Simplicial lists in operad theory I

TL;DR

The paper develops a novel simplicial-list framework to model non-symmetric operads by replacing traditional dendroidal approaches with leveled-tree indexing. The central construction is the list nerve $N^l: \mathsf{Operad} \to \mathsf{sList}$, which is fully faithful, together with a nerve theorem that characterizes when a simplicial list arises from an operad, thereby modeling $\infty$-operads via quasi-categories. The authors prove that $\mathsf{sList}$ is a presheaf category over a base $\Upsilon$ of rooted leveled trees and construct a coherent nerve that yields $\infty$-operads from Kan-enriched operads, while also defining a homology theory for simplicial lists and exploring connections to existing models. These results provide a combinatorial, homotopical framework for non-symmetric operads and point toward extensions to symmetric, properadic, and related higher-algebraic structures, with future work on model structures and comparisons to dendroidal/Segal approaches.

Abstract

We define a category $\mathsf{List}$ whose objects are sets and morphisms are mappings which assign to an element in the domain an ordered sequence (list) of elements in the codomain. We introduce and study a category of simplicial objects $\mathsf{slist}$ whose objects are functors $Δ^{op} \to \mathsf{List}$, which we call simplicial lists, and morphisms are natural transformations which have functions as components. We demonstrate that $\mathsf{sList}$ supports the combinatorics of (non-symmetric) operads by constructing a fully-faithful nerve functor $N^l : \mathsf{Operad} \to \mathsf{sList}$ from the category of operads. This leads to a reasonable model for the theory of non-symmetric $\infty$-operads. We also demonstrate that $\mathsf{sList}$ has the structure of a presheaf category. In particular, we study a subcategory $\mathsf{sList}_{\text{op}}$ of operadic simplicial lists, in which the nerve functor takes values. The latter category is also a presheaf category over a base whose objects may be interpreted as levelled trees. We construct a coherent nerve functor which outputs an $\infty$-operad for each operad enriched in Kan complexes. We also define homology groups of simplicial lists and study first properties.

Theorems & Definitions (34)

• Definition 1.1: The category of listings
• Remark 1.1
• Definition 1.2: Operadic simplicial lists
• Theorem 1.1: Nerve Theorem
• Definition 1.3: Quasi-operad
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.4
• Definition 2.1: Colored operad