The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) (...) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing. (shrink)

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP $_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q) (...) Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP $_1$ theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP $_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP $_1$ and NTP $_2$ theories. (shrink)

Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $ -closed logic ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega $ is $\omega $ -relatively compact iff some $D\in \Omega $ fails to be $\omega _1$ -complete iff ${\mathcal {L}}_\Omega $ does not (...) contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$, if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $, then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact. (shrink)

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this article do not yield (...) new classes of theories. (shrink)

A first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega $ -categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and finite index (...) first-order reducts. Using our classification we show that every structure in $\mathcal {M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal {M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas’ conjecture for the class $\mathcal {M}$. (shrink)

Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.

In this paper, we study a stability transfer theorem in d-tame metric abstract elementary classes, in a similar way as in 2, but using superstability-like assumptions which involves a new independence notion instead of ℵ0-locality.

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable (...) and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický. (shrink)