7:18 AM
Is there any place in the literature on $\infty$-operads where arity truncation of operads is discussed? What I mean is one can consider a version of $\infty$-operads where one only keeps track of operations of arity $\le k$ (for example in Lurie's model by pulling back the fibration over $Fin_{\ast}$ to a fibration over $Fin^{\le s}_{\ast}$) and then ther probably should be a free forgetful adjunction with usual $\infty$-operads etc.
I found some references that talk about specific instances of this. For linear operads and/or one coloured etc. For what I have in mind the relevant context is that of symmetric multi-($\infty$-)categories, so at the level of generality of Lurie's model.
(typo, in the k=s in the above of course :) )
@RuneHaugseng I hope i'm not being too rude by tagging you (if it is I apologize in advance). I feel like if anyone has an answer to this it must be you :).

12 hours later…
6:55 PM
@SaalHardali I know that Duncan Clark, a student of John Harper, is interested in arity truncations and their relation to Goodwillie calculus. As a student of Harper's he's allergic to $\infty$-categories, so there may be a language barrier in reading his work, but I think it's there.