A 2Cat-inspired model structure for double categories

TL;DR

This paper develops a new model structure on double categories that aligns with 2-category homotopy theory, introduces a Gray tensor product variant, and establishes Whitehead theorems linking weak equivalences to invertibility up to equivalence.

Contribution

It constructs a novel model structure on double categories that recovers 2-category homotopy theory and relates to existing models via Quillen adjunctions and equivalences.

Findings

The model structure on DblCat recovers 2-category homotopy theory.

The Gray tensor product variant enables enrichment of DblCat.

Whitehead theorem characterizes weak equivalences as invertible up to pseudo natural equivalence.

Abstract

We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors. Unlike previous model structures for double categories, it recovers the homotopy theory of 2-categories through the horizontal embedding $\mathbb{H}\colon2\mathrm{Cat}\to\mathrm{DblCat}$, which is both left and right Quillen, and homotopically fully faithful. Furthermore, we show that Lack's model structure on $2\mathrm{Cat}$ is both left- and right-induced along $\mathbb{H}$ from our model structure on $\mathrm{DblCat}$. In addition, we obtain a $2\mathrm{Cat}$-enrichment of our model structure on $\mathrm{DblCat}$, by using a variant of the Gray tensor product. Under certain conditions, we prove a Whitehead theorem, characterizing our weak equivalences as the double functors which admit an inverse pseudo double functor up to horizontal pseudo natural equivalence. This retrieves the Whitehead theorem for 2-categories. Analogous statements hold for the category $\mathrm{wkDblCat}_s$ of weak double categories and strict double functors, whose homotopy theory recovers that of bicategories. Moreover, we show that the full embedding $\mathrm{DblCat}\to\mathrm{wkDblCat}_s$ is a Quillen equivalence.