Unital k-Restricted Infinity-Operads

TL;DR

This paper develops a model for unital k-restricted infinity-operads, which are variants of infinity-operads limited to arity ≤ k, and explores their adjunctions and filtrations.

Contribution

It introduces a new model for unital k-restricted infinity-operads using complete Segal presheaves on closed k-dendroidal trees and analyzes their adjunction properties.

Findings

Established a model for unital k-restricted infinity-operads.

Proved the restriction functors admit fully faithful adjoints.

Provided a filtration and co-filtration framework for unital infinity-operads.

Abstract

We study unital $\infty$-operads by their arity restrictions. Given $k \geq 1$, we develop a model for unital $k$-restricted $\infty$-operads, which are variants of $\infty$-operads which has only $(\leq k)$-arity morphisms, as complete Segal presheaves on closed $k$-dendroidal trees, which are closed trees build from corollas with valences $\leq k$. Furthermore, we prove that the restriction functors from unital $\infty$-operads to unital $k$-restricted $\infty$-operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying $k$, the left and right adjoints give a filtration and a co-filtration for any unital $\infty$-operads by $k$-restricted $\infty$-operads.