dc.contributor | Grojnowski, Ian | |

dc.creator | Holstein, Julian Victor Sebastian | |

dc.date.accessioned | 2018-11-24T23:26:14Z | |

dc.date.available | 2014-02-04T11:10:53Z | |

dc.date.available | 2018-11-24T23:26:14Z | |

dc.date.issued | 2014-01-07 | |

dc.identifier | https://www.repository.cam.ac.uk/handle/1810/245136 | |

dc.identifier.uri | http://repository.aust.edu.ng/xmlui/handle/123456789/3792 | |

dc.description.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. | |

dc.language | en | |

dc.publisher | University of Cambridge | |

dc.publisher | Department of Pure Mathematics and Mathematical Statistics | |

dc.rights | http://creativecommons.org/licenses/by-nc-nd/2.0/uk/ | |

dc.rights | Attribution-NonCommercial-NoDerivs 2.0 UK: England & Wales | |

dc.subject | Algebraic topology | |

dc.subject | Category theory | |

dc.title | Morita cohomology | |

dc.type | Thesis | |