Alternative notion to intercategories: part I. A tricategory of double categories

Abstract

This is a first of a series of two papers. Our motive is to tackle the question raised in B\"ohm's "The Gray Monoidal Product of Double Categories" from Applied Categorical Structures: which would be an alternative notion to intercategories of Grandis and Par\'e, so that monoids in B\"ohm's monoidal category $Dbl$ of strict double categories and double pseudo functors be an example of it? Before addressing this question we observe that although bicategories embed into pseudo double categories, this embedding is not monoidal, with the usual notion of a monoidal pseudo double category. We then prove that monoidal bicategories embed into the mentioned monoids of B\"ohm. In order to fit B\"ohm's monoid into an intercategory-type object, we start by upgrading the category $Dbl$ to a 2-category and end up rather with a tricategory $\DblPs$. We propose an alternative definition of intercategories as internal categories in this tricategory, enabling $Dbl$ to be an example of this gadget. The formal definition of a category internal to the (type of a) tricategory (of) $\DblPs$, as well as another important example of these in the literature, we leave for a subsequent paper.