A categbrial study of initiality in uniform topology

Abstract

This thesis consists of two chapters, of which the

first presents a categorial study of the concept of

initiality (also known as projective generation) and the

second gives applications in the.theory of uniform and

quasi-uniform spaces,

The' first three sections of chapter 1 expound basic

aspects of initiality, such as its relation to categorial

limits and to embeddings, the· latter being defined with

respect to a faithful functor to a bicategory. The notion

of a separated object with respect to such a functor is

defined. The main contribution of the chapter lies in the

fourth section, where, given two amnestic functors

L : A-+ B and M : B ~ f, the functors F : B ~ A

with MLF ~ M are studied. These functors were characterized

by Hu'fuek (i.967aJ. Our investigations branch away from

Hu~ek's by proce~ding to further specialization of the type

of functor F. We characterize the functors F for whi~h the

initiality construction depends only on a class of objects

of A. Among these F we study the right inverses of F in

(vi)

detail. The results are then applied in the sixth section

to find conditions for the existence of an adjoint right

inverse of L, in the seventh section of explore the

natural order structure of the class of all right inverses

of L, and in the eighth section to study monoref1e6tors in

B. The chapter ends with a brief investigation of the

lifting of epireflectors from B to A against a functor

L : A. ~ B,

Chapter 2 starts with a minimal self-contained intro~

duction to the categories of the quasi-uniform spaces (Qun),

the bitopological spaces (2 Top), and the quasi-uniformizable

bi topolog ica 1 spaces (Berg) : their class ica 1 analogues a re

We show that the notion of separated object,

abstractly introduced in chapter 1, corresponds in each of

these categories to the epireflective property obtained by

lifting the T

0

-epireflector from Top. We settle a numbe~

of questions concerning right inverses of forgetful functors

among the mentioned categories and among their subcategories

of ~eparated objects. We also "cha~acterize the epimorphisms,

extreme monomorphi~ms and equalizers in these categories.

Thus equipped, we study total boundedness and completeness

in Q.un, and compactness in 2 Top, by regarding these properties

as lifted from Un and Top, respectively; we thereby obtain

some new results and perspectives on quasi-uniform spaces.