Browsing by Author "van Zyl, Phillippus Johannes III"
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- ItemTowards projective set theory(Stellenbosch : Stellenbosch University, 2017-12) van Zyl, Phillippus Johannes III; Janelidze, Zurab; Gray, James Richard Andrew; Rewitzky, Ingrid; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.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.