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archives@tulane.edu / In this thesis, I define and study the foundations of the new framework of graded category theory, which I propose as just one structure that fits under the general banner of what Andree Eheresman has called “dynamic category theory”. Two approaches to defining graded categories are developed and shown to be equivalent formulations by a novel variation on the Grothendieck construction. Various notions of graded categorical constructions are studied within this framework. In particular, the structure of graded categories in general is then further elucidated by studying so-called “variable-object” models, and a version of the Yoneda lemma for graded categories. As graded category theory was originally developed in order to better understand the intuitive notions of absolute and relative cardinality – these notions are applied to the problem of vindicating the Skolemite thesis that “all sets, from an absolute perspective, are countable”. Finally, I discuss some open problems in this framework, discuss some potential applications, and discuss some of the relationships of my approach to existing approaches in the literature. / 1 / Nathan bedell

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The Hochschild-Kostant-Rosenberg theorem, which relates the Hochschild homology of an algebra to its modules of differential $n$-forms, can be considered a benchmark for smoothness of an algebra. This notion is used here in the search for a conception of smoothness formulated in the context of codifferential categories, both commutative and noncommutative. Since it is desirable to adapt such a conception to the noncommutative context, the theory of this domain is developed considerably; a significant result in this direction establishes a connection between noncommutative codifferential categories and commutative ones. This investigation necessitates, then, both the formulation of a notion of smoothness for $T$-algebras in codifferential categories and an adaptation of the Hochschild-Kostant-Rosenberg theorem to a wide variety of contexts which includes noncommutative ones. The former consideration fosters both the notion of a smooth monad, and a formulation of Andr\'e-Quillen homology in codifferential categories; the latter engenders a highly adaptable version of the Hochschild-Kostant-Rosenberg theorem. Specifically, it is shown that for any algebra modality there exists a corresponding Hochschild-Kostant-Rosenberg theorem. This includes a version of the theorem for the free associative algebra monad, the conclusion of which is satisfied by noncommutative smooth algebras.

The main theme of this thesis is the parallel between results in topos theory and the theory of additive functor categories. In chapter 2, we provide a general overview of the topics used in the rest of the thesis. Locally finitely presentable categories are introduced, and their expression as essentially algebraic categories is explained. The theory of localization for toposes and abelian categories is introduced, and it is shown how these localizations correspond to theories in appropriate logics. In chapter 3, we look at conditions under which the category of modules for a ring object R in a topos E is locally finitely presented, or locally coherent. We show that if E is locally finitely presented, then the category of modules is also; however we show that far stronger conditions are required for the category of modules to be locally coherent. In chapter 4, we show that the Krull-Gabriel dimension of a locally coherent abelian category C is equal to the socle length of the lattice of regular localizations of C. This is used to make an analogous definition of Krull-Gabriel dimension for regular toposes, and the value of this dimension is calculated for the classifying topos of the theory of G-sets, where G is a cyclic group admitting no elements of square order. In chapter 5, we introduce a notion of strong flatness for algebraic categories (in the sense studied by Adamek, Rosickey and Vitale). We show that for a monoid M of finite geometric type, or more generally a small category C with the corresponding condition, the category of M-acts, or more generally the category of set-valued functors on C, has strongly flat covers.

To every small category or Grothendieck topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. In this thesis, we investigate this invariant in the particular context of quasi-equational theories, which are multi-sorted equational theories in which operations may be partially de fined. It is known that every such theory T has a classifying topos, which is a topos that classi fies all topos-theoretic models of the theory, and that this classifying topos is in fact equivalent to the covariant presheaf category Sets^fpTmod, with fpTmod being the category of all finitely presented, set-based models of T. We then investigate the isotropy group of this classifying topos of T, which will therefore be a presheaf of groups on fpTmod, and show that it encodes a notion of inner automorphism for the theory. The main technical result of this thesis is a syntactic characterization of the isotropy group of a quasi-equational theory, and we illustrate the usefulness of this characterization by applying it to various concrete examples of quasi-equational theories.

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Names and name-binding are useful concepts in the theory and practice of formal systems. In this thesis we study them in the context of dependent type theory. We propose a novel dependent type theory with primitives for the explicit handling of names. As the main application, we consider programming and reasoning with abstract syntax involving variable binders. Gabbay and Pitts have shown that Fraenkel Mostowski (FM) set theory models a notion of name using which informal work with names and binding can be made precise. They have given a number of useful constructions for working with names and binding, such as a syntax-independent notion of freshness that formalises when two objects do not share names, and a freshness quantifier that simplifies working with names and binding. Related to FM set theory, a number of systems for working with names have been given, among them are the first-order Nominal Logic, the higher-order logic FM-HOL, the Theory of Contexts as well as the programming language FreshML. In this thesis we study how dependent type theory can be extended with primitives for working with names and binding. Our choice of primitives is different from that in FM set theory. In FM set theory the fundamental primitive for working with names is swapping. All other concepts such as \alpha-equivalence classes and binding are constructed from it. For dependent type theory, however, this approach of constructing everything from swapping is not ideal, since it requires us to make strong assumptions on the type theory. For instance, the construction of \alpha-equivalence classes from swapping appears to require quotient types. Our approach is to treat constructions such as \alpha-equivalence classes and name-binding directly, turning them into primitives of the type theory. To do this, it is convenient to take freshness rather than swapping as the fundamental primitive. Based on the close correspondence between type theories and categories, we approach the design of the dependent type theory by studying the categorical structure underlying FM set theory. We start from a monoidal structure capturing freshness. By analogy with the definition of simple dependent sums \Sigma and products \Pi from the cartesian product, we define monoidal dependent sums \Sigma * and products \Pi * from the monoidal structure. For the type of names N, we have an isomorphism \Sigma *_N\iso\Pi *_N generalising the freshness quantifier. We show that this structure includes \alpha-equivalence classes, name binding, unique choice of fresh names as well as the freshness quantifier. In addition to the set theoretic model corresponding to FM set theory, we also give a realizability model of this structure. The semantic structure leads us to a bunched type theory having both a dependent additive context structure and a non-dependent multiplicative context structure. This type theory generalises the simply-typed \alpha\lambda-calculus of O'Hearn and Pym in the additive direction. It includes novel monoidal products \Pi * and sums \Sigma * as well as hidden-name types H for working with names and binding. We give examples for the use of the type theory for programming and reasoning with abstract syntax involving binders. We show that abstract syntax can be handled both in the style of FM set theory and in the style of Weak Higher Order Abstract Syntax. Moreover, these two styles of working with abstract syntax can be mixed, which has interesting applications such as the derivation of a term for the unique choice of new names.

The relational data model uses set theory to provide a formal background, thus ensuring a rigorous mathematical data model with support for manipulation. Newer generation database models are based on the object-oriented paradigm, and so fall short of having such a formal background, especially in some of the more complex data manipulation areas. We use category theory to provide a formalism for object databases, in particular the object-relational model. Our model is known as the Product Model. This thesis will describe our formal model for the key aspects of object databases. In particular, we will examine how the Product Model deals with three of the most important problems inherent in object databases, those of queries, closure and views. As well as this, we investigate the more common database concepts, such as keys, relationships and aggregation. We will illustrate the feasibility of this model, by producing a prototype implementation using PIFDM. PIFDM is a semantic data model database system based on the functional model of Shipman, with object-oriented extensions.

This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.