In classical model theory, Stone spaces of complete syntactic types (roughly, comprehensive syntactic descriptions of potential elements of models) play an important role. In an abstract elementary class, syntactic types are replaced by Galois types (or orbital types) corresponding to orbits in a sufficiently large model under automorphisms fixing its submodels. Here I present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation, extending the classical picture. I 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.

First identification of AECs as special case of accessible categories (discovered independently by Rosický and Beke). Provides a dictionary for translating between model-theoretic and category-theoretic contexts.

Building on the topological picture developed in [1], I introduce a family of rank functions, derived from the Cantor-Bendixson rank on the spaces of Galois types, along with related notions of total transcendence, i.e. ordinal-valuedness of the aforementioned ranks. We focus, in particular, on abstract elementary classes satisfying the tameness property, where the connections between stability and total transcendence are most evident. Using ranks as a measure of complexity, I show that a type over a small model cannot have too many extensions to types over larger models, lest its complexity get too high. This means, in short, an upward transfer of stability: few types over small models implies few types over larger ones. A variety of other rank functions have since been used in this context by Boney and Vasey, among others.

In response to the "Global approach" preprint of Mariano, Villaveces, and Zambrano. While the aforementioned paper develops a category of abstract elementary classes and its properties (i.e. the existence of pullbacks) in the fundamentally syntactic language of institutions, we hew to the path of accessible categories. In particular, we consider several (2-)categories of accessible categories with added structure, including the category of abstract elementary classes, and show that most, if not all, are closed under a suitably 2-categorical notion of limit, i.e. the PIE-limits. We also examine cases in which the functors are not concrete, i.e. do not preserve underlying sets.

Further develops connections between accessible categories and abstract model theory, analyzing properties of a hierarchy of categories extending from the former to the latter. Includes a new, purely category-theoretic proof of Boney's theorem on tameness of AECs under a large cardinal assumption, reducing the problem to the accessibility of (powerful) images of accessible functors under the same large cardinal assumption, 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.

Metric abstract elementary classes (or mAECs) are much like AECs, but their objects are built over complete metric spaces rather than discrete sets, and their operations are required to be (uniformly) continuous. We show that 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 syntactic Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor (a tool for building minimal models with given linear orders as their spines) in case the category is large. For mAECs in particular, arguments refining those in [5] yield a proof that any categorical mAEC, i.e. any mAEC with just one model up to isomorphism of a particular infinite size, has few types over its smaller models.

Two of the most important additional assumptions on abstract classes are that they satisfy the joint embedding property (JEP), i.e. any two models embed into a third, and the amalgamation property (AP), i.e. for any pair of embeddings M₁←K→M₂, there is an N and embeddings M₁→N←M₂ such that the resulting square commutes. Baldwin and Boney have shown that any AEC satisfies AP and JEP under the assumption of a proper class of strongly compact cardinals. Here we generalize this result in two ways: with accessible categories in place of AECs, and almost strongly compact cardinals in place of strongly compact. More importantly, the argument, again by the result of Makkai/Paré used in [5], proceeds entirely through the visual manipulation of (very small) category-theoretic diagrams. Also contains a proof that mAECs satisfy a suitably metric, ε-ized version of tameness under a large cardinal assumption. This requires only a slight tweaking of that in the JSL paper.

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.

The internal size of an object M inside a given category is, roughly, the least infinite cardinal λ such that any morphism from M into the colimit of a λ⁺-directed system factors through one of the components of the system. The existence spectrum of a category is the class of cardinals λ such that the category has an object of internal size λ. We study the existence spectrum in the μ-AECs of [6], which are, up to equivalence of categories, the same as accessible categories with all morphisms monomorphisms. We show for example that, assuming instances of the singular cardinal hypothesis which follow from a large cardinal axiom, μ-AECs which admit intersections have objects of all sufficiently large internal sizes. We also investigate the relationship between internal sizes and cardinalities and analyze a series of examples, including one of Shelah---a certain class of sufficiently-closed constructible models of set theory---which show that the categoricity spectrum can behave very differently depending on whether we look at categoricity in cardinalities or in internal sizes.

The goal of this paper is to develop an axiomatization of (non)forking independence that is purely category-theoretic in nature, in the hopes (at least for me) of making this important notion available to a broader mathematical audience for whom it is likely entirely foreign, or has heretofore been buried in terrifying layers of model-theoretic technicality. We link the existence of a stable independence notion to the existence of effective unions in coregular locally presentable categories (including, e.g. Grothendieck abelian categories), and prove canonicity in accessible categories whose morphisms are mono---a still more general class. The latter result exploits the connection between such accessible categories and mu-AECs (see 6 above), which allows us to leverage extensive portions of the model-theoretic toolkit.

13. Approximations of superstability in concrete accessible categories. With Jiří Rosický. In preparation. (arXiv)

A very technical paper, and potentially the start of a much broader project to generalize the fundamental results in the classification theory of AECs to sufficiently nice accessible categories, or at least to clearly isolate when and where the particular axioms of AECs are truly necessary. In particular, we here generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms, dropping isomorphism-fullness from the usual list of AEC axioms. Among other things, we prove that if any category of this form is categorical in a successor cardinal, directed colimits of saturated objects are themselves saturated: put another way, surprisingly coarse colimits of injectives are themselves injective.

This paper continues an ongoing project examining the effects of large cardinal assumptions in categorical algebra and, in particular, seeking (relatively) concrete mathematical statements equivalent to the existence of various large cardinals. This paper is concerned, in particular, with the relationship between almost measurable cardinals, the closure of images of accessible functors under colimits of chains, and locality of Galois types in abstract classes of structures, ultimately providing an equivalence linking category theory, model theory, and large cardinals. Builds on joint work with Rosický, as well as work of Brooke-Taylor/Rosický and Boney/Unger.

16. Tameness, powerful images, and large cardinals. With Will Boney. Submitted. (arXiv)

17. Weak factorization systems and stable independence. With Jiří Rosický and Sebastien Vasey. Submitted. (arXiv)

18. Hilbert spaces and C*-algebras are not finitely concrete. With Jiří Rosický and Sebastien Vasey. In preparation. (arXiv)

We show that a variety of categories, including Hilbert spaces with linear isometries and commutative unital C*-algebras with *-homomorphisms, cannot be AECs---even allowing for changes of language and underlying sets---and are therefore not first-order axiomatizable.

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.