Masters Degrees (Mathematical Sciences)

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Browsing Masters Degrees (Mathematical Sciences) by Subject "Abelian categories"

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    On a unified categorical setting for homological diagram lemmas
    (Stellenbosch : Stellenbosch University, 2011-12) Michael Ifeanyi, Friday; Janelidze, Zurab; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division of Mathematics.
    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.
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    On extensivity of morphisms in general categories
    (Stellenbosch : Stellenbosch University, 2023-03) Theart, Emma; Hoefnagel, Michael; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH SUMMARY: The notion of an extensive category captures a fundamental property of the category of sets, namely, that coproducts are disjoint and universal. This property may be restricted in several ways, one of which is with respect to morphisms in a category. The resulting notion of “extensive morphism” is the central notion of this thesis. An object is then called “mono-extensive” if every monomorphism into it is extensive. We explore these notions in categories which are far from being extensive. The category Set of pointed sets, for instance, in not extensive (since it is pointed), but a morphism in Set is extensive if and only if it has trivial kernel. In the category of finitely generated abelian groups, we show that a group G is mono-extensive if and only if it is cyclic. This leads to an open question about the category of abelian groups: is an abelian group G mono-extensive if and only if it is locally cyclic? We establish various theoretical results, one of the main results being a characterisation of coextensive categories: a Barr-exact category with global support is coextensive if and only if its monomorphisms are coextensive.