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Duoidal R-Matrices

Tony Zorman

Abstract

In this note, we define an analogue of R-matrices for bialgebras in the setting of a monad that is opmonoidal over two tensor products. Analogous to the classical case, such structures bijectively correspond to duoidal structures on the Eilenberg--Moore category of the monad. Further, we investigate how a cocommutative version of this lifts the linearly distributive structure of a normal duoidal category.

Duoidal R-Matrices

Abstract

In this note, we define an analogue of R-matrices for bialgebras in the setting of a monad that is opmonoidal over two tensor products. Analogous to the classical case, such structures bijectively correspond to duoidal structures on the Eilenberg--Moore category of the monad. Further, we investigate how a cocommutative version of this lifts the linearly distributive structure of a normal duoidal category.

Paper Structure

This paper contains 7 sections, 11 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasitriangularity
  4. Double opmonoidal monads
  5. R-matrices
  6. From R-matrices to duoidal structures and back
  7. Linearly distributive monads

Key Result

Proposition 2.2

Let $(B, \mu, \eta)$ be a monad on a monoidal category $\mathcal{C}$. There exists a bijective correspondence between bimonad structures on $B$ and monoidal structures on $\mathcal{C}^B$ such that the canonical forgetful functor $U^B\colon \mathcal{C}^{B} \longrightarrow \mathcal{C}$ is strict monoi

Figures (6)

  • Figure 1: Verification that $S$ satisfies \ref{['eq:r-matrix-lift']}.
  • Figure 2: Proof that $\xi$ is a morphism of $T$-algebras.
  • Figure 3: Proof that $\xi$ satisfies \ref{['eq:middle-interchange-assoc1']}.
  • Figure 4: The map $R$ satisfies \ref{['eq:r-matrix-lift']}.
  • Figure 5: The left-left linear distributor satisfies \ref{['eq:linearly-distributive-monad-1']}.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Proposition 2.7: lewis72:coher, Aguiar2010
  • Definition 3.1: Aguiar2010
  • Proposition 3.2: booker13:tannak
  • Example 3.3
  • ...and 24 more