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Cyclic operads through modules

Thomas Willwacher

Abstract

We describe a way to compute mapping spaces of cyclic operads through modules. As an application we compute the homotopy automorphism space of the cyclic Batalin-Vilkovisky (Hopf co-)operad.

Cyclic operads through modules

Abstract

We describe a way to compute mapping spaces of cyclic operads through modules. As an application we compute the homotopy automorphism space of the cyclic Batalin-Vilkovisky (Hopf co-)operad.
Paper Structure (36 sections, 20 theorems, 164 equations)

This paper contains 36 sections, 20 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Prerequisites and model categories
  3. Conventions on dg vector spaces
  4. Symmetric sequences and cyclic sequences
  5. Model categories of operads and cooperads
  6. Dg operads
  7. Dg cooperads
  8. Dg Hopf cooperads
  9. Fiber sequences for operad-module pairs
  10. Adjunctions between cyclic and non-cyclic (co)operads
  11. Bar constructions
  12. Cyclic bar and cobar construction
  13. The cobar construction often preserves weak equivalences
  14. Auxiliary symmetric sequences
  15. Comparison of bar constructions
  16. ...and 21 more sections

Key Result

Theorem 1.1

The forgetful functor $F$ is part of a Quillen adjunction For $\mathop{\mathrm{\mathsfit{P}}}\nolimits$ any augmented cyclic operad the derived counit of the adjunction is a weak equivalence.

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 3.1
  • ...and 26 more