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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 *d ^{N}* = 0 rather than

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.

1. Homologically optimal categories of sequences lead to *N*-complexes$(function(){PrimeFaces.cw("OverlayPanel","overlay846371",{id:"formSmash:j_idt553:0:j_idt557",widgetVar:"overlay846371",target:"formSmash:j_idt553:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Simplicial Structure on Complexes$(function(){PrimeFaces.cw("OverlayPanel","overlay846372",{id:"formSmash:j_idt553:1:j_idt557",widgetVar:"overlay846372",target:"formSmash:j_idt553:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Categorification of the ring of cyclotomic integers for products of two primes$(function(){PrimeFaces.cw("OverlayPanel","overlay846373",{id:"formSmash:j_idt553:2:j_idt557",widgetVar:"overlay846373",target:"formSmash:j_idt553:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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urn-nbn$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_j_idt1232",{id:"formSmash:j_idt1232",widgetVar:"widget_formSmash_j_idt1232",showEffect:"fade",hideEffect:"fade",showDelay:500,hideDelay:300,target:"formSmash:altmetricDiv"});});

CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1285",{id:"formSmash:lower:j_idt1285",widgetVar:"widget_formSmash_lower_j_idt1285",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1286_j_idt1288",{id:"formSmash:lower:j_idt1286:j_idt1288",widgetVar:"widget_formSmash_lower_j_idt1286_j_idt1288",target:"formSmash:lower:j_idt1286:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});