Truncated operads and simplicial spaces

It was shown in a recent paper by Boavida de Brito and Weiss that a well-known construction which to a plain (=monochromatic) topological operad associates a topological category and a functor from it to the category of finite sets is homotopically fully faithful, under mild conditions on the operads. The main result here is a generalization of that statement to k-truncated plain topological operads. A k-truncated operad is a weaker version of operad where all operations have arity at most k.