Towards projective set theory

[摘要] ENGLISH ABSTRACT : In this thesis an axiomatic framework is presented which extends the projectivegroup theory introduced by Z Janelidze to also hold for sets. The isomorphismtheorems are reformulated so that they hold for sets. Interestingly, the theoremsdo not hold for a number of null cases, which in this sense makes it a point-freeapproach to set theory-that is, singletons cannot be selected as abstract images ofmorphisms, but they can be studied by factorisation properties. In particular, thisaspect is explained in the last chapter, where a comparison is drawn between theisomorphism theorems here and those for regular categories presented in Tholen'sdoctoral thesis. The proofs are done by means of chasing elements of ΣX, herecalled A-subobjects, forwards and backwards, where ΣX is the fibre at an objectX in C for which the functor G : C −→ Gal is the central object of study in theaxiomatic setting; moreover, the axioms are functorially self-dual for this functor.A minor result on bounded morphisms is included: when a bounded morphism isthe left adjoint of a Galois connection with meets and joins it is equivalent to theFrobenius property for Galois connections.

[发布机构] Stellenbosch University