I am looking for references concerning the following facts, which I believe to be true:

In a Kan complex $K$, can I use "empty cubes" $\partial(\Delta^1)^{(n+1)}$ with the vertex $(0,0,\ldots,0)$ at the basepoint (under the obvious notation $0$ and $1$ for the two $0$-simplices of $\Delta^1$) as representatives for the simplicial homotopy groups of $K$?

More precisely, I would like to have an "algorithm" (i.e. a precise order in which to do the homotopies) to homotope the empty cube to to a single $n$-simplex with the boundary mapping to the basepoint. Moreover, I would also like to have the extension to the whole cube: If the empty cube can be filled then the homotopy class should be trivial, and I would like an extension of the homotopy giving the corresponding homotopy of the $n$-simplex to the basepoint.

Note: I want to do all of this in simplicial sets, so no cubical sets. This question is related but works with a different representation of homotopy classes (and also does not assume $K$ to be Kan). It does not answer my question in a satisfactory way.

Note 2: I am aware that one of the reasons we work with simplices is that for many things they behave "better" than cubes. Unfortunately, I would need what I detailed above for a nice application that I'm afraid would become horribly complicated if I were to use the standard definition of simplicial homotopy groups.


Yes. The map ∂◻^{n+1}→S^n that collapses the complement of a single n-dimensional face of ◻^{n+1} is a simplicial weak equivalence. This implies that the induced map Map(S^n,K)→Map(∂◻^{n+1},K) is a simplicial weak equivalence because both mapping spaces are derived (K is Kan and all simplicial sets are cofibrant).

For an explicit algorithm, consider the inclusions ∂◻^{n+1}→J and S^n→J, where J denotes the pointed join of ∂◻^{n+1} and S^n. These inclusions are acyclic cofibrations of simplicial sets. A map ∂◻^{n+1}→K can be extended to a map J→K by repeatedly filling in horns using the Kan property of K. The resulting map J→K can then be restricted to a map S^n→K, which yields the desired representative for an empty cube. The case of fillable cubes is treated in the same way.

  • $\begingroup$ Thanks! Would you have a reference where this is done/stated? Having something citable would be great! $\endgroup$ Mar 9 '20 at 17:09
  • $\begingroup$ @DanielRobert-Nicoud: The primary reference is Proposition I.11.5 in Goerss and Jardine's Simplicial Homotopy Theory, which shows that the maps that are claimed to be weak equivalences are indeed weak equivalences. $\endgroup$ Mar 9 '20 at 21:23
  • $\begingroup$ Great, thank you! Time to go find my copy of the book! $\endgroup$ Mar 9 '20 at 21:35

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