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N-complexes and Categorification
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0002-3197-6933
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 90
Keyword [en]
Homological algebra, Category theory, Triangulated categories, K-theory, Hopfological algebra
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-260111ISBN: 978-91-506-2476-2 (print)OAI: oai:DiVA.org:uu-260111DiVA: diva2:846374
Public defence
2015-10-02, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2015-09-11 Created: 2015-08-17 Last updated: 2015-09-11
List of papers
1. Homologically optimal categories of sequences lead to N-complexes
Open this publication in new window or tab >>Homologically optimal categories of sequences lead to N-complexes
(English)Manuscript (preprint) (Other academic)
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-260108 (URN)
Available from: 2015-08-17 Created: 2015-08-17 Last updated: 2015-08-17
2. Simplicial Structure on Complexes
Open this publication in new window or tab >>Simplicial Structure on Complexes
(English)Manuscript (preprint) (Other academic)
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-260109 (URN)
Available from: 2015-08-17 Created: 2015-08-17 Last updated: 2015-08-17
3. Categorification of the ring of cyclotomic integers for products of two primes
Open this publication in new window or tab >>Categorification of the ring of cyclotomic integers for products of two primes
(English)Manuscript (preprint) (Other academic)
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-260110 (URN)
Available from: 2015-08-17 Created: 2015-08-17 Last updated: 2015-08-17

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Mirmohades, Djalal

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