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Morita cohomology 10.17863/CAM.16230 Holstein, JVS 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 ˇ 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 ˇ 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. University of Cambridge Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/245136 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/38e69cc9-eb4d-435a-b49b-5a09764ceda5/download 71758d7e7a5c6b8c2a635da11555c941 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d562caf1-d64e-4620-9b21-3b825c3cdbaf/download 835269bda140c10400fe0606a14c3d21 http://creativecommons.org/licenses/by-nc-nd/2.0/uk/ 4902 Mathematical Physics 4904 Pure Mathematics 49 Mathematical Sciences