This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.

We build a model structure from the simple point of departure of a structured interval in a monoidal category — more generally, a structured cylinder and a structured co-cylinder in a category.

The main goal of this thesis is to construct a tensor product on the 2-category BFC-A of braided fusion categories containing a symmetric fusion category A. We achieve this by introducing the new notion of Z(A)-crossed braided categories. These are categories enriched over the Drinfeld centre Z(A) of the symmetric fusion category. We show that Z(A) admits an additional symmetric tensor structure, which makes it into a 2-fold monoidal category. ByTannaka duality, A= Rep(G) (or Rep(G; w)) for a finite group G (or finite super-group (G,w)). Under this identication Z(A) = VectG[G], the category of G-equivariant vector bundles over G, and we show that the symmetric tensor product corresponds to (a super version of) to the brewise tensor product. We use the additional symmetric tensor product on Z(A) to define the composition in Z(A)-crossed braided categories, whereas the usual tensor product is used for the monoidal structure. We further require this monoidal structure to be braided for the switch map that uses the braiding in Z(A). We show that the 2-category Z(A)-XBF is equivalent to both BFC=A and the 2-category of (super)-G-crossed braided categories. Using the former equivalence, the reduced tensor product on BFC-A is dened in terms of the enriched Cartesian product of Z(A)-enriched categories on Z(A)-XBF. The reduced tensor product obtained in this way has as unit Z(A). It induces a pairing between minimal modular extensions of categories having A as their Mueger centre.