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Tape Diagrams for Monoidal Monads

Filippo Bonchi, Cipriano Junior Cioffo, Alessandro Di Giorgio, Elena Di Lavore

Abstract

Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, $\oplus$ and $\otimes$, where $\otimes$ distributes over $\oplus$. However, their applicability is limited to categories where $\oplus$ is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.

Tape Diagrams for Monoidal Monads

Abstract

Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, and , where distributes over . However, their applicability is limited to categories where is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.

Paper Structure

This paper contains 15 sections, 10 theorems, 29 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. Copy Discard and Finite (Co)Product Categories
  4. Rig Categories
  5. Finite Coproduct-Copy Discard Rig categories
  6. Combining Rig Categories with Algebraic Theories
  7. Preliminaries on Algebraic Theories
  8. $\mathbb{T}$-rig categories
  9. $\mathbb{T}$-Copy Discard Rig Categories
  10. Tape Diagrams
  11. Freely Generated Sesquistrict Rig Categories
  12. Tape Diagrams for $\mathbb{T}$-Rig Categories
  13. Tape Diagrams for $\mathbb{T}$-cd Rig Categories
  14. Conclusions
  15. Coherence Axioms

Key Result

Proposition 5

Let $T\colon \mathbf{C}\to \mathbf{C}$ be a symmetric monoidal monad on a cd category $(\mathbf{C},\odot,I, \blacktriangleleft_{}, \hbox{!}_{})$. Then the Kleisli category $(\mathbf{C}_T, \odot_T, I, \blacktriangleleft_{}^\flat, \hbox{!}_{}^\flat)$ is a cd category.

Figures (10)

  • Figure 1: Diagrams for the coherence conditons in \ref{['equation: coherence1']}.
  • Figure 2: Axioms for $\mathbb{T}$-tape diagrams. Here $(t,t') \in E$.
  • Figure 3: The morphism of $\mathbb{T}$-cd rig categories $\left \llbracket - \right \rrbracket_\mathcal{I} \colon \mathbf{T}^{\mathsf{cd}}_{\Gamma,\mathbb{T}} \to \mathbf{C}$ induced by an interpretation $\mathcal{I} = (\alpha_\mathcal{S}, \alpha_\Gamma)$ of the monoidal signature $(\mathcal{S},\Gamma)$ in $\mathbf{C}$. This is the compositional semantics of tapes.
  • Figure 4: Coherence axioms of monoidal categories
  • Figure 5: Coherence axioms of symmetric monoidal categories
  • ...and 5 more figures

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Proposition 5: From cioffogadduccitrotta
  • Definition 6
  • Definition 7
  • Lemma 8
  • Proposition 9
  • Definition 10
  • ...and 20 more