论文标题
双重类别的2CAT风格的模型结构
A 2Cat-inspired model structure for double categories
论文作者
论文摘要
我们在double类别和双函子的类别$ \ mathrm {dblcat} $上构建模型结构。与以前的双类别模型结构不同，它通过嵌入$ \ mathbb {h} \ colon2 \ mathrm {cat} \ to \ mathrm {dblcat} $的水平嵌入$ \ mathbb {h} \ colon2 \ mathrm {cat} $恢复了两类的同义理论。此外，我们证明了$ 2 \ mathrm {cat} $上的Lavy模型结构是从$ \ Mathrm {dblCat} $上的$ \ Mathbb {H} $沿$ \ Mathbb {H} $沿左右诱导的。此外，通过使用灰色张量产品的变体，我们在$ \ mathrm {dblcat} $上获得了$ 2 \ mathrm {cat} $ - 在$ \ mathrm {dblcat} $上富集了我们的模型结构。 在某些条件下，我们证明了一个白头定理，将我们的弱等效量描述为双函子，该双函子接收一个逆伪双函子，直到水平伪自然等效性。这将检索Whitehead定理的两类。 类似的语句适用于类别$ \ mathrm {wkdblcat} _ s $的弱双重类别和严格的双函子，其同质性理论恢复了bicateGories的同质理论。此外，我们表明，完整的嵌入$ \ mathrm {dblcat} \ to \ mathrm {wkdblcat} _s $是quillen等价。
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.
