三范畴的充实和内化，张量范畴的情况和范畴间的替代概念
Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories
摘要：
本文件是处理以下三个问题的结果。首先，我们希望双范畴嵌入到伪双范畴中是monoidal的，如果使用monoidal伪双范畴的通常概念，则不是monoidal的。第二，在\cite{Gabi}中提出了一个问题：哪一个是Grandis和Paré的范畴间的替代概念，所以Böhm的严格双范畴和严格双函子的幺半范畴$(Dbl,\ot)$中的Monoidal范畴和Gray型Monoidal乘积是它的一个例子？我们得到并证明了$(Dbl,\ot)$的么半群结构解决了第一个问题。另一方面，为了解决第二个问题，我们将范畴$Dbl$提升为一个三元范畴$\DblPs$，并提出了一个考虑三元范畴内部范畴的范畴间定义。%，使$(Dbl,\ot)$中的monoid成为此小工具的示例。除了$(Dbl,\ot)$中的幺半群--更重要的是，包含$(Dbl,\ot)$作为子1-范畴的三范畴中的弱伪幺半群--大部分范畴间的例子也是这个小工具的例子，逃脱的是那些依赖于产品在拉回上的松弛性的例子，作为对偶范畴。为了后一个目的，我们定义了三大类（$\DblPs$类型）内部的范畴，这同时服务于我们的第三个动机。即受张量范畴的三范畴和$(1\times 2)$ -范畴的启发，在较弱的条件下，证明了在某些类型的三范畴上富集的范畴可以成为它们内部的范畴.我们说明了这种情况下张量范畴相对于环境的三范畴$2\x\Cat_{wk}$的2-范畴，伪函子，自然变换和修改。
英文摘要：
This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the monoidal structure of $(Dbl,\ot)$ resolves the first question. On the other hand, resolving the second question, we upgrade the category $Dbl$ to a tricategory $\DblPs$ and propose %an alternative definition of intercategories as to consider internal categories in this tricategory. %, enabling monoids in $(Dbl,\ot)$ to be examples of this gadget. Apart from monoids in $(Dbl,\ot)$ - more importanlty, weak pseudomonoids in a tricategory containing $(Dbl,\ot)$ as a sub 1-category - most of the examples of intercategories are also examples of this gadget, the ones that escape are those that rely on laxness of the product on the pullback, as duoidal categories. For the latter purpose we define categories internal to tricategories (of the type of $\DblPs$), which simultaneously serves our third motive. Namely, inspired by the tricategory and $(1\times 2)$-category of tensor categories, we prove under mild conditions that categories enriched over certain type of tricategories may be made into categories internal in them. We illustrate this occurrence for tensor categories with respect to the ambient tricategory $2\x\Cat_{wk}$ of 2-categories, pseudofunctors, pseudonatural transformations and modifications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
