Comparing loose bimodules and double barrels using pseudo-models of enhanced sketches
Jason Brown, Kevin Carlson, Sophie Libkind, David Jaz Myers
TL;DR
The paper addresses the challenge of defining and relating loose bimodules between double categories by presenting two formulations, as pseudo-bimodules and as double barrels, and proving their 2-categorical equivalence via pseudo-models of enhanced sketches. A central methodological contribution is the development of $\mathscr{F}$-sketches and pseudo-models, along with a slice theorem and the Grothendieck-style correspondence between pseudo-F-functors and $\mathscr{F}$-opfibrations, which together yield a robust framework for loose universal properties. The work further introduces model opfibrations, restriction operations for loose bimodules, and the notion of loose adjunctions, showing how these notions support coherent loose limits, adjunctions, and colimits such as van Kampen-type constructions within double categories. Collectively, these results provide a comprehensive foundation for loose bimodule theory, enabling new forms of compositional reasoning in double category theory and potential applications to coupled dynamical systems and related areas.
Abstract
(Pseudo) double categories have two sorts of morphisms: tight ones which compose strictly, and loose ones which compose up to coherent isomorphism. In this paper, we consider bimodules between double categories in the loose direction. We provide two formulation of this concept -- first as pseudo-bimodules between pseudo-categories in the 2-category of categories, and second as double barrels generalizing Joyal's definition of bimodules between categories as functors into the walking arrow -- and prove these two formulations equivalent. In order to prove this equivalence, we define a notion of \emph{pseudo-model} of an enhanced sketch, which may be of independent interest. We then consider some double category theory unlocked by the theory of loose bimodules: loose adjunctions, and loose limits.