论文标题
立方体在$（\ infty，\ infty）$ - 类别中密集
Cubes are dense in $(\infty,\infty)$-categories
论文作者
论文摘要
我们表明，严格的1类别$ \平方$的立方体 - 定义为严格的$ω$ - 类别的完整子类别，其对象的对象是箭头类别的灰色张量 - $（\ infty，1）$ - 类别$ - 类别$ \ mathsf {cat}_Ω不完整的变体。更确切地说，我们表明Joyal的类别$θ$包含在$ \ square $的基本完成中，实际上，$ \ square $的diDempotent完成在悬架和楔形总和下是关闭的。该结果扩展了Campbell和Maehara在Dimension 2中的定理。在坎贝尔和梅哈拉的计划之后，我们将在以后的工作中采用此结果，以对灰色张量的新构建较弱的$（\ infty，n）$类别进行新的结构。
We show that the strict 1-category $\square$ of cubes -- defined to be the full subcategory of strict $ω$-categories whose objects are the Gray tensor powers of the arrow category -- are dense in the $(\infty,1)$-category $\mathsf{Cat}_ω$ of weak $(\infty,\infty)$-categories, in both Rezk-complete and incomplete variants. More precisely, we show that Joyal's category $Θ$ is contained in the idempotent completion of $\square$, and in fact that the idempotent completion of $\square$ is closed under suspensions and wedge sums. This result extends a theorem of Campbell and Maehara in dimension 2. Following Campbell and Maehara's program, we will in future work apply this result to give a new construction of the Gray tensor product of weak $(\infty,n)$-categories.
