Universal theories categorical in power and kappa-generated models

Shelah's eventual categoricity conjecture in universal classes: part I

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…

Categoricity and Universal Classes

Let $(\mathcal{K} ,\subseteq )$ be a universal class with $LS(\mathcal{K})=λ$ categorical in regular $κ>λ^+$ with arbitrarily large models, and let $\mathcal{K}^*$ be the class of all $\mathcal{A}\in\mathcal{K}_{>λ}$ for which there is $\mathcal{B} \in \mathcal{K}_{\geκ}$ such that $\mathcal{A}\subseteq\mathcal{B}$. We prove that $\mathcal{K}^*$ is categorical in every $ξ>λ^+$, $\mathcal{K}_{\ge\beth_{(2^{λ^+})^+}} \subseteq \mathcal{K}^{*}$, and the models of $\mathcal{K}^*_{>λ^+}$ are essentially vector spaces (or trivial i.e. disintegrated).

Universal classes near $\aleph_1$

Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …

On categoricity in successive cardinals

We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal $\mathbb{L}_{ω_1, ω}$ sentence categorical on an end segment of cardinals below $\beth_ω$ must be categorical also everywhere above $\beth_ω$. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.

Universal homogeneous constraint structures and the hom-equivalence classes of weakly oligomorphic structures

We derive a new sufficient condition for the existence of ω-categorical universal structures in classes of relational structures with constraints, augmenting results by Cherlin, Shelah, Chi, and Hubička and Nešetřil. Using this result we show that the hom-equivalence class of any countable weakly oligomorphic structure has up to isomorphism a unique model-complete smallest and greatest element, both of which are ω-categorical. As the main tool we introduce the category of constraint structures, show the existence of universal homogeneous objects, and study their automorphism groups. All constr…

$(\infty,1)$-范畴论中的理解模式

$(\infty,1)$-Categorical Comprehension Schemes

本文在$(\infty,1)$-范畴论中定义并研究理解概念，通过单值元理论实现Bénabou基础，推广小性与相对可定义性，并利用理解模式刻画高阶范畴性质。

无蝶形图与泛型自同构

Bowtie-free graphs and generic automorphisms

本文证明了可数泛型omega范畴无蝶形图允许泛型自同构，并指出该图不可有限齐化。

A unified framework for notions of algebraic theory

Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this paper, we develop a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category,…