  We show that a profinite completion functor for (simplicial or topological)
operads with good homotopical properties can be constructed as a left Quillen
functor from an appropriate model category of infinity-operads to a certain
model category of profinite infinity-operads. The construction is based on a
notion of lean infinity-operad, and we characterize those infinity-operads
weakly equivalent to lean ones in terms of homotopical finiteness properties.
Several variants of the construction are also discussed, such as the cases of
unital (or closed) infinity-operads and of infinity-categories.
