The Q-Shaped Derived Category
Specialeforsvar: Marius Verner Bach Nielsen
Titel: The Q-Shaped Derived Category
Abstract: In this thesis we will for any locally Gorenstein category A with enough projectives and a sufficiently nice kMod- enriched category Q study the category of Q-shaped modules in A. To this category we associate two model structures, one projective and one injective named after their trivially cofibrant and trivially fibrant objects, respectively. These model structures
have the same weak equivalence and the Q-shaped derived category of A is the Kan localization at this class of maps. We show that under appropriate assumptions on Q and A there exists cohomology functors, Hi [q] (−): Q,AMod → A, such that a map is a weak equivalence if and only if it induces an isomorphism on cohomology for every i > 0 and q ∈ Q. All of the above is heavily based on ideas from [HJ21], and in the final two chapters we apply our results to recover most of the results of Holm and Jørgensens paper.
Vejleder: Henrik G. Holm
Censor: Niels Lauritzen, Aarhus Universitet