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Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences

Nicola Gambino, Richard Garner, Christina Vasilakopoulou

Abstract

We extend the arithmetic product of species of structures and symmetric sequences studied by Maia and Mendez and by Dwyer and Hess to coloured symmetric sequences and show that it determines a normal oplax monoidal structure on the bicategory of coloured symmetric sequences. In order to do this, we establish general results on extending monoidal structures to Kleisli bicategories. Our approach uses monoidal double categories, which help us to attack the difficult problem of verifying the coherence conditions for a monoidal bicategory in an efficient way.

Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences

Abstract

We extend the arithmetic product of species of structures and symmetric sequences studied by Maia and Mendez and by Dwyer and Hess to coloured symmetric sequences and show that it determines a normal oplax monoidal structure on the bicategory of coloured symmetric sequences. In order to do this, we establish general results on extending monoidal structures to Kleisli bicategories. Our approach uses monoidal double categories, which help us to attack the difficult problem of verifying the coherence conditions for a monoidal bicategory in an efficient way.
Paper Structure (12 sections, 34 theorems, 114 equations)

This paper contains 12 sections, 34 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Double categories
  3. Maps of double categories
  4. Monoidal double categories
  5. Maps of monoidal double categories
  6. Monoids in monoidal double categories
  7. Double monads
  8. Monoidal double monads
  9. Monoidal Kleisli double categories
  10. The arithmetic product of coloured symmetric sequences
  11. Coherence axioms for an oplax monoidal double category
  12. Proof of \ref{['thm:oplaxmonoidalKlT']}

Key Result

Lemma 2.7

Let $f \colon X \rightarrow X'$ be vertical $1$-cell of a double category $\mathbb{C}$. Giving a companion $(\widehat{f}, p_1, p_2)$ for $f$ is equivalent to giving either of the following:

Theorems & Definitions (100)

  • Definition 2.1: Double category
  • Remark 2.2
  • Example 2.3: Bicategories and monoidal categories as double categories
  • Example 2.4: The double category of matrices
  • Example 2.5: The double category of profunctors
  • Definition 2.6
  • Lemma 2.7
  • Example 2.8
  • Proposition 2.9
  • Definition 3.1: Oplax double functor, double functor
  • ...and 90 more