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A spectral sequence for spaces of maps between operads

Florian Göppl

Abstract

We construct a tower of fibrations approximating the derived mapping space between two simplicially enriched operads subject to mild conditions. The n-th stage of the tower is obtained by neglecting operations with more than n inputs. The main theorem describes the layers of this tower.

A spectral sequence for spaces of maps between operads

Abstract

We construct a tower of fibrations approximating the derived mapping space between two simplicially enriched operads subject to mild conditions. The n-th stage of the tower is obtained by neglecting operations with more than n inputs. The main theorem describes the layers of this tower.

Paper Structure

This paper contains 11 sections, 28 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Operads and dendroidal objects
  3. Operads
  4. Dendroidal sets and spaces
  5. A tower of derived mapping spaces
  6. Construction of the tower
  7. Excision in categories
  8. Boundaries and coboundaries revisited
  9. Dévissage
  10. Application
  11. The derived mapping spaces of little cube operads

Key Result

Lemma 1

(= Lemma TowerConvergence, Corollary corfilt.) The above tower of derived mapping spaces converges, i.e., for all dendroidal spaces $X$ and $Y$ the induced map is a weak homotopy equivalence.

Theorems & Definitions (73)

  • Lemma
  • Theorem
  • Theorem
  • Corollary
  • Definition 2.1.1
  • Example 2.1.2
  • Definition 2.1.3
  • Example 2.1.4
  • Example 2.1.5: The little disks operad
  • Example 2.1.6: The Fulton-MacPherson operad
  • ...and 63 more