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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.

Doctor of Philosophy / Department of Mathematics / David Yetter / This dissertation begins the development of the deformation theorem of monoidal categories which accounts for the function that all arrow-valued operations, composition, the arrow part of the monoidal product, and structural natural transformation are deformed. The first chapter is review of algebra deformation theory. It includes the Hochschild complex of an algebra, Gerstenhaber's deformation theory of rings and algebras, Yetter's deformation theory of a monoidal category, Gerstenhaber and Schack's bialgebra deformation theory and Markl and Shnider's deformation theory for Drinfel'd algebras. The second chapter examines deformations of a small $k$-linear monoidal category. It examines deformations beginning with a naive computational approach to discover that as in Markl and Shnider's theory for Drinfel'd algebras, deformations of monoidal categories are governed by the cohomology of a multicomplex. The standard results concerning first order deformations are established. Obstructions are shown to be cocycles in the special case of strict monoidal categories when one of composition or tensor or the associator is left undeformed. At the end there is a brief conclusion with conjectures.

In this dissertation we generalise the basic theory of Hopf algebras to the context of autonomous pseudomonoids in monoidal bicategories. Autonomous pseudomonoids were introduced in [13] as generalisations of both autonomous monoidal categories and Hopf algebras. Much of the theory of autonomous pseudomonoids developed in [13] was inspired by the example of autonomous (pro)monoidal enriched categories. The present thesis aims to further develop the theory with results inspired by Hopf algebra theory instead. We study three important results in Hopf algebra theory: the so-called 'fundamental theorem of Hopf modules', the 'Drinfel'd quantum double' and its relation with the centre of monoidal categories, and 'Radford's formula'. The basic result of this work is a general fundamental theorem of Hopf modules that establishes conditions equivalent to the existence of a left dualization. With this result as a base, we are able to construct the centre (defined in [83]) and the lax centre of an autonomous pseudomonoid as an Eilenberg-Moore construction for certain monad. As an application we show that the Drinfel'd double of a finite-dimensional Hopf algebra is equivalent to the centre of the associated pseudomonoid. The next piece of theory we develop is a general Radford's formula for autonomous map pseudomonoids formula in the case of a (coquasi) Hopf algebra. We also introduce 'unimodular' autonomous pseudomonoids. In the last part of the dissertation we apply the general theory to enriched categories with a (chosen) class of (co)limits, with emphasis in the case of finite (co)limits. We construct tensor products of such categories by means of pseudo-commutative enriched monads (a slight generalisation of the pseudo-commutative 2-monads of [37], and showing that lax-idempotent 2-monads are pseudo-commutative. Finally we apply the general theory developed for pseudomonoids to deduce the main results of [27].

Extending endofunctors on the category of sets and functions to the category of sets and relations requires one to introduce a certain amount of laxness. This in turn requires us to consider bicategories rather than ordinary categories. The subject of lax extensions of Set-based functors is one of the fundamental components of monoidal topology, an active area of research in categorical algebra. The recent theory of linear bicategories, due to Cockett, Koslowski and Seely, is an extension of the usual notion of bicategory to include a second composition in a way analogous to the two connectives of multiplicative linear logic. It turns out that the category of sets and relations has a second composition making it a linear bicategory. The goal of this thesis is first to define the notion of lax extension of Set-based functors to linear bicategories, and then demonstrate crucial properties of our definition.

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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.

<p dir="ltr">We first introduce several families of monoidal categories with plethysm products as their monoidal products and use this to describe operadic structures as plethysm monoids. In order to link this approach with the classical theory, we give a generalization of the Baez-Dolan plus construction. We then show that an operadic structure can be defined as a plethysm monoid if its associated Feynman category is a plus construction of a unique factorization category.</p>

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In the 1980s and 1990s, Joyal and Street developed a graphical notation for various flavours of monoidal category using graphs drawn in the plane, commonly known as string diagrams. In particular, their work comprised a rigorous topological foundation of the notation. In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notions they called ⊸-schedules, ⊗-schedules and heaps. Schedules described interleavings of plays in games formed using ⊸ and ⊗, and heaps provided pointers used for backtracking. Their definitions were combinatorial in nature, but researchers often draw certain pictures when working in practice. In this thesis, we extend the framework of Joyal and Street to give a formal account of the graphical methods already informally employed by researchers in game semantics. We give a geometric formulation of ⊸-schedules and ⊗-schedules, and prove that the games they describe are isomorphic to those described in Harmer et al.’s terms, and also those given by a more general graphical representation of interleaving across games of multiple components. We further illustrate the value of the geometric methods by demonstrating that several proofs of key properties (such as that the composition of ⊸-schedules is associative) can be made straightforward, reflecting the geometry of the plane, and overstepping some of the cumbersome combinatorial detail of proofs in Harmer et al.’s terms. We further extend the framework of formal plane diagrams to account for the heaps and pointer structures used in the backtracking functors for O and P.