The relative lattice path operad
Alexandre Quesney
TL;DR
The work develops a robust combinatorial and homotopical model for the Swiss Cheese operad $\mathcal{SC}_m$ by introducing the relative lattice path operad $\mathcal{RL}$, a two-colour operad whose condensation via a functor-operad $\xi(\mathcal{O})$ and the coendomorphism $\mathrm{Coend}_{\mathcal{O}}(\delta)$ yields topological and chain models weakly equivalent to $\mathcal{SC}_m$ and $C_*(\mathcal{SC}_m)$. It constructs two cellular decompositions of $\mathcal{SC}_m$ indexed by posets $\mathcal{RK}_m$ and $\mathcal{RK}'_m$, enabling a Berger-style cellular recognition principle and proving that $\mathrm{Coend}_{\mathcal{RL}_m}(\delta)$ and $\mathrm{Coend}_{\mathcal{RL}'_m}(\delta)$ are weakly equivalent to the Swiss Cheese operad for $\delta$ chosen as either topological realization or chains. The paper further introduces relative surjection operads $\mathcal{RS}_m$ and $\mathcal{RS}'_m$ as weak sub-operads of the coend constructions and provides a concrete tree-based description of RL$_2$ and RL$'_2$, yielding algebras that correspond to pairs $(\mathcal{M},\mathcal{Z})$ of a multiplicative operad and a bimodule with an injective map $\iota: \mathcal{M} \to \mathcal{Z}$ (and the analogous $\mathcal{Z}'$ case), thereby connecting operadic actions to relative loop-space structures. Overall, the framework offers a practical, combinatorial route to model and manipulate relative (Swiss Cheese) operads and their actions on cochains and iterated relative loop spaces, with explicit algebraic realizations in low arities.
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.