<p> This THESIS comprises the core of Chapter I and a self-contained excerpt from Chapter II of the author's work "Fibred Categories and the Theory of Structures". As such, it contains a recasting of "categorical algebra" on the (BOURBAKI) set-theoretic frame of GROTHENDIECK-SONNERuniverses, making use of the GROTHENDIECK structural definition of category from the beginning. The principle novelties of the presentation result from the exploitation of an intrinsic construction of the arrow category C^2 of a VL -category C. This construction gives rise to the adjunction of a (canonical) (VL-CAT)-category structure to the couple (C^2, C), for which the consequent category structure supplied the couple (CAT(T,C^2), CAT(T, C)) for each category T, is simply that of natural transformations of functors (which as such are nothing more than functors into the arrow category).</p> / Thesis / Doctor of Philosophy (PhD)

Absolutely pure modules act in ways similar to injective modules. Therefore, through-out this document we investigate many of these properties of absolutely pure modules which are modelled after those similar properties of injective modules. The results we develop can be broken into three categories: localizations, covers and derived functors. We form S1M, an S1R module, for any Rmodule M. We state and prove some known results about localizations. Using these known techniques and properties of localizations, we arrive at conditions on the ring R which make an absolutely pure S1Rmodule into an absolutely pure Rmodule. We then show that under certain conditions, if A is an absolutely pure Rmodule, then S1A will be an absolutely pure S1Rmodule. Also, we dene conditions on the ring R which guarantee that the class of absolutely pure modules will be covering. These include R being left coherent, which we show implies a number of other necessary properties. We also develop derived functors similar to Extn R (whose development uses injective modules). We call these functors Axtn R, prove they are well dened, and develop many of their properties. Then we dene natural maps between Axtn(M;N) and Extn(M;N) and discuss what conditions on M and N guarantee that these maps are isomorphisms.

<p>This thesis presents several module systems, in particular Mei and DMei, designed for mechanized mathematics systems. Mei is a λ-calculus style module system that supports higher-order functors in a natural way. The semantics of functor application is based on substitution. A novel coercion mechanism integrates a parameter passing mechanism based on theory interpretations with simple λ-calculus style higher-order functors. DMei extends Mei by supporting dependent functor types. Mei is the first module system that successfully supports both higher-order functors and a parameter passing mechanism based on theory interpretations.</p> / Thesis / Doctor of Philosophy (PhD)

vii, 34 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study the Bousfield localization functors known as [Special characters omitted], as described in [MahS]. In particular we would like to understand how they interact with suspension and how they stabilize. We prove that suitably connected [Special characters omitted]-acyclic spaces have suspensions which are built out of a particular type n space, which is an unstable analog of the fact that [Special characters omitted]-acyclic spectra are built out of a particular type n spectrum. This theorem follows Dror-Farjoun's proof in the case n = 1 with suitable alterations. We also show that [Special characters omitted] applied to a space stabilizes in a suitable way to [Special characters omitted] applied to the corresponding suspension spectrum. / Committee in charge: Hal Sadofsky, Chairperson, Mathematics; Arkady Berenstein, Member, Mathematics; Daniel Dugger, Member, Mathematics; Dev Sinha, Member, Mathematics; William Rossi, Outside Member, English

Thesis (PhD)--Macquarie University (Division of Information & Communication Sciences, Dept. of Mathematics), 2007. / Thesis by publication. Bibliography: p. 119-123.

Στην παρούσα εργασία παρουσιάζουμε βασικές έννοιες του αντικειμένου της Ομολογιακής Άλγεβρας,όπως αυτές των μακρών ακριβών ακολουθιών, τις επεκτάσεις των modules και τις ομάδες Ext. Στη συνέχεια παρουσιάζουμε τις επεκτάσεις των ομολογιακών και συνομολογιακών συναρτητών, τους παράγωγους συναρτητές, που προκύπτουν μέσω προβολικών και ενριπτικών επιλύσεων αντικειμένων Αβελιανών κατηγοριών. Τέλος χαρακτηρίζουμε τους παράγωγους συναρτητές μέσω της καθολικής τους ιδιότητας. / In this work we present the basic concepts of Homological Algebra, such as the long exact sequences, the extensions of modules and the Ext-groups.Further we present the extensions of the homology and cohomology functors, the derived functors that arise through projective and injective resolutions from objects of an Abelian category. Finally we characterize derived functors through their universal property.

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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.

This thesis has four chapters. After a brief introduction in Chapter 1, the $AJ$-conjecture is introduced in Chapter 2. The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If $K$ satisfies the $AJ$-conjecture, sufficient conditions on $K$ are given for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of Theorem 2.2.1 are satisfied. Chapter 3 is about quantum curves and their relation to the $AJ$ conjecture. The variables $l$ and $m$ of the $A$-polynomial are quantized to operators that act on holomorphic functions. Motivated by a heuristic definition of the Jones polynomial from quantum physics, an annihilator of the Chern-Simons section of the Chern-Simons line bundle is found. For torus knots, it is shown that the annihilator matches with that of the colored Jones polynomial. In Chapter 4, a tangle functor is defined using semicyclic representations of the quantum group $U_q(sl_2)$. The semicyclic representations are deformations of the standard representation used to define Kashaev's invariant for a knot $K$ in $S^3$. It is shown that at certain roots of unity the semicyclic tangle functor recovers Kashaev's invariant.

Thesis by publication. / Thesis (PhD)--Macquarie University (Division of Information & Communication Sciences, Dept. of Mathematics), 2007. / Bibliography: p. 119-123. / Introduction -- Mackey functors on compact closed categories -- Lax braidings and the lax centre -- On centres and lax centres for promonoidal catagories -- Pullback and finite coproduct preserving functors between categories of permutation representations -- Conclusion. / This thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally promonoidal categories ... The third contribution of this thesis is the study of functors between categories of permutation representations. / x,123 p. ill