## Morita cohomology

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##### Authors

Holstein, Julian Victor Sebastian

##### Advisors

Grojnowski, Ian

##### Date

2014-01-07##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Pure Mathematics and Mathematical Statistics

##### Qualification

PhD

##### Language

English

##### Type

Thesis

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Show full item record##### Citation

Holstein, J. V. S. (2014). Morita cohomology (doctoral thesis).

##### Abstract

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.

##### Keywords

Algebraic topology, Category theory

##### Identifiers

This record's URL: http://www.repository.cam.ac.uk/handle/1810/245136

##### Rights

Attribution-NonCommercial-NoDerivs 2.0 UK: England & Wales

Licence URL: http://creativecommons.org/licenses/by-nc-nd/2.0/uk/

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