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Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules

Ricardo Campos, Julien Ducoulombier, Najib Idrissi

TL;DR

This work develops leveled-tree combinatorics to construct explicit cofibrant/fibrant resolutions for (co)operads and (co)operadic (co)bimodules in spectra and CDGAs, extending the Boardman–Vogt and bar–cobar frameworks to leveled settings. The leveled BV resolutions $W_{l}$, leveled bar constructions $ ext{B}_{l}$, and leveled cobar constructions $ ext{Ω}_{l}$ are shown to be isomorphic to their classical counterparts in the operad and cooperad cases, while enabling a compatible treatment of bimodules and cobimodules. Key results include identifications $W_{l} ext{O} \\cong W ext{O}$, $ ext{B}_{l}( ext{O}) \\\cong ext{B}( ext{O})$, $ ext{Ω}_{l} ext{C} \\\cong ext{Ω}( ext{C})$, and fibrant/cofibrantResolution statements for Hopf Λ-cooperads/cobimodules via leveled constructions, together with explicit relations to indecomposable/primitive elements. The framework yields quasi-isomorphisms between leveled BV resolutions and leveled bar–cobar constructions, and provides tools for computing derived mapping spaces (e.g., in embedding problems) through cofree/coinduced structures; the Λ-extensions and simplicial frames further enhance applicability to rational homotopy and deformation problems.

Abstract

We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements.

Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules

TL;DR

This work develops leveled-tree combinatorics to construct explicit cofibrant/fibrant resolutions for (co)operads and (co)operadic (co)bimodules in spectra and CDGAs, extending the Boardman–Vogt and bar–cobar frameworks to leveled settings. The leveled BV resolutions , leveled bar constructions , and leveled cobar constructions are shown to be isomorphic to their classical counterparts in the operad and cooperad cases, while enabling a compatible treatment of bimodules and cobimodules. Key results include identifications , , , and fibrant/cofibrantResolution statements for Hopf Λ-cooperads/cobimodules via leveled constructions, together with explicit relations to indecomposable/primitive elements. The framework yields quasi-isomorphisms between leveled BV resolutions and leveled bar–cobar constructions, and provides tools for computing derived mapping spaces (e.g., in embedding problems) through cofree/coinduced structures; the Λ-extensions and simplicial frames further enhance applicability to rational homotopy and deformation problems.

Abstract

We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements.

Paper Structure

This paper contains 76 sections, 56 theorems, 192 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Summary of results
  4. Categories of trees
  5. Resolutions of operads in spectra
  6. Resolutions of operadic bimodules in spectra
  7. Resolutions of Hopf cooperads
  8. Resolutions of Hopf cooperadic cobimodules
  9. Notations and conventions
  10. Model category structures for (co)operads and (co)bimodules
  11. The model category of 1-reduced $\Lambda$-operads in spectra
  12. The category of spectra
  13. The category of 1-reduced operads
  14. The category of 1-reduced $\Lambda$-operads
  15. The model category of $\Lambda$-bimodules in spectra
  16. ...and 61 more sections

Key Result

Theorem A

The functor $\alpha:\mathbb{L}[n] \rightarrow \mathbb{T}^{\geq 2}[n]$ which forgets the leveled structure and the bivalent vertices is full and surjective on objects. It admits an explicit right inverse $\beta:\mathbb{T}^{\geq 2}[n]\rightarrow \mathbb{L}[n]$ which is faithful and injective on object

Figures (23)

  • Figure 1: Example of a planar $4$-tree.
  • Figure 2: A leveled $6$-tree.
  • Figure 4: Permutations in the category $\mathbb{L}[6]$.
  • Figure 5: Leveled trees.
  • Figure 6: Partial compositions of the family represented in Figure \ref{['famlevtree']}.
  • ...and 18 more figures

Theorems & Definitions (141)

  • Theorem A: Theorem \ref{['pro:now first theorem in intro']}
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem 2.1: Fre
  • Remark 2.2
  • Theorem 2.3: Fre
  • Example 2.4
  • Theorem 2.5: DucoulombierFresseTurchin2019
  • ...and 131 more