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

TL;DR

This paper introduces a new tricategory of double categories and proposes an alternative notion to intercategories, aiming to embed monoids in B"ohm's monoidal category as internal categories within this tricategory.

Contribution

It constructs a tricategory of double categories and defines an alternative to intercategories as internal categories in this tricategory, providing a new framework for monoids in B"ohm's setting.

Findings

Constructed a tricategory blPs of double categories.

Proposed an alternative definition of intercategories as internal categories in blPs.

Showed that Dbl can be an example of this new structure.

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.