A 2Cat-inspired model structure for double categories

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 $\mathbb{H}:2\mathrm{Cat}\to \mathrm{DblCat}$, 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.

arXiv comment42 pages; shortened the exposition, otherwise the same as the previous version

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Cited by in corpus (5)

• A model structure for weakly horizontally invariant double categories
• Bi-initial objects and bi-representations are not so different
• A double $(\infty ,1)$-categorical nerve for double categories
• Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces
• Additivity and Fiber Sequences for Combinatorial K-Theory