Question. Call a cubical model category a model category enriched over cubical sets equipped with a tensor product $X \otimes K$ and a cotensor product $X^K$ where $X$ is an object of the model category and $K$ a cubical set, and satisfying M6 and M7 in the terminology of Hirschhorn's book page 161. Is such a notion used somewhere ? Or maybe with more general presheaf categories modeling homotopy types ? By googling or by searching MatchSciNet or ZentrallBlatt, the answer seems to be negative but maybe I am using the wrong keywords. Or is there a reason for not using them ?
Motivation. I would like to calculate some homotopy function spaces on a model category which is unlikely to be simplicial, but probably cubical in some sense.