[Paper Review] Combinatorial homotopy theory for operads

Jovana Obradović|arXiv (Cornell University)|Jun 14, 2019

Homotopy and Cohomology in Algebraic Topology30 references4 citations

TL;DR

This paper provides a combinatorial characterization of the minimal model $\mathcal{O}_\infty$ for the colored operad $\mathcal{O}$ encoding non-symmetric operads, using hypergraph polytopes as the underlying spaces of operations. It generalizes the $A_\infty$-operad structure via associahedra and extends the construction to the $W$-construction and the minimal model of the cyclic operad $\mathcal{C}$, offering explicit, geometrically meaningful resolutions for strongly homotopy algebras.

ABSTRACT

We introduce an explicit combinatorial characterization of the minimal model ${\cal O}_{\infty}$ of the coloured operad ${\cal O}$ encoding non-symmetric operads. In our description of ${\cal O}_{\infty}$, the spaces of operations are defined in terms of hypergraph polytopes and the composition structure generalizes the one of the $A_{\infty}$-operad. As further generalizations of this construction, we present a combinatorial description of the $W$-construction applied on ${\cal O}$, as well as of the minimal model of the coloured operad ${\cal C}$ encoding non-symmetric cyclic operads.

Motivation & Objective

• To provide an explicit, combinatorial description of the minimal model $\mathcal{O}_\infty$ for the colored operad $\mathcal{O}$ encoding non-symmetric operads.
• To generalize the $A_\infty$-operad structure by realizing its operations through hypergraph polytopes, particularly operadic polytopes derived from rooted trees.
• To construct a combinatorial $W$-construction (Boardman-Vogt resolution) for $\mathcal{O}$ using cubical subdivisions of operadic polytopes.
• To extend the framework to define the minimal model $\mathcal{C}_\infty$ for the colored operad $\mathcal{C}$ encoding non-symmetric cyclic operads, preserving cyclic symmetry in relations.

Proposed method

• The minimal model $\mathcal{O}_\infty$ is constructed using hypergraph polytopes associated with the edge-graphs of rooted trees, where each polytope represents the space of operations.
• The composition structure in $\mathcal{O}_\infty$ generalizes the Stasheff associahedron construction, with operations indexed by hypergraph constructs and tree data.
• The $W$-construction on $\mathcal{O}$ is realized via a cubical subdivision of the operadic polytopes, encoding homotopy-coherent compositions.
• For cyclic operads, the construction is adapted by introducing cyclic operadic trees with a cyclic ordering $\tau$, and defining equivalence classes under cyclic symmetry.
• The differential $d_{\mathcal{O}_\infty}$ is defined by splitting vertices of constructs, preserving the underlying tree and hypergraph structure.
• The minimal model $\mathcal{C}_\infty$ is built as a free operad on equivalence classes of quadruples $({\cal T}, \sigma, \tau, C)$, with composition defined via tree grafting and construct merging.

Experimental results

Research questions

• RQ1 How can the minimal model $\mathcal{O}_\infty$ of the operad $\mathcal{O}$ encoding non-symmetric operads be explicitly described using combinatorial and geometric data?
• RQ2 Can the $W$-construction on $\mathcal{O}$ be realized combinatorially through cubical subdivisions of hypergraph polytopes?
• RQ3 How can the notion of a minimal model be extended to non-symmetric cyclic operads, preserving cyclic symmetry in the relations?
• RQ4 What is the role of hypergraph polytopes—specifically operadic polytopes derived from rooted trees—in encoding higher homotopy structures for operads?
• RQ5 Can the resulting minimal models be shown to be quasi-isomorphic to the original operads while maintaining a minimal differential structure?

Key findings

• The minimal model $\mathcal{O}_\infty$ is explicitly realized as a dg operad whose operations are indexed by hypergraph constructs on the edge-graph of a rooted tree, with composition generalizing the associahedron structure.
• The $W$-construction on $\mathcal{O}$ is combinatorially described as a cubical subdivision of the operadic polytopes, providing a resolution of $\mathcal{O}$.
• The minimal model $\mathcal{C}_\infty$ for the cyclic operad $\mathcal{C}$ is constructed as a free operad on equivalence classes of cyclic operadic trees with hypergraph constructs, preserving cyclic symmetry.
• The differential $d_{\mathcal{C}_\infty}$ acts by splitting vertices of constructs and maps the empty construct to zero, ensuring minimality.
• The operad $\mathcal{C}_\infty$ is proven to be a minimal model for $\mathcal{C}$, with the differential supported only on decomposable elements.
• The construction establishes $\mathcal{C}_\infty$ as a strict infinity operad, with a geometric and combinatorial foundation rooted in hypergraph polytopes.