No description available.

No description available.

Classical and quantum Chern-Simons with gauge group U(1)N were classified by Belov and Moore in [BM05]. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary (2 + 1)-dimensional TQFTs and modular tensor categories. We study group categories and extend them slightly to produce modular tensor categories that correspond to toral Chern-Simons. Group categories have been widely studied in other contexts in the literature [FK93],[Qui99],[JS93],[ENO05],[DGNO07]. The main result is a proof that the associated projective representation of the mapping class group is isomorphic to the one from toral Chern-Simons. We also remark on an algebraic theorem of Nikulin that is used in this paper. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

For a given abelian category o/, a category E is formed by considering exact sequences of o/. If one imposes the condition that a split sequence be regarded as the zero object, then the resulting sequence category E/S is shown to be abelian. The intrinsic algebraic structure of E/S is examined and related to the theory of coherent functors and functor rings. E/S is shown to be the natural setting for the study of pure and copure sequences and the theory is further developed by introducing repure sequences. The concept of pure semi-simple categories is examined in terms of E/S. Localization with respect to pure sequences is developed, leading to results concerning the existence of algebraically compact objects. The final topic is a study of the simple sequences and their relationship to almost split exact sequences. / Science, Faculty of / Mathematics, Department of / Graduate

This paper establishes a basis for abelian categories, then gives the statement and proof of two equivalent definitions of an abelian category, the development of the basic theory of such categories, and the proof of some theorems involving this basic theory.

No description available.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis. / AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaves [61]-62).

The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.