Philosophiae Doctor - PhD / In a category C with a proper (E; M)-factorization system for morphisms, we further investigate
categorical topogenous structures and demonstrate their prominent role played
in providing a uni ed approach to the theory of closure, interior and neighbourhood operators.
We then introduce and study an abstract notion of C asz ar's syntopogenous structure
which provides a convenient setting to investigate a quasi-uniformity on a category. We
demonstrate that a quasi-uniformity is a family of categorical closure operators. In particular,
it is shown that every idempotent closure operator is a base for a quasi-uniformity.
This leads us to prove that for any idempotent closure operator c (interior i) on C there
is at least a transitive quasi-uniformity U on C compatible with c (i). Various notions of
completeness of objects and precompactness with respect to the quasi-uniformity de ned
in a natural way are studied.
The great relationship between quasi-uniformities and closure operators in a category
inspires the investigation of categorical quasi-uniform structures induced by functors. We
introduce the continuity of a C-morphism with respect to two syntopogenous structures
(in particular with respect to two quasi-uniformities) and utilize it to investigate the quasiuniformities
induced by pointed and copointed endofunctors. Amongst other things, it
is shown that every quasi-uniformity on a re
ective subcategory of C can be lifted to a
coarsest quasi-uniformity on C for which every re
ection morphism is continuous. The
notion of continuity of functors between categories endowed with xed quasi-uniform
structures is also introduced and used to describe the quasi-uniform structures induced
by an M- bration and a functor having a right adjoint.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uwc/oai:etd.uwc.ac.za:11394/7081 |
| Date | January 2019 |
| Creators | Iragi, Minani |
| Contributors | Holgate, David B., Razafindrakoto, Ando |
| Publisher | University of the Western Cape |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Rights | University of the Western Cape |
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