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Operads and equivariance

Alexander Corner, Nick Gurski

Abstract

Operads were originally defined by May to have right actions of the symmetric groups, but later formulations have also used no groups actions at all or group actions by such families as the braid groups. We call such families action operads, as they are the algebraic objects that encode parametrized group actions on operads. In Part I of this paper, we study the basic algebra of action operads $Λ$ and the $Λ$-operads they act upon. In Part II, we study $Λ$-operads in the 2-category of small categories.

Operads and equivariance

Abstract

Operads were originally defined by May to have right actions of the symmetric groups, but later formulations have also used no groups actions at all or group actions by such families as the braid groups. We call such families action operads, as they are the algebraic objects that encode parametrized group actions on operads. In Part I of this paper, we study the basic algebra of action operads and the -operads they act upon. In Part II, we study -operads in the 2-category of small categories.
Paper Structure (17 sections, 65 theorems, 292 equations)

This paper contains 17 sections, 65 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Background: Operads
  4. Action Operads
  5. Examples
  6. Action Operads as Extensions
  7. Presentations for Action Operads
  8. L-Operads and their Algebras
  9. The Substitution Product
  10. Background: Group Actions and 2-limits
  11. Background: $2$-monads and their Algebras
  12. L-Operads in Cat as $2$-monads
  13. Coherence
  14. Group Actions and Cartesian $2$-monads
  15. Action Operads as Clubs
  16. ...and 2 more sections

Key Result

Proposition 3.16

[prop]prop:cat-of-sym-op There is a category with

Theorems & Definitions (242)

  • Definition 2.10: (Underlying permutation)
  • Remark 2.12: (Left action of symmetric groups on tuples)
  • Definition 2.13: (Block sum)
  • Remark 2.14
  • Definition 2.15: (Duplication)
  • Remark 2.16
  • Remark 2.17
  • Definition 3.1: (Symmetric operad)
  • Remark 3.3
  • Remark 3.5
  • ...and 232 more