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Eilenberg-Moore Bicategories for Opmonoidal Pseudomonads

Adrian Miranda

Abstract

We analyse compatibility between monads and monoidal structures in the two-dimensional setting. We describe sufficient conditions for monoidal structures to lift to the Eilenberg-Moore pseudoalgebras. We then extend these results to braids, syllapses and symmetries. To achieve these results we define the Gray-tensor product of pseudomonads, and examine its interaction with the Eilenberg-Moore construction.

Eilenberg-Moore Bicategories for Opmonoidal Pseudomonads

Abstract

We analyse compatibility between monads and monoidal structures in the two-dimensional setting. We describe sufficient conditions for monoidal structures to lift to the Eilenberg-Moore pseudoalgebras. We then extend these results to braids, syllapses and symmetries. To achieve these results we define the Gray-tensor product of pseudomonads, and examine its interaction with the Eilenberg-Moore construction.
Paper Structure (34 sections, 30 theorems, 30 equations)

This paper contains 34 sections, 30 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Context and goals
  3. Summary of main results and contributions
  4. Key ideas and techniques
  5. Strictification
  6. Alternative perspectives on compatibility between structures
  7. Ambistrict $2$-cells
  8. $\mathbf{Gray}$-tensor products of $2$-natural transformations
  9. Faithfulness on $2$-cells of $U^S: \mathcal{A}^S \to \mathcal{A}$
  10. Conventions
  11. Diagrams and notation
  12. Size
  13. Review of symmetric opmonoidal monads
  14. Perspectives on compatibility between pseudomonads and monoidal structures
  15. Opmonoidal pseudomonads as pseudomonads in the tricategory of monoidal bicategories
  16. ...and 19 more sections

Key Result

Theorem 3.1.5

Theorems & Definitions (89)

  • Remark 2.0.1
  • Remark 2.0.2
  • Remark 3.0.1
  • Definition 3.1.2
  • Remark 3.1.3
  • Definition 3.1.4
  • Theorem 3.1.5
  • Definition 3.2.1
  • Example 3.2.2
  • Proposition 3.2.3
  • ...and 79 more