Unital k-Restricted Infinity-Operads
Amartya Shekhar Dubey, Yu Leon Liu
TL;DR
This work develops a unital ∞-operad framework refined by arity restrictions. By modeling unital k-restricted ∞-operads as complete Segal presheaves on closed k-dendroidal trees Ω^c_{≤k}, the authors prove that restriction functors admit fully faithful left and right adjoints, given by left and right Kan extensions that preserve completeness. They establish a canonical filtration and cofiltration O → L_kO and O → R_kO, provide explicit multi-ary morphism descriptions (via colimits and limits over ≤k-ary data), and connect the constructions to familiar operadic families (A_∞ and E_n via Fulton–MacPherson models). The results supply a robust arity-controlled approach to unital ∞-operads and open avenues for extending to non-closed trees and lifting obstruction theories in operadic contexts.
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.