Monads and limits in bicategories of circuits

TL;DR

This paper explores monads within a bicategory of circuits, providing explicit descriptions of free monads and analyzing the existence of double limits, thereby connecting automata theory with bicategory theory.

Contribution

It introduces a novel description of monads in a bicategory of circuits using semifree bicrossed products and examines the structure of double limits in this context.

Findings

Explicit description of free monads via bicrossed products

Characterization of monad maps related to automata structures

Analysis of double limits in the bicategory of circuits

Abstract

We study monads in the (pseudo-)double category $\mathbf{KSW}(\mathcal{K})$ where loose arrows are Mealy automata valued in an ambient monoidal category $\mathcal{K}$, and the category of tight arrows is $\mathcal{K}$. Such monads turn out to be elegantly described through instances of semifree bicrossed products (bicrossed products of monoids, in the sense of Zappa-Sz\'ep-Takeuchi, where one factor is a free monoid). This result which gives an explicit description of the `free monad' double left adjoint to the forgetful functor. (Loose) monad maps are interesting as well, and relate to already known structures in automata theory. In parallel, we outline what double co/limits exist in $\mathbf{KSW}(\mathcal{K})$ and express in a synthetic language, based on double category theory, the bicategorical features of Katis-Sabadini-Walters `bicategory of circuits'.