Browsing by Author "Theart, Emma"

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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.