Unital k-Restricted Infinity-Operads

Abstract

We study unital $\infty$-operads by their arity restrictions. Given $k\ge 1$, we develop a model for unital $k$-restricted $\infty$-operads, which are variants of $\infty$-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 $\infty$-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.

19 pages, comments welcome; Small Error in Sec 3.1 fixed