Unital k-Restricted Infinity-Operads

We study unital $\mathrm{\infty }$-operads by their arity restrictions. Given $k\ge 1$, we develop a model for unital $k$-restricted $\mathrm{\infty }$-operads, which are variants of $\mathrm{\infty }$-operads which has only $(\le k)$-arity morphisms, as complete Segal presheaves on closed $k$-dendroidal trees, which are closed trees build from corollas with valences $\le k$. Furthermore, we prove that the restriction functors from unital $\mathrm{\infty }$-operads to unital $k$-restricted $\mathrm{\infty }$-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 $\mathrm{\infty }$-operads by $k$-restricted $\mathrm{\infty }$-operads.