On extensivity of morphisms in general categories

Simple item page
dc.contributor.advisorHoefnagel, Michaelen_ZA
dc.contributor.authorTheart, Emmaen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2023-03-06T17:27:45Z
dc.date.accessioned2023-05-18T07:15:08Z
dc.date.available2023-03-06T17:27:45Z
dc.date.available2023-05-18T07:15:08Z
dc.date.issued2023-03
dc.descriptionThesis (MSc)--Stellenbosch University, 2023.en_ZA
dc.description.abstractENGLISH 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.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: Die begrip van ’n ekstensiewe kategorie beskryf ’n fundamentele eienskap van die kategorie van versamelings, naamlik dat dit disjunk en universeel is. Hierdie eienskap kan op verskeie maniere beperk word, waarvan een ten opsigte van morfismes in ’n kategorie is. Die gevolglike begrip van ‘n “ekstensiewe morfisme” is die sentrale idee van hierdie tesis. ’n Objek word dan “monoekstensief” genoem indien elke monomorfisme in die objek in ekstensief is. Ons ondersoek hierdie konsepte in kategoriee wat nie noodwendig ekstensief is nie. Die kategorie Set van gepunte versamelings, byvoorbeeld, is nie ekstensief nie (aangesien dit ’n nulobjek het), maar ’n morfisme in Set is ekstensief as en slegs as sy kern triviaal is. In die kategorie van eindig gegenereerde abelse groepe wys ons dat ’n groep G mono-ekstensief is as en slegs as dit siklies is. Dit lei tot ’n ope vraag met betrekking tot die kategorie van abelse groepe: is ’n abelse groep G mono-ekstensief as en slegs as dit lokaal siklies is? Ons bevestig verskeie teoretiese resultate, met een van die hoof resultate ’n karakterisering van ko-ekstensiewe kategoriee: ’n Barr-presiese kategorie met globale ondersteuning is ko-ekstensief as en slegs as al sy monomorfismes ko-ekstensief is.af_ZA
dc.description.versionMasters
dc.format.extentv, 86 pages
dc.identifier.urihttp://hdl.handle.net/10019.1/127305
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch University
dc.rights.holderStellenbosch University
dc.subject.lcshMorphisms (Mathematics)en_ZA
dc.subject.lcshCategories (Mathematics)en_ZA
dc.subject.lcshAbelian categoriesen_ZA
dc.subject.nameUCTD
dc.titleOn extensivity of morphisms in general categoriesen_ZA
dc.typeThesisen_ZA
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
theart_extensivity_2023.pdf
Size:
1.41 MB
Format:
Adobe Portable Document Format
Description:
Download
Collections
Masters Degrees (Mathematical Sciences)