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Monads and limits in bicategories of circuits

Fosco Loregian

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-Szép-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'.

Monads and limits in bicategories of circuits

Abstract

We study monads in the (pseudo-)double category where loose arrows are Mealy automata valued in an ambient monoidal category , and the category of tight arrows is . Such monads turn out to be elegantly described through instances of semifree bicrossed products (bicrossed products of monoids, in the sense of Zappa-Szép-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 and express in a synthetic language, based on double category theory, the bicategorical features of Katis-Sabadini-Walters `bicategory of circuits'.
Paper Structure (14 sections, 12 theorems, 9 equations)

This paper contains 14 sections, 12 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The pseudo double category Mly
  3. The structure of Mly
  4. Monoidality
  5. Co/monads in Mly
  6. Modules over a monad
  7. Double monad maps
  8. Tight co/limits in Mly
  9. Some genuinely double co/limits
  10. Companions and conjoints
  11. From the universal fibration to the pseudo double category of Mealy automata, and back
  12. Future work
  13. Paralipomena and proofs
  14. Proofs

Key Result

Proposition 2.1

If $\mathbf{D}$ is a monoidal double category, of which the monoidal constraints have loosely strong companions, then its underlying bicategory [of loose morphisms, N/A] $\clH(\mathbf{D})$ is a monoidal bicategory. If $\mathbf{D}$ is braided or symmetric, so is $\clH(\mathbf{D})$.

Theorems & Definitions (42)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Definition 2.2: Monad in a pseudo double category
  • Definition 2.3: Monad in $\ref{['cart_dbl_mly']}$
  • Remark 2.4: Canonical extensions of $d,s$
  • Corollary 2.5: Monads as matching pairs
  • ...and 32 more