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.
Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-260111 |
Date | January 2015 |
Creators | Mirmohades, Djalal |
Publisher | Uppsala universitet, Algebra och geometri, Uppsala : Matematiska institutionen |
Source Sets | DiVA Archive at Upsalla University |
Language | English |
Detected Language | English |
Type | Doctoral thesis, comprehensive summary, info:eu-repo/semantics/doctoralThesis, text |
Format | application/pdf |
Rights | info:eu-repo/semantics/openAccess |
Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 90 |
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