N-complexes and Categorification
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
This thesis consists of three papers about N-complexes and their uses in categorification. N-complexes are generalizations of chain complexes having a differential d satisfying dN = 0 rather than d2 = 0. Categorification is the process of finding a higher category analog of a given mathematical structure.
Paper I: We study a set of homology functors indexed by positive integers a and b and their corresponding derived categories. We show that there is an optimal subcategory in the domain of every functor given by N-complexes with N = a + b.
Paper II: In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of N-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this in general and present evidence that the axioms of triangulated categories have a simplicial origin.
Paper III: Let n be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of n:th cyclotomic integers.
Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen , 2015. , 20 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 90
Homological algebra, Category theory, Triangulated categories, K-theory, Hopfological algebra
Algebra and Logic
Research subject Mathematics
IdentifiersURN: urn:nbn:se:uu:diva-260111ISBN: 978-91-506-2476-2OAI: oai:DiVA.org:uu-260111DiVA: diva2:846374
2015-10-02, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Oppermann, Steffen, Professor
Mazorchuk, Volodymyr, Professor
List of papers