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Let $\mathbb{E}_d$ denote the little discs operad for $1 \le d \le \infty$and let $\mathcal{C}$ be an $\infty$-category all of whose mapping spaces are$n$-truncated. We prove that when considering $\mathbb{E}_d$-monoids in$\mathcal{C}$, all coherence diagrams of arity $>n+3$ are redundant. Moregenerally, for an $\infty$-operad $\mathcal{O}$ we bound the arity of therelevant coherence diagrams in terms of the connectivity of certain operadicpartition complexes associated to $\mathcal{O}$.

Arity Approximation of $\infty$-Operads