Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories

Abstract

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 Par\'e, so that
monoids in B\"ohm'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
to consider internal categories in this tricategory. %, enabling monoids in $(Dbl,\ot)$ to be examples of this gadget.
For this purpose we define categories internal to tricategories (of the type of $\DblPs$), which simultaneously serves our third motive.
Apart from monoids in $(Dbl,\ot)$ - more importantly, weak pseudomonoids in a tricategory containing $(Dbl,\ot)$ as a sub 1-category -
most of the examples of intercategories are also examples of this new gadget. The ones that escape are duoidal categories and Gray categories,
as their monoidal product induces a lax double functor on the Cartesian product.
What our third motive concerns, 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.