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Operadic Deformation Theory

Ricardo Campos, Albin Grataloup

Abstract

This is a survey on recent progress in algebraic deformation theory and the application of algebraic operads to its study. We review the classical homotopical tools in the theory of algebraic operads, namely Koszul duality. We give concrete examples of applications of such tools to various flavours of problems related to deformations of algebraic structures. We also study formal moduli problems and related notions from the operadic point of view.

Operadic Deformation Theory

Abstract

This is a survey on recent progress in algebraic deformation theory and the application of algebraic operads to its study. We review the classical homotopical tools in the theory of algebraic operads, namely Koszul duality. We give concrete examples of applications of such tools to various flavours of problems related to deformations of algebraic structures. We also study formal moduli problems and related notions from the operadic point of view.
Paper Structure (39 sections, 69 theorems, 160 equations)

This paper contains 39 sections, 69 theorems, 160 equations.

Table of Contents

  1. Algebraic Operad Theory
  2. Algebraic Operads and Cooperads
  3. Algebraic Structures over an Operad or Cooperad
  4. A short digression on the universal enveloping algebra
  5. Classical Constructions for Algebraic Operads
  6. Model Categorical Aspects
  7. Model categories and homotopical algebra
  8. Model structures for (co)algebras
  9. Model structure on operads
  10. Koszul Duality
  11. Bar-Cobar Adjunction for Algebras
  12. A comment on operads as algebras
  13. Algebraic Deformation Theory
  14. Algebraic Structures Up to Homotopy
  15. Homotopy algebras and morphisms
  16. ...and 24 more sections

Key Result

Proposition 1.6

For $\mathscr{D} = \mathbb{N}^\sim$ and $\mathscr{V} = \mathrm{Mod}_k$, we obtain: where $\mathrm{Ind}_{\Sigma_p \times \Sigma_q}^{\Sigma_n}$ denotes the inductionRecall that given a morphism of associative algebras $f\colon A\to B$, we have an induction/extension of scalars functor $B\otimes_A - \colon \mathop{\mathrm{Mod}}\nolimits_A \to \mathop{\mathrm{Mod}}\nolimits_B$, left a

Theorems & Definitions (137)

  • Definition 1.1
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • Definition 1.12
  • Proposition 1.15
  • Definition 1.17
  • ...and 127 more