There is an ongoing massive effort by many researchers to link category theory and geometry, especially homotopy coherence and categorical coherence. This constitutes just a part of the broad undertaking known as categorification as described by Baez and Dolan. This effort has as a partial goal that of understanding the categories and functors that correspond to loop spaces and their associated topological functors. Progress towards this goal has been advanced greatly by the recent work of Balteanu, Fiedorowicz, Schwänzl, and Vogt who show a direct correspondence between k–fold monoidal categories and k–fold loop spaces through the categorical nerve. This thesis pursues the hints of a categorical delooping that are suggested when enrichment is iterated. At each stage of successive enrichments, the number of monoidal products seems to decrease and the categorical dimension to increase, both by one. This is mirrored by topology. When we consider the loop space of a topological space, we see that paths (or 1–cells) in the original are now points (or objects) in the derived space. There is also automatically a product structure on the points in the derived space, where multiplication is given by concatenation of loops. Delooping is the inverse functor here, and thus involves shifting objects to the status of 1–cells and decreasing the number of ways to multiply. Enriching over the category of categories enriched over a monoidal category is defined, for the case of symmetric categories, in the paper on A∞–categories by Lyubashenko. It seems that it is a good idea to generalize his definition first to the case of an iterated monoidal base category and then to define V–(n + 1)–categories as categories enriched over V–n–Cat, the (k−n)–fold monoidal strict (n+1)–category of V–n–categories where k<n ∈ N. We show that for V k–fold monoidal the structure of a (k−n)–fold monoidal strict (n + 1)–category is possessed by V–n–Cat. / Ph. D.

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We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object, so-called near-group categories. Data describing the fusion rule is reduced to an abelian group G and a nonnegative integer k. Conditions are given, in terms of G and k, for the existence or nonexistence of coherent associative structures for such fusion rules (ie, solutions to MacLane's pentagon equation). An explicit construction of matrix solutions to the pentagon equations is given for the cases where we establish existence, and classification of the distinct solutions is carried out partially. Many of these associative structures also support (braided) commutative and tortile structures and we indicate when the additional structures are possible. Small examples are presented in detail suitable for use in computational applications. / Ph. D.

This thesis is divided into two parts. The rst part studies homological algebra of quasihereditary algebras, with the underlying theme being to understand properties of the Yoneda algebra of standard modules. We will rst show how homological properties of a quasi-hereditary algebra are carried over to its tensor products and wreath products. We then determine the extgroups between indecomposable standard modules of a Cubist algebra of Chuang and Turner. We will also determine generators, hence the quiver, of the Yoneda algebra of standard modules for the rhombal algebras of Peach. We also obtain a higher multiplication vanishing condition for certain rhombal algebras. The second part of this thesis studies the notion of simple-minded systems, introduced by Koenig and Liu. Such systems were designed to generate the stable module categories of artinian algebras by extension, in the same way as the sets of simple modules. We classify simple-minded systems for representation- nite self-injective algebras, and establish connections of them to various notions in combinatorics and related derived categories. We also look at the notion of simple-minded systems de ned on triangulated categories, and obtain some classi cation results using a connection between the simple-minded systems of a triangulated category and of its orbit category.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: While the notions of open and closed subsets in a topological space are dual to each other, they take on another meaning when points and complements are no longer available. Closure operators have been extensively used to study topological notions on categories. Though this has recovered a fair amount of topological results and has brought an economy of e ort and insight into Topology, it is thought that certain properties, such as convergence, are naturally associated with neighbourhoods. On the other hand, it is interesting enough to investigate certain notions, such as that of closed maps, which in turn are naturally associated with closure by means of neighbourhoods. We propose in this thesis a set of axioms for neighbourhoods and test them with the properties of connectedness and compactness. / AFRIKAANSE OPSOMMING: Al is die twee konsepte van oop en geslote subversamelings in 'n topologiese ruimte teenoorgesteldes van mekaar, verander hul betekenis wanneer punte en komplemente nie meer ter sprake is nie. Die gebruik van afsluitingsoperatore is alreeds omvattend in die studie van topologiese konsepte in kategorieë, toegepas. Alhoewel 'n redelike aantal topologiese resultate, groeiende belangstelling en groter insig tot Topologie die gevolg was, word daar geglo dat seker eienskappe, soos konvergensie, op 'n natuurlike wyse aan omgewings verwant is. Nietemin is dit van belang om sekere eienskappe, soos geslote afbeeldings, wat natuurlik verwant is aan afsluiting, te bestudeer. In hierdie proefskrif stel ons 'n aantal aksiomas oor omgewings voor en toets dit gevolglik met die eienskappe van samehangendheid en kompaktheid.

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Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaf 49). Also available on the World Wide Web.

Thesis--University of Florida. / Manuscript copy. Vita. Description based on print version record. Bibliography: leaf 67.

Thesis--University of Florida. / Description based on print version record. Manuscript copy. Vita. Bibliography: leaf 60.