2020/04/29 by Moser, Lyne, Sarazola, Maru, Verdugo, Paula · 2 citations
#18D20 #18N10 #18N40 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2004.14233
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 ℍ\colon2Cat\toDblCat, 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 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. This retrieves the Whitehead theorem for 2-categories. Analogous statements hold for the category wkDblCats of weak double categories and strict double functors, whose homotopy theory recovers that of bicategories. Moreover, we show that the full embedding DblCat\towkDblCats is a Quillen equivalence.