A model structure for weakly horizontally invariant double categories

TL;DR

This paper develops a new model structure on double categories, characterizes weakly horizontally invariant double categories, and explores their homotopical properties and relations to 2-categories.

Contribution

It constructs a model structure on double categories with specific fibrations and weak equivalences, and relates it to 2-category theory via a homotopical embedding.

Findings

The model structure is monoidal with respect to Gray tensor product.

The functor $ ext{H}^{ ext{simeq}}$ is right Quillen and homotopically fully faithful.

A Whitehead Theorem characterizes weak equivalences with fibrant source.

Abstract

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.