A morphism of \(\infty\)-operads is \(n\)-surjective (resp. \(n\)-faithful) if and only if its image under the composite functor \(\mathrm{Op}\hookrightarrow(\mathrm{Cat}_\infty)_{/\mathrm{Fin}_*} \xrightarrow{{\textup{fgt}}} \mathrm{Cat}_\infty\) is so (in the sense of definition 5.3.1).
Proof.
The Segal conditions for \(\infty\)-operads imply that surjectivity on underlying functors is equivalent to surjectivity on \(\infty\)-categories of colors. Moreover, the hom-spaces in an \(\infty\)-operad are disjoint unions of (finite) products of multi-hom spaces, and these operations both preserve the class of \((n-1)\)-connected (resp. \((n-1)\)-truncated) morphisms of spaces. (To see that products preserve \((n-1)\)-connectedness (resp. \((n-1)\)-truncatedness), note that this notion is determined fiberwise, that fibers of a product of morphisms of spaces are computed factorwise since limits commute with limits, and that \((n-1)\)-truncated (resp. \((n-1)\)-connected) spaces are stable under products.) ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2