they may not be so easy to compute from a knowledge of the large model. But then few have worked with these models!

One moral is that these models don’r arise directly from a pointed space, but you have to construct a “resolution of the space by $n$-cubes of fibrations” (Richard Steiner has a nice account of this).

Perhaps the idea of cubical resolutions can be more generally applicable!

]]>I’ve discussed this further in a talk “The intuitions for cubical sets in nonabelian algebraic topology”. I gave at the IHP in Paris in June, 2014, available on my preprint page. http://pages.bangor.ac.uk/~mas010/brownpr.html

]]>PS What I think is wrong with your argument above is that Omega’s perfect predictive capacity means that the situation cannot be modeled by loading BoxB before you choose. I would say that a more correct model would be to first give you a limited time with BoxA and leave BoxB empty until after you have elected to forego BoxA. (After all, it is hard to predict the future but the past is often easier – except, apparently, for the case of predicting who will have been first to make this observation.)

]]>