Repository logo
 

Morita cohomology


Loading...
Thumbnail Image

Type

Change log

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 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.

Description

Date

Advisors

Qualification

Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge

Rights and licensing

Except where otherwised noted, this item's license is described as Attribution-NonCommercial-NoDerivs 2.0 UK: England & Wales