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Towards projective set theory
Thesis (MSc)--Stellenbosch University, 2017
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2017
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| _version_ | 1867614054352158720 |
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| access_status_str | Open Access |
| author | Van Zyl, Phillippus Johannes III |
| author2 | Janelidze, Zurab |
| author_browse | Janelidze, Zurab Van Zyl, Phillippus Johannes III |
| author_facet | Janelidze, Zurab Van Zyl, Phillippus Johannes III |
| author_sort | Van Zyl, Phillippus Johannes III |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2017 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/102962 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:45:56.159Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2017 |
| publishDateRange | 2017 |
| publishDateSort | 2017 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/102962 Towards projective set theory Van Zyl, Phillippus Johannes III Janelidze, Zurab Gray, James Richard Andrew Rewitzky, Ingrid Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Axioms Category of sets Projective set theory Projective group theory Galois connection Zero morphism Homomorphism theorems UCTD Thesis (MSc)--Stellenbosch University, 2017 ENGLISH ABSTRACT : In this thesis an axiomatic framework is presented which extends the projective group theory introduced by Z Janelidze to also hold for sets. The isomorphism theorems are reformulated so that they hold for sets. Interestingly, the theorems do not hold for a number of null cases, which in this sense makes it a point-free approach to set theory—that is, singletons cannot be selected as abstract images of morphisms, but they can be studied by factorisation properties. In particular, this aspect is explained in the last chapter, where a comparison is drawn between the isomorphism theorems here and those for regular categories presented in Tholen’s doctoral thesis. The proofs are done by means of chasing elements of ΣX, here called A-subobjects, forwards and backwards, where ΣX is the fibre at an object X in C for which the functor G : C −→ Gal is the central object of study in the axiomatic setting; moreover, the axioms are functorially self-dual for this functor. A minor result on bounded morphisms is included: when a bounded morphism is the left adjoint of a Galois connection with meets and joins it is equivalent to the Frobenius property for Galois connections. AFRIKAANSE OPSOMMING : In hierdie tesis word ’n aksiomatiese raamwerk ontwikkel en uiteengesit om die projektiewe groepsleer van Z Janelidze uit te brei om ook versamelings in the sluit. Die isomorfismestellings word hergeformuleer sodat dit ook vir versamelings geldig is. Hierdie proses het die interessante gevolg dat die stellings vir ’n klas van nul gevalle nie geldig is nie en in daardie sin kan mens hiérdie benadering sien as ’n puntvrye versamelingsleer. ’n Enkele punt kan nie vasgevang word as die abstrakte beeld van ’n morfisme nie, maar kan wel bestudeer word op grond van faktoriseringseienskappe. In besonder word hierdie aspek verduidelik in die laaste hoofstuk, waar ’n vergelyking getref word met die isomorfismestellings vir reëlmatige kategorieë voorgesit in Tholen se doktorale tesis. Die bewyse van die stellings word by wyse van elemente van ΣX, hier genoem A-subvoorwerpe, vorentoe en agtertoe aan te volg, waar ΣX die beeld van ’n voorwerp X in ’n kategorie C is vir die funktor G : C −→ Gal, ’n sentrale struktuur waarvoor die aksiomas funktoriaal self-duaal is. ’n Kort resultaat vir begrensde morfismes word ingesluit wat sê dat wanneer ’n begrensde morfisme ’n linker adjunk van ’n Galois konneksie met infima en suprema is, is dit ekwivalent aan die Frobenius eienskap vir Galois konneksies. 2017-11-27T12:55:59Z 2017-12-11T11:17:49Z 2017-11-27T12:55:59Z 2017-12-11T11:17:49Z 2017-12 Thesis http://hdl.handle.net/10019.1/102962 en_ZA Stellenbosch University viii, 99 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Axioms Category of sets Projective set theory Projective group theory Galois connection Zero morphism Homomorphism theorems UCTD Van Zyl, Phillippus Johannes III Towards projective set theory |
| title | Towards projective set theory |
| title_full | Towards projective set theory |
| title_fullStr | Towards projective set theory |
| title_full_unstemmed | Towards projective set theory |
| title_short | Towards projective set theory |
| title_sort | towards projective set theory |
| topic | Axioms Category of sets Projective set theory Projective group theory Galois connection Zero morphism Homomorphism theorems UCTD |
| url | http://hdl.handle.net/10019.1/102962 |
| work_keys_str_mv | AT vanzylphillippusjohannesiii towardsprojectivesettheory |