Traditional recursion theory is the study of computable functions on the natural numbers. This thesis considers recursion theory in first-order arithmetical theories and categories, thus expanding the work of Ritchie and Young, Lambek, Scott, and Hofstra. We give a complete characterisation of the representability of computable functions in arithmetical theories, paying attention to the differences between intuitionistic and classical theories and between theories with and without induction. When considering recursion theory from a category-theoretic perspective, we examine syntactic categories of arithmetical theories. In this setting, we construct a strong parameterised natural numbers object and give necessary and sufficient conditions to construct a Turing category associated to an intuitionistic arithmetical theory with induction.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22980号 / 理博第4657号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 長谷川 真人, 准教授 照井 一成, 准教授 河村 彰星 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer categeory. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras gl_m|n(A) with entries in A. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford superalgebra, respectively. We include background on monoidal supercategories and Frobenius superalgebras and discuss some possible further directions.

No description available.

Given a symmetric Frobenius superalgebra A equipped with a compatible involution, we define the associated Frobenius Brauer category B(A) and affine Frobenius Brauer category AB(A), generalizing the plain Brauer category B and affine Brauer category AB. We define the orthosymplectic Lie superalgebra osp m|2n(A) and a functor from B(A) to osp m|2n(A)-mod, the category of supermodules over osp m|2n(A). We also define a functor from AB(A) to the endofunctor supercategory of osp m|2n(A)-mod.We prove that these two functors are well-defined and use the former functor to prove a basis result for B(A, δ), a specialized version of B(A). Prior to defining these categories and functors, we provide the background information on super-mathematics and Frobenius superalgebras needed to understand the new results.

<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>

Understanding the action of an autoequivalence on a triangulated category is generally a very difficult problem. If one can find a stability condition for which the autoequivalence is "compatible", one can explicitly write down the action of this autoequivalence. In turn, the now understood autoequivalence can provide ways of extracting geometric information from the stability condition. In this thesis, we elaborate on what it means for an autoequivalence and stability condition to be "compatibile" and derive a sufficiency criterion. / text

In thinking about quantum causality one would like to approach rigorous QFT from outside the perspective of QFT, which one expects to recover only in a specific physical domain of quantum gravity. This thesis considers issues in causality using Category Theory, and their application to field theoretic observables. It appears that an abstract categorical Machian principle of duality for a ribbon graph calculus has the potential to incorporate the recent calculation of particle rest masses by Brannen, as well as the Bilson-Thompson characterisation of the particles of the Standard Model. This thesis shows how Veneziano n point functions may be recovered in such a framework, using cohomological techniques inspired by twistor theory and recent MHV techniques. This distinct approach fits into a rich framework of higher operads, leaving room for a generalisation to other physical amplitudes. The utility of operads raises the question of a categorical description for the underlying physical logic. We need to consider quantum analogues of a topos. Grothendieck's concept of a topos is a genuine extension of the notion of a space that incorporates a logic internal to itself. Conventional quantum logic has yet to be put into a form of equal utility, although its logic has been formulated in category theoretic terms. Axioms for a quantum topos are given in this thesis, in terms of braided monoidal categories. The associated logic is analysed and, in particular, elements of linear vector space logic are shown to be recovered. The usefulness of doing so for ordinary quantum computation was made apparent recently by Coecke et al. Vector spaces underly every notion of algebra, and a new perspective on it is therefore useful. The concept of state vector is also readdressed in the language of tricategories.

No description available.

A fundamental component of theoretical computer science is the application of logic. Logic provides the formalisms by which we can model and reason about computational questions, and novel computational features provide new directions for the development of logic. From this perspective, the unusual features of quantum computation present both challenges and opportunities for computer science. Our existing logical techniques must be extended and adapted to appropriately model quantum phenomena, stimulating many new theoretical developments. At the same time, tools developed with quantum applications in mind often prove effective in other areas of logic and computer science. In this thesis we explore logical aspects of this fruitful source of ideas, with category theory as our unifying framework. Inspired by the success of diagrammatic techniques in quantum foundations, we begin by demonstrating the effectiveness of string diagrams for practical calculations in category theory. We proceed by example, developing graphical formulations of the definitions and proofs of many topics in elementary category theory, such as adjunctions, monads, distributive laws, representable functors and limits and colimits. We contend that these tools are particularly suitable for calculations in the field of coalgebra, and continue to demonstrate the use of string diagrams in the remainder of the thesis. Our coalgebraic studies commence in chapter 3, in which we present an elementary formulation of a representation result for the unitary transformations, following work developed in a fibrational setting in [Abramsky, 2010]. That paper raises the question of what a suitable "fibred coalgebraic logic" would be. This question is the starting point for our work in chapter 5, in which we introduce a parameterized, duality based frame- work for coalgebraic logic. We show sufficient conditions under which dual adjunctions and equivalences can be lifted to fibrations of (co)algebras. We also prove that the semantics of these logics satisfy certain "institution conditions" providing harmony between syntactic and semantic transformations. We conclude by studying the impact of parameterization on another logical aspect of coalgebras, in which certain fibrations of predicates can be seen as generalized invariants. Our focus is on the lifting of coalgebra structure along a fibration from the base category to an associated total category of predicates. We show that given a suitable parameterized generalization of the usual liftings of signature functors, this induces a "fibration of fibrations" capturing the relationship between the two different axes of variation.