Abstract
dc:description.abstractThis 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.
Degree
thesis:*- Name dc:type.qualificationname
- Doctor of Philosophy (PhD)
- Level dc:type.qualificationlevel
- Doctoral
- Grantor dc:publisher.institution
- University of Cambridge
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Holstein, JVS
Subjects
dc:subject × 3Rights
dc:rightsIdentifiers
dc:identifier.*- DOI dc:identifier.doi
- https://doi.org/10.17863/CAM.16230
- OAI identifier oai:identifier
- oai:www.repository.cam.ac.uk:1810/245136