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What is a natural definition of a cubical map between cubical complexes? What is its geometric realization?

I found some definitions, such as here or here, where a cubical map between cubical complexes $f: Q_1\to Q_2$ was defined so that each cube (represented by a set of vertices) is mapped to a subset of the vertices of some $q\in Q_2$ (not necessarily to a cube) and adjacent vertices are mapped to vertices of distance at most one. (Or something similar in the other reference.)

Apparently, one can then define an induced map on chains/cochains, homology etc. Does a cubical map (in the above, or a different sense) have a natural geometric realization as a map between the underlying topological spaces (as it is in the case of simplicial maps)?

I'm aware of the notion of cubical set which is a set of abstract cubes endowed with boundary- and degeneracy-operators. Apart of that, there are several additional structures on cubical sets, such as connections, but it looks complicated.

Therefore I'm asking if there is something easier and still strong enough for representing simple continuous maps as cubical maps (homotopic to the original one, in some sense). I would like to have some nice analogy of "simplicial approximations of continuous maps" in the cubical world.

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I think one can give definition of cubical maps of cubical complexes in a similar way to simplicial maps between simplicial sets. Cubical maps between cubical sets is a map of vertices which is compatible with face and degeneracy maps.

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  • $\begingroup$ Sure, you can do it. The problem is that it's very restrictive and there are very few such maps. For example, if you model a sphere as the boundary of a cube and want to approximate a sphere-value map $f: X\to S^n$, then, unlike in the simplicial case, it is not clear whether any kind of "cubical approximation up to homotopy" via subdivision of $X$ works... $\endgroup$ – Peter Franek Apr 29 '16 at 10:32

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