Truncated operads and simplicial spaces

TL;DR

This paper generalizes a known homotopical property of topological operads to k-truncated operads, which restricts operations to arities up to k, expanding the understanding of their associated categorical structures.

Contribution

It extends the homotopical full faithfulness result from plain operads to k-truncated operads, broadening the applicability of the previous framework.

Findings

Homotopically fully faithful functor for k-truncated operads

Generalization of operad-category correspondence

Enhanced understanding of truncated operad structures

Abstract

It was shown in a recent paper by Boavida de Brito and Weiss that a well-known construction which to a plain (=monochromatic) topological operad associates a topological category and a functor from it to the category of finite sets is homotopically fully faithful, under mild conditions on the operads. The main result here is a generalization of that statement to k-truncated plain topological operads. A k-truncated operad is a weaker version of operad where all operations have arity at most k.