论文标题
弱水平不变双类的模型结构
A model structure for weakly horizontally invariant double categories
论文作者
论文摘要
我们在双类别的类别$ \ mathrm {dblcat} $和双函子上构建模型结构，其琐碎的纤维是在对象上溢出的双函子，在水平和垂直形态上充满了对象，并且完全忠于正方形。并且其纤毛物体是弱水平不变的双重类别。 We show that the functor $\mathbb H^{\simeq}\colon \mathrm{2Cat}\to \mathrm{DblCat}$, a more homotopical version of the usual horizontal embedding $\mathbb H$, is right Quillen and homotopically fully faithful when considering Lack's model structure on $ \ mathrm {2cat} $。特别是，$ \ mathbb h^{\ simeq} $展示了$ \ mathbb h $的级别纤维替代品。此外，从$ \ Mathbb H^{\ simeq} $沿$ \ Mathrm {2CAT} $上的Lack Model结构沿$ \ Mathbb H^{\ simeq} $右引起。 我们还表明，相对于Böhm的灰色张量产品，这种模型结构是单体的。最后，我们证明了一个白头定理，该定理表征了弱源的弱等价，是双函子，该双函子接收伪逆至水平伪自然等效性。
We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories. We show that the functor $\mathbb H^{\simeq}\colon \mathrm{2Cat}\to \mathrm{DblCat}$, a more homotopical version of the usual horizontal embedding $\mathbb H$, is right Quillen and homotopically fully faithful when considering Lack's model structure on $\mathrm{2Cat}$. In particular, $\mathbb H^{\simeq}$ exhibits a levelwise fibrant replacement of $\mathbb H$. Moreover, Lack's model structure on $\mathrm{2Cat}$ is right-induced along $\mathbb H^{\simeq}$ from the model structure for weakly horizontally invariant double categories. We also show that this model structure is monoidal with respect to Böhm's Gray tensor product. Finally, we prove a Whitehead Theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudo inverse up to horizontal pseudo natural equivalence.
