A model structure for weakly horizontally invariant double categories

We construct a model structure on the category \(\mathrm{DblCat}\) of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories. We show that the functor \(\mathbb H^{\simeq}\colon \mathrm{2Cat}\to \mathrm{DblCat}\), a more homotopical version of the usual horizontal embedding \(\mathbb H\), is right Quillen and homotopically fully faithful when considering Lack's model structure on \(\mathrm{2Cat}\). In particular, \(\mathbb H^{\simeq}\) exhibits a levelwise fibrant replacement of \(\mathbb H\). Moreover, Lack's model structure on \(\mathrm{2Cat}\) is right-induced along \(\mathbb H^{\simeq}\) from the model structure for weakly horizontally invariant double categories. We also show that this model structure is monoidal with respect to B\"ohm's Gray tensor product. Finally, we prove a Whitehead Theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudo inverse up to horizontal pseudo natural equivalence.