We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the ∞-categorical approach to ind- and pro-categories in an appendix.

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 ∞-operads to a certain model category of profinite ∞-operads. The construction is based on a notion of lean ∞-operad, and we characterize those ∞-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) ∞-operads and of ∞-categories. 

We use the machinery of Paper I to give an alternative proof of one of the main results of [ABS21]. This result states that the category of noncommutative CW-spectra can be modelled as the category of spectral presheaves on a certain category M, whose objects can be thought of as “suspension spectra of matrix algebras”. The advantage of our proof is that it mainly relies on well-known results on (stable) model categories.

We describe simple criteria under which a given functor is naturally equivalent to an enriched one. We do this for several bases of enrichment, namely (pointed) simplicial sets, (pointed) topological spaces and orthogonal spectra. We also describe a few corollaries, such as a result on simplicial Dwyer-Kan localizations that may be of independent interest.

We describe the Goodwillie calculus of functors between categories of enriched diagrams and generalize Goodwillie’s classification of n-homogeneous functors to this setting. Using this classification, we construct an operad structure on the derivatives of the identity functor from Top_∗^C to itself that directly generalizes Ching’s operad structure on the derivatives of the identity functor from Top_∗ to itself.