We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle.

We highlight connections between accessible categories and abstract elementary classes (AECs), and provide a dictionary for translating properties and results between the two contexts. We also illustrate a few applications of purely category-theoretic methods to the study of AECs, with model-theoretically novel results. In particular, the category-theoretic approach yields two surprising consequences: a structure theorem for categorical AECs, and a partial stability spectrum for weakly tame AECs.

We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition known as tameness, where the connections between stability and total transcendence are most evident. As a byproduct, we obtain a partial upward stability transfer result for tame abstract elementary classes stable in a cardinal λ with λ to the ℵ₀ larger than λ.

We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of "limit type" (specifically, PIE limits) which generalizes the approach of Mariano, Villaveces and Zambrano away from the syntactically oriented framework of institutions. Moreover, we provide a broader view of this closure phenomenon, considering a variety of categories of accessible categories with additional structure, and relaxing the assumption that the morphisms are concrete functors.

We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption, reducing the question to the accessibility of the powerful image of a natural functor and invoking an old result of Makkai and Paré. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

We introduce μ-Abstract Elementary Classes (μ-AECs) as a broad framework for model theory that includes complete boolean algebras and metric spaces, and begin to develop their classification theory. Moreover, we note that μ-AECs correspond precisely to accessible categories in which all morphisms are monomorphisms, and begin the process of reconciling these divergent perspectives: for example, the preliminary classification-theoretic results for μ-AECs transfer directly to accessible categories with monomorphisms.

We show that metric abstract elementary classes (mAECs) are, in the sense of [5], coherent accessible categories with directed colimits, with concrete ℵ₁-directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC, an AEC-like category in which only the κ-directed colimits need be concrete, and begin to develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [5] yield a proof that any categorical mAEC is μ-d-stable in many cardinals below the categoricity cardinal.

We present several new model-theoretic applications of the fact, due to Makkai and Paré, that under the assumption that there exists a proper class of almost strongly compact cardinals, the powerful image of any accessible functor is accessible. In particular, we generalize to the context of accessible categories the recent Hanf number computations of Baldwin and Boney, namely that in an abstract elementary class (AEC) if the joint embedding and amalgamation properties hold for models of size up to a sufficiently large cardinal, then they hold for models of arbitrary size. Moreover, we prove that, under the above-mentioned large cardinal assumption, every metric AEC is strongly d-tame, strengthening a result of Boney and Zambrano and pointing the way to further generalizations. In each case, the argument reduces to simple, graphical computations in suitable diagram categories.

We exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.

Working in the context of μ-abstract elementary classes (μ-AECs)---or, equivalently, accessible categories with all morphisms monomorphisms---we examine two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary results on the existence and categoricity spectra of μ-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in μ-AECs.

Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in μ-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible---and useful---to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a category and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory.

An accessible category is, roughly, a category with all sufficiently directed colimits, in which every object can be resolved as a directed system of "small" subobjects. Such categories admit a purely category-theoretic replacement for cardinality: the internal size. Generalizing results and methods from arXiv:1708.06782, we examine set-theoretic problems related to internal sizes and prove several Löwenheim-Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large accessible category has an object in all internal sizes of high-enough cofinality. We also introduce the notion of a filtrable accessible category---one in which any object can be represented as the colimit of a chain of strictly smaller objects---and examine the conditions under which an accessible category is filtrable.

We generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms. In particular, we prove that if any category of this form is categorical in a successor, directed colimits of saturated objects are themselves saturated.

Through careful analysis of an argument of Brooke-Taylor and Rosický, we show that the powerful image of any accessible functor is closed under colimits of κ-chains, κ a sufficiently large almost measurable cardinal. This condition on powerful images, by methods resembling those of Lieberman and Rosický, implies κ-locality of Galois types. As this, in turn, implies sufficient measurability of κ, via a paper of Boney and Unger, we obtain an equivalence: a purely category-theoretic characterization of almost measurable cardinals.

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of Boney/Unger, Brooke-Taylor/Rosický, Lieberman, and Lieberman/Rosický.

We exhibit a bridge between the theory of weak factorization systems, a categorical concept used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the cofibrantly generated weak factorization systems (those that are generated by a set, rather than a class, of morphisms) are exactly those that give rise to stable independence notions. This two-way connection yields a powerful new tool to build tame and stable abstract elementary classes. In particular, we generalize a construction of Baldwin/Eklof/Trlifaj to prove that the category of flat modules with flat monomorphisms has a stable independence notion, and explain how this connects to the fact that every module has a flat cover.

We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital C*-algebras with *-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.

We consider the behavior of Galois types in abstract elementary classes (AECs), and introduce several new techniques for use in the analysis of the associated stability spectra. More broadly, we develop novel perspectives on AECs---topological and category-theoretic---from which these techniques flow, and which hold considerable promise as lines of future investigation. After a presentation of the preliminaries in Chapter 2, we give a method of topologizing sets of Galois types over structures in AECs with amalgamation. The resulting spaces---analogues of the Stone spaces of syntactic types---support, among other things, natural correspondences between their topological properties and semantic properties of the AEC (tameness, for example, emerges as a separation principle). In Chapter 4, we note that the newfound topological structure yields a family of Morley-like ranks, along with a new notion of total transcendence. We show that in tame AECs, total transcendence follows from stability in certain cardinals, and that total transcendence, in turn, allows us to bound the number of types over large models. This leads to several upward stability transfer results, one of which generalizes a result of Baldwin, Kueker and VanDieren. The same analysis works in weakly tame AECs provided that they are also weakly stable, a notion that arises in the context of accessible categories. In Chapter 5, we analyze the category-theoretic structure of AECs, and give an axiomatization of AECs as accessible subcategories of their ambient categories of structures. We also give a dictionary for translating notions from the theory of accessible categories into the language of AECs, and vice versa. Weak stability occurs in any accessible category---hence in any AEC---and, since this is what we require to conclude stability in weakly tame AECs, we get the beginnings of a stability spectrum in this context. We close with a curious result: an equivalence between the class of large structures in a categorical AEC and a category of sets with actions of the monoid of endomorphisms of the categoricity structure, effectively reducing the AEC to a simple concrete category.

We enquire into the categorical semantics of the simply typed λ-calculus, establishing the strong completeness of fibrational poset semantics and proving, moreover, that every λ-theory has a representation of this form. The program of this thesis falls naturally into four parts. In Chapter 1, the syntax of the simply-typed λ-calculus is introduced by way of the type-free calculus, which is then shown to be a special case. Chapter 2 discusses the notions of semantic completeness that will be employed throughout, and presents a first pass at a workable system of semantics, first in the category of cartesian closed categories, then in the presheaves on such categories. In Chapter 3, the recent sheaf representation theorem due to Steven Awodey is introduced, along with a sketch of its proof. Categories of fibrations over partially ordered sets are examined in Chapter 4, and it is proven that the class of models in such categories constitutes strongly complete semantics for the simply-typed λ-calculus.