# Categoricity and multidimensional diagrams

Saharon Shelah<EMAIL_ADDRESS>http://shelah.logic.at Einstein
Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 91904, Israel, and Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019, USA  and  Sebastien Vasey<EMAIL_ADDRESS>http://math.harvard.edu/~sebv/ Department of Mathematics
Harvard University
Cambridge, Massachusetts, USA  Dedicated to the memory of Oren Kolman.

(Date:
AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52,
03C55, 03C75, 03E05, 03E55.)

###### Abstract.

We study multidimensional diagrams in independent amalgamation in the
framework of abstract elementary classes (AECs). We use them to prove the
eventual categoricity conjecture for AECs, assuming a large cardinal axiom.
More precisely, we show assuming the existence of a proper class of strongly
compact cardinals that an AEC which has a single model of _some_ high-enough
cardinality will have a single model in _any_ high-enough cardinal. Assuming a
weak version of the generalized continuum hypothesis, we also establish the
eventual categoricity conjecture for AECs with amalgamation.

###### Key words and phrases:

abstract elementary classes, good frames, categoricity, forking,
multidimensional diagrams, excellence

The first author would like to thank the Israel Science Foundation for partial
support of this research (Grant No. 242/03).

Research partially supported by European Research Council grant 338821, and by
National Science Foundation grant no: 136974.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and semilattices
  3. 3 Abstract classes and skeletons
  4. 4 Compact abstract elementary classes
  5. 5 Combinatorics of abstract classes
  6. 6 Good frames
  7. 7 Two-dimensional independence notions
  8. 8 Multidimensional independence
  9. 9 Finite combinatorics of multidimensional independence
  10. 10 The multidimensional amalgamation properties
  11. 11 Continuous independence relations
  12. 12 Building primes
  13. 13 Excellent classes
  14. 14 Excellence and categoricity: the main theorems

##  1\. Introduction

###  1.1. General background and motivation

One of the most important result of modern model theory is Morley’s
categoricity theorem [Mor65]: a countable first-order theory categorical in
_some_ uncountable cardinal is categorical in _all_ uncountable cardinals.
Here, we call a theory (or more generally a class of structures) _categorical_
in a cardinal $\lambda$ if it has (up to isomorphism) exactly one model of
cardinality $\lambda$. Morley’s theorem generalizes the fact that
algebraically closed fields of characteristic zero are categorical in all
uncountable cardinals.

The machinery developed in the proof of Morley’s theorem has proven to be
important, both for pure model theory and for applications to other fields of
mathematics (on the former, see e.g. the first author’s book [She90]; on the
latter, see e.g. [Bou99]). Specifically, the proof of Morley’s theorem shows
that there is a notion of independence generalizing algebraic independence (as
well as linear independence) for the models of the theory in question. This
was later precisely generalized and studied by the first author, who called
this notion _forking_. Forking has now become a central concept of model
theory.

It is natural to look for extensions of Morley’s theorem, the real goal being
to look for generalizations of _forking_ to other setups. The first author has
established [She74] that for _any_ (not necessarily countable) first-order
theory $T$, if $T$ is categorical in _some_ $\mu>|T|$, then $T$ is categorical
in _all_ $\mu^{\prime}>|T|$. One possible next step would be to prove versions
of Morley’s theorem for infinitary logics, like
$\mathbb{L}_{\omega_{1},\omega}$ (countably-many conjunctions are allowed
within a single formula). Indeed, already in the late seventies the first-
author conjectured (see [She83a, Conjecture 2]):

######  Conjecture 1.1 (Categoricity conjecture for
$\mathbb{L}_{\omega_{1},\omega}$, strong version).

Let $\psi$ be an $\mathbb{L}_{\omega_{1},\omega}$-sentence. If $\psi$ is
categorical in _some_ $\mu\geq\beth_{\omega_{1}}$, then $\psi$ is categorical
in _all_ $\mu^{\prime}\geq\beth_{\omega_{1}}$.

Forty years down the road, and despite a lot of partial approximations (see
the history given in the introduction of [Vas17d]), Conjecture 1.1 is still
open. The main difficulty is that the compactness theorem fails, hence one has
to work much more locally. The interest is that one can derive weak version of
compactness from the categoricity assumption itself, often combined with large
cardinals or combinatorial set-theoretic assumptions. Thus any proof is likely
to exhibit a rich interplay between set theory and model theory.

The threshold $\beth_{\omega_{1}}$ appears in Conjecture 1.1 because it is
provably best possible (for lower thresholds, there is a standard
counterexample due to Morley, see [AV, 4.1]). Nevertheless, the spirit of the
conjecture is captured by the following eventual version:

######  Conjecture 1.2 (Categoricity conjecture for
$\mathbb{L}_{\omega_{1},\omega}$, eventual version).

There is a threshold cardinal $\theta$ such that if $\psi$ is an
$\mathbb{L}_{\omega_{1},\omega}$-sentence categorical in _some_
$\mu\geq\theta$, then $\psi$ is categorical in _all_ $\mu^{\prime}\geq\theta$.

###  1.2. Eventual categoricity from large cardinals

Conjecture 1.2 is also still open. In the present paper, we prove it assuming
that a large cardinal axiom holds. More precisely and generally:

Corollary 14.7. Let $\kappa$ be a strongly compact cardinal and let $\psi$ be
an $\mathbb{L}_{\kappa,\omega}$-sentence. If $\psi$ is categorical in _some_
$\mu\geq\beth_{\left(2^{\kappa}\right)^{+}}$, then $\psi$ is categorical in
_all_ $\mu^{\prime}\geq\beth_{\left(2^{\kappa}\right)^{+}}$.

This solves Question 6.14(1) in the first author’s list of open problems
[She00] and improves on a result of Makkai and the first author [MS90] who
proved the statement of Corollary 14.7 under the additional assumption that
the initial categoricity cardinal $\mu$ is a _successor_ cardinal. We see this
as quite a strong assumption, since it allows one to work directly with models
of a single cardinality $\mu_{0}$ (where $\mu=\mu_{0}^{+}$). The gap between
[MS90] and Corollary 14.7 is further reflected by the several additional
levels of sophistication that we use to prove the latter. For example, we work
in the framework of abstract elementary classes (AECs). This was introduced by
the first author [She87] as a general semantic framework encompassing in
particular classes axiomatized by “reasonable” logics. The following version
of Conjecture 1.2 has been stated there [She09a, N.4.2]:

######  Conjecture 1.3 (Eventual categoricity conjecture for AECs).

An AEC categorical in _some_ high-enough cardinal is categorical in _all_
high-enough cardinals.

In this paper, we more generally prove Conjecture 1.3 assuming a large
cardinal axiom (a proper class of strongly compact cardinals). Again,
Conjecture 1.3 was known in the special case where the starting categoricity
cardinal is a successor (a result of Boney [Bon14]). The second author had
also proven it for AECs closed under intersections [Vas17d, 1.7].

###  1.3. Multidimensional diagrams

The proof of Corollary 14.7 only tangentially uses large cardinals. We have
already mentioned that one key step in the proof is the move to AECs: this is
done because of generality, but also because the framework has a lot of
closure, allowing us for example to take subclasses of saturated models and
still remain inside an AEC. The other key step is the use of multidimensional
diagrams in independent amalgamation. Essentially, the idea is to generalize
[She83a, She83b] (which discussed $\mathbb{L}_{\omega_{1},\omega}$ only) to
AECs. This was started in the first author’s two volume book [She09a, She09b],
where the central concept of a _good $\lambda$-frame_ was introduced. Roughly,
an AEC $\mathfrak{K}$ has a good $\lambda$-frame if it has several basic
structural properties in $\lambda$ (amalgamation, no maximal models, and
stability), and also has a superstable-like notion of forking defined for
models of cardinality $\lambda$. The idea here is that everything is assumed
only for models of cardinality $\lambda$. The program, roughly, is to develop
the theory of good frames so that one can start with a good $\lambda$-frame,
extend it to a good $\lambda^{+}$-frame, and keep going until the structure of
the whole class is understood.

In order to carry out this program in practice, one has to prove that the good
$\lambda$-frame has more structural properties. Specifically, one first proves
(under some conditions) that a notion of _nonforking amalgamation_ can be
defined (initially, the definition of a good frame only requires a notion of
nonforking for types of single elements, but here we get one for types of
models). Thus one has at least a _two-dimensional_ notion of nonforking (it
applies to squares of models). From this, one seeks to define a three-
dimensional notion (for cubes of models), and then inductively an
$n$-dimensional notion. It turns out that once one has a good $n$-dimensional
notion for all $n<\omega$ (still for models of cardinality $\lambda$), then
one can lift this to the whole class. In [She83a, She83b], an
$\mathbb{L}_{\omega_{1},\omega}$ sentence with all the good $n$-dimensional
properties was called _excellent_ , so we naturally call an AEC with those
properties an _excellent_ AEC. It turns out that this is truly a
generalization of [She83a, She83b], and in fact another consequence of the
methods of this paper is Theorem 14.12, which generalizes [She83a, She83b] to
any $\operatorname{PC}_{\aleph_{0}}$-AEC (and hence in particular to
$\mathbb{L}_{\omega_{1},\omega}(Q)$, see [She09a, II.3.5]; it had been a
longstanding open question whether such a generalization was possible).

The use of multidimensional diagrams has had several more applications. For
example, the _main gap theorem_ of the first author [She85] is a theorem about
countable first-order theory but was proven using multidimensional diagrams.
Thus we believe that the systematic study of multidimensional diagrams
undertaken in this paper is really its key contribution (in fact, a main gap
theorem for certain AECs now also seems within reach - the axioms of [GL05]
hold inside ${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}}$ if
$\mathfrak{K}$ is an excellent class, see Section 13 \- but we leave further
explorations to future work). Let us note that multidimensional diagrams for
AECs were already studied in [She09a, III.12], but we do not rely on this
here. See later for how the present paper relates to this reference.

###  1.4. Eventual categoricity in AECs with amalgamation

We mention another application of multidimensional diagram, which is really
why this paper was started. In [She09a, Theorem IV.7.12], it was asserted that
the eventual categoricity conjecture holds for AECs with amalgamation,
assuming a weak version of the generalized continuum hypothesis (GCH).
However, the proof relied on a claim of [She09a, III.12] that was not proven
there, but promised to appear in [She]. See also [Vas17b, Section 11] for an
exposition of the proof, modulo the claim.

We prove this missing claim in the present paper (Corollary 14.4). As a
consequence, we obtain the consistency of the eventual categoricity conjecture
in AECs with amalgamation:

Corollary 14.10. Assume $2^{\theta}<2^{\theta^{+}}$ for all cardinals
$\theta$. Let $\mathfrak{K}$ be an AEC with amalgamation. If $\mathfrak{K}$ is
categorical in _some_
$\mu\geq\beth_{\left(2^{\aleph_{\text{LS}(\mathfrak{K})^{+}}}\right)^{+}}$,
then $\mathfrak{K}$ is categorical in _all_
$\mu^{\prime}\geq\beth_{\left(2^{\aleph_{\text{LS}(\mathfrak{K})^{+}}}\right)^{+}}$.

Note that a similar-looking result starting with categoricity in a _successor_
cardinal was established (in ZFC) in [She99]. There however, only a downward
transfer was obtained (i.e. nothing was said about categoricity _above_ the
starting categoricity cardinal $\mu$). Moreover the threshold cardinal was
bigger. After the present paper was written, the second author [Vas]
established that the threshold cardinal in Corollary 14.10 can be lowered
further111The reasons for the appearance of the strange cardinal
$\beth_{\left(2^{\aleph_{\text{LS}(\mathfrak{K})^{+}}}\right)^{+}}$ are
perhaps best explained by the proof itself. The short version is that
$\aleph_{\text{LS}(\mathfrak{K})^{+}}$ is the minimal limit cardinal with
cofinality greater than $\text{LS}(\mathfrak{K})$., all the way to
$\beth_{\left(2^{\text{LS}(\mathfrak{K})}\right)^{+}}$.

Note also that is is known (see for example [MS90, 1.13] or [Bon14, 7.3]) that
AECs categorical above a strongly compact cardinal have (eventual)
amalgamation. Thus Corollary 14.10 can be seen as a generalization of
Corollary 14.7. However, the two results are formally incompatible since there
are no cardinal arithmetic assumptions in the hypotheses of Corollary 14.7. By
a result of Kolman and the first author [SK96], some amount of amalgamation
already follows from categoricity above a _measurable_ cardinal. In fact, this
amount of amalgamation is enough to carry out the proof of Corollary 14.10,
see Theorem 14.9. Thus the present paper also establishes that Conjecture 1.3
is consistent with a proper class of measurable cardinals.222In fact, the
machinery of this paper was designed to use a measurable rather than a
strongly compact in Corollary 14.7. However, some technical points remain on
how exactly to do this, so for simplicity we focus on the strongly compact
case and leave the measurable case to future works.

###  1.5. Other approaches to eventual categoricity

The eventual categoricity conjecture is known to hold under several additional
assumptions (in addition to those already mentioned). Grossberg and VanDieren
[GV06c, GV06a] established an _upward_ categoricity transfer from categoricity
in a successor in AECs (with amalgamation) that they called _tame_ (we use the
term “local” here, preferred by the first author). Roughly, orbital types in
such AECs are determined by their restrictions to small sets. This is a very
desirable property which is known to follow from large cardinals [Bon14].

Recently, the second author [Vas17d, Vas17e] established the eventual
categoricity conjecture for _universal classes_ : classes of models
axiomatized by a universal $\mathbb{L}_{\infty,\omega}$ sentence (or
alternatively, classes of models closed under isomorphism, substructure, and
unions of chains). The proof is in ZFC and the threshold is low (not a large
cardinal). Further, the starting cardinal does not need to be a successor. To
prove this result, the second author established eventual categoricity more
generally in AECs that have amalgamation are local, and in addition have
_primes_ over models of the form $Ma$ [Vas18].

Now, it is folklore that any excellent AEC (for the purpose of this
discussion, an AEC satisfying strong multidimensional amalgamation properties)
should have amalgamation, be local, and have primes. Thus the main
contribution of the present paper is to establish excellence for the setups
mentioned above (large cardinals or amalgamation with weak GCH). Then one can
essentially just quote the categoricity transfer of the second author.

Note however that even with additional locality assumptions, the known
categoricity transfers for AECs are proven using local approaches (mainly good
frames or a close variant333See [Vas17b] on how the Grossberg-VanDieren result
can be proven using good frames.), and transferring the local structure upward
using the locality assumptions. The present paper takes the local approach
further by using it to _prove_ locality conditions, through excellence. Of
course, in the present paper we pay a price: cardinal arithmetic or large
cardinal assumptions. It remains open whether paying such a price is
necessary, or whether excellence (and therefore eventual categoricity) can be
proven in ZFC.

###  1.6. How to prove excellence

While the present paper is admittedly complex, relying on several technical
frameworks to bootstrap itself, we can readily describe the main idea of the
proof of excellence. A starting point is a result of the first author [She09a,
I.3.8]: assuming $2^{\lambda}<2^{\lambda^{+}}$ (the weak diamond, see [DS78]),
an AEC categorical in $\lambda$ and $\lambda^{+}$ must have amalgamation in
$\lambda$. This can be rephrased as follows: the $1$-dimensional amalgamation
property in $\lambda$ holds provided that a strong version of the
$0$-dimensional amalgamation property holds in $\lambda^{+}$ (we think of the
$0$-dimensional amalgamation property as joint embedding).

A key insight (a variant of which already appears in [She09a, III.12.30]) is
that this argument can be repeated by looking at _diagrams_ (or _systems_) of
models rather than single models (we do not quite get an AEC, but enough of
the properties hold). We obtain, roughly, that $(n+1)$-dimensional
amalgamation in $\lambda$ follows from a strong version of $n$-dimensional
amalgamation in $\lambda^{+}$. Since we are talking about _nonforking_
amalgamation, we call these properties “uniqueness” properties rather than
“amalgamation” properties, but the idea remains. Parametrizing (and forgetting
about the “strong” part), we have that $(\lambda^{+},n)$-uniqueness implies
$(\lambda,n+1)$-uniqueness.

Now, it was already known how to obtain $(\lambda,1)$, and even
$(\lambda,2)$-uniqueness. In fact, it is known (under reasonable assumptions)
how to obtain a $\lambda$ such that $(\lambda^{+n},2)$-uniqueness holds for
all $n<\omega$ (such a setup is called an _$\omega$ -successful frame_).
Assuming $2^{\lambda^{+m}}<2^{\lambda^{+(m+1)}}$ for all $m<\omega$, we obtain
from the earlier discussion used with each $\lambda^{+m}$ that
$(\lambda^{+m},3)$-uniqueness holds for all $m<\omega$. Continuing like this,
we get $(\lambda^{+m},n)$-uniqueness for all $n,m<\omega$.

The implementation of this sketch is complicated by the fact that there are
other properties to consider, including extension properties. Further, we have
to ensure a _strong_ uniqueness property (akin to categoricity) in the
successor cardinal, so the induction is much more complicated.

Forgetting this for the moment, let us explain how, assuming large cardinals,
one can remove the cardinal arithmetic assumption. For this we assume more
generally that we have an increasing sequence
$\langle\lambda_{m}:n<\omega\rangle$ of cardinals such that for all
$m<\omega$, $2^{<\lambda_{m+1}}=2^{\lambda_{m}}<\lambda_{m+1}$ (in other
words, $\lambda_{m+1}$ is least such that
$2^{\lambda_{m}}<2^{\lambda_{m+1}}$). Let
$\lambda_{\omega}:=\sup_{m<\omega}\lambda_{m}$. Assume that
$(\lambda,2)$-uniqueness holds for all
$\lambda\in[\lambda_{0},\lambda_{\omega})$. In the previous setup, we had that
$\lambda_{m+1}=\lambda_{m}^{+}$. Now, as in [SV99, 1.2.4], we can generalize
the first author’s aforementioned result on getting amalgamation from two
successive categoricity to: whenever $\lambda<\lambda^{\prime}$ and
$2^{\lambda}=2^{<\lambda^{\prime}}<2^{\lambda^{\prime}}$ and categoricity in
$\lambda^{\prime}$ implies there exists an amalgamation base in
$[\lambda,\lambda^{\prime})$. Essentially, in some
$\mu\in[\lambda,\lambda^{\prime})$, we will have amalgamation. The
$n$-dimensional version of this will essentially be that if
$(\lambda^{\prime},n)$-uniqueness holds, then $(\mu,n+1)$-uniqueness holds for
_some_ $\mu\in[\lambda,\lambda^{\prime})$.

So far, we have not used large cardinals. However, we use them to “fill in the
gaps” and make sure that we have amalgamation in every cardinals below $\mu$.
So let $\mu_{0}<\mu$. Fix a triple $(M_{0},M_{1},M_{2})$ of size $\mu_{0}$ to
amalgamate. Since we have large cardinals, we can take ultraproducts by a
sufficiently complete ultrafilters to obtain an extension
$(M_{0}^{\prime},M_{1}^{\prime},M_{2}^{\prime})$ of this triple that has size
$\mu$. Now using amalgamation in $\mu$ we can amalgamate this new triple, and
hence get an amalgam of the original triple as well. The suitable
$n$-dimensional version of this argument can be carried out, and we similarly
obtain $(\lambda,n)$-uniqueness for all
$\lambda\in[\lambda_{0},\lambda_{\omega})$ and $n<\omega$.

###  1.7. Structure of the paper

We assume that the reader has a solid knowledge of AECs, including [Bal09]
(and preferably at least parts of the first three chapters of [She09a]) but
also the more recent literature. We have tried to repeat the relevant
background but still, to keep the paper to a manageable size we had to heavily
quote and use existing arguments.

Schematically, for each $n$ section $n$ discusses a certain framework, call it
framework $n$. Framework $n$ is more powerful than framework $n-1$. Section
$n$ discusses how to get into framework $n$, starting with some instances of
framework $n-1$. Then enough properties of framework $n$ are also proven so
that one can develop the properties of framework $n+1$ in the next section. We
have tried to minimize the cross-section dependencies: the reader can often
forget about previous framework and focus on the current one. At the end of
the paper, an index will also help the reader getting back to previous
definitions.

Specifically, the paper is organized as follows. After some notational
preliminaries, AECs and more generally abstract classes are discussed (Section
3). Then a special framework that will be used to deal with large cardinals,
compact AECs, is introduced (Section 4). We do some combinatorics, in
particular the weak diamond argument alluded to earlier, in Section 5. The
next framework, good frames, is discussed in Section 6. We then move on to
two-dimensional independence notions (Section 7), and multidimensional
independence notions (Section 8). Some properties of multidimensional
independence notions are then proven that are purely combinatorial, in the
sense of only requiring finitely many steps (no arguments involving unions of
chains and resolutions), see Sections 9, 10. One then moves on to looking at
multidimensional independence notions with also requirements on chains of
systems (Section 11). After a section on primes (Section 12), we discuss
excellent AECs (Section 13) and conclude with the main theorems of this paper
(Section 14).

Each section begins with a short overview of its organization and main
results.

###  1.8. Acknowledgments

We thank Will Boney and Adi Jarden for comments that helped improve the paper.

##  2\. Preliminaries and semilattices

Our notation is standard, and essentially follows [She09a] and [SV]. For a
structure $M$, we write $|M|$ for its universe and $\|M\|$ for its
cardinality. We imitate this convention for ordered classes (see Definition
3.1 and Notation 3.2): $\mathfrak{K}$ will denote a pair
$(K,\leq_{\mathfrak{K}})$, where $K$ is a class of structures and
$\leq_{\mathfrak{K}}$ is a partial order on $K$. We will sometimes write
$|\mathfrak{K}|$ for $K$, but may identify $\mathfrak{K}$ and $K$ when there
is no danger of confusion.

We write $[A]^{\mu}$ (respectively $[A]^{<\mu}$) for the set of all subsets of
$A$ of cardinality $\mu$ (respectively strictly less than $\mu$).

Recall that a (meet) _semilattice_ is a partial order $(I,\leq)$ so where
every two elements have a meet (i.e. a greatest lower bound). We will often
identify $I$ with $(I,\leq)$. We write $s\land t$ for the meet of two elements
$s,t\in I$. In this paper, $I$ will almost always be finite, and in this case
we write $\bot$ for $\land I$ (when $I$ is not empty), the least element of
$I$. We will also use interval notation: for $u\leq v$ both in $I$, we write
$[u,v]$ for the partial order with universe $\\{w\in I\mid u\leq w\leq v\\}$,
similarly for $(u,v)$, $(u,\infty)$, and other interval notations.

When $I$ and $J$ are semilattices, we write $I\subseteq J$ to mean that $I$ is
a _subsemilattice_ of $J$. In particular, the meet operation in $I$ and $J$
coincides. Notice that any initial segment of a semilattice is a
subsemilattice.

Given $I$ and $J$ partial orders, we write $I\times J$ for their product
ordered lexicographically: $(i_{1},j_{1})\leq(i_{2},j_{2})$ if and only if
$i_{1}\leq i_{2}$ and $j_{1}\leq j_{2}$. When $I$ and $J$ are semilattices,
$I\times J$ is a semilattice: the meet is given by the meet of each
coordinate.

For $u$ a set, we write $\mathcal{P}(u)$ for the semilattice
$(\mathcal{P}(u),\subseteq)$, where the meet operation is given by $s\land
t=s\cap t$ and $\bot=\emptyset$. We write $\mathcal{P}^{-}(u)$ for the
semilattice on $\mathcal{P}(u)\backslash\\{u\\}$ induced by $\mathcal{P}(u)$.
Very often, we will have $u=n=\\{0,1,\ldots,n-1\\}$. More generally, the
semilattices we will work with will usually be finite initial segment of
$\mathcal{P}(\omega)$.

Note that for any set $u$, $\mathcal{P}(u)\cong\mathcal{P}(|u|)$, and
$\mathcal{P}^{-}(u)\cong\mathcal{P}^{-}(|u|)$. Moreover,
$\mathcal{P}(n)\times\mathcal{P}(m)\cong\mathcal{P}(n+m)$ for $n,m<\omega$.

##  3\. Abstract classes and skeletons

The definition of an abstract class gives the minimal properties we would like
any class of structures discussed in this paper to satisfy. We also discuss
the notion of a _skeleton_ of an abstract class: a subclass that captures many
of the essential properties of the original class but may be more manageable.

We study special kind of abstract classes: _fragmented AECs_ are the main new
concept: they are AECs, except that the chain axiom is weakened so that chains
that jump cardinalities may not be closed under unions. Class of saturated
models in a superstable first-order theory are an example, and we will use
them extensively to deal with such setups.

We finish this section with a review of some standard definitions and results
on AECs.

Let us start with the definition of an abstract class (due to Rami Grossberg).

######  Definition 3.1.

An _abstract class_ is a pair $\mathfrak{K}=(K,\leq_{\mathfrak{K}})$ such that
$K$ is a class of structures in a fixed (here always finitary) vocabulary
$\tau=\tau(\mathfrak{K})$, $\leq_{\mathfrak{K}}$ is a partial order on $K$
extending $\tau$-substructure, and $\mathfrak{K}$ is closed under
isomorphisms.

We often do not distinguish between $\mathfrak{K}$ and its underlying class,
writing for example “$M\in\mathfrak{K}$”. When we need to be precise, we will
use the following notation:

######  Notation 3.2.

Let $\mathfrak{K}=(K,\leq_{\mathfrak{K}})$ be an abstract class.

  1. (1)

We write $|\mathfrak{K}|$ for the underlying class $K$ of $\mathfrak{K}$.

  2. (2)

Let $K_{0}\subseteq K$ be a class of structures which is closed under
isomorphisms. We let $\mathfrak{K}\upharpoonright
K_{0}:=(K_{0},\leq_{\mathfrak{K}}\upharpoonright K_{0})$, where
$\leq_{\mathfrak{K}}\upharpoonright K_{0}$ denotes the restriction of
$\leq_{\mathfrak{K}}$ to $K_{0}$.

We will also use the following standard notation:

######  Notation 3.3.

Let $\mathfrak{K}$ be an abstract class and let $\Theta$ be a class of
cardinals. We write $\mathfrak{K}_{\Theta}$ for the abstract class
$\mathfrak{K}\upharpoonright\\{M\in\mathfrak{K}\mid\|M\|\in\Theta\\}$. That
is, it is the abstract class whose underlying class is the class of models of
$\mathfrak{K}$ of size in $\Theta$, ordered by the restriction of the ordering
of $\mathfrak{K}$. We write $\mathfrak{K}_{\lambda}$ instead of
$\mathfrak{K}_{\\{\lambda\\}}$, $\mathfrak{K}_{\leq\lambda}$ instead of
$\mathfrak{K}_{[0,\lambda]}$, etc.

We define amalgamation, orbital types, stability, etc. for abstract classes as
in the preliminaries of [Vas16c] but we mostly use the notation and
terminology of [She09a]. In particular we write
$\mathbf{tp}_{\mathfrak{K}}(\bar{b}/A;N)$ for the orbital type of $\bar{b}$
over $A$ in $N$ as computed inside $\mathfrak{K}$. Usually, $\mathfrak{K}$
will be clear from context, so we will omit it.

The following locality notion (also called tameness) will play an important
role. It is implicit already in [She99], but is it considered as a property of
AECs per say by Grossberg and VanDieren in [GV06b].

######  Definition 3.4.

Let $\mathfrak{K}$ be an abstract class and let $\kappa$ be an infinite
cardinal.

We say that $\mathfrak{K}$ is _$( <\kappa)$-local_ if for any
$M\in\mathfrak{K}$ and any $p,q\in\mathscr{S}(M)$, if $p\upharpoonright
A=q\upharpoonright A$ for any $A\in[M]^{<\kappa}$, then $p=q$. We say that
$\mathfrak{K}$ is _$\kappa$ -local_ if it is $(<\kappa^{+})$-local.

We will also use the following standard notions of saturation:

######  Definition 3.5.

Let $\mathfrak{K}$ be an abstract class and let $M\leq_{\mathfrak{K}}N$ both
be in $\mathfrak{K}$.

  1. (1)

We say that $N$ is _universal over $M$_ if whenever $N_{0}$ is such that
$M\leq_{\mathfrak{K}}N_{0}$ and $\|N_{0}\|=\|M\|$, we have that there exists
$f:N_{0}\xrightarrow[M]{}N$.

  2. (2)

We say that $N$ is _$(\lambda,\delta)$ -brimmed over $M$_ if there exists an
increasing continuous chain $\langle M_{i}:i\leq\delta\rangle$ in
$\mathfrak{K}_{\lambda}$ such that $M_{0}=M$, $M_{\delta}=M$, and $M_{i+1}$ is
universal over $M_{i}$ for all $i<\delta$.

  3. (3)

We say that $N$ is _brimmed over $M$_ if it is $(\lambda,\delta)$-brimmed over
$M$ for some $\lambda$ and $\delta$.

  4. (4)

We say that $N$ is _brimmed over $A$_ (for $A\subseteq|N|$ a set) if there
exists $M^{\prime}\in\mathfrak{K}$ such that $M^{\prime}\leq_{\mathfrak{K}}N$,
$A\subseteq|M^{\prime}|$, and $N$ is brimmed over $M^{\prime}$.

  5. (5)

We say that $N$ is _brimmed_ if it is brimmed over some $M^{\prime}$.

  6. (6)

We say that $M$ is _$\lambda$ -model-homogeneous_ if whenever
$M_{0}\leq_{\mathfrak{K}}N_{0}$ are both in $\mathfrak{K}$ with
$M_{0}\leq_{\mathfrak{K}}M$ and $\|N_{0}\|<\lambda$, there exists
$f:N_{0}\xrightarrow[M_{0}]{}M$. We say that $M$ is _model-homogeneous_ if it
is $\|M\|$-model-homogeneous.

  7. (7)

We say that $M$ is _$\lambda$ -saturated_ if for any
$M_{0}\leq_{\mathfrak{K}}M$ with $M_{0}\in\mathfrak{K}_{<\lambda}$, any
$p\in\mathscr{S}(M_{0})$ is realized inside $M$. We say that $M$ is
_saturated_ if it is $\|M\|$-saturated.

We give the definition of a few more useful general concepts:

######  Definition 3.6.

The _domain_ $\operatorname{dom}(\mathfrak{K})$ of an abstract class
$\mathfrak{K}$ is the class of cardinals $\lambda$ such that
$\mathfrak{K}_{\lambda}\neq\emptyset$.

######  Definition 3.7.

A _sub-abstract class_ of an abstract class
$\mathfrak{K}=(K,\leq_{\mathfrak{K}})$ is an abstract class
$\mathfrak{K}^{\ast}=(K^{\ast},\leq_{\mathfrak{K}^{\ast}})$ such that
$K^{\ast}\subseteq K$ and for $M,N\in K^{\ast}$,
$M\leq_{\mathfrak{K}^{\ast}}N$ implies $M\leq_{\mathfrak{K}}N$.

The definition of a skeleton is due to the second author [Vas16a, 5.3]. We
have further weakened the requirement on chains appearing there.

######  Definition 3.8.

A _skeleton_ of an abstract class $\mathfrak{K}$ is a sub-abstract class
$\mathfrak{K}^{\ast}$ of $\mathfrak{K}$ such that:

  1. (1)

For any $M\in\mathfrak{K}$, there exists $N\in\mathfrak{K}^{\ast}$ with
$M\leq_{\mathfrak{K}}N$.

  2. (2)

For any $M,N\in\mathfrak{K}^{\ast}$ with $M\leq_{\mathfrak{K}}N$, there exists
$N^{\prime}\in\mathfrak{K}^{\ast}$ such that
$M\leq_{\mathfrak{K}^{\ast}}N^{\prime}$ and
$N\leq_{\mathfrak{K}^{\ast}}N^{\prime}$.

Examples of skeletons will be given later (see Fact 3.24). A simple example
the reader can keep in mind is the class of $\aleph_{0}$-saturated models of a
first-order theory, ordered either by elementary substructure or by being
“universal over”.

The following apparent strengthening of the definition of a skeleton will be
useful.

######  Lemma 3.9.

Let $\mathfrak{K}^{\ast}$ be a skeleton of $\mathfrak{K}$. Let
$M\in\mathfrak{K}$, $n<\omega$, and let $\langle M_{i}:i<n\rangle$ be a (not
necessarily increasing) sequence of elements of $\mathfrak{K}^{\ast}$ such
that $M_{i}\leq_{\mathfrak{K}}M$ for all $i<n$. Then there exists
$N\in\mathfrak{K}^{\ast}$ such that $M\leq_{\mathfrak{K}}N$ and
$M_{i}\leq_{\mathfrak{K}^{\ast}}N$ for all $i<n$.

###### Proof.

We work by induction on $n$. If $n=0$, use (1) in the definition of a skeleton
to find $N\in\mathfrak{K}^{\ast}$ with $M\leq_{\mathfrak{K}}N$. Assume now
that $n=m+1$. By the induction hypothesis, let $N_{0}\in\mathfrak{K}^{\ast}$
be such that $M\leq_{\mathfrak{K}}N_{0}$ and
$M_{i}\leq_{\mathfrak{K}^{\ast}}N_{0}$ for all $i<m$. Now apply (2) in the
definition of a skeleton, with $M,N$ there standing for $M_{m},N_{0}$ here. We
get $N\in\mathfrak{K}^{\ast}$ such that $N_{m}\leq_{\mathfrak{K}^{\ast}}N$ and
$N_{0}\leq_{\mathfrak{K}^{\ast}}N$. By transitivity of
$\leq_{\mathfrak{K}^{\ast}}$, we have that $M_{i}\leq_{\mathfrak{K}^{\ast}}N$
for all $i<m$, so $N$ is as desired. ∎

We now give the definition of several properties that an abstract class may
have. In particular, it could be a fragmented AEC, or even just an AEC.

######  Definition 3.10.

Let $\mathfrak{K}$ be an abstract class.

  1. (1)

Let $\text{LS}(\mathfrak{K})$ be the least cardinal
$\lambda\geq|\tau(\mathfrak{K})|+\aleph_{0}$ such that for any
$M\in\mathfrak{K}$ and any $A\subseteq|M|$, there is $M_{0}\in\mathfrak{K}$
with $M_{0}\leq_{\mathfrak{K}}M$, $A\subseteq|M_{0}|$, and
$\|M_{0}\|\leq|A|+\lambda$. When such a $\lambda$ does not exist, we write
$\text{LS}(\mathfrak{K})=\infty$.

  2. (2)

Let $[\lambda_{1},\lambda_{2})$ be an interval of cardinals and let $\delta$
be a limit ordinal. We say that $\mathfrak{K}$ is
_$([\lambda_{1},\lambda_{2}),\delta)$ -continuous_ if whenever $\langle
M_{i}:i<\delta\rangle$ is an increasing chain in
$\mathfrak{K}_{[\lambda_{1},\lambda_{2})}$, then
$M_{\delta}:=\bigcup_{i<\delta}M_{i}$ is such that $M_{\delta}\in\mathfrak{K}$
and $M_{0}\leq_{\mathfrak{K}}M_{\delta}$. We say that $\mathfrak{K}$ is
_$[\lambda_{1},\lambda_{2})$ -continuous_ if it is
$([\lambda_{1},\lambda_{2}),\delta)$-continuous for all limit ordinals
$\delta$. We say that $\mathfrak{K}$ is _continuous_ if it is
$\operatorname{dom}(\mathfrak{K})$-continuous.

  3. (3)

We say that $\mathfrak{K}$ is _coherent_ if whenever
$M_{0},M_{1},M_{2}\in\mathfrak{K}$, if
$|M_{0}|\subseteq|M_{1}|\subseteq|M_{2}|$, $M_{0}\leq_{\mathfrak{K}}M_{2}$,
and $M_{1}\leq_{\mathfrak{K}}M_{2}$, then $M_{0}\leq_{\mathfrak{K}}M_{1}$.

  4. (4)

$\mathfrak{K}$ is a _fragmented abstract elementary class (AEC)_ if it
satisfies:

    1. (a)

(Coherence) $\mathfrak{K}$ is coherent.

    2. (b)

(Löwenheim-Skolem-Tarski axiom) $\text{LS}(\mathfrak{K})<\infty$.

    3. (c)

(Restricted chain axioms) Let $\langle M_{i}:i<\delta\rangle$ be an increasing
chain in $\mathfrak{K}$. Let $M_{\delta}:=\bigcup_{i<\delta}M_{i}$.

      1. (i)

_If_ $\|M_{i}\|=\|M_{\delta}\|$ for some $i<\delta$, then:

        1. (A)

$M_{\delta}\in\mathfrak{K}$.

        2. (B)

$M_{0}\leq_{\mathfrak{K}}M_{\delta}$.

      2. (ii)

(Smoothness) If $N\in\mathfrak{K}$ is such that $M_{i}\leq_{\mathfrak{K}}N$
for all $i<\delta$, then $M_{\delta}\in\mathfrak{K}$ and
$M_{\delta}\leq_{\mathfrak{K}}N$.

  5. (5)

$\mathfrak{K}$ is an _abstract elementary class (AEC)_ if it is a continuous
fragmented AEC.

  6. (6)

Define also the notion of a _[very] weak [fragmented] AEC_ , where “weak”
means that the smoothness axiom may not hold and “very weak” means that the
coherence axiom may also not hold.

The following are examples of fragmented AECs. The generalization of the first
one to AECs will play an important role in the present paper. The second will
not be studied here, but we give it as an additional motivation.

######  Example 3.11.

Let $T$ be a superstable first-order theory and let $\mathfrak{K}$ be its
class of saturated models, ordered by elementary substructure. Then
$\mathfrak{K}$ is a fragmented AEC with $\text{LS}(\mathfrak{K})\leq 2^{|T|}$.

######  Example 3.12.

Let $\phi$ be a complete $\mathbb{L}_{\omega_{1},\omega}(Q)$-sentence ($Q$ is
the quantifier “there exists uncountably many”) and let $\Phi$ be a countable
fragment of $\mathbb{L}_{\omega_{1},\omega}(Q)$ containing $\phi$. Assume that
for every $\psi(\bar{x})\in\Phi$, there exists a relation $R_{\psi}(\bar{x})$
such that $\psi\models\forall\bar{x}\psi(\bar{x})\leftrightarrow
R_{\psi}(\bar{x})$. Let $K$ be the class of models $M$ of
$\mathbb{L}_{\omega,\omega}$ consequences of $\phi$ which also model $\psi$ if
$M$ uncountable. Order it by $M\leq_{\mathfrak{K}}N$ if and only if $M\preceq
N$ and for any $\psi(x,\bar{y})\in\Phi$, if $M\models R_{\neg
Qx\psi(x,\bar{y})}[\bar{a}]$, then $\psi(M,\bar{a})=\psi(N,\bar{a})$. Then
$\mathfrak{K}$ is a fragmented AEC with $\text{LS}(\mathfrak{K})=\aleph_{0}$.

In any AEC, models of big cardinality can be resolved as an increasing union
of smaller models. We make this into a definition:

######  Definition 3.13.

Let $\mathfrak{K}$ be an abstract class and let $M_{0},M\in\mathfrak{K}$ with
$M_{0}<_{\mathfrak{K}}M$. We say that $M$ is _$\delta$ -resolvable over
$M_{0}$_ if there exists a strictly increasing continuous chain $\langle
N_{i}:i\leq\delta\rangle$ such that:

  1. (1)

$N_{0}=M_{0}$, $N_{\delta}=M$.

  2. (2)

$\|M_{i}\|=\|M_{0}\|+|i|$ for all $i\leq\delta$.

We say that $M$ is _resolvable over $M_{0}$_ if it is $\delta$-resolvable over
$M_{0}$ for some limit ordinal $\delta$. We say that $M$ is _resolvable_ if it
is resolvable over $M_{0}$ for every $M_{0}\in\mathfrak{K}$ with
$M_{0}\leq_{\mathfrak{K}}M$ and $\|M_{0}\|<\|M\|$. We say that $\mathfrak{K}$
is _resolvable_ if any $M\in\mathfrak{K}$ is resolvable.

######  Remark 3.14.

For any fragmented AEC $\mathfrak{K}$,
$\mathfrak{K}_{\geq\text{LS}(\mathfrak{K})}$ is resolvable (this uses that we
do not require $\|M_{i}\|=\|M_{\delta}\|$ in the smoothness axiom).

It is known that any class of structures is contained in a smallest AEC.
Indeed, by [She09a, I.2.19]:

######  Fact 3.15.

Let $\\{\mathfrak{K}_{i}:i\in I\\}$ be a non-empty collection of AECs, all in
the same vocabulary $\tau$. Then $\bigcap_{i\in I}\mathfrak{K}_{i}$ (defined
as $(\bigcap_{i\in I}|\mathfrak{K}_{i}|,\bigcap_{i\in
I}\leq_{\mathfrak{K}_{i}})$) is an AEC.

So:

######  Fact 3.16.

For any abstract class $\mathfrak{K}$, there exists a unique smallest AEC
$\mathfrak{K}^{\ast}$ such that $\mathfrak{K}$ is a sub-abstract class of
$\mathfrak{K}^{\ast}$.

###### Proof.

Take the intersection of the collection of all AECs that include
$\mathfrak{K}$ as a sub-abstract class. Note that this collection is non-
empty: it contains the AEC of all $\tau(\mathfrak{K})$-structures, ordered
with substructure. ∎

######  Definition 3.17.

Let $\mathfrak{K}$ be an abstract class. We call the AEC $\mathfrak{K}^{\ast}$
given by Fact 3.16, the AEC _generated by $\mathfrak{K}$_.

AECs are uniquely determined by their restrictions of size $\lambda$:

######  Fact 3.18 ([She09a, II.1.23]).

Let $\mathfrak{K}^{1}$ and $\mathfrak{K}^{2}$ be AECs with
$\lambda:=\text{LS}(\mathfrak{K}^{1})=\text{LS}(\mathfrak{K}^{2})$. If
$\mathfrak{K}_{\leq\lambda}^{1}=\mathfrak{K}_{\leq\lambda}^{2}$, then
$\mathfrak{K}^{1}=\mathfrak{K}^{2}$.

We now look into how much of this uniqueness is carried over in fragmented
AECs. We will specifically look at totally categorical fragmented AECs (this
is not such a strong assumption, since classes of saturated models are
examples).

First, only part of the fragmented AEC suffices in order to specify the AEC it
generates.

######  Lemma 3.19.

Let $\mathfrak{K}$ be a fragmented AEC with
$\mathfrak{K}_{<\text{LS}(\mathfrak{K})}=\emptyset$. Then the AEC generated by
$\mathfrak{K}$ is the same as the AEC generated by
$\mathfrak{K}_{\text{LS}(\mathfrak{K})}$.

###### Proof.

Let $\lambda:=\text{LS}(\mathfrak{K})$. Let $\mathfrak{K}^{1}$ be the AEC
generated by $\mathfrak{K}$ and let $\mathfrak{K}^{2}$ be the AEC generated by
$\mathfrak{K}_{\leq\lambda}$. Since $\text{LS}(\mathfrak{K})=\lambda$, it is
easy to check that $\text{LS}(\mathfrak{K}^{1})=\lambda$. Of course, we also
have that $\text{LS}(\mathfrak{K}^{2})=\lambda$. Moreover,
$\mathfrak{K}_{\lambda}^{1}=\mathfrak{K}_{\lambda}^{2}$. By Fact 3.18 and
Lemma 3.19, $\mathfrak{K}^{1}=\mathfrak{K}^{2}$. ∎

Toward studying categoricity in fragmented AECs, we look at homogeneous
models. As in [She09a, I.2.5], we have:

######  Fact 3.20.

Let $\mathfrak{K}$ be a fragmented AEC and let
$M_{0}\leq_{\mathfrak{K}}M_{\ell}$, $\ell=1,2$. If
$\|M_{1}\|=\|M_{2}\|>\text{LS}(\mathfrak{K})$ and both $M_{1}$ and $M_{2}$ are
model-homogeneous, then $M_{1}\cong_{M_{0}}M_{2}$.

We obtain:

######  Theorem 3.21.

Let $\mathfrak{K}$ be a fragmented AEC with amalgamation and
$\mathfrak{K}_{<\text{LS}(\mathfrak{K})}=\emptyset$. Assume that for every
$\lambda\in\operatorname{dom}(\mathfrak{K})$, $\mathfrak{K}$ is categorical in
$\lambda$ and stable in $\lambda$. Then any
$M\in\mathfrak{K}_{>\text{LS}(\mathfrak{K})}$ is model-homogeneous.

###### Proof.

Let
$\lambda\in\operatorname{dom}(\mathfrak{K})\cap(\text{LS}(\mathfrak{K}),\infty)$
and let $M\in\mathfrak{K}_{\lambda}$. Pick $M_{0}\leq_{\mathfrak{K}}N_{0}$ in
$\mathfrak{K}_{<\lambda}$ with $M_{0}\leq_{\mathfrak{K}}M$. By categoricity
and stability in $\lambda$, $M$ is $(\lambda,\lambda)$-brimmed. If $\lambda$
is regular, this means that we can find $M^{0}\in\mathfrak{K}_{\lambda}$ such
that $M_{0}\leq_{\mathfrak{K}}M^{0}$ and $M$ is universal over $M^{0}$. Now by
amalgamation, there exists $N^{0}\in\mathfrak{K}_{\lambda}$ and
$f:N_{0}\xrightarrow[M_{0}]{}N^{0}$ such that $M^{0}\leq_{\mathfrak{K}}N^{0}$.
Now by universality let $g:N^{0}\xrightarrow[M^{0}]{}M$. Then $gf$ embeds
$N_{0}$ into $M$ over $M_{0}$.

If $\lambda$ is singular, let $\mu:=\|N_{0}\|$. Pick
$M^{0}\leq_{\mathfrak{K}}M$ with $M^{0}\in\mathfrak{K}_{\mu^{+}}$ and $M^{0}$
containing $M_{0}$. By the previous case, $N_{0}$ embeds into $M^{0}$ (and
hence into $M$) over $M_{0}$, as desired. ∎

######  Theorem 3.22 (Canonicity of categorical fragmented AECs).

Let $\mathfrak{K}^{1}$ and $\mathfrak{K}^{2}$ be fragmented AECs with
$\Theta:=\operatorname{dom}(\mathfrak{K}^{1})=\operatorname{dom}(\mathfrak{K}^{2})$
and $\text{LS}(\mathfrak{K}^{1})=\text{LS}(\mathfrak{K}^{2})$. Suppose that
for $\ell=1,2$, $\mathfrak{K}^{\ell}$ has amalgamation and is stable and
categorical in every $\lambda\in\Theta$. If
$\mathfrak{K}_{\leq\text{LS}(\mathfrak{K}^{1})}^{1}=\mathfrak{K}_{\leq\text{LS}(\mathfrak{K}^{2})}^{2}$,
then $\mathfrak{K}^{1}=\mathfrak{K}^{2}$.

###### Proof.

Without loss of generality,
$\mathfrak{K}_{<\text{LS}(\mathfrak{K}^{\ell})}^{\ell}=\emptyset$ for
$\ell=1,2$. Let $\lambda>\text{LS}(\mathfrak{K})$ be such that
$\lambda\in\Theta$ and assume inductively that we know
$\mathfrak{K}_{<\lambda}^{1}=\mathfrak{K}_{<\lambda}^{2}$. We have that the
model $M^{\ell}$ of cardinality $\lambda$ in $\mathfrak{K}^{\ell}$ is model-
homogeneous (Theorem 3.21) for $\ell=1,2$. By categoricity and the inductive
assumption, there exists $M_{\ell}\in\mathfrak{K}^{\ell}$ of cardinality
strictly less than $\lambda$ with $M_{\ell}\leq_{\mathfrak{K}}M^{\ell}$ and
$f:M_{1}\cong M_{2}$. Now use the proof of Fact 3.20 (the usual back and forth
argument) to conclude that $M^{1}\cong M^{2}$. This implies that
$|\mathfrak{K}_{\lambda}^{1}|=|\mathfrak{K}_{\lambda}^{2}|$. To see that
$\mathfrak{K}_{\lambda}^{1}=\mathfrak{K}_{\lambda}^{2}$ (that is, the
orderings also coincide), observe that the AECs generated by
$\mathfrak{K}^{1}$ and $\mathfrak{K}^{2}$ must be the same (Fact 3.18 and
Lemma 3.19), hence the orderings must coincide on all models there. ∎

######  Definition 3.23.

Let $\mathfrak{K}$ be an abstract class.

  1. (1)

We write ${\mathfrak{K}}^{\lambda\text{-sat}}$ for the abstract class
$\mathfrak{K}\upharpoonright\\{M\in\mathfrak{K}\mid M\text{ is
}\lambda\text{-saturated}\\}$ of $\lambda$-saturated models in $\mathfrak{K}$
ordered by the restriction of the ordering of $\mathfrak{K}$.

  2. (2)

We write $\mathfrak{K}^{\text{sat}}$ for the abstract class
$\mathfrak{K}\upharpoonright\\{M\in\mathfrak{K}\mid M\text{ is saturated}\\}$.

  3. (3)

We write $\mathfrak{K}^{\text{brim}}$ for the abstract class with
$|\mathfrak{K}^{\text{brim}}|=\\{M\in\mathfrak{K}\mid M\text{ is brimmed}\\}$
ordered by $M\leq_{\mathfrak{K}^{\text{brim}}}N$ if and only if
$M\leq_{\mathfrak{K}}N$ and one of the following conditions holds:

    1. (a)

$M=N$.

    2. (b)

$\|M\|<\|N\|$.

    3. (c)

$N$ is brimmed over $M$ (so $\|M\|=\|N\|$).

It is known [Vas17c, 5.7] that an AEC with amalgamation categorical in a high-
enough cardinal will satisfy a weak version of locality, and unions of chains
of $\lambda$-saturated models will be $\lambda$-saturated there. This gives
several examples of fragmented AECs and skeletons. Note that these results
have a long history (detailed at the reference), with contributions of, among
others, both the first and second author.

######  Fact 3.24 (Structure of categorical AECs with amalgamation).

Let $\mathfrak{K}$ be an AEC with arbitrarily large models. Let
$\mu>\text{LS}(\mathfrak{K})$. Assume that $\mathfrak{K}$ is categorical in
$\mu$ and $\mathfrak{K}_{<\mu}$ has amalgamation and no maximal models.

  1. (1)

If $\mu\geq h(\text{LS}(\mathfrak{K}))$, then there exists
$\chi\in(\text{LS}(\mathfrak{K}),\mu)$ such that
$\mathfrak{K}^{\text{sat}}_{(\chi,\mu)}$ is $\chi$-local.

  2. (2)

Let $\lambda\leq\mu$. If $\text{cf}(\lambda)\geq\text{LS}(\mathfrak{K})^{+}$,
then there exists $\chi\in(\text{LS}(\mathfrak{K}),\lambda)$ such that
$\mathfrak{K}^{\text{sat}}_{(\chi,\lambda)}$ is $\chi$-local.

  3. (3)

For any $\lambda\in(\text{LS}(\mathfrak{K}),\mu]$,
${\mathfrak{K}}^{\lambda\text{-sat}}$ is an AEC with Löwenheim-Skolem-Tarski
number $\lambda$. In particular,
$\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\mu]}$ is a fragmented
AEC with Löwenheim-Skolem-Tarski number $\text{LS}(\mathfrak{K})^{+}$.
Moreover, $\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\mu]}$ is
categorical in every $\lambda\in(\text{LS}(\mathfrak{K}),\mu)$ and it is a
skeleton of $\mathfrak{K}_{(\text{LS}(\mathfrak{K}),\mu]}$.

  4. (4)

Let $\Theta:=[\text{LS}(\mathfrak{K}),\mu)$. Then:

     * •

$\mathfrak{K}^{\text{brim}}_{\Theta}$ is a fragmented very weak AEC which is
$[\text{LS}(\mathfrak{K}),\lambda)$-continuous and categorical in $\lambda$
for every $\lambda\in\Theta$.

     * •

$\mathfrak{K}^{\text{brim}}_{\Theta}$ is a skeleton of
$\mathfrak{K}_{\Theta}$.

     * •

$|\mathfrak{K}^{\text{brim}}_{(\text{LS}(\mathfrak{K}),\mu)}|=|\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\mu)}|$
and $\mathfrak{K}^{\text{brim}}_{(\text{LS}(\mathfrak{K}),\mu)}$ is a sub-
abstract class of $\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\mu)}$.

Regarding categoricity transfers we will use the following two known results.
The first can be traced back to the presentation theorem of the first author.
See [Vas17b, 9.2] for a full proof.

######  Fact 3.25 (Morley’s omitting type theorem for AECs).

Let $\mathfrak{K}$ be an AEC with amalgamation and let
$\mu>\text{LS}(\mathfrak{K})$. If every model in $\mathfrak{K}_{\mu}$ is
$\text{LS}(\mathfrak{K})^{+}$-saturated, then there exists
$\chi<h(\text{LS}(\mathfrak{K}))$ such that every model in
$\mathfrak{K}_{\geq\chi}$ is $\text{LS}(\mathfrak{K})^{+}$-saturated.

The second is due to the second author. It is a consequence of an omitting
type theorem of the first author (see the reference below for a history).

######  Fact 3.26 ([Vas17b, 9.8]).

Let $\mathfrak{K}$ be an $\text{LS}(\mathfrak{K})$-local AEC with
amalgamation, and arbitrarily large models. If $\mathfrak{K}$ is categorical
in _some_ $\mu>\text{LS}(\mathfrak{K})$, then $\mathfrak{K}$ is categorical in
a proper class of cardinals.

##  4\. Compact abstract elementary classes

Compact AECs are an axiomatization of the AECs “essentially below $\kappa$”
introduced by Boney [Bon14, 2.10]. They encompass classes of models of an
$\mathbb{L}_{\kappa,\omega}$ sentence, $\kappa$ a strongly compact, and more
generally AECs closed under $\kappa$-complete ultraproducts, $\kappa$ a
strongly compact cardinal below (yes, below!) the Löwenheim-Skolem-Tarski
number. The reader who does not want to process yet another abstract
definition can with little loss think of $\mathfrak{K}_{\geq\kappa}$, where
$\mathfrak{K}$ is an AEC and $\kappa>\text{LS}(\mathfrak{K})$ is a strongly
compact cardinal444With a little bit of care, one could adapt the definitions
to work also with _almost_ strongly compact cardinals, see [BTR17]. Since
there are no obvious benefits, we have not bothered..

######  Definition 4.1.

Let $\mathfrak{K}$ be an AEC and let $\kappa$ be a strongly compact cardinal
(we allow $\kappa=\aleph_{0}$). We call $\mathfrak{K}$ _$\kappa$ -compact_ if:

  1. (1)

$\kappa\leq\text{LS}(\mathfrak{K})$ and
$\mathfrak{K}_{<\text{LS}(\mathfrak{K})}=\emptyset$.

  2. (2)

$\mathfrak{K}$ is closed under $\kappa$-complete ultraproducts. More
precisely, if $U$ is a $\kappa$-complete ultrafilter on some index set $I$ and
$\langle M_{i}:i\in I\rangle$ is a sequence in $\mathfrak{K}$, then
$\prod_{i\in I}M_{i}/U$ is in $\mathfrak{K}$. Moreover if $\langle N_{i}:i\in
I\rangle$ is another sequence in $\mathfrak{K}$ and
$M_{i}\leq_{\mathfrak{K}}N_{i}$ for all $i\in I$, then $\prod_{i\in
I}M_{i}/U\leq_{\mathfrak{K}}\prod_{i\in I}N_{i}/U$.

We call $\mathfrak{K}$ _compact_ if it is $\kappa$-compact for some strongly
compact cardinal $\kappa$.

Immediately from the definition, we have:

######  Remark 4.2.

Let $\mathfrak{K}$ be a $\kappa$-compact AEC.

  1. (1)

$\mathfrak{K}$ has no maximal models (just take an ultrapower).

  2. (2)

For any cardinal $\lambda$, $\mathfrak{K}_{\geq\lambda}$ is a $\kappa$-compact
AEC.

  3. (3)

The conclusion of [Bon14, 4.3] holds.

The two main examples are:

######  Fact 4.3.

  1. (1)

Let $\kappa$ be a strongly compact cardinal (we allow $\kappa=\aleph_{0}$) and
let $T$ be a theory in $\mathbb{L}_{\kappa,\omega}$. Let $\Phi$ be a fragment
containing $T$. Let $\mathfrak{K}$ be the class of models of $T$, ordered by
$M\leq_{\mathfrak{K}}N$ if and only if $M\preceq_{\Phi}N$. Then
$\mathfrak{K}_{\geq\kappa}$ is a $\kappa$-compact AEC with
$\text{LS}(\mathfrak{K}_{\geq\kappa})\leq|\Phi|+|\tau(\Phi)|+\kappa$.

  2. (2)

Let $\mathfrak{K}$ be an AEC and let $\kappa>\text{LS}(\mathfrak{K})$ be a
strongly compact cardinal. Then $\mathfrak{K}_{\geq\kappa}$ is
$\kappa$-compact.

The following is an important property of compact AECs. Using the terminology
of [Bon14, 3.3], it should be called “fully $(<\kappa)$-tame and short over
the empty set”. We prefer the shorter “$(<\kappa)$-short” here.

######  Definition 4.4.

We say that an abstract class $\mathfrak{K}$ is _$( <\kappa)$-short_ if for
any $M_{1},M_{2}\in\mathfrak{K}$, any ordinal $\alpha$, and any
$\bar{a}_{\ell}\in{}^{\alpha}M_{\ell}$, $\ell=1,2$, if
$\mathbf{tp}(\bar{a}_{1}\upharpoonright
I/\emptyset;M_{1})=\mathbf{tp}(\bar{a}_{2}\upharpoonright I/\emptyset;M_{2})$
for any $I\in[\alpha]^{<\kappa}$, then
$\mathbf{tp}(\bar{a}_{1}/\emptyset;M_{1})=\mathbf{tp}(\bar{a}_{2}/\emptyset;M_{2})$.
We say that $\mathfrak{K}$ is _$\kappa$ -short_ if it is
$(<\kappa^{+})$-short. We say that $\mathfrak{K}$ is _short_ if it is
$(<\kappa)$-short for some $\kappa$.

######  Remark 4.5 ([Bon14, 3.4]).

If an abstract class $\mathfrak{K}$ is $(<\kappa)$-short, then it is
$(<\kappa)$-local (put an enumeration of $M$ as part of both $\bar{b}_{1}$ and
$\bar{b}_{2}$).

######  Fact 4.6 ([Bon14, 4.5]).

If $\mathfrak{K}$ is a $\kappa$-compact AEC, then $\mathfrak{K}$ is
$(<\kappa)$-short.

Finally, we note that categorical compact AECs have amalgamation:

######  Fact 4.7.

Let $\mathfrak{K}$ be a compact AEC. If $\mathfrak{K}$ is categorical in some
$\mu>\text{LS}(\mathfrak{K})$, then $\mathfrak{K}$ has amalgamation.

###### Proof.

As in [SK96], $\mathfrak{K}_{<\mu}$ has amalgamation. Now as in the proof of
[BB17, 2.3], we can use sufficiently complete fine ultrafilters to push
amalgamation up and get that $\mathfrak{K}$ has amalgamation. ∎

Thus we obtain that categorical compact AECs already have quite a lot of
structure:

######  Corollary 4.8.

Let $\mathfrak{K}$ be a compact AEC. If $\mathfrak{K}$ is categorical in
_some_ $\mu>\text{LS}(\mathfrak{K})$, then $\mathfrak{K}$ has amalgamation, no
maximal models, is $(<\text{LS}(\mathfrak{K}))$-short, and is categorical in a
proper class of cardinals.

###### Proof.

Say $\mathfrak{K}$ is $\kappa$-compact. By Fact 4.7, $\mathfrak{K}$ has
amalgamation. By Remark 4.2, $\mathfrak{K}$ has no maximal models. By Fact
4.6, $\mathfrak{K}$ is $(<\kappa)$-short, hence by Remark 4.5 it is
$\text{LS}(\mathfrak{K})$-local. Now apply Fact 3.26. ∎

##  5\. Combinatorics of abstract classes

In addition to a few technical lemmas, the main result of this section is a
generalization of the “amalgamation from categoricity and weak diamond”
theorem of the first author [She09a, I.3.8]. The proof is essentially the
same, but the axioms of AECs that were not used are dropped (as in the study
of nice categories in [GS83, §4]), and an additional parametrization in the
spirit of classification theory over a predicate [PS85, She86] is introduced:
we consider the $\phi$-amalgamation property to mean essentially that we can
amalgamate while fixing the set defined by the formula $\phi$.

First, we recall a generally useful folklore fact about increasing continuous
chains reflecting on a club:

######  Lemma 5.1.

Let $\lambda$ be a regular uncountable cardinal. Let $\langle
A_{i}:i\leq\lambda\rangle$, $\langle B_{i}:i\leq\lambda\rangle$ be increasing
continuous sequences of sets such that:

  1. (1)

$A_{\lambda}=B_{\lambda}$.

  2. (2)

For all $i<\lambda$, $|A_{i}|+|B_{i}|<\lambda$.

Then $C:=\\{i<\lambda\mid A_{i}=B_{i}\\}$ is club.

###### Proof.

$C$ is clearly closed by continuity of the chains. To see unboundedness, let
$i<\lambda$. We build $\langle j_{k}:k<\omega\rangle$ an increasing sequence
of ordinals below $\lambda$ such that $j_{0}=i$, and for all $k<\omega$,
$B_{j_{k}}\subseteq A_{j_{k+1}}$ and $A_{j_{k}}\subseteq B_{j_{k+1}}$. This is
straightforward using regularity of $\lambda$, $A_{\lambda}=B_{\lambda}$,
$|A_{j}|+|B_{j}|<\lambda$ for all $j$, and the fact that the chains are
increasing. Now by continuity of the chains, $j:=\sup_{k<\omega}j_{k}$ is in
$C$, as desired. ∎

We use this to prove a result about chain of structures that are not even
necessarily ordered by substructure:

######  Definition 5.2.

For $M$ and $N$ $\tau$-structures, $M\subseteq^{\ast}N$ ($M$ is a _weak
substructure of $N$_) if $|M|\subseteq|N|$ and for all relations symbols $R$
in $\tau$, $R^{M}\subseteq R^{N}$.

######  Lemma 5.3.

Let $\lambda$ be a regular uncountable cardinal. Let $\tau$ be a vocabulary
and let $\langle M_{i}:i\leq\lambda\rangle$, $\langle
N_{i}:i\leq\lambda\rangle$ be $\subseteq^{\ast}$-increasing continuous
sequences of $\tau$-structures. Assume:

  1. (1)

$M_{\lambda}=N_{\lambda}$.

  2. (2)

$|\tau|<\lambda$.

  3. (3)

$\|M_{i}\|+\|N_{i}\|<\lambda$ for all $i<\lambda$.

Then the set $C:=\\{i<\lambda\mid M_{i}=N_{i}\\}$ is club.

###### Proof.

Expanding $M_{i}$ and $N_{i}$ if necessary, we may assume without loss of
generality that there is a unary predicate $P$ in $\tau$ such that
$P^{M_{i}}=M_{i}$ and $P^{N_{i}}=N_{i}$ for all $i<\lambda$. For each
$R\in\tau$, the set $C_{R}:=\\{i<\lambda\mid R^{M_{i}}=R^{N_{i}}\\}$ is club
by Lemma 5.1. Now observe that $C=\bigcap_{R\in\tau}C_{R}$. ∎

The following technical criteria for being resolvable (Definition 3.13) is
used in the proof of Lemma 11.9.

######  Lemma 5.4.

Let $\mathfrak{K}$ be an abstract class. Assume:

  1. (1)

$\text{LS}(\mathfrak{K})<\infty$,
$\mathfrak{K}_{<\text{LS}(\mathfrak{K})}=\emptyset$.

  2. (2)

For any $\lambda\in\operatorname{dom}(\mathfrak{K})$, $\mathfrak{K}$ is
$[\text{LS}(\mathfrak{K}),\lambda)$-continuous.

  3. (3)

(Coherence for models of different sizes) For
$M_{0},M_{1},M_{2}\in\mathfrak{K}$ with $M_{0}\leq_{\mathfrak{K}}M_{2}$,
$M_{1}\leq_{\mathfrak{K}}M_{2}$, $|M_{0}|\subseteq|M_{1}|$, _and_
$\|M_{0}\|<\|M_{1}\|<\|M_{2}\|$, we have that $M_{0}\leq_{\mathfrak{K}}M_{1}$.

  4. (4)

(Smoothness for big extensions) If $\langle M_{i}:i\leq\delta\rangle$ is
increasing and $N\in\mathfrak{K}$ is such that $M_{i}\leq_{\mathfrak{K}}N$ for
all $i<\delta$ and $\|M_{\delta}\|<\|N\|$, then
$M_{\delta}\leq_{\mathfrak{K}}N$.

  5. (5)

(Resolvability for successors) If $\lambda\geq\text{LS}(\mathfrak{K})$,
$M\leq_{\mathfrak{K}}N$ are such that $M\in\mathfrak{K}_{\lambda}$ and
$N\in\mathfrak{K}_{\lambda^{+}}$, then there exists $\langle
M_{i}:i<\lambda^{+}\rangle$ increasing continuous in $\mathfrak{K}_{\lambda}$
such that $M_{0}=M$ and $\bigcup_{i<\lambda^{+}}M_{i}=N$.

Then $\mathfrak{K}$ is resolvable.

###### Proof.

Let $M_{0}\leq_{\mathfrak{K}}M$ be given with $\|M_{0}\|<\|M\|$. Write
$\lambda:=\|M_{0}\|$, $\mu:=\|M\|$. We show by induction on $\mu$ that $M$ is
resolvable over $M$. If $\mu=\lambda^{+}$, this is the assumption. Assume now
that $\lambda^{+}<\mu$. If $\mu$ is a successor, say $\mu_{0}^{+}$, use that
$\text{LS}(\mathfrak{K})\leq\lambda$ to pick $M_{1}$ such that
$M_{1}\leq_{\mathfrak{K}}M$, $|M_{0}|\subseteq|M_{1}|$, and
$\lambda<\|M_{1}\|=\mu_{0}$. By coherence for models of different sizes,
$M_{0}\leq_{\mathfrak{K}}M_{1}$. By the induction hypothesis, there is a
resolution $\langle M_{0,i}:i<\delta_{0}\rangle$ of $M_{1}$ over $M_{0}$. By
assumption, there is also a resolution $\langle M_{1,i}:i<\delta_{1}\rangle$
of $M$ over $M_{1}$. Now concatenate these two resolutions.

Thus we can assume that $\mu$ is limit. Let $\delta:=\text{cf}(\mu)$ and let
$\langle A_{i}:i<\delta\rangle$ be an increasing sequence of sets such that
$|A_{i}|<\mu$ for all $i<\delta$ and $\bigcup_{i<\mu}A_{i}=M$. We build
$\langle M_{i}:i<\delta\rangle$ increasing continuous such that for all
$i<\delta$:

  1. (1)

$M_{0}=M$.

  2. (2)

$A_{i}\subseteq|M_{i+1}|$ for all $i<\delta$.

  3. (3)

$\|M_{i}\|<\|M_{i+1}\|<\|M\|$.

This is possible using the coherence and smoothness assumptions. This is
enough. At the end, $\bigcup_{i<\delta}M_{i}=M$. We can then take a resolution
of $M_{i+1}$ over $M_{i}$ for each $i<\delta$ (using the induction
hypothesis), and concatenate all these resolutions to obtain the final desired
resolution. ∎

We will use the following form of the weak diamond principle (the proof is the
same as in [DS78, 6.1]; or see the appendix of [She98]).

######  Fact 5.5.

Let $\lambda$ be an infinite cardinal and suppose there is an infinite
cardinal $\lambda_{0}<\lambda$ such that
$2^{\lambda_{0}}=2^{<\lambda}<2^{\lambda}$. Then ($\lambda$ is regular
uncountable and) for every sequence $\langle
f_{\eta}\in{}^{\lambda}\lambda:\eta\in{}^{\lambda}2\rangle$, there exists
$\eta\in{}^{\lambda}2$ such that the set

$S_{\eta}:=\\{\delta<\lambda\mid\exists\nu\in{}^{\lambda}2:f_{\eta}\upharpoonright\delta=f_{\nu}\upharpoonright\delta,\eta\upharpoonright\delta=\nu\upharpoonright\delta,\eta\upharpoonright(\delta+1)\neq\nu\upharpoonright(\delta+1)\\}$

is stationary.

The following notions relativize the usual amalgamation, categoricity, and
universality properties to a formula $\phi$:

######  Definition 5.6.

Let $\mathfrak{K}$ be an abstract class in a vocabulary
$\tau=\tau(\mathfrak{K})$ and let $\phi(x)$ be a quantifier-free
$\mathbb{L}_{\omega,\omega}(\tau)$-formula.

  1. (1)

For $M,N\in\mathfrak{K}$, we say $M$ and $N$ are _$\phi$ -equal_ if
$\phi(M)=\phi(N)$ and
$R^{M}\upharpoonright\phi(M)=R^{N}\upharpoonright\phi(N)$ for every relation
and function symbols $R$ of $\tau$. In other words, the (partial)
$\tau$-structures induced by $\phi$ on $M$ and $N$ are equal.

  2. (2)

A _$\phi$ -span_ is a triple $(M_{0},M_{1},M_{2})$ such that
$M_{0}\leq_{\mathfrak{K}}M_{\ell}$, $\ell=1,2$, and $M_{1}$ and $M_{2}$ are
$\phi$-equal.

  3. (3)

A _$\phi$ -amalgam_ of a $\phi$-span $(M_{0},M_{1},M_{2})$ is a triple
$(N,f_{1},f_{2})$ such that $N\in\mathfrak{K}$ and
$f_{\ell}:M_{\ell}\xrightarrow[M_{0}]{}N$ are such that
$f_{1}\upharpoonright\phi(M_{1})=f_{2}\upharpoonright\phi(M_{2})$.

  4. (4)

We say that $M$ is a _$\phi$ -amalgamation base (in $\mathfrak{K}$)_ if
$M\in\mathfrak{K}$ and every $\phi$-span $(M,M_{1},M_{2})$ has a
$\phi$-amalgam.

  5. (5)

We say that $\mathfrak{K}$ has the _$\phi$ -amalgamation property_ if every
$M\in\mathfrak{K}$ is a $\phi$-amalgamation base.

  6. (6)

We say that $N\in\mathfrak{K}$ is _$\phi$ -universal_ if whenever
$M\in\mathfrak{K}$ and $f:\phi[M]\cong\phi[N]$, there exists $g:M\rightarrow
N$ which extends $f$.

  7. (7)

We say that $\mathfrak{K}$ has _$\phi$ -uniqueness_ if whenever
$M_{0},M_{1},M_{2}\in\mathfrak{K}$ are such that
$M_{0}<_{\mathfrak{K}}M_{\ell}$, $\ell=1,2$, there is
$f:\phi(M_{1})\cong_{\phi(M_{0})}\phi(M_{2})$.

  8. (8)

We say that $\mathfrak{K}$ is _$\phi$ -categorical_ if whenever
$M,N\in\mathfrak{K}$, we have that $\phi(M)\cong\phi(N)$.

######  Remark 5.7.

  1. (1)

Setting $\phi(x)$ to be $x\neq x$, we get back the usual definitions of the
amalgamation property and of a universal model.

  2. (2)

Setting $\phi(x)$ to be $x=x$, we get back the usual definition of
categoricity.

######  Theorem 5.8.

Let $\mathfrak{K}$ and $\mathfrak{K}^{\ast}$ be abstract classes such that
$|\mathfrak{K}|=|\mathfrak{K}^{\ast}|$ and $M\leq_{\mathfrak{K}}N$ implies
$M\leq_{\mathfrak{K}^{\ast}}N$. Let $\phi$ be a quantifier-free
$\mathbb{L}_{\omega,\omega}(\tau(\mathfrak{K}))$-formula, and let
$\Theta=[\lambda_{1},\lambda_{2}]$ be an interval of cardinals. Assume
$2^{\lambda_{1}}=2^{<\lambda_{2}}<2^{\lambda_{2}}$. If:

  1. (1)

$\mathfrak{K}$ is $[\lambda_{1},\lambda_{2})$-continuous.

  2. (2)

There is a $\phi$-universal model in $\mathfrak{K}_{\lambda_{2}}^{\ast}$.

  3. (3)

$\mathfrak{K}_{\lambda_{2}}$ is $\phi$-categorical.

  4. (4)

For any $\lambda\in[\lambda_{1},\lambda_{2})$:

    1. (a)

$\mathfrak{K}_{\lambda}$ has $\phi$-uniqueness.

    2. (b)

For any $\phi$-span $\bar{M}=(M_{0},M_{1},M_{2})$ in $\mathfrak{K}_{\lambda}$,
if $\bar{M}$ has a $\phi$-amalgam in $\mathfrak{K}^{\ast}$, then $\bar{M}$ has
a $\phi$-amalgam in $\mathfrak{K}_{\lambda}$.

Then for any $M\in\mathfrak{K}_{[\lambda_{1},\lambda_{2})}$, there exists
$N\in\mathfrak{K}_{[\lambda_{1},\lambda_{2})}$ such that
$M\leq_{\mathfrak{K}}N$ and $N$ is a $\phi$-amalgamation base in
$\mathfrak{K}_{\|N\|}$.

###### Proof.

Suppose not. We build an increasing continuous tree $\langle
M_{\eta}:\eta\in{}^{\leq\lambda_{2}}2\rangle$ such that for all
$\eta\in{}^{<\lambda_{2}}2$:

  1. (1)

$M_{\langle\rangle}=M$.

  2. (2)

$M_{\eta}\in\mathfrak{K}_{\lambda_{1}+\ell(\eta)}$.

  3. (3)

$|M_{\eta}|\subseteq\lambda_{2}$.

  4. (4)

$\phi(M_{\eta})=\phi(M_{\nu})$ for all $\nu\in{}^{\ell(\eta)}2$.

  5. (5)

In $\mathfrak{K}_{\lambda_{1}+\ell(\eta)}$, $(M_{\eta},M_{\eta\smallfrown
0},M_{\eta\smallfrown 1})$ is a $\phi$-span that has no $\phi$-amalgam.

This is possible: use the assumption that the conclusion fails to take care of
requirement (5), and use $\phi$-uniqueness and some renaming to take care of
requirements (3) and (4). This is enough: it is easy to check that we must
have that $M_{\eta}<_{\mathfrak{K}}M_{\eta\smallfrown\ell}$ for all $\ell\in
2$ and $\eta\in{}^{<\lambda_{2}}2$. Thus
$M_{\eta}\in\mathfrak{K}_{\lambda_{2}}$ for each $\eta\in{}^{\lambda_{2}}2$.

Let $A$ be the $\tau(\mathfrak{K})$-structure induced by $\phi(M_{\eta})$ for
some (or any, see requirement (4)) $\eta\in{}^{\lambda_{2}}2$. Let
$\mathfrak{C}\in\mathfrak{K}_{\lambda_{2}}^{\ast}$ be $\phi$-universal in
$\mathfrak{K}_{\lambda_{2}}^{\ast}$. By $\phi$-categoricity, pick
$h:A\cong\phi(N)$. For each $\eta\in{}^{\lambda_{2}}2$, pick
$g_{\eta}:M_{\eta}\rightarrow\mathfrak{C}$ a $\mathfrak{K}^{\ast}$-embedding
extending $h$. We have that $|M_{\eta}|\subseteq\lambda_{2}$, so extend each
$g_{\eta}$ arbitrarily to $f_{\eta}\in{}^{\lambda_{2}}\lambda_{2}$.

By Fact 5.5, there exists $\eta\in{}^{\lambda_{2}}2$ such that the set
$S_{\eta}$ described there is stationary. Let
$C_{\eta}:=\\{\delta<\lambda_{2}\mid|M_{\eta\upharpoonright\delta}|\subseteq\delta\\}$.
This is a club (by Lemma 5.1 applied to $\langle
M_{\eta\upharpoonright\delta}:\delta<\lambda_{2}\rangle$, $\langle
M_{\eta\upharpoonright\delta}\cap\delta:\delta<\lambda_{2}\rangle$), so let
$\delta\in S_{\eta}\cap C_{\eta}$. We then must have that there is
$\nu\in{}^{\lambda_{2}}2$ such that
$\eta_{0}:=\eta\upharpoonright\delta=\nu\upharpoonright\delta$,
$\eta\upharpoonright(\delta+1)\neq\nu\upharpoonright(\delta+1)$, and (in
particular), $g_{\eta}$ and $g_{\nu}$ agree on $M_{\eta_{0}}$. Moreover,
$g_{\eta}$ and $g_{\nu}$ also extend $h$, so must in particular agree on
$\phi(M_{\eta\upharpoonright(\delta+1)})$. This means that
$(N,g_{\eta}\upharpoonright
M_{\eta\upharpoonright(\delta+1)},g_{\nu}\upharpoonright
M_{\nu\upharpoonright(\delta+1)})$ is a $\phi$-amalgam of
$(M_{\eta},M_{\eta\upharpoonright(\delta+1)},M_{\nu\upharpoonright(\delta+1)})$
in $\mathfrak{K}^{\ast}$. By assumption (4b), this means there is such a
$\phi$-amalgam in $\mathfrak{K}_{\lambda_{1}+\delta}$, a contradiction to
requirement (5). ∎

######  Remark 5.9.

Setting $\mathfrak{K}=\mathfrak{K}^{\ast}$,
$\lambda_{1}=\text{LS}(\mathfrak{K})=\lambda$,
$\lambda_{2}=\text{LS}(\mathfrak{K})^{+}$, $\phi(x):=x\neq x$, we get back the
following result of the first author [She09a, I.3.8]: if an AEC $\mathfrak{K}$
with $\lambda:=\text{LS}(\mathfrak{K})$ is categorical in $\lambda$, has a
universal model in $\lambda^{+}$, and $2^{\lambda}<2^{\lambda^{+}}$, then
$\mathfrak{K}$ has amalgamation in $\lambda$.

##  6\. Good frames

Good frames are a local notion of “bare bone” superstability, introduced by
the first author [She09a, Chapter II]. Essentially, an AEC $\mathfrak{K}$ has
a _good $\lambda$-frame_ if it looks superstable in $\lambda$ in the sense
that $\mathfrak{K}_{\lambda}$ has some reasonable structural properties (like
amalgamation), and there is a forking-like notion for types of singletons over
models.

In this section, we give the definition we will use and state some fundamental
results (most of them known or folklore) about good frames.

The definition we give here allows good frames in several cardinals (as in
[Vas16b, 2.21]) but using fragmented AECs (so also allowing “shrinking
frames”, [Vas17b, Appendix A]). The frames in this paper will always be type-
full (i.e. all nonalgebraic types will be basic), so we will drop the
adjective and ignore basic types in the definition.

######  Definition 6.1.

A _(type-full) good frame_ is a pair $\mathfrak{s}=(\mathfrak{K},F)$, where:

  1. (1)

$\mathfrak{K}$ is a fragmented AEC such that:

    1. (a)

$\mathfrak{K}\neq\emptyset$.

    2. (b)

$\mathfrak{K}_{<\text{LS}(\mathfrak{K})}=\emptyset$.

    3. (c)

$\mathfrak{K}$ has amalgamation, joint embedding, and no maximal models.

    4. (d)

For every $M\in\mathfrak{K}$, there is $N\in\mathfrak{K}$ such that $N$ is
universal over $M$ and $\|N\|=\|M\|$.

  2. (2)

$F$ is a binary relation taking as input pairs $(p,M)$, where $p$ is an
orbital type and $M$ is a model in $\mathfrak{K}$. We write _$p$ does not fork
over $M$_ (or _$p$ does not $\mathfrak{s}$-fork over $M$_) instead of $F(p,M)$
and require that $F$ satisfies the following:

    1. (a)

If $p$ does not fork over $M$, then $p\in\mathscr{S}(N)$ for some
$N\geq_{\mathfrak{K}}M$.

    2. (b)

Invariance: if $f:N\cong N^{\prime}$ and $p\in\mathscr{S}(N)$ does not fork
over $M$, then $f(p)$ does not fork over $f[M]$.

    3. (c)

Monotonicity: if $p\in\mathscr{S}(N)$ does not fork over $M$ and
$M\leq_{\mathfrak{K}}M^{\prime}\leq_{\mathfrak{K}}N$, then $p\upharpoonright
M^{\prime}$ does not fork over $M$ and $p$ does not fork over $M^{\prime}$.

    4. (d)

Disjointness: if $p\in\mathscr{S}(N)$ does not fork over $M$, then
$p\upharpoonright M$ is algebraic if and only if $p$ is algebraic.

    5. (e)

Extension: for any $M\leq_{\mathfrak{K}}N$ and any $p\in\mathscr{S}(M)$ there
exists $q\in\mathscr{S}(N)$ such that $q$ extends $p$ and $q$ does not fork
over $M$.

    6. (f)

Uniqueness: for any $M\leq_{\mathfrak{K}}N$ and any $p,q\in\mathscr{S}(N)$, if
both $p$ and $q$ do not fork over $M$ and $p\upharpoonright M=q\upharpoonright
M$, then $p=q$.

    7. (g)

Local character: If $\langle M_{i}:i\leq\delta\rangle$ is increasing
continuous in $\mathfrak{K}$ and $p\in\mathscr{S}(M_{\delta})$, then there
exists $i<\delta$ such that $p$ does not fork over $M_{i}$.

    8. (h)

Symmetry: If $p=\mathbf{tp}(a/N;N^{\prime})$ does not fork over $M$ and
$b\in|N|$, then there exists $M^{\prime},N^{\prime\prime}\in\mathfrak{K}$ such
that $N^{\prime}\leq_{\mathfrak{K}}N^{\prime\prime}$,
$M\leq_{\mathfrak{K}}M^{\prime}$, $a\in M^{\prime}$, and
$\mathbf{tp}(b/M^{\prime};N^{\prime\prime})$ does not fork over $M$.

We write $\mathfrak{K}_{\mathfrak{s}}$ for the class of the frame
$\mathfrak{s}$. We say that $\mathfrak{s}$ is _on $\mathfrak{K}^{\ast}$_ if
$\mathfrak{K}_{\mathfrak{s}}=\mathfrak{K}^{\ast}$. The _domain_ of a good
frame $\mathfrak{s}$ is the domain of $\mathfrak{K}_{\mathfrak{s}}$ (see
Definition 3.6). For $\lambda\in\operatorname{dom}(\mathfrak{s})$, we say that
$\mathfrak{s}$ is _categorical in $\lambda$_ if $\mathfrak{K}_{\mathfrak{s}}$
is categorical in $\lambda$. We say that $\mathfrak{s}$ is _categorical_ if it
is categorical in _all_ $\lambda\in\operatorname{dom}(\mathfrak{s})$. We also
define restrictions to smaller classes of models such as
$\mathfrak{s}_{\lambda}$ in the natural way. We say that $\mathfrak{s}$ is a
_good $\lambda$-frame_ if $\mathfrak{s}=\mathfrak{s}_{\lambda}$.

We will use the following construction of a good frame:

######  Fact 6.2.

Let $\mathfrak{K}$ be an AEC with arbitrarily large models. Let
$\mu>\text{LS}(\mathfrak{K})$ and let $\theta\geq\mu$. Assume that
$\mathfrak{K}_{<\theta}$ has amalgamation and no maximal models. Assume
further that $\mathfrak{K}$ is categorical in $\mu$ and
$\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\theta)}$ is
$\text{LS}(\mathfrak{K})$-local. Then there is a (categorical) good frame on
$\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\theta)}$.

###### Proof.

By [Vas17c, 5.7(1)], $\mathfrak{K}$ is $\text{LS}(\mathfrak{K})$-superstable
and has $\text{LS}(\mathfrak{K})$-symmetry. The result now follows from the
proof of [Vas17b, A.3]. ∎

######  Remark 6.3.

Combining local character, transitivity, and uniqueness, we have that whenever
$\mathfrak{s}$ is a good frame, then $\mathfrak{K}_{\mathfrak{s}}$ is
$\min(\operatorname{dom}(\mathfrak{s}))$-local.

Regarding disjointness, it follows from the other properties if the frame is
categorical:

######  Lemma 6.4.

If $\mathfrak{s}$ satisfies all the properties of good frames, except perhaps
disjointness, and $\mathfrak{s}$ is categorical, then $\mathfrak{s}$ satisfies
disjointness as well.

###### Proof.

By the conjugation property [She09a, III.1.21] (whose proof never uses
disjointness). ∎

We now state and prove canonicity of the framework. First, frames with the
same restriction in their low cardinals are the same:

######  Lemma 6.5.

Let $\mathfrak{s}$ and $\mathfrak{t}$ be good frames with
$\mathfrak{K}_{\mathfrak{s}}=\mathfrak{K}_{\mathfrak{t}}$. Let
$\lambda:=\min(\operatorname{dom}(\mathfrak{s}))$. If
$\mathfrak{s}_{\lambda}=\mathfrak{t}_{\lambda}$, then
$\mathfrak{s}=\mathfrak{t}$.

###### Proof.

Let $\mathfrak{K}:=\mathfrak{K}_{\mathfrak{s}}=\mathfrak{K}_{\mathfrak{t}}$.
Let $M\leq_{\mathfrak{K}}N$ be in $\mathfrak{K}$ and let $p\in\mathscr{S}(N)$.
Assume that $p$ does not $\mathfrak{s}$-fork over $M$. We show that $p$ does
not $\mathfrak{t}$-fork over $M$, and the converse is symmetric. First, by
local character and transitivity there exists $M_{0}\in\mathfrak{K}_{\lambda}$
such that $p$ does not $\mathfrak{s}$-fork over $M_{0}$. In particular (by
monotonicity), for every $N_{0}\in\mathfrak{K}_{\lambda}$ with
$M_{0}\leq_{\mathfrak{K}}N_{0}\leq_{\mathfrak{K}}N$, $p\upharpoonright N_{0}$
does not $\mathfrak{s}$-fork over $M_{0}$. Since
$\mathfrak{s}_{\lambda}=\mathfrak{t}_{\lambda}$, $p\upharpoonright N_{0}$ does
not $\mathfrak{t}$-fork over $M_{0}$. Now pick
$N_{0}^{\prime}\in\mathfrak{K}_{\lambda}$ such that $p$ does not
$\mathfrak{t}$-fork over $N_{0}^{\prime}$ and (by monotonicity), enlarge it so
that $M_{0}\leq_{\mathfrak{K}}N_{0}^{\prime}$. Then by transitivity, $p$ does
not $\mathfrak{t}$-fork over $M_{0}$, hence (by monotonicity) over $M$, as
desired. ∎

######  Remark 6.6.

Disjointness and symmetry are not used in the proof. Regarding local
character, we only use that any type does not fork over a model of size
$\lambda$.

######  Fact 6.7 (Canonicity of categorical good frames).

Let $\mathfrak{s}$ and $\mathfrak{t}$ be categorical good frames with
$\operatorname{dom}(\mathfrak{s})=\operatorname{dom}(\mathfrak{t})$. Let
$\lambda:=\min(\operatorname{dom}(\mathfrak{s}))$. If
$\left(\mathfrak{K}_{\mathfrak{s}}\right)_{\lambda}=\left(\mathfrak{K}_{\mathfrak{t}}\right)_{\lambda}$,
then $\mathfrak{s}=\mathfrak{t}$.

###### Proof.

By [Vas16a, 9.7], $\mathfrak{s}_{\lambda}=\mathfrak{t}_{\lambda}$. By
canonicity of categorical fragmented AECs (Theorem 3.22),
$\mathfrak{K}_{\mathfrak{s}}=\mathfrak{K}_{\mathfrak{t}}$. By Lemma 6.5,
$\mathfrak{s}=\mathfrak{t}$. ∎

Categoricity of a good frame may seem a strong assumption. However, we have:

######  Fact 6.8.

Let $\mathfrak{s}$ be a good frame and let
$\lambda:=\min(\operatorname{dom}(\mathfrak{s}))$. If $\mathfrak{s}$ is
categorical in $\lambda$, then there is a categorical good frame
$\mathfrak{t}$ with
$\operatorname{dom}(\mathfrak{t})=\operatorname{dom}(\mathfrak{s})$ and
$\left(\mathfrak{K}_{\mathfrak{t}}\right)_{\lambda}=\left(\mathfrak{K}_{\mathfrak{s}}\right)_{\lambda}$.

###### Proof.

By the proof of [Vas17b, A.2]. ∎

We recall the following result about categoricity transfers in good frames:

######  Fact 6.9.

Let $\mathfrak{s}$ be a good frame with fragmented AEC $\mathfrak{K}$. Let
$\Theta:=\operatorname{dom}(\mathfrak{K})$, let $\lambda:=\min(\Theta)$, and
let $\mu\in\Theta$. Assume that $\mathfrak{K}$ is categorical in $\lambda$.

Let $\mathfrak{K}^{\ast}$ be the AEC generated by $\mathfrak{K}_{\mu}$. If
$\mathfrak{K}^{\ast}$ is categorical in $\mu^{+}$, then:

  1. (1)

$\mathfrak{K}$ is $\theta$-continuous for every $\theta\in\Theta$.

  2. (2)

$\mathfrak{K}$ is categorical in every $\mu^{\prime}\in\Theta$.

###### Proof.

By Fact 6.8, we might as well assume that $\mathfrak{s}$ is categorical. We
are then in the setup of [Vas17b, Theorem A.9], which gives the result we
want. ∎

We will also use the following upward frame transfer:

######  Definition 6.10 ([She09a, II.2.4,II.2.5]).

Let $\mathfrak{s}$ be a good $\lambda$-frame and let $\mathfrak{K}$ be the AEC
generated by $\mathfrak{K}_{\mathfrak{s}}$. We let
$\mathfrak{s}^{up}:=(\mathfrak{K},F)$, where $F$ is the following binary
relation on pairs $(p,M)$, with $p$ is an orbital type and $M$ is a model in
$\mathfrak{K}$: $F(p,M)$ if and only if $p\in\mathscr{S}(N)$ for some
$M\leq_{\mathfrak{K}}N$, and there exists
$M_{0}\in\mathfrak{K}_{\mathfrak{s}}$ such that $M_{0}\leq_{\mathfrak{K}}M$
and for all $N_{0}\in\mathfrak{K}_{\mathfrak{s}}$ with
$M_{0}\leq_{\mathfrak{K}}N_{0}\leq_{\mathfrak{K}}N$, $p\upharpoonright N_{0}$
does not $\mathfrak{s}$-fork over $M_{0}$.

######  Fact 6.11.

Let $\mathfrak{s}$ be a good $\lambda$-frame.

  1. (1)

[She09a, II.2.11] $\mathfrak{s}^{up}$ satisfies all the axioms from the
definition of a type-full good frame, except for (1c),(1d), (2e), (2f), and
(2h) in Definition 6.1.

  2. (2)

[BV17, 6.9] If $\mathfrak{K}_{\mathfrak{s}^{up}}$ is $\lambda$-local and has
amalgamation, then $\mathfrak{s}^{up}$ is a good frame.

##  7\. Two-dimensional independence notions

While good frames describe a forking-like relation for types of _singletons_
over models, two-dimensional independence notions describe a forking-like
relations for types of models over models. At that point, it seems more
convenient to think of such a relation as a $4$-ary relation on squares of
models. Squares in nonforking amalgamation form the simplest nontrivial
example of an independent system (a concept defined in the next section).

Here, we state the important properties of two-dimensional independence
relations. We look also at their canonicity (Theorem 7.14). Two key questions
are when a two-dimensional independence relation can be built from a good
frame, and when a two-dimensional independence relation for models of size
$\lambda$ implies the existence of a good $\lambda^{+}$-frame (these themes
are present already in [She09a, II, III]). Frames that are well-behaved in
these respects are called _extendible_ (Definition 7.15). A frame that can be
extended $\omega$-many steps is called _$( <\omega)$-extendible_. These
correspond (but are slightly more convenient to work with than) the
$\omega$-successful good frames in [She09a, III]. We give conditions under
which such frames exist, both in the compact (Fact 7.21) and non-compact (Fact
7.16) cases. For the latter result, the weak diamond (a weakening of the
generalized continuum hypothesis) is assumed.

######  Definition 7.1.

Let $\mathfrak{K}$ be an abstract class. A _two-dimensional independence
relation (or notion) on $\mathfrak{K}$_ is a $4$-ary relation
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ on $\mathfrak{K}$ satisfying:

  1. (1)

$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(M_{0},M_{1},M_{2},M_{3})$ implies
$M_{0}\leq_{\mathfrak{K}}M_{\ell}\leq_{\mathfrak{K}}M_{3}$ for $\ell=1,2$. We
write $M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ instead of
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(M_{0},M_{1},M_{2},M_{3})$.

  2. (2)

If $M_{0}\leq_{\mathfrak{K}}M_{\ell}\leq_{\mathfrak{K}}M_{3}$, $\ell=1,2$ and
$f:M_{3}\rightarrow M_{3}^{\prime}$ is a $\mathfrak{K}$-embedding, then
$M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ if and only if
$f[M_{1}]\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{f[M_{0}]}^{M_{3}^{\prime}}f[M_{2}]$.

  3. (3)

Monotonicity: if $M_{1}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ and
$M_{0}\leq_{\mathfrak{K}}M_{1}^{\prime}\leq_{\mathfrak{K}}M_{1}$, then
$M_{1}^{\prime}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$.

  4. (4)

Disjointness: if $M_{1}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$, then $M_{1}\cap
M_{2}=M_{0}$.

  5. (5)

Symmetry: if $M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$, then
$M_{2}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{1}$.

  6. (6)

Transitivity: if $M_{1}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ and
$M_{3}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{2}}^{M_{5}}M_{4}$, then
$M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{5}}M_{4}$.

  7. (7)

Extension: whenever $M_{0}\leq_{\mathfrak{K}}M_{\ell}$, $\ell=1,2$, there
exists $M_{3}\in\mathfrak{K}$ and
$f_{\ell}:M_{\ell}\xrightarrow[M_{0}]{}M_{3}$ such that
$f_{1}[M_{1}]\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}f_{2}[M_{2}]$.

  8. (8)

Uniqueness: whenever $M_{1}^{\ell}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}^{\ell}}^{M_{3}^{\ell}}M_{2}^{\ell}$
for $\ell=0,1$ and $f_{k}:M_{k}^{0}\cong M_{k}^{1}$, $k<3$ are such that
$f_{0}\subseteq f_{1}$, $f_{0}\subseteq f_{2}$, then there exists
$M_{3}^{2}\in\mathfrak{K}$ with $M_{3}^{1}\leq_{\mathfrak{K}}M_{3}^{2}$ and
and $f_{3}:M_{3}^{0}\rightarrow M_{3}^{2}$ such that $f_{k}\subseteq f_{3}$
for all $k<3$.

######  Definition 7.2.

Let $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ be a two-dimensional independence
notion on $\mathfrak{K}$. If $\mathfrak{s}$ is a good frame on $\mathfrak{K}$,
we say that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ _respects $\mathfrak{s}$_ if whenever
$M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ and
$a\in|M_{1}|$, we have that $\mathbf{tp}(a/M_{2};M_{3})$ does not
$\mathfrak{s}$-fork over $M_{0}$.

As in Lemma 6.4, note that disjointness is not really needed in some cases:

######  Lemma 7.3.

Assume that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ satisfies all the properties of a
two-dimensional independence notion on $\mathfrak{K}$, except perhaps for
disjointness. Assume that $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ respects a good frame $\mathfrak{s}$
on $\mathfrak{K}$. Then $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ satisfies disjointness.

###### Proof.

Because $\mathfrak{s}$ satisfies disjointness. ∎

It is sometimes useful to extend $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ to take sets on the left and right
hand sides. This is the content of the next definition, versions of which were
already considered by both the first author [She09a, II.6.35] and the second
author (in joint work with Boney, Grossberg, and Kolesnikov) [BGKV16]. At this
level of generality, this is studied in recent work of Lieberman, Rosický, and
the second author [LRV, 8.2].

######  Definition 7.4.

Let $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ be a two-dimensional independence
relation on $\mathfrak{K}$.

  1. (1)

Define a $4$-ary relation $\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}$ as follows:
$A\overset{N}{\underset{M}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$ holds if and only if
$M\leq_{\mathfrak{K}}N$, $A,B\subseteq|N|$, and there exists
$M_{1},M_{2},M_{3}\in\mathfrak{K}$ such that $N\leq_{\mathfrak{K}}M_{3}$,
$A\subseteq|M_{1}|$, $B\subseteq|M_{2}|$, and $M_{1}\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$.

  2. (2)

For $p$ an orbital type, we say that $p$ _does not fork over $M$_ if
$M\in\mathfrak{K}$, and there exists $b,A,M_{3}$ such that
$p=\mathbf{tp}(b/A;M_{3})$ and
$b\overset{M_{3}}{\underset{M}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}A$.

  3. (3)

Define a binary relation $F=F(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$ taking as input pairs $(p,M)$, where
$p$ is an orbital type and $M$ is a model in $\mathfrak{K}$ as follows: $pFM$
if $p\in\mathscr{S}(N)$ for some $N\geq_{\mathfrak{K}}M$ and $p$ does not fork
over $M$. We let $\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits):=(\mathfrak{K},F(\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits))$.

Properties of $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ generalize to
$\overline{\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}$ as follows:

######  Fact 7.5 ([LRV, 8.4,8.5]).

Let $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ be a two-dimensional independence
relation on $\mathfrak{K}$.

  1. (1)

Let $M_{0}\leq_{\mathfrak{K}}M_{\ell}\leq_{\mathfrak{K}}M_{3}$ for $\ell=1,2$.
Then $M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ if and only if
$M_{1}\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}M_{2}$.

  2. (2)

(Preservation under $\mathfrak{K}$-embeddings) Given
$M_{0}\leq_{\mathfrak{K}}M_{3}$, $A,B\subseteq UM_{3}$, and
$f:M_{3}\rightarrow N_{3}$, we have that
$A\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$ if and only if
$f[A]\overset{N_{3}}{\underset{f[M_{0}]}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}f[B]$.

  3. (3)

(Monotonicity) If
$A\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$ and $A_{0}\subseteq A$,
$B_{0}\subseteq B$, then
$A_{0}\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B_{0}$.

  4. (4)

(Normality) $A\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$ if and only if
$AM_{0}\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}BM_{0}$.

  5. (5)

(Base monotonicity) If
$A\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$,
$M_{0}\leq_{\mathfrak{K}}M_{2}\leq_{\mathfrak{K}}M_{3}$, and $UM_{2}\subseteq
B$, then $A\overset{M_{3}}{\underset{M_{2}}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$.

  6. (6)

(Extension) Whenever $M\leq_{\mathfrak{K}}N$ and
$p\in\mathscr{S}^{<\infty}(M)$, there exists $q\in\mathscr{S}^{<\infty}(N)$
extending $p$ such that $q$ does not fork over $M$.

  7. (7)

(Symmetry) $A\overset{N}{\underset{M}{\overline{\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$ holds if and only if
$B\overset{N}{\underset{M}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}A$ holds.

  8. (8)

(Uniqueness) Given $p,q\in\mathscr{S}^{<\infty}(B;N)$ with
$M\leq_{\mathfrak{K}}N$ and $UM\subseteq B\subseteq UN$, if $p\upharpoonright
M=q\upharpoonright M$ and $p$, $q$ do not fork over $M$, then $p=q$.

  9. (9)

(Transitivity) If $M_{0}\leq_{\mathfrak{K}}M_{2}\leq_{\mathfrak{K}}M_{3}$,
$A\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}M_{2}$ and
$A\overset{M_{3}}{\underset{M_{2}}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$, then
$A\overset{M_{3}}{\underset{M_{0}}{\overline{\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits}}}B$.

In a general two-dimensional independence relation $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$, $\mathfrak{s}(\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$ may not induce a good frame (because
e.g. such a relation also exists in strictly stable first-order theories). We
call the ones that do (and satisfy a few more convenient properties) _good_ :

######  Definition 7.6.

A two-dimensional independence notion $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ on $\mathfrak{K}$ is _good_ if it
satisfies the following properties:

  1. (1)

$\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$ is a good frame on $\mathfrak{K}$
(in particular, $\mathfrak{K}$ is a fragmented AEC).

  2. (2)

Long transitivity: if $\delta$ is a limit ordinal, $\langle
M_{i}:i\leq\delta\rangle$, $\langle N_{i}:i\leq\delta\rangle$ are increasing
continuous in $\mathfrak{K}$ and $N_{i}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{i}}^{N_{i+1}}M_{i+1}$ for all
$i<\delta$, then $N_{0}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N_{\delta}}M_{\delta}$.

  3. (3)

Local character: if $M\leq_{\mathfrak{K}}N$ and $A\subseteq|N|$, there exists
$M_{0}\leq_{\mathfrak{K}}N_{0}$ such that $N_{0}\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M$, $A\subseteq|N_{0}|$,
and $\|N_{0}\|\leq|A|+\text{LS}(\mathfrak{K})$.

Note that the local character property of two-dimensional independence notions
is vacuous in case the relation is on $\mathfrak{K}_{\lambda}$. In this case,
the following replacement is useful (this is related to the definition of
successful $\text{good}^{+}$ in [She09a, III], see [BV, 2.14]; the “reflects
down” terminology appears in [Vas17b, 3.7(2)]):

######  Definition 7.7.

A good two-dimensional independence notion $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ on $\mathfrak{K}$ _reflects down_ if
for any $\lambda\in\operatorname{dom}(\mathfrak{K})$ and any two increasing
continuous chains $\langle M_{i}:i<\lambda^{+}\rangle$, $\langle
N_{i}:i<\lambda^{+}\rangle$ in $\mathfrak{K}_{\lambda}$, there is a club
$C\subseteq\lambda^{+}$ such that for any $i<j$ in $C$,
$N_{i}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{i}}^{N_{j}}M_{j}$.

######  Remark 7.8.

If $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is a good two-dimensional
independence notion on $\mathfrak{K}$ and
$\lambda,\lambda^{+}\in\operatorname{dom}(\mathfrak{K})$, then by local
character $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits\upharpoonright\mathfrak{K}_{\lambda}$
reflects down.

We will use the following very useful fact about good two-dimensional
independence notions: a union of independent brimmed squares is brimmed, in
the following sense:

######  Fact 7.9 ([She09a, II.6.29]).

Let $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ be a good two-dimensional
independence notion on $\mathfrak{K}$. Let $\langle M_{i}:i\leq\delta\rangle$,
$\langle N_{i}:i\leq\delta\rangle$ be increasing continuous in $\mathfrak{K}$
such that $N_{i}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{i}}^{N_{i+1}}M_{i+1}$ for all
$i<\delta$. If $N_{i+1}$ is brimmed over $M_{i}\cup N_{i}$ (see Definition
3.5(4)) for all $i<\delta$, then $N_{\delta}$ is brimmed over $M_{\delta}\cup
N_{0}$.

Regarding brimmed models, we also have that one can resolve them in a nice
way:

######  Fact 7.10 ([She09a, III.1.17]).

Let $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ be a good two-dimensional
independence notion on $\mathfrak{K}$. Let
$\lambda\in\operatorname{dom}(\mathfrak{K})$ be such that
$\lambda^{+}\in\operatorname{dom}(\mathfrak{K})$. Let
$M,N\in\mathfrak{K}_{\lambda^{+}}$ be brimmed such that $N$ is brimmed over
$M$. Then there exists $\langle M_{i}:i<\lambda^{+}\rangle,\langle
N_{i}:i<\lambda^{+}\rangle$ increasing continuous resolutions of $M$ and $N$
in $\mathfrak{K}_{\lambda}$ such that $M_{i}$ is brimmed and $N_{i}$ is
brimmed over $M_{i}$ for all $i<\lambda^{+}$.

Being good and reflecting down is useful, but one sometimes want more than the
long transitivity property. This is the content of the next definition:

######  Definition 7.11.

A two-dimensional independence notion $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ on $\mathfrak{K}$ is _very good_ if
it is good, reflects down, and satisfies in addition _strong continuity_ :
whenever $\langle M_{\ell}^{i}:i\leq\delta\rangle$ are increasing continuous
in $\mathfrak{K}$, $\ell<4$, and $M_{1}^{i}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}^{i}}^{M_{3}^{i}}M_{2}^{i}$ for
all $i<\delta$, then $M_{1}^{\delta}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}^{\delta}}^{M_{3}^{\delta}}M_{2}^{\delta}$.

We now proceed to show that very good categorical two-dimensional independence
notions are canonical. This is essentially [BGKV16], but since the setup here
is not as global as there, we use a slightly different road.

######  Lemma 7.12.

If $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}$ and $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$ are very good two-dimensional
independence notions on $\mathfrak{K}$ and $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}\upharpoonright\mathfrak{K}_{\lambda}=\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}\upharpoonright\mathfrak{K}_{\lambda}$
for all $\lambda\in\operatorname{dom}(\mathfrak{K})$, then
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$.

###### Proof.

Assume that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}(M_{0},M_{1},M_{2},M_{3})$. We show
that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}(M_{0},M_{1},M_{2},M_{3})$, and the
converse will be symmetric. We proceed by induction on $\|M_{3}\|$. If
$\|M_{0}\|=\|M_{3}\|$, then by assumption $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}(M_{0},M_{1},M_{2},M_{3})$. Assume
now that $\|M_{0}\|<\|M_{3}\|$. Let $\delta:=\|M_{3}\|$. For $\ell=1,2,3$,
build $\langle M_{\ell}^{i}:i\leq\delta\rangle$ increasing continuous such
that for all $i\leq\delta$:

  1. (1)

$\|M_{\ell}^{i}\|=\|M_{0}\|+|i|$ for $\ell=1,2,3$.

  2. (2)

$nf^{1}(M_{0},M_{1}^{i},M_{2}^{i},M_{3}^{i})$.

  3. (3)

$M_{\ell}^{\delta}=M_{\ell}$ for $\ell=1,2,3$.

This is possible using monotonicity and the fact that $\mathfrak{K}$ is a
fragmented AEC. This is enough: by the induction hypothesis,
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}(M_{0},M_{1}^{i},M_{2}^{i},M_{3}^{i})$
for all $i<\delta$. By strong continuity, $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}(M_{0},M_{1},M_{2},M_{3})$, as
desired. ∎

######  Fact 7.13.

Let $\mathfrak{K}$ be fragmented AEC and let
$\lambda\geq\text{LS}(\mathfrak{K})$. Let $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}$ and $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$ be two good two-dimensional
independence notions on $\mathfrak{K}_{\lambda}$ which reflect down. If
$\mathfrak{K}$ is categorical in $\lambda$, then $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$.

###### Proof.

Let $\mathfrak{s}^{\ell}:=\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{\ell})$. By canonicity of categorical
good frames (Fact 6.7), $\mathfrak{s}^{1}=\mathfrak{s}^{2}$, so write
$\mathfrak{s}:=\mathfrak{s}^{1}$. By [Vas17b, 3.11], $\mathfrak{s}$ has the
existence property for uniqueness triples (see [She09a, II.5.3(3)]). By
[She09a, II.6.3(3)], $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$. ∎

######  Theorem 7.14 (Canonicity).

If $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}$ and $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$ are two very good two-dimensional
independence notions on $\mathfrak{K}$ and $\mathfrak{K}$ is categorical in
every $\lambda\in\operatorname{dom}(\mathfrak{K})$, then $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}$.

###### Proof.

By Fact 7.13, $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{1}\upharpoonright\mathfrak{K}_{\lambda}=\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits^{2}\upharpoonright\mathfrak{K}_{\lambda}$
for every $\lambda\in\operatorname{dom}(\mathfrak{K})$. Now apply Lemma 7.12.
∎

The next topic is when (very) good two-dimensional independence notions exist.
It is natural to construct them from good frames, since we already have
sufficient conditions for the existence of good frames (Fact 6.2). A good
frame that can be extended to a good two-dimensional independence relation
which reflects down will be called _extendible_ (this is essentially
equivalent to “successful $\text{good}^{+}$” in [She09a, III.1], but has a
shorter definition):

######  Definition 7.15.

We say a good frame $\mathfrak{s}$ is _extendible_ if there is a good two-
dimensional independence notion $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ on $\mathfrak{K}_{\mathfrak{s}}$
which reflects down. We say that $\mathfrak{s}$ is _very good_ if in addition
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is very good.

We will use the following sufficient condition for a good frame to be
extendible:

######  Fact 7.16.

Let $\mathfrak{s}$ be a good frame and let
$\lambda\in\operatorname{dom}(\mathfrak{s})$. If
$2^{\lambda}<2^{\lambda^{+}}$, $\mathfrak{s}$ is categorical in $\lambda$, and
$\lambda^{+}\in\operatorname{dom}(\mathfrak{s})$, then
$\mathfrak{s}_{\lambda}$ is extendible.

###### Proof.

By [Vas17b, E.8] (the main idea of the argument is due to the first author,
see [She09a, p. 798]), $\mathfrak{s}_{\lambda}$ has what is called the
existence property for uniqueness triples. By [She09a, II.6.34], there is a
two-dimensional independence relation $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ on $\mathfrak{K}_{\lambda}$ which
satisfies long transitivity and respects $\mathfrak{s}_{\lambda}$. Now, any
time a type does not fork $\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$-fork, this implies that it does not
$\mathfrak{s}_{\lambda}$-fork as $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ respects $\mathfrak{s}_{\lambda}$.
But we know that $\mathfrak{s}$ satisfies uniqueness and
$\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$ satisfies extension, see Fact 7.5,
so by the proof of [BGKV16, 4.1],
$\mathfrak{s}=\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$. We have shown that
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is good (since we only consider it as
a relation on $\mathfrak{K}_{\lambda}$, local character is not relevant).

It remains to see that $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ reflects down. Since $\mathfrak{s}$
is a good frame and $\lambda^{+}\in\operatorname{dom}(\mathfrak{s})$, we have
that $\left(\mathfrak{K}_{\mathfrak{s}}\right)_{[\lambda,\lambda^{+}]}$ is
$\lambda$-local (Remark 6.3). Thus we can apply [Jar16, 7.15] to obtain that
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ reflects down, as desired. ∎

Note also that being extendible implies that the frame itself can be extended
(this is the main idea of the end of Chapter II in [She09a]):

######  Fact 7.17.

If $\mathfrak{s}$ is an extendible categorical good $\lambda$-frame, then
there exists a (unique) categorical good frame $\mathfrak{t}$ such that
$\operatorname{dom}(\mathfrak{t})=\\{\lambda,\lambda^{+}\\}$ and
$\mathfrak{t}_{\lambda}=\mathfrak{s}_{\lambda}$.

###### Proof.

Let $\mathfrak{K}$ be the AEC generated by $\mathfrak{K}_{\mathfrak{s}}$. By
[She09a, III.1.6(2)] (or see [JS13, 10.1.9]), there is a good
$\lambda^{+}$-frame $\mathfrak{s}^{+}$ on
$\mathfrak{K}^{\text{sat}}_{\lambda^{+}}$. Since by [She09a, III.1.10],
$\mathfrak{K}^{\text{sat}}_{\lambda^{+}}$ is $\lambda$-local, nonforking in
$\mathfrak{s}^{+}$ must be generated by nonforking in $\mathfrak{s}$ (i.e.
$\mathfrak{s}^{+}=\mathfrak{s}^{up}\upharpoonright\mathfrak{K}^{\text{sat}}_{\lambda^{+}}$).
Thus letting
$\mathfrak{t}:=\mathfrak{s}^{up}\upharpoonright(\mathfrak{K}^{\text{sat}}_{\lambda^{+}}\cup\mathfrak{K}_{\lambda})$,
we get the result. ∎

######  Definition 7.18.

For $\mathfrak{s}$ an extendible categorical good $\lambda$-frame, we write
$\mathfrak{s}^{+}$ for $\mathfrak{t}_{\lambda^{+}}$, where $\mathfrak{t}$ is
as given by Fact 7.17.

Since we now know how to extend a frame to the next cardinal, it makes sense
to define when one can do this successively:

######  Definition 7.19 ([She09a, III.1.12]).

Let $\mathfrak{s}$ be a categorical good $\lambda$-frame. We define by
induction on $n<\omega$ what it means for $\mathfrak{s}$ to be $n$-extendible
as well as a good $\lambda^{+n}$-frame $\mathfrak{s}^{+n}$ as follows:

  1. (1)

$\mathfrak{s}$ is always $0$-extendible and $\mathfrak{s}^{+0}=\mathfrak{s}$.

  2. (2)

$\mathfrak{s}$ is $(n+1)$-extendible if it is $n$-extendible and
$\mathfrak{s}^{+n}$ is extendible.

  3. (3)

If $\mathfrak{s}$ is $(n+1)$-extendible, let
$\mathfrak{s}^{+(n+1)}:=\left(\mathfrak{s}^{+n}\right)^{+}$.

We say that $\mathfrak{s}$ is _$( <\omega)$-extendible_ if $\mathfrak{s}$ is
$n$-extendible for all $n<\omega$.

When a good $\lambda$-frame is $(<\omega)$-extendible, then after taking its
successor a few times, it becomes very good. Moreover after this is done one
can “connect” forking between the cardinals, getting a good
$[\lambda,\lambda^{<\omega})$-frame:

######  Fact 7.20.

Let $\mathfrak{s}$ be a categorical good $\lambda$-frame.

  1. (1)

If $\mathfrak{s}$ is $\omega$-successful and $\text{good}^{+}$ (in the sense
of [She09a, III.1]), then $\mathfrak{s}$ is $(<\omega)$-extendible.

  2. (2)

If $\mathfrak{s}$ is $4$-extendible, then $\mathfrak{s}^{+3}$ is very good.

  3. (3)

If $\mathfrak{s}$ is very good and $2$-extendible, then $\mathfrak{s}^{+}$ is
very good.

  4. (4)

If $\mathfrak{s}$ is very good and $(<\omega)$-extendible, then there exists a
(unique) very good categorical frame $\mathfrak{t}$ such that
$\operatorname{dom}(\mathfrak{t})=[\lambda,\lambda^{+\omega})$ and
$\mathfrak{t}_{\lambda^{+n}}=\mathfrak{s}^{+n}$ for all $n<\omega$.

###### Proof.

  1. (1)

Essentially follows from [BV, 2.14].

  2. (2)

By [She09a, III.8.19] (see the proof of [Vas16a, 12.14]).

  3. (3)

Also by [She09a, III.8.19].

  4. (4)

Let
$\mathfrak{t}:=\mathfrak{s}^{up}\upharpoonright(\mathfrak{K}_{\mathfrak{s}}\cup\mathfrak{K}^{\text{sat}}_{[\lambda^{+},\lambda^{+\omega})})$.
As in the proof of Fact 7.17, we have enough locality for nonforking to
connect, so this works (see also [Vas17b, Appendix A]).

∎

While in general, it is not known how to build extendible good frames without
using the weak diamond (Fact 7.16), in the compact case it is possible:

######  Fact 7.21.

Let $\mathfrak{K}$ be a compact AEC. If $\mathfrak{K}$ is categorical in some
$\mu>\text{LS}(\mathfrak{K})$, then there exists a (unique) categorical very
good frame on $\mathfrak{K}^{\text{sat}}_{\geq\text{LS}(\mathfrak{K})^{+6}}$.

###### Proof.

Let $\kappa$ be such that $\mathfrak{K}$ is $\kappa$-compact. By Corollary
4.8, $\mathfrak{K}$ has amalgamation, no maximal models, is $(<\kappa)$-short,
and is categorical in a proper class of cardinals. By essentially the main
result of [Vas16a], and more precisely by [Vas17d, A.16],
${\mathfrak{K}}^{\lambda\text{-sat}}$ is what is called there fully good.
Here, we have set
$\lambda:=\left(\text{LS}(\mathfrak{K})^{<\kappa}\right)^{+5}$. Since $\kappa$
is strongly compact, we have by a result of Solovay (see the proof of [Jec03,
20.8]) that $\text{LS}(\mathfrak{K})^{<\kappa}\leq\text{LS}(\mathfrak{K})^{+}$
(of course this also holds if $\kappa=\aleph_{0}$). Thus
$\lambda\leq\text{LS}(\mathfrak{K})^{+6}$. Checking the definition of fully
good in [Vas16a, 8.4], we see that this implies that
$\mathfrak{K}^{\text{sat}}_{\geq\text{LS}(\mathfrak{K})^{+6}}$ carries a very
good two-dimensional independence relation, as desired. ∎

##  8\. Multidimensional independence

We define here the main notion of this paper: multidimensional independence
relations. This section contains mostly definitions and easy lemmas. A result
of importance is how to build multidimensional independence relations from the
two-dimensional relations (Theorem 8.20).

###  8.1. Systems

We start by defining what is meant by a system of models.

######  Definition 8.1.

For $\mathfrak{K}$ an abstract class and $I=(I,\leq)$ a partial order, an
_$(I,\mathfrak{K})$ -system_ is a sequence $\mathbf{m}=\langle M_{u}:u\in
I\rangle$ such that $u\leq v$ implies $M_{u}\leq_{\mathfrak{K}}M_{v}$. A
_$\mathfrak{K}$ -system_ is an $(I,\mathfrak{K})$-system for some $I$. For
$\mathbf{m}$ a $\mathfrak{K}$-system, we write $I(\mathbf{m})$ for the unique
$I$ such that $\mathbf{m}$ is an $(I,\mathfrak{K})$-system. When
$\mathfrak{K}$ is clear from context, we omit it from all the above
definitions.

We will often be interested in systems where all extensions are strict. In
fact, a strengthening of this, being proper, is more useful (one can see it as
an abstract version of Definition 3.5(4), when we take $A$ to consist of the
union of several models):

######  Definition 8.2.

Let $\mathbf{m}=\langle M_{u}:u\in I\rangle$ be an $I$-system. Let
$\mathfrak{K}^{\ast}$ be a skeleton of $\mathfrak{K}$ (the reader can think of
$\mathfrak{K}^{\ast}=\mathfrak{K}$ at first reading).

  1. (1)

Let $u\in I$. We say that $u$ is _$\mathfrak{K}^{\ast}$ -proper in
$\mathbf{m}$_ if $M_{u}\in\mathfrak{K}^{\ast}$ and there exists
$N\in\mathfrak{K}^{\ast}$ such that:

    1. (a)

For all $v\in I$, $v<u$ implies
$M_{v}\leq_{\mathfrak{K}}N<_{\mathfrak{K}^{\ast}}M_{u}$.

    2. (b)

For all $v\in I$, if $v<u$ and $M_{v}\in\mathfrak{K}^{\ast}$, then
$M_{v}\leq_{\mathfrak{K}^{\ast}}N$.

  2. (2)

For $I_{0}$ a sub-order of $I$, we say that $\mathbf{m}$ is
_$(I_{0},\mathfrak{K}^{\ast})$ -proper_ if every $u\in I_{0}$ is
$\mathfrak{K}^{\ast}$-proper in $\mathbf{m}$. When
$\mathfrak{K}^{\ast}=\mathfrak{K}$ and $I_{0}=I$, we omit them.

Note that a system where all extensions are strict may not be proper:

######  Example 8.3.

Let $\mathfrak{K}$ be the AEC of all infinite sets, ordered by subset. Let
$A\subseteq B$ be two countably infinite sets with $B\backslash A$ infinite.
Partition $B\backslash A$ into two infinite sets $C_{1}$ and $C_{2}$ and let
$A_{\ell}:=A\cup C_{\ell}$. Then the $\mathcal{P}(2)$-system
$(A,A_{1},A_{2},B)$ is not proper but all the extensions are strict.

The next two lemmas are easy properties of proper systems: being proper does
not depend on the exact indexing set, and proper systems can be built.

######  Lemma 8.4.

Let $I_{0}\subseteq I_{1}\subseteq I$ all be partial orders and let
$\mathbf{m}$ be an $I$-system. Let $\mathfrak{K}^{\ast}$ be a skeleton of
$\mathfrak{K}$. If $\mathbf{m}$ is $(I_{0},\mathfrak{K}^{\ast})$-proper, then
$\mathbf{m}\upharpoonright I_{1}$ is $(I_{0},\mathfrak{K}^{\ast})$-proper.

###### Proof.

Straightforward. ∎

######  Lemma 8.5.

Let $\mathfrak{K}^{\ast}$ be a skeleton of $\mathfrak{K}$. Let
$\mathbf{m}=\langle M_{u}:u\in I\rangle$ be an $I$-system with $I$ a finite
partial order and let $v$ be maximal in $I$. If $\mathfrak{K}$ has no maximal
models, then there exists $N\in\mathfrak{K}$ such that
$M_{v}\leq_{\mathfrak{K}}N$ and $v$ is $\mathfrak{K}^{\ast}$-proper in
$\mathbf{m}\upharpoonright(I\backslash\\{v\\})\smallfrown\langle N\rangle$.

###### Proof.

Using that $\mathfrak{K}$ has no maximal models, let $M\in\mathfrak{K}$ be
such that $M_{v}<_{\mathfrak{K}}M$. Now apply Lemma 3.9 with $M$, $\langle
M_{i}:i<n\rangle$ there standing for $M$, $\\{M_{u}\in\mathfrak{K}^{\ast}\mid
u<v\\}$ here. ∎

It is now time to define how systems can relate to each other, in particular
how they can be isomorphic, extensions, etc.

######  Definition 8.6.

  1. (1)

Let $I\subseteq J$ be partial orders and let $\mathbf{m}=\langle M_{u}:u\in
J\rangle$ be a $J$-system. Then $\mathbf{m}\upharpoonright I:=\langle
M_{u}:u\in I\rangle$.

  2. (2)

For $\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in I\rangle$, $\ell=1,2$ both
$I$-systems, say $\mathbf{m}_{1}\leq_{\mathfrak{K}}\mathbf{m}_{2}$ if
$M_{u}^{1}\leq_{\mathfrak{K}}M_{u}^{2}$ for all $u\in I$. We say that
$\mathbf{m}_{2}$ is a _disjoint_ extension of $\mathbf{m}_{1}$ (written
$\mathbf{m}_{1}\leq_{\mathfrak{K}}^{d}\mathbf{m}_{2}$) if
$\mathbf{m}_{1}\leq_{\mathfrak{K}}\mathbf{m}_{2}$ and $M_{u}^{2}\cap
M_{v}^{1}\subseteq M_{u}^{1}$ for all $u,v\in I$.

  3. (3)

[She09a, III.12.14(4)] For $k<\omega$, $\mathbf{m}_{\ell}=\langle
M_{u}^{\ell}:u\in I\rangle$, $\ell<k$ all $I$-systems with
$\mathbf{m}_{0}\leq_{\mathfrak{K}}\mathbf{m}_{1}\leq_{\mathfrak{K}}\ldots\leq_{\mathfrak{K}}\mathbf{m}_{k-1}$,
let
$\mathbf{m}:=\mathbf{m}_{0}\ast\mathbf{m}_{1}\ast\ldots\ast\mathbf{m}_{k-1}$
be the system defined as follows:

    1. (a)

It is indexed by the partial order $J:=I\times k$ ordered lexicographically,
so $\mathbf{m}=\langle M_{u}:u\in J\rangle$.

    2. (b)

For all $u\in I$ and $i<k$, $M_{(u,i)}=M_{u}^{i}$.

  4. (4)

For $\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in I\rangle$, $\ell=1,2$ both
$I$-systems, we say $f$ is a _system embedding_ from $\mathbf{m}_{1}$ to
$\mathbf{m}_{2}$ and write $f:\mathbf{m}_{1}\rightarrow\mathbf{m}_{2}$ if
$f=\langle f_{u}:u\in I\rangle$, for all $u\in I$, $f_{u}$ is a
$\mathfrak{K}$-embedding from $M_{u}^{1}$ to $M_{u}^{2}$, and $u\leq v$
implies $f_{u}\subseteq f_{v}$.

  5. (5)

For $\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in I\rangle$, $\ell=1,2$ both
$I$-systems, we say $f$ is an _isomorphism_ from $\mathbf{m}_{1}$ to
$\mathbf{m}_{2}$ and write $f:\mathbf{m}_{1}\cong\mathbf{m}_{2}$ if $f=\langle
f_{u}:u\in I\rangle$ is a system embedding from $\mathbf{m}_{1}$ to
$\mathbf{m}_{2}$ and for all $u\in I$, $f_{u}$ is an isomorphism from
$M_{u}^{1}$ onto $M_{u}^{2}$.

  6. (6)

For $\pi:I_{1}\cong I_{2}$ an isomorphism of partial orders,
$\mathbf{m}=\langle M_{u}:u\in I_{1}\rangle$ an $I_{1}$-system, let
$\pi(\mathbf{m})$ denote the $I_{2}$-system $\langle M_{\pi^{-1}(v)}:v\in
I_{2}\rangle$.

  7. (7)

A system $\mathbf{m}=\langle M_{u}:u\in I\rangle$ is called _disjoint_ if
whenever $u,v,w,u^{\ast}\in I$ are such that $u\leq v\leq u^{\ast}$ and $u\leq
w\leq u^{\ast}$, then $M_{u}\subseteq M_{v}\cap M_{w}$.

  8. (8)

A system $\mathbf{m}=\langle M_{u}:u\in I\rangle$ is called _fully disjoint_
if whenever $u,v,w\in I$ are such that $u\leq v$ and $u\leq w$, then
$M_{u}\subseteq M_{v}\cap M_{w}$.

######  Remark 8.7.

Every disjoint system is isomorphic to a fully disjoint system.

It is often useful to code systems as an abstract class:

######  Definition 8.8.

Let $\mathfrak{K}$ be an abstract class and let $I$ be a partial order. The
_vocabulary of $(\mathfrak{K},I)$-systems_ is
$\tau^{I}:=\tau(\mathfrak{K})\cup\\{P_{i}:i\in I\\}$, where each $P_{i}$ is a
new unary predicate. The _abstract class of $(\mathfrak{K},I)$-systems_ is the
abstract class $\mathfrak{K}^{I}=(K^{I},\leq_{\mathfrak{K}^{I}})$ defined as
follows:

  1. (1)

$K^{I}$ consists of all the $\tau^{I}$-structures $M$ such that $\langle
M^{P_{i}}:i\in I\rangle$ forms a fully disjoint $(\mathfrak{K},I)$-system (we
see $M^{P_{i}}$ as a $\tau(\mathfrak{K})$-structure). We identify the elements
of $K^{I}$ with the corresponding systems.

  2. (2)

$\mathbf{m}_{1}\leq_{\mathfrak{K}^{I}}\mathbf{m}_{2}$ if and only if
$\mathbf{m}_{1}\leq_{\mathfrak{K}}^{d}\mathbf{m}_{2}$ (see Definition 8.6(2)).

###  8.2. Multidimensional independence relations

Multidimensional independence relations consist of systems indexed by a
certain class of semilattices. For reasons that will become apparent, we
require that this class has a certain amount of closure:

######  Definition 8.9.

A class $\mathcal{I}$ of semilattices is called _closed_ if:

  1. (1)

$\mathcal{I}$ is closed under isomorphisms.

  2. (2)

$\mathcal{I}$ is closed under taking initial segments.

  3. (3)

For any $I\in\mathcal{I}$ and any $u\in I\backslash\\{\bot\\}$,
$[u,\infty)_{I}\times\\{0,1\\}\in\mathcal{I}$.

The main examples of a closed class of semilattice are:

######  Lemma 8.10.

Let $n<\omega$ and let $\mathcal{I}$ be the class of all initial semilattices
isomorphic to an initial segments of $\mathcal{P}(n)$. Then $\mathcal{I}$ is
closed.

###### Proof.

The third condition of Definition 8.9 is the only non-trivial one. Let
$I\in\mathcal{I}$, and identify it with an initial segment of
$\mathcal{P}(n)$. Let $u\in I\backslash\\{\bot\\}$ (so $u\neq\emptyset$). Then
$[u,\infty)_{I}$ is the set of all subsets of $\mathcal{P}(n)$ extending $u$.
As a semilattice, it is isomorphic to $\mathcal{P}(n\backslash
u)\cong\mathcal{P}(n-|u|)$. Thus
$[u,\infty)_{I}\times\\{0,1\\}\cong\mathcal{P}(n-|u|)\times\mathcal{P}(1)\cong\mathcal{P}(n-|u|+1)$,
which since $|u|\geq 1$ is an initial segment of $\mathcal{P}(n)$. ∎

A multidimensional independence relation consists of an abstract class
$\mathfrak{K}$, a closed class of finite semilattices, and a class of systems
indexed by these semilattices. Several properties are required, akin to the
requirements in the definition of a two-dimensional independence relation. A
complication arises in the monotonicity properties: there are several and may
not always seem natural. Nevertheless they will hold for the relation we
construct and are all used in the next section. Note that we do not know
whether a multidimensional independence relations are canonical (i.e. whether
under reasonable conditions there can be at most one satisfying this list of
axiom).

######  Definition 8.11.

A _multidimensional independence relation (or notion)_ is a triple
$\mathfrak{i}=(\mathfrak{K},\mathcal{I},\operatorname{NF})$, where
$\mathfrak{K}=\mathfrak{K}_{\mathfrak{i}}$ is an abstract class,
$\mathcal{I}=\mathcal{I}_{\mathfrak{i}}$ is a class of semilattices, and
$\operatorname{NF}=\operatorname{NF}_{\mathfrak{i}}$ is a class of
$(\mathfrak{K},I)$-systems with $I\in\mathcal{I}$. We call the members of
$\operatorname{NF}$ _independent systems_. We require the following
properties:

  1. (1)

$\mathcal{I}$ is a _closed_ class of _finite_ semilattices.

  2. (2)

Invariance: If $\mathbf{m}$ and $\mathbf{m}^{\prime}$ are systems and
$f:\mathbf{m}\cong\mathbf{m}^{\prime}$, then $\operatorname{NF}(\mathbf{m})$
if and only if $\operatorname{NF}(\mathbf{m}^{\prime})$.

  3. (3)

Nontriviality: $\operatorname{NF}\neq\emptyset$ and for every
$I\in\mathcal{I}$ and every $M\in\mathfrak{K}$, there exists an independent
$I$-system containing $M$.

  4. (4)

Disjointness: Every independent system is disjoint (see Definition 8.6(7)).

  5. (5)

Symmetry: If $I_{1},I_{2}\in\mathcal{I}$, $\pi:I_{1}\cong I_{2}$ is an
isomorphism, and $\mathbf{m}$ is an $I_{1}$-system, then
$\operatorname{NF}(\mathbf{m})$ if and only if
$\operatorname{NF}(\pi(\mathbf{m}))$, where $\pi(\mathbf{m})$ is the
$I_{2}$-system naturally induced by $\pi$ (see Definition 8.6(6)).

  6. (6)

Transitivity: If
$\mathbf{m}_{0}\leq_{\mathfrak{K}}\mathbf{m}_{1}\leq_{\mathfrak{K}}\mathbf{m}_{2}$,
$\mathbf{m}_{0}\ast\mathbf{m}_{1}$ is independent, and
$\mathbf{m}_{1}\ast\mathbf{m}_{2}$ is independent, then
$\mathbf{m}_{0}\ast\mathbf{m}_{2}$ is independent.

  7. (7)

Monotonicity 1: If $\mathbf{m}$ is an $I$-system,
$\operatorname{NF}(\mathbf{m})$, and $J\subseteq I$ is such that
$J\in\mathcal{I}$, then $\operatorname{NF}(\mathbf{m}\upharpoonright J)$.

  8. (8)

Monotonicity 2: If $\mathbf{m}=\langle M_{u}:u\in I\rangle$ is an $I$-system,
$v$ is a maximal element in $I$, and $M_{v}\leq_{\mathfrak{K}}N$, then
$\mathbf{m}$ is independent if and only if $\langle M_{u}:u\in
I\backslash\\{v\\}\rangle\smallfrown\langle N\rangle$ is independent.

  9. (9)

Monotonicity 3: If $\mathbf{m}$ is an $I$-system, then $\mathbf{m}$ is
independent if and only if for any maximal $v\in I$,
$\mathbf{m}\upharpoonright\\{u\in I:u\leq v\\}$ is independent.

  10. (10)

Monotonicity 4: Let $I_{1},I_{2}\subseteq I$ all be in $\mathcal{I}$. Let
$\mathbf{m}$ be an $I$-system. If:

    1. (a)

$I_{1}$ is an initial segment of $I$.

    2. (b)

$I_{1}\cap I_{2}$ is cofinal in $I_{1}$.

    3. (c)

$I_{1}\cup I_{2}=I$.

    4. (d)

$\mathbf{m}\upharpoonright I_{1}$ and $\mathbf{m}\upharpoonright I_{2}$ are
independent.

Then $\mathbf{m}$ is independent.

  11. (11)

Monotonicity 5: Let $I\in\mathcal{I}$, let $u\in I\backslash\\{\bot\\}$, and
let $I_{0}:=[u,\infty)_{I}$. Let $\mathbf{m}$, $\mathbf{m}^{\prime}$ be
$I$-systems, $\mathbf{m}_{0}:=\mathbf{m}\upharpoonright I_{0}$ be independent,
and $\mathbf{m}_{0}^{\prime}:=\mathbf{m}^{\prime}\upharpoonright I_{0}$ be
such that $\mathbf{m}_{0}\leq_{\mathfrak{K}}\mathbf{m}_{0}^{\prime}$ and
$\mathbf{m}_{0}\ast\mathbf{m}_{0}^{\prime}$ is independent. Assume that
$\mathbf{m}\upharpoonright(I\backslash
I_{0})=\mathbf{m}^{\prime}\upharpoonright(I\backslash I_{0})$. Then
$\mathbf{m}$ is independent if and only if $\mathbf{m}^{\prime}$ is
independent.

The natural restrictions can be defined on a multidimensional independence
relation:

######  Definition 8.12.

Let $\mathfrak{i}=(\mathfrak{K},\mathcal{I},\operatorname{NF})$ be a
multidimensional independence relation.

  1. (1)

For $\mathcal{J}\subseteq\mathcal{I}$ a class of partial orders, we let
$\mathfrak{i}\upharpoonright\mathcal{J}$ denote the multidimensional
independence relation
$\mathfrak{j}=(\mathfrak{K},\mathcal{J},\operatorname{NF}\upharpoonright\mathcal{J})$,
where $\operatorname{NF}\upharpoonright\mathcal{J}$ denotes the class of
systems in $\operatorname{NF}$ that are $J$-systems for $J\in\mathcal{J}$.

  2. (2)

For $\mathfrak{K}^{\ast}$ a sub-abstract class of $\mathfrak{K}$, let
$\mathfrak{i}\upharpoonright\mathfrak{K}^{\ast}$ denote the multidimensional
independence relation
$\mathfrak{j}=(\mathfrak{K}^{\ast},\mathcal{I},\operatorname{NF}\upharpoonright\mathfrak{K}^{\ast})$,
where $\operatorname{NF}\upharpoonright\mathfrak{K}^{\ast}$ denotes the class
of systems in $\operatorname{NF}$ that are also $\mathfrak{K}^{\ast}$-systems.
We may write $\mathfrak{i}_{\lambda}$ instead of
$\mathfrak{i}\upharpoonright\mathfrak{K}_{\lambda}$.

The next lemma shows that, under some closure conditions on $\mathcal{I}$,
transitivity follows from the other axioms:

######  Lemma 8.13.

Let $\mathfrak{i}$ be a multidimensional independence relation. Let
$\mathbf{m}_{0},\mathbf{m}_{1},\mathbf{m}_{2}$ be $I$-systems with
$I\in\mathcal{I}_{\mathfrak{i}}$. If
$\mathbf{m}_{0}\leq_{\mathfrak{K}_{\mathfrak{i}}}\mathbf{m}_{1}\leq_{\mathfrak{K}_{\mathfrak{i}}}\mathbf{m}_{2}$,
$\mathbf{m}_{0}\ast\mathbf{m}_{1}$ is independent,
$\mathbf{m}_{1}\ast\mathbf{m}_{2}$ is independent, and $I\times
3\in\mathcal{I}_{\mathfrak{i}}$, then
$\mathbf{m}_{0}\ast\mathbf{m}_{1}\ast\mathbf{m}_{2}$ is independent. In
particular, if $\mathcal{I}_{\mathfrak{i}}$ is closed under products the
transitivity axiom follows from the others.

###### Proof.

Say the $\mathbf{m}_{\ell}$’s are $I$-systems. Let $J:=I\times 3$. Let
$\mathbf{m}:=(\mathbf{m}_{0}\ast\mathbf{m}_{1}\ast\mathbf{m}_{2}$. We check
that $\mathbf{m}$ is independent by using monotonicity 4: let
$I_{1}:=I\times\\{0,1\\}$ and let $I_{2}:=I\times\\{1,2\\}$. We have that
$\mathbf{m}\upharpoonright I_{1}=\mathbf{m}_{1}\ast\mathbf{m}_{2}$ and
$\mathbf{m}\upharpoonright I_{2}=\mathbf{m}_{2}\ast\mathbf{m}_{3}$. Both are
independent by assumption. Moreover, $I_{1}\cup I_{2}=J$, $I_{1}$ is an
initial segment of $J$, and $I_{1}\cap I_{2}=I\times\\{1\\}$ is cofinal in
$I_{1}$. Thus monotonicity 4 implies that $\mathbf{m}$ is independent. The “in
particular” part follows from monotonicity 1. ∎

One wants to order systems so that if $\mathbf{m}_{2}$ extends
$\mathbf{m}_{1}$, then the entire diagram that this forms is independent. This
is formalized by looking at $\mathbf{m}_{1}\ast\mathbf{m}_{2}$:

######  Definition 8.14.

Let $\mathfrak{i}$ be a multidimensional independence relation. We define a
binary relation $\leq_{\mathfrak{i}}$ on $\operatorname{NF}_{\mathfrak{i}}$ as
follows: $\mathbf{m}_{1}\leq_{\mathfrak{i}}\mathbf{m}_{2}$ if and only if
$\mathbf{m}_{1}\leq_{\mathfrak{K}}\mathbf{m}_{2}$ (so in particular
$I(\mathbf{m}_{1})=I(\mathbf{m}_{2})$) and
$\operatorname{NF}_{\mathfrak{i}}(\mathbf{m}_{1}\ast\mathbf{m}_{2})$. Let
$\mathfrak{K}_{\mathfrak{i},I}$ be the sub-abstract class of