Classification theory and the number of non-isomorphic models

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Abstract And Concrete Categories The Joy Of Cats

Category-theoretic aspects of abstract elementary classes

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois (orbital) type of length less than a fixed cardinal $$\kappa $$?. We show:

Theorem 0.1 (The semantic---syntactic correspondence) An AEC K is fully $$({<}\kappa )$$(<?)-tame and type short if and only if Galois types are syntactic in the Galois Morleyization.

This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are:

Theorem 0.2 Let K be a $$\text {LS}(K)$$LS(K)-tame AEC with amalgamation. The following are equivalent:

(1)K is Galois stable in some $$\lambda \ge \text {LS}(K)$$??LS(K).

(2)K does not have the order property (defined in terms of Galois types).

(3)There exist cardinals $$\mu $$μ and $$\lambda _0$$?0 with $$\mu \le \lambda _0 < \beth _{(2^{\text {LS}(K)})^+}$$μ≤?0<?(2LS(K))+ such that K is Galois stable in any $$\lambda \ge \lambda _0$$???0 with $$\lambda = \lambda ^{<\mu }$$?=?<μ.

Theorem 0.3 Let K be a fully $$({ \text {LS}(K)$$?=??>LS(K). If K is Galois stable, then the class of $$\kappa $$?-Galois saturated models of K admits an independence notion ($$({<}\kappa )$$(<?)-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.

Infinitary stability theory

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an order-property, we consider {\bf Definition.} Let $M_0 \prec N$ be models from $K$ and $A$ be a set. We say that the Galois-type of $A$ over $M$ \emph{does not fork over $M_0$} iff for all small $a \in A$ and all small $N^- \prec N$, we have that Galois-type of $a$ over $N^-$ is realized in $M_0$. Assuming property (E) (see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a \big cardinal". Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [BGKV] it is established that this notion of non-forking is the only independence relation possible.

Forking in Short and Tame AECs

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. 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.

Classification theory for accessible categories

We show that metric abstract elementary classes are coherent
accessible categories with directed colimits, with concrete
$\aleph_1$-directed colimits and concrete monomorphisms. More
broadly, we define a notion of $\kappa$-concrete Abstract
Elementary Class and 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.

Metric abstract elementary classes as accessible 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.

Universal abstract elementary classes and locally multipresentable categories

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.

Weak factorization systems and stable independence

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