Cooking up model structures on ind- and pro-categories

Abstract: This licentiate thesis consists of three papers related to model structures on ind- and pro-categories.In Paper I a general method for constructing simplicial model structures on ind- and pro-categories is described. This method is particularly useful for constructing ``profinite'' analogues of known model categories. The construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but it can also be applied to construct many interesting new model structures. In addition, some general properties of this method are studied, such as its functorial behaviour and its relation to Bousfield localization. The construction is compared to the ∞-categorical approach to ind- and pro-categories in an appendix.In Paper II, it is shown 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 method for constructing this model category of profinite ∞-operads and the profinite completion functor is similar to the method described in Paper I, but there are a few subtle differences that make this construction more involved. In understanding the model structure for profinite ∞-operads, an important role is played by the so-called lean ∞-operads. It is shown that these lean ∞-operads can, up to homotopy, be characterized by certain homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) ∞-operads and of ∞-categories.In Paper III, the general method from Paper I is used to give an alternative proof of a result by Arone, Barnea and Schlank. This result states that the stabilization of the category of noncommutative CW-complexes can be modelled as the category of spectral presheaves on a certain category M. The advantage of this alternative proof is that it mainly relies on well-known results on (stable) model categories.