Model categories, pro-categories and functors

Abstract: This thesis consists of five papers. The first three are concerned with various model structures on ind- and pro-categories, while the last two are concerned with the homotopy theory of functors.In Paper I, a general method for constructing simplicial model structures on ind- and pro-categories is described and its basic properties are studied. This method is particularly useful for constructing "profinite" analogues of known model categories. It recovers various known model structures and also constructs many interesting new model structures. 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 infinity-operads to a certain model category of profinite infinity-operads. The construction of the latter model category is inspired by the method described in Paper I, but there are a few subtle differences that make its construction more involved.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.In Paper IV, the question of whether an ordinary functor between enriched categories is equivalent to an enriched functor is addressed. This is done for several types of enrichments: namely when the base of enrichment is (pointed) topological spaces, (pointed) simplicial sets or orthogonal spectra. Simple criteria are obtained under which this question has a positive answer.In Paper V, the Goodwillie calculus of functors between categories of enriched diagram spaces is described. It is shown that the layers of the Goodwillie tower are classified by certain types of diagrams in spectra, directly generalizing Goodwillie's original classification. Using this classification, an operad structure on the derivatives of the identity functor is constructed that generalizes an operad structure originally constructed by Ching.