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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.

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    $\begingroup$ The category of cubical sets has the disadvantage that the monoidal product that you want is not the cartesian product, but a new one which is not symmetric. To my knowledge, most of the literature on enriched model categories assumes that the enriching category is symmetric. $\endgroup$ – Tyler Lawson Jun 29 '16 at 18:55
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    $\begingroup$ The general notion of an enriched model category specializes to the one you're looking for if one enriches over the model category of cubical sets (with or without connections). (It's not necessary for the monoidal structure on the enriching category to be symmetric.) The relevant theory also immediately implies that homotopy function spaces can be computed by deriving the enriched hom. $\endgroup$ – Dmitri Pavlov Jun 30 '16 at 17:13
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    $\begingroup$ There are many different things that "cubical set" can mean. Some of them do have a symmetric tensor product. $\endgroup$ – Mike Shulman Jun 30 '16 at 19:51
  • $\begingroup$ The book "Nonabelian Algebraic topology: filtered space, crossed complexes, cubical homotopy groupoids" (EMS, 2011) uses cubical methods which are essential for the main results, which give an accounts of the border between homology and homotopy theory without setting up singular homology theory. Cubes are better than simplices for discussing homotopies, and "algebraic inverses to subdivisions". $\endgroup$ – Ronnie Brown Mar 31 '18 at 14:32

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