# Representing simplicial homotopy classes by empty cubes

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.