Show simple item record

Morita cohomology

dc.contributorGrojnowski, Ian
dc.creatorHolstein, Julian Victor Sebastian
dc.date.accessioned2018-11-24T23:26:14Z
dc.date.available2014-02-04T11:10:53Z
dc.date.available2018-11-24T23:26:14Z
dc.date.issued2014-01-07
dc.identifierhttps://www.repository.cam.ac.uk/handle/1810/245136
dc.identifier.urihttp://repository.aust.edu.ng/xmlui/handle/123456789/3792
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 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.
dc.languageen
dc.publisherUniversity of Cambridge
dc.publisherDepartment of Pure Mathematics and Mathematical Statistics
dc.rightshttp://creativecommons.org/licenses/by-nc-nd/2.0/uk/
dc.rightsAttribution-NonCommercial-NoDerivs 2.0 UK: England & Wales
dc.subjectAlgebraic topology
dc.subjectCategory theory
dc.titleMorita cohomology
dc.typeThesis


Files in this item

FilesSizeFormatView
Morita Cohomology.pdf473.6Kbapplication/pdfView/Open

This item appears in the following Collection(s)

Show simple item record