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Thesis--M.A., University of Hong Kong. / Typewritten.

published_or_final_version / Mathematics / Master / Master of Arts

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To complete a category is to embed it into a larger one which is closed under a given type of limits (or colimits). A fairly classical construction scheme, in which the objects of the new category are defined as "formal limits" of diagrams of the given one, is carried out in two specific examples. / In the case of the exact completion (')C of a finitely complete category C, we look at "formal coequalizers" of equivalence spans of C, which end up being the quotients (in (')C) of the corresponding equivalence relation. The objects of (')C are hence objects of C together with an equivalence span, and its morphisms are equivalence classes of "compatible" morphisms of C. Thus constructed, (')C is an exact category whose structure is studied in detail in Chapter I, as well as the universal property it satisfies. / The second example is the category pro-C of pro-objects, or "formal cofiltered limits" of C. Its objects are cofiltered diagrams of C, but can be characterized in various ways. The known properties of pro-C are summarized, extended and used to lift to pro-C some regularity and exactness properties of C (Chapters II & III). / The category of finite sets is an interesting example: its category of pro-objects is equivalent to the category of Stone spaces which is not exact, but whose exact completion in the sense of {9} is the category of compact Hausdorff spaces.

The idea of an operational category over A generalizes the notions of tripleable and equational category over A, and also the dual notions of cotripleable and coequational category. An operational category, U:D (--->) A is given by a presentation ((theta),H) (UNFORMATTED TABLE FOLLOWS) / D C('T) / C('(theta)) / A C('B) / H*(TABLE ENDS) / where (theta) is a functor bijective on objects and D is a specified pullback. R:Op(A) (--->) Cat/A is defined as the category of operational categories (and functors) with given presentations. Another category, Op(,o)(A) over Cat/A of operational categories with standard presentations is also defined. There is a fixed theory (theta)(,o), employed in every standard presentation. Op(,o)(A) is a retract of Op(A) over Cat/A: (UNFORMATTED TABLE FOLLOWS) / i / Op(,o)(A) Op(A) / s / R(,o) R / Cat/A(TABLE ENDS) / i.e. every operational category (and functor) has a standard presentation (but not s(REVTURNST)i!). Also R(,o) has a left adjoint L(,o) and Op(,o)(A) is complete. Finally, there is a category of algebras, S(,*)-Alg over Cat/A such that Op(,o)(A) (DBLTURN) S(,*)-Alg over Cat/A. Thus, the operational categories can be determined by their internal structure, without reference to any presentation. Some properties of operational categories and some special cases are also examined.

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