A model structure for weakly horizontally invariant double categories

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