AI 中文总结
本文在$\boldsymbol{Cat}$和$2\boldsymbol{Cat}$上构造新对称幺半结构，利用分解系统建立模型结构，通过弱对称幺半Quillen等价关联Gray与灵活张量积。
AI 中文摘要
本文在范畴$\boldsymbol{Cat}$上引入一种新的对称幺半结构，称为“图张量积”，其单位元为终范畴。该张量积可分解为趣味张量积与笛卡尔积的中间形式，建立起这两种经典幺半结构的联系。我们将此构造推广到2范畴$2\boldsymbol{Cat}$上的对称幺半结构，其单位元仍为$D^0$，并在2范畴的趣味张量积与笛卡尔积间给出类似分解。利用$2\boldsymbol{Cat}$上的$(\text{bo},\text{lff})$分解系统，我们在$2\boldsymbol{Cat}$上构造了一种新的对称幺半闭模型结构，其张量积在灵活2范畴的子范畴上限制为笛卡尔积。最后，我们证明该对称幺半模型结构构成一个弱对称幺半Quillen等价的正方形，关联了Gray张量积与灵活张量积。
英文摘要
In this paper, we introduce a new symmetric monoidal structure on $\mathbf{Cat}$, called the \emph{graph tensor product}, with unit given by the terminal category. This tensor product falls in the middle of a factorization between the funny tensor product and the Cartesian product, giving a factorization connecting these two classical monoidal structures. We extend this construction to a symmetric monoidal structure on $2\mathbf{Cat}$, again with unit $D^0$, which provides an analogous factorization between the funny tensor product and the Cartesian product of $2$-categories. Using the $(\mathrm{bo},\mathrm{lff})$ factorization system on $2\mathbf{Cat}$, we construct a new symmetric monoidal closed model structure on $2\mathbf{Cat}$ whose tensor product restricts to the Cartesian product on the subcategory of flexible $2$-categories. Finally, we prove that this symmetric monoidal model structure fits into a square of weak symmetric monoidal Quillen equivalences relating the Gray tensor product and the flexible tensor product.
