[Paper Review] The relative lattice path operad

Alexandre Quesney|arXiv (Cornell University)|Nov 18, 2015

Advanced Topics in Algebra24 references3 citations

TL;DR

This paper constructs a combinatorial set-level operad, $ρ\mathcal{L}$, as a model for the Swiss Cheese operad, and uses Batanin-Berger's condensation process to derive weakly equivalent topological and chain operads. The key result is that the condensed operad $Σρ\mathcal{L}_{m}$ is weakly equivalent to the topological and chain Swiss Cheese operads $σ\mathcal{C}_{m}$ and $C_*(\sigma\mathcal{C}_{m})$, respectively, providing algebraic models for actions of 2-fold loop spaces on 2-fold relative loop spaces.

ABSTRACT

We construct a set-theoretic coloured operad that may be thought of as a combinatorial model for the Swiss Cheese operad. This is the relative (or Swiss Cheese) version of the lattice path operad constructed by Batanin and Berger. By adapting their condensation process we obtain a topological (resp. chain) operad that we show to be weakly equivalent to the topological (resp. chain) Swiss Cheese operad.

Motivation & Objective

• To provide a combinatorial, set-level model for the Swiss Cheese operad using lattice path structures.
• To apply Batanin-Berger's condensation process to this model to obtain weakly equivalent topological and chain operads.
• To construct algebraic models for actions of 2-fold loop spaces on 2-fold relative loop spaces via the resulting operads.
• To generalize Berger's cell decomposition of the little cubes operad to the Swiss Cheese setting.
• To establish a recognition principle for relative loop space actions using a poset operad $ρ\mathcal{K}_{m}$.

Proposed method

• Construct a 2-coloured operad $ρ\mathcal{L}$ in sets as a combinatorial model for the Swiss Cheese operad, based on lattice paths.
• Apply the condensation functor $F$ to $ρ\mathcal{L}_{m}$ using a cosimplicial object $δ$ in a symmetric monoidal category with zero object.
• Use two specific cosimplicial objects: $δ_{\text{Top}}$ for topological operads and $δ_{\mathbb{Z}}$ for chain complexes.
• Define a cellular decomposition of $σ\mathcal{C}_{m}$ using a poset operad $ρ\mathcal{K}_{m}$, generalizing Berger’s decomposition of the little cubes operad.
• Establish a zig-zag of weak equivalences between $σ\mathcal{C}_{m}$ and the classifying operad of $ρ\mathcal{K}_{m}$, proving that any $ρ\mathcal{K}_{m}$-cellular operad is weakly equivalent to $σ\mathcal{C}_{m}$.
• Construct explicit operations on the cobar construction $ΩB$ and relative cobar construction $Ω(B,C)$ to define an action of the condensed chain operad $Σρ\mathcal{L}_{2}$ and its suboperad $ρ\mathcal{S}_{2}$.

Experimental results

Research questions

• RQ1 Can a combinatorial set-level operad be constructed to model the Swiss Cheese operad?
• RQ2 Does the condensation process applied to this model yield weakly equivalent topological and chain operads to the standard Swiss Cheese operad?
• RQ3 Can the resulting operads provide algebraic models for actions of 2-fold loop spaces on 2-fold relative loop spaces?
• RQ4 Is there a cellular decomposition of the Swiss Cheese operad that generalizes Berger’s decomposition of the little cubes operad?
• RQ5 Can the relative cobar construction be endowed with an $ρ\mathcal{S}_{2}$-algebra structure via explicit operations?

Key findings

• The operad $Σρ\mathcal{L}_{m}(\delta_{\text{Top}})$ is weakly equivalent to the topological Swiss Cheese operad $σ\mathcal{C}_{m}$.
• The operad $Σρ\mathcal{L}_{m}(\delta_{\mathbb{Z}})$ is weakly equivalent to the chain Swiss Cheese operad $C_*(\u03c3\mathcal{C}_{m})$.
• The chain operad $Σρ\mathcal{L}_{2}(\delta_{\mathbb{Z}})$ admits a weakly equivalent suboperad $ρ\mathcal{S}_{2}$, which is a relative version of the surjection operad.
• Any topological $ρ\mathcal{K}_{m}$-cellular operad is weakly equivalent to the Swiss Cheese operad $σ\mathcal{C}_{m}$, establishing a recognition principle.
• The pair $(\underline{\Omega}_{u}B, \underline{\Omega}_{u}(B,C))$ is an algebra over the condensed chain operad $Σρ\mathcal{L}_{2}(\delta_{\mathbb{Z}})$ for a unital/counital bialgebra $B$ and $B$-comodule $C$ in unital algebras.
• The pair $(\underline{\Omega}C_*^1M, \underline{\Omega}(C_*^1M, C_*^1N))$ admits an action of the operad $ρ\mathcal{S}_{2}$ via explicitly defined operations on the cobar constructions.