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A 2Cat-inspired model structure for double categories. (English. French summary) Zbl 1498.18031

Summary: We construct a model structure on the category 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 \(\mathbb{H}: 2\text{Cat}\to\text{DblCat}\), which is both left and right Quillen, and homotopically fully faithful. Furthermore, we show that Lack’s model structure on 2Cat is both left- and right-induced along \(\mathbb{H}\) from our model structure on DblCat. In addition, we obtain a 2Cat-enrichment of our model structure on 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.

MSC:

18N10 2-categories, bicategories, double categories
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