$\mathfrak{K}^{I}$ (Definition 8.8) consisting of all independent $I$-systems
and ordered by $\leq_{\mathfrak{i}}$.

One may want in addition to require that the system
$\mathbf{m}_{1}\ast\mathbf{m}_{2}$ is _proper_ :

######  Definition 8.15.

Let $\mathfrak{i}$ be a multidimensional independence relation. Let
$I\in\mathcal{I}_{\mathfrak{i}}$. We let
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},I}$ be the class of _proper_
independent $I$-systems ordered by
$\mathbf{m}_{1}\leq_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i},I}}\mathbf{m}_{2}$
if and only if either $\mathbf{m}_{1}=\mathbf{m}_{2}$ or
$\mathbf{m}_{1}\ast\mathbf{m}_{2}$ is a _proper_ independent system.

Absent from the properties in Definition 8.11 were any kind of extension or
uniqueness. We define these now. Note that existence says there _is_ a system,
extension that an _existing_ system can be extended. Strong uniqueness will
hold only in classes of brimmed models, where also the extensions are brimmed:
it says in particular that there is only one proper system indexed by a given
lattice. Uniqueness is the property that will hold in the original class: it
says that each independent system has at most one completion, in the sense
that any two completions amalgamate.

######  Definition 8.16.

Let $\mathfrak{i}=(\mathfrak{K},\mathcal{I},\operatorname{NF})$ be a
multidimensional independence relation. The following are additional
properties that $\mathfrak{i}$ may have:

  1. (1)

Existence: For any $I\in\mathcal{I}$ there exists a proper independent
$I$-system.

  2. (2)

Extension: For any $I,J\in\mathcal{I}$ with $I$ an initial segment of $J$, if
$\mathbf{m}$ is an independent $I$-system, then there exists a $(J\backslash
I)$-proper independent $J$-system $\mathbf{m}^{\prime}$ such that
$\mathbf{m}\cong\mathbf{m}^{\prime}\upharpoonright I$.

  3. (3)

Strong uniqueness: Let $\mathbf{m}_{1}$, $\mathbf{m}_{2}$ be independent
$J$-systems and let $I$ be an initial segment of $J$ with $I\in\mathcal{I}$.
Let $f:\mathbf{m}_{1}\upharpoonright I\cong\mathbf{m}_{2}\upharpoonright I$.
If $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ are $(J\backslash I)$-proper
(Definition 8.2), then there exists $g:\mathbf{m}_{1}\cong\mathbf{m}_{2}$
extending $f$.

  4. (4)

Uniqueness: Let $\mathbf{m}_{1}$, $\mathbf{m}_{2}$ be independent $J$-systems
and let $I$ be an initial segment of $J$ with $J=I\cup\\{v\\}$, $I<v$. Let
$f:\mathbf{m}_{1}\upharpoonright I\cong\mathbf{m}_{2}\upharpoonright I$. Then
there exists an independent $J$-system $\mathbf{m}$ and systems embeddings
(see Definition 8.6(4) $g_{\ell}:\mathbf{m}_{\ell}\rightarrow\mathbf{m}$,
$\ell=1,2$ such that $g_{\ell}$ extends $f$ for $\ell=1,2$.

Any reasonable multidimensional independence relation induces a two-
dimensional independence relation:

######  Definition 8.17.

Let $\mathfrak{i}$ be a multidimensional independence relation with
$\mathcal{P}(2)\in\mathcal{I}_{\mathfrak{i}}$. Define 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$}}}\limits=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i})$ by
$M_{1}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{M_{3}}M_{2}$ if and only if
$(M_{0},M_{1},M_{2},M_{3})$ (seen as a system indexed by $\mathcal{P}(2)$) is
an independent system.

######  Lemma 8.18.

If $\mathfrak{i}$ is a multidimensional independence relation with existence,
extension, and uniqueness such that
$\mathcal{P}(2)\in\mathcal{I}_{\mathfrak{i}}$, 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$}}}\limits(\mathfrak{i})$ (from Definition 8.17)
is a two-dimensional independence notion on $\mathfrak{K}_{\mathfrak{i}}$.

###### Proof.

The least trivial axioms to prove are monotonicity and symmetry. Symmetry can
be obtained by using the symmetry property of multidimensional independence
relations with the automorphism of $\mathcal{P}(2)$ swapping $\\{0\\}$ and
$\\{1\\}$. Monotonicity actually follows from the other axioms, see [LRV,
3.23]. ∎

###  8.3. Building a multidimensional independence relation

Starting from a two-dimensional independence relation, one can also build a
multidimensional independence relation in the natural way:

######  Definition 8.19.

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 an abstract class $\mathfrak{K}$. Let
$\mathfrak{i}=\mathfrak{i}({\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits})$ be the following multidimensional
independence relation:

  1. (1)

$\mathfrak{K}_{\mathfrak{i}}=\mathfrak{K}$.

  2. (2)

$\mathcal{I}_{\mathfrak{i}}$ is the class of all finite semilattices.

  3. (3)

For $I\in\mathcal{I}_{\mathfrak{i}}$ and $\mathbf{m}=\langle M_{u}:u\in
I\rangle$ an $I$-system, $\mathbf{m}\in\operatorname{NF}_{\mathfrak{i}}$ if
and only if whenever $u_{1},u_{2},v\in I$ and $u_{1}\leq v$, $u_{2}\leq v$,
then $M_{u_{1}}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land
u_{2}}}^{M_{v}}M_{u_{2}}$.

######  Theorem 8.20.

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 two-dimensional independence
notion on an abstract class $\mathfrak{K}$ with no maximal models, then
$\mathfrak{i}=\mathfrak{i}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$ is a multidimensional independence
notion such 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$}}}\limits(\mathfrak{i})=\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$. Moreover, $\mathfrak{i}$ restricted
to the class of initial segments of $\mathcal{P}(2)$ has existence, extension,
and uniqueness.

###### Proof.

Most of the axioms follow directly from the definitions. We only prove:

  * •

Monotonicity 4: Let $I_{1},I_{2}$, $\mathbf{m}=\langle M_{u}:u\in I\rangle$ be
as in the statement of monotonicity 4. Let $u_{1},u_{2},v\in I$ with
$u_{1}\leq v$, $u_{2}\leq v$. We have to see that
$M_{u_{1}}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land
u_{2}}}^{M_{v}}M_{u_{2}}$. If both $u_{1}$ and $u_{2}$ are in $I_{1}$, or both
are in $I_{2}$, then this is immediate from the assumptions that
$\mathbf{m}\upharpoonright I_{1}$ and $\mathbf{m}\upharpoonright I_{2}$ are
both independent (and preservation 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$}}}\limits$ under $\mathfrak{K}$-embeddings).
Assume now that one is in $I_{1}\backslash I_{2}$ and the other in
$I_{2}\backslash I_{1}$. By symmetry, without loss of generality $u_{1}\in
I_{1}\backslash I_{2}$, $u_{2}\in I_{2}\backslash I_{1}$.

By assumption, $I_{1}\cap I_{2}$ is maximal in $I$ and $I=I_{1}\cup I_{2}$, so
there exists $u^{\prime}\in I_{2}$ such that $u_{1}\leq u^{\prime}$. Let
$u_{1}^{\prime}:=\land\\{u^{\prime}\in I_{2}\cap I_{1}\mid u_{1}\leq
u^{\prime}\\}$. In other words, $u_{1}^{\prime}$ is the minimal element of
$I_{1}\cap I_{2}$ such that $u_{1}\leq u_{1}^{\prime}$. By minimality, we must
have that $u_{1}^{\prime}\leq v$ (otherwise, consider
$u_{1}^{\prime\prime}:=u_{1}^{\prime}\land v$; since $I_{1}$ is an initial
segment, this is in $I_{1}\land I_{2}$ and is still above $u_{1}$). By the
previous paragraph, we have that $M_{u_{1}^{\prime}}\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}^{\prime}\land
u_{2}}}^{M_{v}}M_{u_{2}}$. Also by the previous paragraph (recalling that
$I_{1}$ is an initial segment, so $u_{1}^{\prime}\land u_{2}\in I_{1}\cap
I_{2}$), $M_{u_{1}}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land u_{1}^{\prime}\land
u_{2}}}^{M_{u_{1}^{\prime}}}M_{u_{1}^{\prime}\land u_{2}}$. By transitivity
for $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$, $M_{u_{1}}\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land u_{1}^{\prime}\land
u_{2}}}^{M_{v}}M_{u_{2}}$. Now note that $u_{1}\leq u_{1}^{\prime}$ by
assumption, hence $u_{1}\land u_{1}^{\prime}\land u_{2}=u_{1}\land u_{2}$, as
desired.

  * •

Monotonicity 5: Fix the data given by the statement of monotonicity 5. Write
$\mathbf{m}=\langle M_{u}:u\in I\rangle$ and $\mathbf{m}^{\prime}=\langle
M_{u}^{\prime}:u\in I\rangle$.

    * –

We first show that $\mathbf{m}^{\prime}$ independent implies that $\mathbf{m}$
is independent. Let $u_{1},u_{2},u_{3}\in I$ be such that $u_{1},u_{2}\leq
u_{3}$. We want to see that $M_{u_{1}}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land
u_{2}}}^{M_{u_{3}}}M_{u_{2}}$. There are several cases:

      * *

If $u_{1},u_{2}\notin I_{0}$, then $u_{1}\land u_{2}\notin I_{0}$ (as
$I\backslash I_{0}\subseteq I$), hence the result follows directly from the
assumption that $M_{u_{1}}^{\prime}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land
u_{2}}^{\prime}}^{M_{u_{3}}^{\prime}}M_{u_{2}}^{\prime}$ and ambient
monotonicity.

      * *

If $u_{1},u_{2}\in I_{0}$, then apply transitivity.

      * *

If $u_{1}\in I_{0}$, $u_{2}\notin I_{0}$, and $u_{1}\land u_{2}\in I_{0}$,
apply transitivity again.

      * *

If $u_{1}\in I_{0}$, $u_{2}\notin I_{0}$, and $u_{1}\land u_{2}\notin I_{0}$,
use monotonicity.

    * –

Assume now that $\mathbf{m}$ is independent. Let $u_{1},u_{2},u_{3}\in I$ be
such that $u_{1},u_{2}\leq u_{3}$. We want to see that
$M_{u_{1}}^{\prime}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u_{1}\land
u_{2}}^{\prime}}^{M_{u_{3}}^{\prime}}M_{u_{2}}^{\prime}$ and we again split
into cases. The cases $u_{1},u_{2}\notin I_{0}$ and $u_{1},u_{2}\in I_{0}$ are
similar to before. If $u_{1}\in I_{0}$, $u_{2}\notin I_{0}$, then $u_{3}\in
I_{0}$ since $I_{0}$ is and end segment. Thus we can apply transitivity and we
are done.

  * •

Transitivity: follows from Lemma 8.13.

That $\mathfrak{i}$ restricted to the class of initial segments of
$\mathcal{P}(2)$ has existence, extension, and uniqueness follows from the
corresponding 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$}}}\limits$: we only have to note that a
$\mathcal{P}(2)$-system $\langle M_{u}:u\in\mathcal{P}(2)\rangle$ is
independent if and only if $M_{\\{0\\}}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{\emptyset}}^{M_{2}}M_{\\{1\\}}$,
by the symmetry axiom 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$}}}\limits$. ∎

##  9\. Finite combinatorics of multidimensional independence

In this section, we assume:

######  Hypothesis 9.1.

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

We consider properties of $\mathfrak{i}$ that are finitary in nature. That is,
they do not depend on any kind of closure under unions of chains. This allows
us to work at a high level of generality (for example, $\mathfrak{K}$ is just
assumed to be an abstract class). A crucial question is whether properties
(such as extension or uniqueness) that hold for brimmed systems (that is, for
systems consisting of brimmed models, ordered by being “brimmed over”) will
also hold for the not necessarily brimmed other systems. We prove general
positive results in this direction here (Theorems 9.12 and 9.15). The
statements use the notion of a skeleton. This is also a place where we use
that $\mathcal{I}$ is closed (Definition 8.9), as well as the monotonicity
axioms in Definition 8.11.

We start by proving a few easy consequences of the definition of a
multidimensional independence relation. First, $\mathcal{P}(1)$-systems are
trivial:

######  Lemma 9.2.

$M\leq_{\mathfrak{K}}N$ both be in $\mathfrak{K}$. If
$\mathcal{P}(1)\in\mathcal{I}$, then $\langle M,N\rangle$ is an independent
$I$-system.

###### Proof.

By nontriviality, there is an independent $\mathcal{P}(1)$-system $\mathbf{m}$
containing $M$. Write $\mathbf{m}=\langle M,N^{\prime}\rangle$. By
monotonicity 2, $\langle M,M\rangle$ is independent. By monotonicity 2 again,
$\langle M,N\rangle$ is independent. ∎

It is also easy to see that existence is weaker than extension:

######  Lemma 9.3.

If $\mathfrak{i}$ has extension, then $\mathfrak{i}$ has existence.

###### Proof.

Let $I\in\mathcal{I}$. Start with the empty system and extend it to a proper
independent $I$-system. ∎

Uniqueness is also weaker than strong uniqueness:

######  Lemma 9.4.

If $\mathfrak{i}$ has strong uniqueness and $\mathfrak{K}$ has no maximal
models, then $\mathfrak{i}$ has uniqueness.

###### Proof.

Given two models $M_{1},M_{2}$ to amalgamate over an independent system
$\mathbf{m}$, take a strict extension $M_{\ell}^{\prime}$ of $M_{\ell}$ and
use strong uniqueness to see that $M_{1}^{\prime}$ is isomorphic to
$M_{2}^{\prime}$ over $\mathbf{m}$, hence $M_{1}$ and $M_{2}$ amalgamate over
$\mathbf{m}$. ∎

###  9.1. Skeletons of independence relations

It is natural to consider what happens to $\mathfrak{i}$ when restricting it
to a skeleton of $\mathfrak{K}$. We overload terminology and also call such
restrictions _skeletons_ of $\mathfrak{i}$:

######  Definition 9.5.

Let $\mathfrak{i}$ and $\mathfrak{i}^{\ast}$ be independence relations. We say
that $\mathfrak{i}^{\ast}$ is a _skeleton_ of $\mathfrak{i}$ if:

  1. (1)

$\mathfrak{K}_{\mathfrak{i}^{\ast}}$ is a skeleton of
$\mathfrak{K}_{\mathfrak{i}}$.

  2. (2)

$\mathfrak{i}^{\ast}=\mathfrak{i}\upharpoonright\mathfrak{K}_{\mathfrak{i}^{\ast}}$.

######  Remark 9.6.

If $\mathfrak{K}^{\ast}$ is a skeleton of $\mathfrak{K}$, then it is
straightforward to check that $\mathfrak{i}\upharpoonright\mathfrak{K}^{\ast}$
will be an independence relation.

We will consider the following localized versions of extension and uniqueness:

######  Definition 9.7.

Let $\mathbf{m}$ be an independent $I$-system and let $\mathfrak{K}^{\ast}$ be
a skeleton of $\mathfrak{K}$.

  1. (1)

We call $\mathbf{m}$ a _$\mathfrak{K}^{\ast}$ -extension base in
$\mathfrak{i}$_ if for any $J\in\mathcal{I}$ such that $J=I\cup\\{v\\}$ and
$I<v$, there exists an independent $J$-system $\mathbf{m}^{\prime}$ such that
$v$ is $\mathfrak{K}^{\ast}$-proper in $\mathbf{m}^{\prime}$ and
$\mathbf{m}\cong\mathbf{m}^{\prime}\upharpoonright I$. When
$\mathfrak{K}^{\ast}=\mathfrak{K}$ and $\mathfrak{i}$ is clear from context,
we omit them and call $\mathbf{m}$ an _extension base_.

  2. (2)

We call $\mathbf{m}$ a _$\mathfrak{K}^{\ast}$ -strong uniqueness base in
$\mathfrak{i}$_ if for any $J\in\mathcal{I}$ such that $J=I\cup\\{v\\}$ and
$I<v$ and any two independent $J$-systems $\mathbf{m}_{1},\mathbf{m}_{2}$ such
that $\mathbf{m}_{1}\upharpoonright I=\mathbf{m}_{2}\upharpoonright
I=\mathbf{m}$ and $v$ is $\mathfrak{K}^{\ast}$-proper in both $\mathbf{m}_{1}$
and $\mathbf{m}_{2}$, we have that
$\mathbf{m}_{1}\cong_{\mathbf{m}}\mathbf{m}_{2}$. As before, when
$\mathfrak{K}^{\ast}=\mathfrak{K}$ and $\mathfrak{i}$ is clear from context,
we omit them.

  3. (3)

We call $\mathbf{m}$ a _uniqueness base in $\mathfrak{i}$_ if for any
$J\in\mathcal{I}$ such that $J=I\cup\\{v\\}$ and $I<v$ and any two independent
$J$-systems $\mathbf{m}_{1},\mathbf{m}_{2}$ such that
$\mathbf{m}_{1}\upharpoonright I=\mathbf{m}_{2}\upharpoonright I=\mathbf{m}$,
we have that there is an $\mathfrak{i}$-independent $J$-system
$\mathbf{m}^{\ast}$ and
$f_{\ell}:\mathbf{m}_{\ell}\xrightarrow[\mathbf{m}]{}\mathbf{m}^{\ast}$
systems embeddings for $\ell=1,2$. When $\mathfrak{i}$ is clear from context,
we omit it.

It turns out that the parameter $\mathfrak{K}^{\ast}$ in the definition of a
$\mathfrak{K}^{\ast}$-extension base can safely be omitted. This will be used
without comments:

######  Lemma 9.8.

Let $\mathbf{m}$ be an $\mathfrak{i}$-independent system and let
$\mathfrak{K}^{\ast}$ be a skeleton of $\mathfrak{K}$. Let
$\mathfrak{i}^{\ast}:=\mathfrak{i}\upharpoonright\mathfrak{K}^{\ast}$.

  1. (1)

$\mathbf{m}$ is a $\mathfrak{K}$-extension base in $\mathfrak{i}$ if and only
if $\mathbf{m}$ is a $\mathfrak{K}^{\ast}$-extension base in $\mathfrak{i}$.

  2. (2)

If $\mathbf{m}$ is $\mathfrak{i}^{\ast}$-independent, then $\mathbf{m}$ is a
$\mathfrak{K}^{\ast}$-extension base in $\mathfrak{i}^{\ast}$ if and only if
$\mathbf{m}$ is a $\mathfrak{K}^{\ast}$-extension base in $\mathfrak{i}$.

###### Proof.

Use monotonicity 2 and the proof of Lemma 8.5. ∎

Of course, having extension is the same as all independent systems being
extension bases, and similarly for the other properties:

######  Lemma 9.9.

$\mathfrak{i}$ has extension [uniqueness] [strong uniqueness] if and only if
every independent system is an extension [uniqueness] [strong uniqueness]
base.

###### Proof.

Straightforward. ∎

Note also that the uniqueness properties transfers down from $\mathfrak{i}$ to
$\mathfrak{i}^{\ast}$:

######  Lemma 9.10.

Let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$. If $\mathfrak{i}$
has [strong] uniqueness, then $\mathfrak{i}^{\ast}$ has [strong] uniqueness.

###### Proof.

Straightforward using the basic properties of a skeleton. ∎

The extension property also transfers. In fact, more can be said:

######  Lemma 9.11.

Let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$. Write
$\mathfrak{K}:=\mathfrak{K}_{\mathfrak{i}}$,
$\mathfrak{K}^{\ast}:=\mathfrak{K}_{\mathfrak{i}^{\ast}}$. Assume that
$\mathfrak{i}$ has extension. Let $I,J\in\mathcal{I}$ with $I$ an initial
segment of $J$. Let $\mathbf{m}$ be an $\mathfrak{i}$-independent $I$-system.
There exists an $\mathfrak{i}$-independent $J$-system $\mathbf{m}^{\prime}$
such that:

  1. (1)

$\mathbf{m}\cong\mathbf{m}^{\prime}\upharpoonright J$.

  2. (2)

$\mathbf{m}^{\prime}$ is $(J\backslash I)$-$\mathfrak{K}^{\ast}$-proper (see
Definition 8.2).

In particular, $\mathfrak{i}^{\ast}$ has extension.

###### Proof.

As in the proof of Lemma 8.5. ∎

We now turn to the harder problem of getting the properties in $\mathfrak{i}$
if we know them in the skeleton. For extension, this can be done provided that
the underlying class is the same (the ordering may be different). The example
to keep in mind is $\mathfrak{K}$ being the class of saturated models of
cardinality $\lambda$ in a given AEC, and $\mathfrak{K}^{\ast}$ consisting of
the same class, but ordered by “being brimmed over”. Note however that the
theorem below also has content when $\mathfrak{i}=\mathfrak{i}^{\ast}$: it
says that it is enough to prove that _proper_ systems are extension bases.

######  Theorem 9.12.

Let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$ such that
$|\mathfrak{K}_{\mathfrak{i}^{\ast}}|=|\mathfrak{K}_{\mathfrak{i}}|$. If every
proper $\mathfrak{i}^{\ast}$-independent system is an extension base, then
$\mathfrak{i}$ has extension.

###### Proof.

Write $\mathfrak{K}:=\mathfrak{K}_{\mathfrak{i}}$,
$\mathfrak{K}^{\ast}:=\mathfrak{K}_{\mathfrak{i}^{\ast}}$. For $\mathbf{m}$ an
$\mathfrak{i}$-independent $I$-system, let $k(\mathbf{m})$ be the number of
$u\in I$ which are _not_ $\mathfrak{K}^{\ast}$-proper in $\mathbf{m}$. We
prove that every $\mathfrak{i}$-independent system $\mathbf{m}$ is an
extension base by induction on $k(\mathbf{m})$.

Let $\mathbf{m}$ be an independent $I$-system and let
$J=I\cup\\{v\\}\in\mathcal{I}$, with $I<v$. If $k(\mathbf{m})=0$, then
$\mathbf{m}$ is an extension base by assumption. Assume now that
$k(\mathbf{m})>0$. Let $u\in I$ be such that $u$ is _not_
$\mathfrak{K}^{\ast}$-proper in $\mathbf{m}$. Note that $u\neq\bot$. Let
$I_{0}:=[u,\infty)_{I}$ and let $\mathbf{m}_{0}:=\mathbf{m}\upharpoonright
I_{0}$. Note that:

  1. (1)

$I_{0}\subseteq I$ and $I_{0}\times\\{0,1\\}\in\mathcal{I}$ (by (1) in the
definition of a multidimensional independence relation), hence also
$I_{0}\in\mathcal{I}$ (it is isomorphic to an initial segment of
$I_{0}\times\\{0,1\\}$).

  2. (2)

$I\backslash I_{0}$ is an initial segment of $I$, hence is in $\mathcal{I}$.

  3. (3)

$k(\mathbf{m}_{0})<k(\mathbf{m})$, since $u$ is minimal in $I_{0}$, hence,
since $|\mathfrak{K}^{\ast}|=|\mathfrak{K}|$, $\mathfrak{K}^{\ast}$-proper in
$\mathbf{m}_{0}$ (we are also using that properness is preserved when passing
to subsystems, i.e. Lemma 8.4).

Let $\mathbf{m}_{0}^{\prime}$ be an $\mathfrak{i}^{\ast}$-independent proper
$I_{0}$-system such that
$\mathbf{m}_{0}\leq_{\mathfrak{i}}\mathbf{m}_{0}^{\prime}$. To see that this
exists, we apply (the proof of) Lemma 9.11, where
$\mathfrak{i},\mathfrak{i}^{\ast},I,J$ there stand for
$\mathfrak{i}\upharpoonright\operatorname{NF}^{\ast}$,
$\mathfrak{i}^{\ast}\upharpoonright\operatorname{NF}^{\ast},I_{0}\times\\{0\\},I_{0}\times\\{0,1\\}$
here, and we have set $\operatorname{NF}^{\ast}$ to be the class of
$\mathfrak{i}$-independent systems $\mathbf{m}^{\ast}$ with
$k(\mathbf{m}^{\ast})<k(\mathbf{m})$. We obtain an
$(I_{0}\times\\{1\\},\mathfrak{K}^{\ast})$-proper independent system
$\mathbf{m}^{\prime}$ with $\mathbf{m}^{\prime}\upharpoonright
I_{0}\times\\{0\\}=\mathbf{m}_{2}$ (we identify $I_{0}$ with
$I_{0}\times\\{0\\}$). Now let
$\mathbf{m}_{0}^{\prime}:=\mathbf{m}\upharpoonright I_{2}\times\\{1\\}$.

Write $\mathbf{m}_{0}^{\prime}=\langle N_{u}:u\in I_{2}\rangle$. For $u\in I$,
let $M_{u}^{\prime}:=M_{u}$ if $u\notin I_{2}$ and $M_{u}^{\prime}=N_{u}$
otherwise. Let $\mathbf{m}^{\prime}:=\langle M_{u}^{\prime}:u\in I\rangle$. By
monotonicity 5, $\mathbf{m}^{\prime}$ is $\mathfrak{i}$-independent. Moreover,
$k(\mathbf{m}^{\prime})<k(\mathbf{m})$ because $u$ is proper in
$\mathbf{m}^{\prime}$. By the induction hypothesis, $\mathbf{m}^{\prime}$ must
be an extension base. Let $\mathbf{m}^{\prime\prime}$ be an independent
$J$-system such that $v$ is proper in $\mathbf{m}^{\prime\prime}$ and there is
$f:\mathbf{m}^{\prime}\cong\mathbf{m}^{\prime\prime}\upharpoonright I$. Let
$\mathbf{m}^{\prime\prime\prime}$ be the unique $J$-system so that
$\mathbf{m}^{\prime\prime\prime}\upharpoonright\\{v\\}=\mathbf{m}^{\prime\prime}\upharpoonright\\{v\\}$
and
$f\upharpoonright\mathbf{m}\cong\mathbf{m}^{\prime\prime\prime}\upharpoonright
I$. By monotonicity 5, $\mathbf{m}^{\prime\prime\prime}$ is independent, and
hence witnesses that $\mathbf{m}$ is an extension base. ∎

This implies a way to obtain extension from existence, provided that the
skeleton satisfies strong uniqueness (in case
$\mathfrak{K}_{\mathfrak{i}^{\ast}}$ consists of brimmed models, ordered by
being brimmed over, it turns out that in reasonable cases strong uniqueness
will hold in $\mathfrak{i}^{\ast}$, though there is no hope of it holding in
$\mathfrak{i}$).

######  Corollary 9.13.

Let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$ with
$|\mathfrak{K}_{\mathfrak{i}^{\ast}}|=|\mathfrak{K}_{\mathfrak{i}}|$. If
$\mathfrak{i}^{\ast}$ has strong uniqueness and existence, then $\mathfrak{i}$
has extension.

###### Proof.

By Theorem 9.12, it suffices to show that every proper
$\mathfrak{i}^{\ast}$-independent $I$-system $\mathbf{m}$ is an extension
base. So let $J=I\cup\\{v\\}\in\mathcal{I}$ with $I<v$. By existence, there is
a proper $\mathfrak{i}^{\ast}$-independent $J$-system $\mathbf{m}^{\ast}$. By
strong uniqueness, $m^{\ast}\upharpoonright I$ is isomorphic to $\mathbf{m}$,
and clearly $\mathbf{m}^{\ast}\upharpoonright I$ is an extension base, so
$\mathbf{m}$ must be one as well. ∎

We now turn to studying under what conditions uniqueness in a skeleton implies
uniqueness in the original independence relation. We are unable to prove the
full analog of Theorem 9.12, so we make the additional assumption that
$\mathcal{I}$ is closed under certain products. The following is the key
lemma:

######  Lemma 9.14.

Let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$. Assume that
$\mathfrak{i}$ has extension. Let $J\in\mathcal{I}_{\mathfrak{i}}$ be such
that $J=I\cup\\{v\\}$, for $I<v$. If
$J\times\\{0,1\\}\in\mathcal{I}_{\mathfrak{i}}$ and any proper
$\mathfrak{i}^{\ast}$-independent $I$-system is a uniqueness base, then any
$\mathfrak{i}$-independent $I$-system is a uniqueness base.

###### Proof.

Let $\mathbf{m}=\langle M_{u}:u\in I\rangle$ be an $\mathfrak{i}$-independent
$I$-system. Let $\mathbf{m}_{1},\mathbf{m}_{2}$ both be
$\mathfrak{i}$-independent $J$-systems, $\mathbf{m}_{\ell}=\langle
M_{u}^{\ell}:u\in J\rangle$, $\ell=1,2$ and without loss of generality
$\mathbf{m}_{1}\upharpoonright I=\mathbf{m}_{2}\upharpoonright I=\mathbf{m}$.
We want to find an amalgam of $M_{v}^{1}$ and $M_{v}^{2}$ fixing $\mathbf{m}$.

Let $J^{\prime}:=J\times\\{0,1\\}$, $I^{\prime}:=I\times\\{0,1\\}$. Identify
$J$ with $J\times\\{0\\}$. By Lemma 9.11, we can find an
$I^{\prime}$-independent system $\mathbf{m}^{\prime}$ such that
$\mathbf{m}^{\prime}\upharpoonright I=\mathbf{m}$ which is
$(I^{\prime}\backslash I)$-$\mathfrak{K}^{\ast}$-proper. For $\ell=1,2$, let
$\mathbf{m}_{\ell}^{\prime}$ be the $I^{\prime}\cup\\{(v,0)\\}$-system
obtained by adding $M_{v}^{\ell}$ to $\mathbf{m}^{\prime}$. By monotonicity 2
and 3, $\mathbf{m}_{\ell}^{\prime}$ is still independent. By extension and
some renaming, one can find a $J^{\prime}$-independent system
$\mathbf{m}_{\ell}^{\prime\prime}$ such that
$\mathbf{m}_{\ell}^{\prime\prime}\upharpoonright(I^{\prime}\cup\\{(v,0)\\})=\mathbf{m}_{\ell}^{\prime}$.
Moreover, we can arrange that $\mathbf{m}_{\ell}^{\prime\prime}$ is
$(J^{\prime}\backslash J)$-$\mathfrak{K}^{\ast}$-proper. Now let
$J^{\ast}:=J\times\\{1\\}$, $I^{\ast}:=I\times\\{1\\}$. We have that
$\mathbf{m}^{\ast}:=\mathbf{m}_{1}^{\prime\prime}\upharpoonright
I^{\ast}=\mathbf{m}_{2}^{\prime\prime}\upharpoonright I^{\ast}$. Moreover for
$\ell=1,2$,
$\mathbf{m}_{\ell}^{\ast}:=\mathbf{m}_{\ell}^{\prime\prime}\upharpoonright
J^{\ast}$ is a proper $\mathfrak{i}^{\ast}$-independent system by
construction. Thus it is a uniqueness base, so $\mathbf{m}_{1}^{\ast}$ and
$\mathbf{m}_{2}^{\ast}$ amalgamate over $\mathbf{m}^{\ast}$. This implies that
$M_{v}^{1}$ and $M_{v}^{2}$ amalgamate over $\mathbf{m}$, as desired. ∎

######  Theorem 9.15.

Assume that $\mathcal{I}_{\mathfrak{i}}$ is closed under products. Let
$\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$ with
$|\mathfrak{K}_{\mathfrak{i}}|=|\mathfrak{K}_{\mathfrak{i}^{\ast}}|$. If
$\mathfrak{i}^{\ast}$ has strong uniqueness and existence, then $\mathfrak{i}$
has uniqueness and extension.

###### Proof.

By Corollary 9.13, $\mathfrak{i}$ has extension. We have to see that for every
$I\in\mathcal{I}$, every $\mathfrak{i}$-independent $I$-system is a uniqueness
base but this is immediate from Lemma 9.14. ∎

The following more local result will be used in the proof of Theorem 10.16. It
says roughly that if a uniqueness base is changed by making one of the models
on the boundary smaller, then it is still a uniqueness base. A version of it
appears in [She09a, III.12.26].

######  Lemma 9.16.

Let $\mathbf{m}=\langle M_{u}:u\in I\rangle$ be an independent $I$-system. Let
$u^{\ast}\in I$ be maximal. Let $N_{u^{\ast}}\in\mathfrak{K}$ be such that
$M_{u^{\ast}}\leq_{\mathfrak{K}}N_{u^{\ast}}$. Let
$\mathbf{m}^{\ast}:=\mathbf{m}\upharpoonright(I\backslash\\{u^{\ast}\\})\smallfrown
N_{u^{\ast}}$. 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$}}}\limits(\mathfrak{i})$ has extension and
$\mathbf{m}^{\ast}$ is a uniqueness base, then $\mathbf{m}$ is a uniqueness
base.

###### Proof.

Let $J=I\cup\\{v\\}\in\mathcal{I}$, $I<v$. Let $\mathbf{m}_{\ell}=\langle
M_{u}^{\ell}:u\in J\rangle$ for $\ell=1,2$ be independent systems with
$\mathbf{m}_{\ell}\upharpoonright I=\mathbf{m}$ for $\ell=1,2$. We want to
amalgamate $M_{v}^{1}$ and $M_{v}^{2}$ over $\mathbf{m}$. 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$}}}\limits:=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i})$.

Using the extension property for $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$, find $f_{\ell},N_{v}^{\ell}$,
$\ell=1,2$ such that
$f_{\ell}:N_{u^{\ast}}\xrightarrow[M_{u^{\ast}}]{}N_{v}^{\ell}$ and
$M_{v}^{\ell}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{u^{\ast}}}^{N_{v}^{\ell}}f_{\ell}[N_{u^{\ast}}]$.
Let $\mathbf{m}_{\ell}^{\prime}:=\langle(M_{u}^{\ell})^{\prime}:u\in J\rangle$
be defined by $(M_{u}^{\ell})^{\prime}:=M_{u}$ if $u\in
I\backslash\\{u^{\ast}\\}$,
$(M_{u^{\ast}}^{\ell})^{\prime}:=f_{\ell}[N_{u^{\ast}}]$, and
$(M_{v}^{\ell})^{\prime}:=N_{v}^{\ell}$. By monotonicity 4,
$\mathbf{m}_{\ell}^{\prime}$ is an independent system.

Let $f:=f_{2}f_{1}^{-1}\upharpoonright(M_{u^{\ast}}^{1})^{\prime}$. Then
$f:\mathbf{m}_{1}^{\prime}\upharpoonright I\cong\mathbf{m}_{2}^{\prime}I$ and
since $\mathbf{m}_{1}^{\prime}\cong\mathbf{m}^{\ast}$ and $\mathbf{m}^{\ast}$
is a uniqueness base, we know that there exists $g$ extending $f$ amalgamating
$N_{v}^{1}$ and $N_{v}^{2}$. Such an $f$ also amalgamates $M_{v}^{1}$ and
$M_{v}^{2}$ over $\mathbf{m}$, as desired. ∎

##  10\. The multidimensional amalgamation properties

Throughout this section, we continue to assume:

######  Hypothesis 10.1.

$\mathfrak{i}=(\mathfrak{K},\mathcal{I},\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$ is a multidimensional independence
relation.

We study the existence and uniqueness properties restricted to systems indexed
by $\mathcal{P}(n)$, for some $n<\omega$. We show (Lemma 10.9, Theorem 10.10)
that once we have those, we can (under reasonable conditions), obtain the
properties for systems indexed by other sets. We also give a relationship
between extension and existence using strong uniqueness, akin to Corollary
9.13 but stronger: we only use strong uniqueness for smaller systems. This is
Theorem 10.11. Finally, we give sufficient conditions for the $n$-dimensional
properties in terms of amalgamation properties inside classes of
$n$-dimensional systems. The most important of these result is Theorem 10.16.

It will be convenient to restrict oneself to systems indexed by finite initial
segments of $\mathcal{P}(\omega)$. Independence relations considering only
such systems will be called _okay_. We then define the $n$-dimensional
properties by saying that they must hold for systems indexed by initial
segments of $\mathcal{P}(n)$.

######  Definition 10.2.

  1. (1)

For $n<\omega$, let $\mathcal{I}_{n}$ be the class of all partial orders
isomorphic to an initial segment of $\mathcal{P}(n)$. Let
$\mathcal{I}_{<\omega}:=\bigcup_{n<\omega}\mathcal{I}_{n}$.

  2. (2)

We say that $\mathfrak{i}$ is _okay_ if
$\mathcal{I}\subseteq\mathcal{I}_{<\omega}$. We say that $\mathfrak{i}$ is
_$n$ -okay_ if it is okay and $\mathcal{P}(n)\in\mathcal{I}_{\mathfrak{i}}$.

  3. (3)

For $P$ a property from Definition 8.16 and $n<\omega$, we say that
$\mathfrak{i}$ has _$n$ -$P$_ if $\mathfrak{i}$ is $n$-okay and
$\mathfrak{i}\upharpoonright\mathcal{I}_{n}$ has $P$. We say that
$\mathfrak{i}$ has _$( <n)$-$P$_ if it has $m$-$P$ for all $m<n$.

  4. (4)

For $P$ a property, $\lambda$ an infinite cardinal and $n<\omega$, we say that
$\mathfrak{i}$ has _$(\lambda,n)$ -$P$_ if
$\mathfrak{i}\upharpoonright\mathfrak{K}_{\lambda}$ has $n$-$P$. We say that
$\mathfrak{i}$ has _$(\lambda, <n)$-$P$_ if it has $(\lambda,m)$-$P$ for all
$m<n$. Similarly define variations like the $(\Theta,n)$-$P$, for $\Theta$ an
interval of cardinals.

######  Remark 10.3.

Since $\mathfrak{i}$ is closed under initial segments, if $\mathfrak{i}$ is
$n$-okay then $\mathcal{I}_{n}\subseteq\mathcal{I}$, so $\mathfrak{i}$ is
$m$-okay for any $m<n$.

Note that $\mathfrak{i}\upharpoonright\mathcal{I}_{<\omega}$ is always an okay
independence relation, so replacing $\mathfrak{i}$ by
$\mathfrak{i}\upharpoonright\mathcal{I}_{<\omega}$, we assume for the rest of
this section that $\mathfrak{i}$ is okay:

######  Hypothesis 10.4.

$\mathfrak{i}$ is _okay_.

It is easy to characterize the low-dimensional amalgamation properties in
terms of familiar properties:

######  Lemma 10.5.

  1. (1)

$\mathfrak{i}$ has $0$-existence if and only if $\mathfrak{i}$ has
$0$-extension if and only if $\mathfrak{K}\neq\emptyset$.

  2. (2)

$\mathfrak{i}$ has strong $0$-uniqueness if and only if $\mathfrak{K}$ has at
most one model up to isomorphism.

  3. (3)

$\mathfrak{i}$ has $0$-uniqueness if and only if $\mathfrak{K}$ has joint
embedding.

  4. (4)

$\mathfrak{i}$ has $1$-extension if and only if $\mathfrak{i}$ is $1$-okay,
$\mathfrak{K}$ is not empty, and $\mathfrak{K}$ has no maximal models.

  5. (5)

$\mathfrak{i}$ has $1$-uniqueness if and only if $\mathfrak{i}$ is $1$-okay
and $\mathfrak{K}$ has amalgamation and joint embedding.

  6. (6)

If $\mathfrak{i}$ has $2$-extension, then $\mathfrak{K}$ has disjoint
amalgamation.

###### Proof.

Immediate. ∎

Interestingly, strong $1$-uniqueness together with uniqueness already implies
strong uniqueness. In case we are considering classes of brimmed models
ordered by being brimmed over, strong $1$-uniqueness is exactly the uniqueness
of brimmed models, a key property studied in many papers [SV99, Van06, Van13,
GVV16, Van16].

######  Lemma 10.6.

If $\mathfrak{i}$ has strong $1$-uniqueness and uniqueness, then
$\mathfrak{i}$ has strong uniqueness.

###### Proof.

We use Lemma 9.9. Let $\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in J\rangle$,
$\ell=1,2$ be independent $J$-systems. Let $v$ be a top element of $J$ and let
$I:=J\backslash\\{v\\}$. Assume that
$\mathbf{m}:=\mathbf{m}_{1}\upharpoonright I=\mathbf{m}_{2}\upharpoonright I$
and $v$ is proper in both $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$. We show that
$M_{v}^{1}$ and $M_{v}^{2}$ are isomorphic over $\mathbf{m}$. By definition of
being proper, we can pick $N_{v}^{\ell}<_{\mathfrak{K}}M_{v}^{\ell}$ so that
$M_{u}^{\ell}\leq_{\mathfrak{K}}N_{v}^{\ell}$, $\ell=1,2$. By uniqueness,
there exists $M\in\mathfrak{K}$ and $f_{\ell}:N_{v}^{\ell}\rightarrow M$ with
$f_{\ell}$ fixing $\mathbf{m}$, $\ell=1,2$. If $|J|\leq 2$, there is nothing
to prove so assume that $|J|>2$. Then $M_{v}^{1}$ must be a proper extension
of $M_{u}^{1}$ for each $u<v$. By strong $0$-uniqueness, $\mathfrak{K}$ must
have no maximal models. Thus we can pick $N\in\mathfrak{K}$ with
$M<_{\mathfrak{K}}N$. By strong $1$-uniqueness and some renaming there exists
$g_{\ell}:M_{v}^{\ell}\cong N$, with $g_{\ell}$ extending $f_{\ell}$,
$\ell=1,2$. In the end, $g:=g_{2}^{-1}g_{1}$ is the desired isomorphism. ∎

We now want to show that it is enough to prove that
$\mathcal{P}^{-}(n)$-systems are extension bases to get $n$-extension. To this
end, the following invariance of a semilattice will be useful:

######  Definition 10.7.

For $I\in\mathcal{I}_{<\omega}$ and $u\in I$, let $n(I,u)$ be the cardinality
of $f(u)$, where $f:I\rightarrow\mathcal{P}(\omega)$ is any semilattice
embedding such that $f[I]$ is an initial segment of $\mathcal{P}(\omega)$. Let
$n(I)$ be the maximum of $n(I,u)$ for all $u\in I$. When $I$ is empty, we
specify $n(\emptyset)=-1$.

######  Lemma 10.8.

Let $n<\omega$ and $I,J\in\mathcal{I}_{<\omega}$ be non-empty.

  1. (1)

If $I\subseteq J$, then $n(I)\leq n(J)$.

  2. (2)

$n(\mathcal{P}(n))=n$.

  3. (3)

$n(\mathcal{P}^{-}(n))=n-1$.

  4. (4)

$n(I\times\\{0,1\\})=n(I)+1$.

  5. (5)

If $I\in\mathcal{I}_{n}$, then $n(I)\leq n$.

###### Proof.

Straightforward. ∎

######  Lemma 10.9.

Let $I,J\in\mathcal{I}_{<\omega}$ and let $I$ be an initial segment of $J$.
Let $n:=n(J)$.

  1. (1)

If for every $m\leq n$, every proper $\mathcal{P}^{-}(m)$-system is an
extension base, then every proper independent $I$-system can be extended to a
proper independent $J$-system.

  2. (2)

If for every $m\leq n$, every proper $\mathcal{P}^{-}(m)$-system is a strong
uniqueness base, then for any proper independent $J$-systems
$\mathbf{m}_{1},\mathbf{m}_{2}$ and any isomorphism
$f:\mathbf{m}_{1}\upharpoonright I\cong\mathbf{m}_{2}\upharpoonright I$, there
is an isomorphism $g:\mathbf{m}_{1}\cong\mathbf{m}_{2}$ extending $f$.

###### Proof.

Identify $J$ with an initial segment of $\mathcal{P}(\omega)$.

  1. (1)

Let $\mathbf{m}$ be a proper independent $I$-system. Let $u\in J\backslash I$
be such that $J_{0}:=I\cup\\{u\\}$ is an initial segment of $J$. It is enough
to show that $\mathbf{m}$ can be extended to a proper independent
$J_{0}$-system. Let $m:=|u|$. Note that $m\leq n$. Let $\pi:u\rightarrow m$ be
a bijection. Since $J_{0}$ is an initial segment of $\mathcal{P}(\omega)$, We
have that $\pi$ induces an isomorphism from $(-\infty,u]=\mathcal{P}(u)$ onto
$\mathcal{P}(m)$. In particular, $\mathbf{m}\upharpoonright\mathcal{P}^{-}(u)$
is an extension base. Now extend it to $\mathcal{P}(u)$ and use monotonicity
to argue that this induces an extension of $\mathbf{m}$ to $J_{0}$.

  2. (2)

Similar.

∎

######  Theorem 10.10.

Let $n<\omega$ and assume that $\mathfrak{i}$ is $n$-okay. If for every $m\leq
n$, every proper $\mathcal{P}^{-}(m)$-system is an extension base, then
$\mathfrak{i}$ has $n$-extension.

###### Proof.

Directly from Theorem 9.12 and Lemma 10.9. ∎

The following result will be useful to obtain extension from existence and
strong uniqueness for lower-dimensional systems:

######  Theorem 10.11.

Let $n<\omega$ and assume that $\mathfrak{i}$ is $n$-okay. If $\mathfrak{i}$
has strong $(<n)$-uniqueness, then $\mathfrak{i}$ has $n$-existence if and
only if $\mathfrak{i}$ has $n$-extension.

###### Proof.

If $\mathfrak{i}$ has $n$-extension, then by Lemma 9.3 it always has
$n$-existence (this does not use strong uniqueness). Assume now that
$\mathfrak{i}$ has $n$-existence. By Theorem 10.10, it suffices to show that
for every $m\leq n$, every proper $\mathcal{P}^{-}(m)$-system is an extension
base. Let $m\leq n$ and let $\mathbf{m}$ be a proper
$\mathcal{P}^{-}(m)$-system. By $n$-existence, there is a proper
$\mathcal{P}(m)$-system $\mathbf{m}^{\prime}$. By strong $(<n)$-uniqueness and
Lemma 10.9,
$\mathbf{m}^{\prime}\upharpoonright\mathcal{P}^{-}(m)\cong\mathbf{m}$. Since
$\mathbf{m}^{\prime}\upharpoonright\mathcal{P}^{-}(m)$ is an extension base,
so is $\mathbf{m}$. ∎

We would like to establish a criteria to obtain uniqueness from a kind of
amalgamation in the class
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathcal{P}(n)}$ (Definition
8.15). It turns out that a slightly finer version of
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathcal{P}(n)}$ is useful for
this purpose: the idea is that we require the interior of the system to be
brimmed but allow the boundary to not extend the rest in brimmed way (of
course this is formulated abstractly using skeletons).

######  Definition 10.12.

Let $n<\omega$ and assume that $\mathfrak{i}$ is $n$-okay. Let
$\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$. Let $K^{\ast}$ be the
class of proper $\mathfrak{i}$-independent $\mathcal{P}(n)$-systems
$\mathbf{m}$ such that $\mathbf{m}\upharpoonright\mathcal{P}^{-}(n)$ is
$\mathfrak{i}^{\ast}$-independent. Let
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
be the abstract class whose underlying class is $K^{\ast}$ and whose ordering
is
$\mathbf{m}_{1}\leq_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}}\mathbf{m}_{2}$
if and only if $\mathbf{m}_{1}=\mathbf{m}_{2}$ or whenever
$\mathbf{m}_{\ell}=\mathbf{m}_{\ell}^{-}\smallfrown M_{\ell}$ with
$\mathbf{m}_{\ell}^{-}:=\mathbf{m}_{\ell}\upharpoonright\mathcal{P}^{-}(n)$,
$\ell=1,2$, we have that $\mathbf{m}_{1}\leq_{\mathfrak{i}}\mathbf{m}_{2}$,
$M_{1}$ and $M_{2}$ are in $\mathfrak{K}_{\mathfrak{i}^{\ast}}$,
$M_{1}<_{\mathfrak{K}_{\mathfrak{i}^{\ast}}}M_{2}$, and
$\mathbf{m}_{1}\upharpoonright\mathcal{P}^{-}(n)<_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},\mathcal{P}^{-}(n)}}\mathbf{m}_{2}\upharpoonright\mathcal{P}^{-}(n)$.

######  Remark 10.13.

We have that
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},\mathcal{P}(n)}\subseteq\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}\subseteq\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathcal{P}(n)}$.
In particular,
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i},\mathcal{P}(n)}=\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathcal{P}(n)}$.

######  Definition 10.14.

For $n<\omega$, we let $\phi_{n}(x)$ be the formula in the vocabulary of
systems (Definition 8.8) which holds inside a $\mathcal{P}(n)$-system $\langle
M_{u}:u\in\mathcal{P}(n)\rangle$ if and only if $a\in\bigcup_{u\subsetneq
n}M_{u}$.

The following easy relationships hold between the $n$-dimensional properties
and classes of proper $n$-dimensional systems. See Definition 5.6 for the
definitions of $\phi$-categoricity, $\phi$-amalgamation, $\phi$-uniqueness,
etc.

######  Lemma 10.15.

Let $n<\omega$. Assume that $\mathfrak{i}$ is $(n+1)$-okay and let
$\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$.

  1. (1)

For any
$\mathbf{m}_{1}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$,
there exists
$\mathbf{m}_{2}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},\mathcal{P}(n)}$
such that $\mathbf{m}_{1}\leq_{\mathfrak{K}_{\mathfrak{i}}}\mathbf{m}_{2}$ and
$\mathbf{m}_{1}\upharpoonright\mathcal{P}^{-}(n)=\mathbf{m}_{2}\upharpoonright\mathcal{P}^{-}(n)$.

  2. (2)

If $\mathfrak{i}^{\ast}$ has strong $(<n)$-uniqueness, then
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
is $\phi_{n}$-categorical.

  3. (3)

If $\mathfrak{i}^{\ast}$ has strong $n$-uniqueness, then any
$\mathbf{m}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},\mathcal{P}(n)}$
is $\phi_{n}$-universal in
$(|\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}|,\leq_{\mathfrak{K}_{\mathfrak{i}}}^{d})$.

  4. (4)

If $\mathfrak{i}^{\ast}$ has strong $n$-uniqueness, then
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
has $\phi_{n}$-uniqueness.

  5. (5)

Let $\bar{\mathbf{m}}=(\mathbf{m}_{0},\mathbf{m}_{1},\mathbf{m}_{2})$ be a
$\phi$-span in
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$,
and write $\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in\mathcal{P}(n)\rangle$,
$\ell=0,1,2$. Then $\bar{\mathbf{m}}$ has a $\phi$-amalgam in
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
if and only if there exists $N\in\mathfrak{K}_{\mathfrak{i}}$ and
$f_{\ell}:M_{\mathcal{P}(n)}^{\ell}\xrightarrow[\mathbf{m}_{0}]{}N$,
$\ell=1,2$, such that $f_{1}\upharpoonright M_{u}^{1}=f_{2}\upharpoonright
M_{u}^{2}$ for all $u\in\mathcal{P}^{-}(n)$.

  6. (6)

If $\mathbf{m}$ is a $\phi_{n}$-amalgamation base in
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$,
then for any
$\mathbf{m}^{\prime}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
with
$\mathbf{m}\leq_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}}\mathbf{m}^{\prime}$,
we have that
$(\mathbf{m}\ast\mathbf{m}^{\prime})\upharpoonright\mathcal{P}^{-}(n+1)$ is a
uniqueness base in $\mathfrak{i}$.

###### Proof.

  1. (1)

By the basic properties of a skeleton.

  2. (2)

Since $\mathfrak{i}^{\ast}$ has strong $(<n)$-uniqueness, any two
$\mathfrak{i}^{\ast}$-independent $\mathcal{P}^{-}(n)$-systems are isomorphic.

  3. (3)

By the first part and strong $n$-uniqueness.

  4. (4)

Expanding the definitions, it suffices to show that if $\mathbf{m}_{1}$,
$\mathbf{m}_{2}$ are two $\mathcal{P}^{-}(n+1)$-systems such that
$\mathbf{m}_{1}\upharpoonright\mathcal{P}^{-}(n)=\mathbf{m}_{2}\upharpoonright\mathcal{P}^{-}(n)$,
then there is an isomorphism $f:\mathbf{m}_{1}\cong\mathbf{m}_{2}$ fixing
$\mathbf{m}_{1}\upharpoonright\mathcal{P}^{-}(n)$. This is implied by strong
$n$-uniqueness, because $n(\mathcal{P}^{-}(n+1))=n$.

  5. (5)

By monotonicity.

  6. (6)

Expand the definitions.

∎

The following result (used in the proof of Lemma 11.16) will be crucial: under
the assumption of density of certain amalgamation bases in the class of
$(n+1)$-dimensional systems, it implies that the _whole_ independence relation
has $(n+1)$-uniqueness. More precisely, we also assume that $n\geq 2$, work
inside a skeleton, and assume that the skeleton has strong $n$-uniqueness and
$(n+1)$-extension (in the case of interest, these will hold).

######  Theorem 10.16.

Let $n\in[2,\omega)$. Let $\mathfrak{i}$ be a multidimensional independence
notion and let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$. If:

  1. (1)

$\mathfrak{i}^{\ast}$ has $(n+1)$-extension and strong $n$-uniqueness.

  2. (2)

For any
$\mathbf{m}_{0}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},\mathcal{P}(n)}$,
there exists a $\phi_{n}$-amalgamation base
$\mathbf{m}_{1}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
such that
$\mathbf{m}_{0}\leq_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}}\mathbf{m}_{1}$.

Then $\mathfrak{i}^{\ast}$ has $(n+1)$-uniqueness.

###### Proof.

By Theorem 9.12, $\mathfrak{i}$ has $(n+1)$-extension. By Lemma 9.14, the
proof of Lemma 10.6, and Lemma 10.9, it suffices to show that every proper
$\mathfrak{i}^{\ast}$-independent $\mathcal{P}^{-}(n+1)$-system is a
uniqueness base. Note that $n(\mathcal{P}^{-}(n+1))=n$, so by Lemma 10.9
again, any two proper $\mathfrak{i}^{\ast}$-independent
$\mathcal{P}^{-}(n+1)$-systems are isomorphic. Thus it suffices to show that
_some_ proper $\mathfrak{i}^{\ast}$-independent $\mathcal{P}^{-}(n+1)$-system
is a uniqueness base.

Pick any
$\mathbf{m}_{0}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},\mathcal{P}(n)}$
(exists by extension in $\mathfrak{i}^{\ast}$). By assumption, there is a
$\phi_{n}$-amalgamation base $\mathbf{m}_{1}$ in
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
which extends $\mathbf{m}_{0}$. By $(n+1)$-extension, there exists
$\mathbf{m}_{2}\in\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
such that
$\mathbf{m}_{1}<_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}}\mathbf{m}_{2}$.
By Lemma 10.15(6),
$\mathbf{m}:=(\mathbf{m}_{1}\ast\mathbf{m}_{2})\upharpoonright\mathcal{P}^{-}(n+1)$
is a uniqueness base. Write $\mathbf{m}=\langle
M_{u}:u\in\mathcal{P}^{-}(n+1)\rangle$ and let $\mathbf{m}^{\prime}=\langle
M_{u}^{\prime}:u\in\mathcal{P}^{-}(n+1)\rangle$ be defined as follows: let
$I_{0}:=\mathcal{P}^{-}(n+1)\backslash\\{n\\}$,
$\mathbf{m}^{\prime}\upharpoonright
I_{0}:=(\mathbf{m}_{0}\ast\mathbf{m}_{2})\upharpoonright I_{0}$,
$M_{n}^{\prime}=M_{n}$. It is easy to check that $\mathbf{m}^{\prime}$ is an
independent system and a uniqueness base as well: anytime a model $N$ extends
all the elements of $\mathbf{m}^{\prime}$, it extends all the elements of
$\mathbf{m}$.

Now let $\mathbf{m}^{\ast}$ be a $\mathcal{P}^{-}(n+1)$-system defined as
follows: $\mathbf{m}^{\ast}\upharpoonright
I_{0}=\mathbf{m}^{\prime}\upharpoonright I_{0}$ and
$\mathbf{m}^{\ast}\upharpoonright\\{n\\}=\mathbf{m}_{0}\upharpoonright\\{n\\}$.
Note that then
$\mathbf{m}^{\ast}\upharpoonright\mathcal{P}(n)=\mathbf{m}_{0}$, and it is
easy to check that $\mathbf{m}^{\ast}$ is a proper
$\mathfrak{i}^{\ast}$-independent system. Moreover,
$\mathbf{m}^{\ast}\upharpoonright\\{n\\}\leq_{\mathfrak{K}}\mathbf{m}^{\prime}\upharpoonright\\{n\\}$
by definition of $\mathbf{m}^{\prime}$, so by Lemma 9.16, $\mathbf{m}^{\ast}$
is a uniqueness base, as desired. ∎

##  11\. Continuous independence relations

This section is central. We define here the conditions a multidimensional
independence relation should satisfy to be well-behaved with respect to
increasing chains. In particular, it should be closed under union of those. We
call such multidimensional independence relations _very good_. They are the
ones we really want to study. We prove the _construction theorem_ (Theorem
11.3), which says that from a very good two-dimensional independence relation,
we can build a very good multidimensional independence relation.

We study an important subclass of independent systems: the _brimmed_ ones
(Definition 11.7). We show that they are quite well-behaved under unions and
resolvability (Lemma 11.9). There are some technical issues here, since we do
not know that the class of $n$-dimensional brimmed systems is closed under
unions until we have proven $(n+1)$-dimensional uniqueness, but we want to use
the former in the proof of the latter. We use the class
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
(Definition 10.12) to get around this.

We prove the $n$-dimensional properties by studying the class of brimmed
systems and using the weak diamond (or a strongly compact cardinal). Crucial
is the _stepping up lemma_ (Lemma 11.16) which shows how to go from the
$n$-dimensional properties to the $(n+1)$-dimensional properties (for brimmed
systems). It leads directly to the _brimmed excellence theorem_ (Theorem
11.17) giving sufficient conditions for the $n$-dimensional properties. Of
course, once we have the properties for brimmed systems we will be able to get
them for regular systems using the results of Section 9.

The first definition is the multidimensional analog of Definition 7.11.

######  Definition 11.1.

We say that a multidimensional independence relation $\mathfrak{i}$ has
_strong continuity_ if whenever $\langle\mathbf{m}_{i}:i\leq\delta\rangle$ is
a $\leq_{\mathfrak{K}}$-increasing continuous chain of
$(I,\mathfrak{K}_{\mathfrak{i}})$-systems and $\mathbf{m}_{i}$ is independent
for all $i<\delta$, then $\mathbf{m}_{\delta}$ is independent.

######  Definition 11.2.

Let $\mathfrak{i}$ be a multidimensional independence relation. We say that
$\mathfrak{i}$ is _very good_ if:

  1. (1)

$\mathfrak{i}$ is $n$-okay for all $n<\omega$ (so
$\mathcal{I}_{\mathfrak{i}}=\mathcal{I}_{<\omega}$), see Definition 10.2.

  2. (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$}}}\limits(\mathfrak{i})$ is very good.

  3. (3)

$\mathfrak{i}$ has strong continuity.

  4. (4)

For any $I\in\mathcal{I}$, both $\mathfrak{K}_{\mathfrak{i},I}$ and the class
$\mathfrak{K}^{\ast}$ of independent $I$-system ordered by
$\leq_{\mathfrak{K}}^{d}$ (see Definition 8.6(2)) are fragmented AECs.
Moreover the AECs they generate have the same models555This is a
multidimensional analog of reflecting down (Definition 7.7)..

Very good multidimensional independence relations are obtained by taking a
very good two-dimensional independence notion and applying the construction
described in Definition 8.19 (and proven to have some good properties in
Theorem 8.20).

######  Theorem 11.3 (The construction theorem).

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 very good two-dimensional
independence notion on $\mathfrak{K}$, then $\mathfrak{i}(\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)\upharpoonright\mathcal{I}_{<\omega}$
(see Definition 10.2) is a very good multidimensional independence notion.

###### Proof.

Let $\mathfrak{i}:=\mathfrak{i}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)\upharpoonright\mathcal{I}_{<\omega}$.
By Theorem 8.20, $\mathfrak{i}$ is a multidimensional independence notion. By
definition, $\mathfrak{i}$ is $n$-okay for all $n<\omega$ 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$}}}\limits({\mathfrak{i}})=\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ (see the statement of Theorem 8.20),
which is very good by assumption. It also follows from the definition of
$\mathfrak{i}$ that it has strong continuity, since $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ has it. It remains to see that for
any $I\in\mathcal{I}$, both $\mathfrak{K}_{\mathfrak{i},I}$ and
$\mathfrak{K}^{\ast}$ (the class of independent $I$-systems ordered by
$\leq_{\mathfrak{K}}^{d}$) are fragmented AECs that generate the same models.
Now $\mathfrak{K}$ is a fragmented AEC because $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is very good. Thus by strong
continuity it follows immediately that $\mathfrak{K}^{\ast}$ is a fragmented
AEC, and that $\mathfrak{K}_{\mathfrak{i},I}$ is a fragmented AEC similarly
follows from the definition 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$}}}\limits$. It remains to check that those two
classes generate the same class of models. Let $\lambda:=\min(\Theta)$, let
$\mathfrak{K}^{1}$ be the AEC generated by $\mathfrak{K}_{\mathfrak{i},I}$,
and let $\mathfrak{K}^{2}$ be the AEC generated by $\mathfrak{K}^{\ast}$. Note
that $|\mathfrak{K}_{\lambda}^{1}|=|\mathfrak{K}_{\lambda}^{2}|$, and
$\leq_{\mathfrak{K}_{\lambda}^{1}}=\leq_{(\mathfrak{K}_{\mathfrak{i},I})_{\lambda}}$,
$\leq_{\mathfrak{K}^{2}}=\leq_{\mathfrak{K}}^{d}$. Since
$\leq_{\mathfrak{K}^{\ast}}$ extends $\leq_{\mathfrak{K}_{\mathfrak{i},I}}$,
we have that $|\mathfrak{K}^{1}|\subseteq|\mathfrak{K}^{2}|$. To see the other
direction, let $\mathbf{m}\in\mathfrak{K}^{2}$. We claim that there exists
$\mathbf{m}_{0}$ in $\mathfrak{K}_{\lambda}^{1}$ such that for any subset $A$
of $\mathbf{m}$ of size $\lambda$ there is
$\mathbf{m}_{1}\in\mathfrak{K}_{\lambda}^{1}$ with
$\mathbf{m}_{0}\leq_{\mathfrak{K}^{1}}\mathbf{m}_{1}$,
$\mathbf{m}_{1}\leq_{\mathfrak{K}^{2}}\mathbf{m}$, and $\mathbf{m}_{1}$
containing $A$. This will suffice by the usual directed system argument.
Suppose that the claim fails. We build
$\langle\mathbf{m}_{i}:i<\lambda^{+}\rangle$
$\leq_{\mathfrak{K}^{2}}$-increasing continuous in
$\mathfrak{K}_{\lambda}^{1}$ such that for all $i<\lambda$,
$\mathbf{m}_{i}\not\leq_{\mathfrak{K}^{1}}\mathbf{m}_{i+1}$. This is possible
by failure of the claim. This is enough: we know 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 very good, hence reflects down.
From the definition of $\mathfrak{i}$, we get that there must be a club of
$i<\lambda^{+}$ such that
$\mathbf{m}_{i}\leq_{\mathfrak{i},I}\mathbf{m}_{i+1}$, a contradiction. ∎

Note that a very good multidimensional independence relation has underlying
class a fragmented AEC, which may possibly “shrink” (e.g. it could be a class
of saturated models). The next lemma shows that the AEC generated by this
fragmented AEC also carries a multidimensional independence relation (which is
quite nice, but may fail to be very good because of the requirement 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$}}}\limits(\mathfrak{i})$ be very good, hence has
extension and uniqueness):

######  Definition 11.4.

We call a multidimensional independence relation $\mathfrak{i}$ _almost very
good_ if:

  1. (1)

$\mathfrak{i}_{\min(\operatorname{dom}(\mathfrak{i}))}$ is very good.

  2. (2)

$\mathfrak{i}$ satisfies all the properties in the definition of very good,
except 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$}}}\limits(\mathfrak{i})$ may not be very good.

######  Lemma 11.5.

Let $\mathfrak{i}$ be a very good multidimensional independence notion. Let
$\lambda:=\min(\operatorname{dom}(\mathfrak{K}_{\mathfrak{i}}))$. Then there
exists a (unique) multidimensional independence relation
$\mathfrak{i}^{\prime}$ such that:

  1. (1)

$\lambda=\min(\operatorname{dom}(\mathfrak{K}_{\mathfrak{i}^{\prime}}))$ and
$\mathfrak{i}_{\lambda}=\mathfrak{i}_{\lambda}^{\prime}$.

  2. (2)

$\mathfrak{i}^{\prime}$ is almost very good.

###### Proof.

Let $\mathfrak{K}$ be the AEC generated by
$\left(\mathfrak{K}_{\mathfrak{i}}\right)_{\lambda}$. More generally, for each
$I\in\mathcal{I}$, let $\mathfrak{K}_{I}$ be the AEC generated by
$\left(\mathfrak{K}_{\mathfrak{i},I}\right)_{\lambda}$. We let
$\mathfrak{i}^{\prime}$ be defined as follows:
$\mathfrak{K}_{\mathfrak{i}^{\prime}}=\mathfrak{K}$,
$\mathcal{I}_{\mathfrak{i}^{\prime}}=\mathcal{I}_{\mathfrak{i}}$, and for any
$(\mathfrak{K},I)$-system $\mathbf{m}$, $\mathbf{m}$ is
$\mathfrak{i}^{\prime}$-independent if and only if
$\mathbf{m}\in\mathfrak{K}_{I}$. It is a straightforward (but long) to check
that $\mathfrak{i}^{\prime}$ is indeed the desired multidimensional
independence relation. ∎

Note that if we have all the $n$-dimensional properties at a fixed cardinal
$\lambda$, we can transfer them up. More precisely:

######  Lemma 11.6.

Let $\mathfrak{i}$ be an almost very good multidimensional independence
relation with domain $[\lambda_{1},\lambda_{2}]$ (for
$\lambda_{1}<\lambda_{2}$) and let $n<\omega$.

  1. (1)

If $\mathfrak{i}$ has $([\lambda_{1},\lambda_{2}),n+1)$-extension, then
$\mathfrak{i}$ has $([\lambda_{1},\lambda_{2}],n)$-extension.

  2. (2)

If $\mathfrak{i}$ has $([\lambda_{1},\lambda_{2}),n+1)$-uniqueness and
$([\lambda_{1},\lambda_{2}),n+1)$-extension, then $\mathfrak{i}$ has
$([\lambda_{1},\lambda_{2}],n)$-uniqueness.

In particular, if $\mathfrak{i}$ has $(\lambda_{1},<\omega)$-extension and
$(\lambda_{1},<\omega)$-uniqueness, then $\mathfrak{i}$ has extension and
uniqueness.

###### Proof.

This is quite standard, and the definitions are tailored to make this work, so
we do not replay the arguments here. See [She83b, §5]. ∎

Brimmed systems are candidates for strong uniqueness, so are an important
object of study:

######  Definition 11.7.

Let $\mathfrak{i}$ be a very good multidimensional independence relation. We
define a multidimensional independence relation $\mathfrak{i}^{\text{brim}}$
as follows:

  1. (1)

$\mathfrak{K}_{\mathfrak{i}^{\text{brim}}}$ consist of
$\mathfrak{K}_{\mathfrak{i}}^{\text{brim}}$, the class of brimmed models in
$\mathfrak{K}_{\mathfrak{i}}$, ordered by being brimmed over (see Definition
3.23).

  2. (2)

$\mathcal{I}_{\mathfrak{i}^{\text{brim}}}=\mathcal{I}_{\mathfrak{i}}$.

  3. (3)

$\operatorname{NF}_{\mathfrak{i}^{\text{brim}}}=\operatorname{NF}_{\mathfrak{i}}\upharpoonright\mathfrak{K}_{\mathfrak{i}^{\text{brim}}}$.

We call a proper $\mathfrak{i}^{\text{brim}}$-independent system a _brimmed_
$\mathfrak{i}$-independent system.

######  Definition 11.8.

Let $\mathfrak{i}$ be a very good multidimensional independence relation and
let $\mathfrak{i}^{\ast}$ be a skeleton of $\mathfrak{i}$. For
$I\in\mathcal{I}_{\mathfrak{i}}$, we let
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i},\mathfrak{i}^{\ast},I}$ be
the sub-abstract class of
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},I}$
(Definition 10.12) with the same ordering but consisting only of the systems
where all models have the same cardinality. Similarly define
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\ast},I}$ to be the sub-
abstract class of $\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\ast},I}$
(Definition 8.15) consisting of systems with all models of the same
cardinality.

The class of brimmed systems satisfies the following properties:

######  Lemma 11.9.

Let $\mathfrak{i}$ be a very good multidimensional independence relation with
domain $\Theta$.

  1. (1)

$\mathfrak{i}^{\text{brim}}$ is a skeleton of $\mathfrak{i}$.

  2. (2)

For any $\lambda\in\Theta$, $\mathfrak{i}^{\text{brim}}$ has strong
$(\lambda,2)$-uniqueness and $(\lambda,2)$-existence.

  3. (3)

For any subinterval $\Theta_{0}$ of $\Theta$,
$\left(\mathfrak{i}_{\Theta_{0}}\right)^{\text{brim}}=\left(\mathfrak{i}^{\text{brim}}\right)_{\Theta_{0}}$.

  4. (4)

For any $I\in\mathcal{I}_{\mathfrak{i}}$,
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},I}$ is a
resolvable abstract class with LST number $\min(\Theta)$.

  5. (5)

Let $\Theta^{-}:=\\{\lambda\in\Theta\mid\lambda^{+}\in\Theta\\}$.

    1. (a)

Let $I\in\mathcal{I}_{\mathfrak{i}}$ and let $n:=n(I)$. If for any
$\lambda\in\Theta^{-}$, we have that $\mathfrak{i}^{\text{brim}}$ has strong
$(\lambda,n+1)$-uniqueness and $(\lambda,n+1)$-extension, then
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\ast},I}$ is
$\Theta^{-}$-continuous.

    2. (b)

Let $n<\omega$. If for any $\lambda\in\Theta^{-}$, we have that
$\mathfrak{i}^{\text{brim}}$ has strong $(\lambda,n)$-uniqueness and
$(\lambda,n)$-extension, then
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i},\mathfrak{i}^{\ast},\mathcal{P}(n)}$
is $\Theta^{-}$-continuous.

###### Proof.

  1. (1)

Directly from the definitions and the (easy to check) fact that
$\mathfrak{K}_{\mathfrak{i}}^{\text{brim}}$ is a skeleton of
$\mathfrak{K}_{\mathfrak{i}}$.

  2. (2)

Because $\mathfrak{i}$ has it.

  3. (3)

Immediate from the definitions.

  4. (4)

We leave it to the reader to verify that the LST number is $\min(\Theta)$. We
prove that $\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},I}$
is resolvable. We check that the conditions of Lemma 5.4 are satisfied. All
the conditions are easy, except for resolvability for successors. So let
$\mathbf{m}\in\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},I}$
have cardinality $\lambda^{+}$ and let
$\mathbf{m}_{0}\leq_{\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},I}}\mathbf{m}_{1}$
have cardinality $\lambda$, with $\lambda,\lambda^{+}\in\Theta$. Since
$\mathfrak{K}_{\mathfrak{i},I}$ is a fragmented AEC, it is resolvable, hence
there exists $\langle\mathbf{m}_{i}:i<\lambda^{+}\rangle$ a resolution of
$\mathbf{m}$ over $\mathbf{m}_{0}$ in $\mathfrak{K}_{\mathfrak{i},I}$. Write
$\mathbf{m}_{i}=\langle M_{u}^{i}:u\in I\rangle$. What are we missing to make
this a resolution in
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},I}$? We want
the $\mathbf{m}_{i}$’s to be brimmed, and in fact we also want
$\mathbf{m}_{i}\ast\mathbf{m}_{i+1}$ to be brimmed. From the properties of
being brimmed, it is in fact enough to show that there is a club $C$ of
$\lambda^{+}$ such that $\langle\mathbf{m}_{i}:i\in C\rangle$ is an increasing
chain in $\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},I}$.
To get this club, first get a club $C_{1}$ (using Lemma 5.3) such that for
$i<j$ both in $C_{1}$, $M_{u}^{i}$ is brimmed and $M_{u}^{j}$ is brimmed over
$M_{u}^{i}$ (this is possible since $M_{u}^{\lambda^{+}}$ is brimmed, hence
saturated). Next, use Fact 7.10 on each pair of models in $\mathbf{m}$ and
intersect clubs to get that on a club $C$, for all $i\in C$, $M_{v}^{i}$ is
brimmed over $M_{u}^{i}$ whenever $u<v$. This is not enough to make
$\mathbf{m}_{i}$ brimmed, as we need for each $u\in I$ an intermediate $M$
such that $M_{u}^{i}$ is brimmed over $M$ and $M$ is brimmed over each
$M_{v}^{i}$, $v<u$. Now by assumption $\mathbf{m}_{\lambda^{+}}$ is brimmed,
so for each $u\in I$ there exists $M_{u}^{\ast}$ with $M_{u}^{\ast}$ brimmed
over $M_{v}^{\lambda^{+}}$ and $M_{u}^{\lambda^{+}}$ brimmed over
$M_{u}^{\ast}$. Resolve $\mathbf{m}_{\lambda^{+}}\smallfrown M_{u}^{\ast}$ as
before to obtain a club $C_{u}$ and a resolution $\langle M_{u}^{\ast,i}:i\in
C_{u}\rangle$ of $M_{u}^{\ast}$ such that $M_{u}^{\ast,i}$ is brimmed over
$M_{v}^{i}$ for each $v<u$ and $M_{u}^{i}$ is brimmed over $M_{u}^{\ast,i}$.
Note that we can further require that for $i<j$ both in $C_{u}$,
$M_{u}^{\ast,j}$ is brimmed over $M_{u}^{\ast,i}$, hence the corresponding
part of the system $\mathbf{m}_{i}\ast m_{j}$ is also brimmed. Now intersect
all the $C_{u}$’s and intersect further with $C_{1}$ to get the desired club.

  5. (5)

    1. (a)

Let $\langle\mathbf{m}_{i}:i<\delta\rangle$ be an increasing chain in
$\left(\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\ast},I}\right)_{\Theta^{-}}$.
Let $\mathbf{m}_{\delta}:=\bigcup_{i<\delta}\mathbf{m}_{i}$. We have that
$\mathbf{m}_{\delta}$ is an independent system, the only issue is to prove
that it is brimmed (that is, that the extensions between the models are
brimmed). Once this is done, it will immediately follow that
$\mathbf{m}_{\delta}$ extends $\mathbf{m}_{0}$. Write $\mathbf{m}_{i}=\langle
M_{u}^{i}:u\in I\rangle$. By Fact 7.9, we directly have that $M_{v}^{\delta}$
is brimmed over $M_{u}^{\delta}$ for each $u<v$. However, we need more: for a
fixed $u\in I$, we want $M_{u}^{\delta}$ to be brimmed over
$\bigcup_{v<u}M_{v}^{\delta}$. Recall (Definition 3.5(4)) that this means that
there is $M$ such that $M$ is brimmed over each $M_{v}^{\delta}$, $v<u$, and
$M_{u}^{\delta}$ is brimmed over $M$. In fact, it suffices to find $M^{\ast}$
such that $M$ extends each $M_{v}^{\delta}$, $v<u$ and $M_{u}^{\delta}$ is
brimmed over $M^{\ast}$ (then pick $M$ brimmed over $M^{\ast}$ such that
$M_{u}^{\delta}$ is brimmed over $M$).

We proceed by induction on $\delta$. So we can assume that
$\delta=\text{cf}(\delta)$ and the chain
$\langle\mathbf{m}_{i}:i<\delta\rangle$ is continuous. Fix $u\in I$ and write
$A_{i}:=\bigcup_{v<u}M_{v}^{i}$. We build $\langle M_{i}:i\leq\delta\rangle$
an increasing continuous chain of brimmed models such that for all $i<\delta$:

      1. (i)

$M_{i+1}$ is brimmed over $M_{i}$.

      2. (ii)

$A_{i}\subseteq|M_{i}|$.

      3. (iii)

$M_{u}^{i}$ is brimmed over $M_{i}$.

      4. (iv)

$M_{u}^{i}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{i}}^{M_{u}^{i+1}}M_{i+1}$ (where
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i})$).

This is enough: by Fact 7.9, $M_{\delta}$ is as desired. This is possible: the
base case uses that $\mathbf{m}_{0}$ is itself a brimmed system. At limits,
take unions (and use Fact 7.9). For the successor step, assume we are given
$i<\delta$. Let $I_{0}:=\\{v\in I\mid v\leq u\\}$, and let
$J:=I_{0}\cup\\{v^{\ast}\\}$, where $v^{\ast}$ is a new element such that
$v^{\ast}<u$ and $v<v^{\ast}$ for all $v\in I$ with $v<u$.

Let $\mathbf{m}_{i}^{\ast}=\langle M_{v}^{\ast}:v\in J\rangle$ be the
$J$-system such that $\mathbf{m}_{i}^{\ast}\upharpoonright
I_{0}=\mathbf{m}_{i}\upharpoonright I_{0}$ and $M_{v^{\ast}}^{\ast}=M_{i}$. It
is easy to check that this is a brimmed independent system. We now use
$(n+1)$-extension to build a brimmed independent $J$-system
$\mathbf{m}^{\ast\ast}$ with
$\mathbf{m}^{\ast}\leq_{\mathfrak{K}}{\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},J}}\mathbf{m}^{\ast}$.
We do this in two steps: first we build an $I$-system extending
$\mathbf{m}^{\ast}\upharpoonright I$ (using extension on $I\times\\{0,1\\}$,
note that $n(I\times\\{0,1\\})=n(I)+1=n+1$), then using two-dimensional
extension to complete this to an extension $\mathbf{m}^{\ast\ast}$. Now by
strong $(n+1)$-uniqueness (again applied in two steps), we must have that
there is $f:\mathbf{m}^{\ast\ast}\cong_{\mathbf{m}_{i}}\mathbf{m}_{i+1}$. Let
$M_{i+1}:=f[M_{v^{\ast}}^{\ast\ast}]$, where $M_{v^{\ast}}^{\ast\ast}$ is the
$v^{\ast}$-indexed element of $\mathbf{m}^{\ast\ast}$.

    2. (b)

Immediate from the previous part applied to $I=\mathcal{P}^{-}(n)$.

∎

As hinted at before, brimmed models are well-behaved with respect to the
$n$-dimensional properties: strong uniqueness is equivalent to uniqueness, and
existence is equivalent to extension (assuming lower-dimensional uniqueness).

######  Lemma 11.10.

Let $\mathfrak{i}$ be a very good independence relation. Let
$\lambda\in\operatorname{dom}(\mathfrak{i})$ and let $n<\omega$. Then:

  1. (1)

$\mathfrak{i}^{\text{brim}}$ has strong $(\lambda,n)$-uniqueness if and only
if $\mathfrak{i}^{\text{brim}}$ has $(\lambda,n)$-uniqueness.

  2. (2)

If $\mathfrak{i}^{\text{brim}}$ has $(\lambda,<n)$-uniqueness, then
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n)$-existence if and only if
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n)$-extension.

###### Proof.

  1. (1)

By Lemma 10.6 and Lemma 11.9(2).

  2. (2)

By Theorem 10.11 and the previous part.

∎

In compact classes, one can take an ultraproduct of brimmed systems and it
will still be brimmed. In fact, we will see that the existence of such a
uniform extension is very powerful, so we call independence relations with
this property _nice_ :

######  Definition 11.11.

Let $\mathfrak{i}$ be a very good multidimensional independence relation. For
$\mathbf{m}_{1},\mathbf{m}_{2}\in\mathfrak{K}_{\mathfrak{i},I}$, say with
$\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in I\rangle$ we say that
$\mathbf{m}_{2}$ is _brimmed over $\mathbf{m}_{1}$_ if
$\mathbf{m}_{1}\leq_{\mathfrak{i}}\mathbf{m}_{2}$ and $M_{u}^{2}$ is brimmed
over $M_{u}^{1}$ for all $u\in I$.

######  Definition 11.12.

A very good multidimensional independence relation $\mathfrak{i}$ is _nice_
if:

  1. (1)

$\mathfrak{K}_{\mathfrak{i}}$ is categorical in every
$\lambda\in\operatorname{dom}(\mathfrak{i})$.

  2. (2)

For any $n<\omega$ and any two $\mathbf{m}_{1}$,
$\mathbf{m}_{2}\in\mathfrak{K}_{\mathfrak{i},\mathcal{P}(n)}$ with
$\mathbf{m}_{1}\upharpoonright\mathcal{P}^{-}(n)=\mathbf{m}_{2}\upharpoonright\mathcal{P}^{-}(n)$,
there exists independent $\mathbf{m}_{1}^{\prime},\mathbf{m}_{2}^{\prime}$
such that:

    1. (a)

$\mathbf{m}_{\ell}^{\prime}$ is brimmed over $\mathbf{m}_{\ell}$.

    2. (b)

$\mathbf{m}_{1}^{\prime}\upharpoonright\mathcal{P}^{-}(n)=\mathbf{m}_{2}^{\prime}\upharpoonright\mathcal{P}^{-}(n)$.

    3. (c)

If $\mathbf{m}_{\ell}=\langle M_{u}^{\ell}:u\in\mathcal{P}(n)\rangle$,
$\mathbf{m}_{\ell}^{\prime}=\langle N_{u}^{\ell}:u\in\mathcal{P}(n)\rangle$,
then $\|M_{u}^{\ell}\|=\|N_{u}^{\ell}\|$ for all $u\in\mathcal{P}(n)$ and
$\ell=1,2$.

Compact AECs have a nice independence notion:

######  Theorem 11.13.

Let $\mathfrak{K}$ be a compact AEC. If $\mathfrak{K}$ is categorical in some
$\mu>\text{LS}(\mathfrak{K})$, then there exists a _nice_ very good
multidimensional independence notion $\mathfrak{i}$ with
$\mathfrak{K}_{\mathfrak{i}}=\mathfrak{K}^{\text{sat}}_{\geq\text{LS}(\mathfrak{K})^{+6}}$.

###### Proof.

By Fact 7.21, there is a categorical very good frame $\mathfrak{s}$ on
$\mathfrak{K}^{\text{sat}}_{\geq\text{LS}(\mathfrak{K})^{+6}}$. 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 the very good two-dimensional
independence notion associated with $\mathfrak{s}$. Let
$\mathfrak{i}:=\mathfrak{i}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits)$. By Theorem 11.3, $\mathfrak{i}$ is
very good, we only have to check that it is nice. From the definition,
$\mathfrak{K}_{\mathfrak{i}}$ is indeed categorical in all cardinals, so we
have to check the second clause in the definition of being nice. Let
$\mathbf{m}_{1},\mathbf{m}_{2}$ be as there. Write $\mathbf{m}_{\ell}=\langle
M_{u}^{\ell}:u\in\mathcal{P}(n)\rangle$. Using [Bon14, 4.11], we can build a
$\kappa$-complete ultrafilter $U$ such that for each $u\in\mathcal{P}(n)$ and
$\ell=1,2$, the $U$-ultrapower $N_{u}^{\ell}$ of $M_{u}^{\ell}$ (seen as an
extension of $M_{u}^{\ell}$) is brimmed over $M_{u}^{\ell}$. Let
$\mathbf{m}_{\ell}^{\prime}:=\langle N_{u}^{\ell}:u\in\mathcal{P}(n)\rangle$.
We show that $\mathbf{m}_{\ell}\leq_{\mathfrak{i}}\mathbf{m}_{\ell}^{\prime}$.
It is then not difficult to show, using a downward Löwenheim-Skolem-like
argument, that one can take appropriate submodels of the $N_{u}^{\ell}$’s to
be of the same size as the $M_{u}^{\ell}$’s and satisfy the requirements.

Essentially by [BG17, BGKV16] (and more precisely by the proof of [BGKV16,
6.8]), the two-dimensional independence relation on $\mathfrak{i}$ is given by
$\kappa$-coheir (see [BG17, 3.2]). Thus it follows from the claim before the
proof of [BG17, 8.2] and some forking calculus that
$\mathbf{m}_{\ell}\leq_{\mathfrak{i}}\mathbf{m}_{\ell}^{\prime}$. ∎

To find a $(\lambda,n+1)$-system, one can simply resolve a
$(\lambda^{+},n)$-system. This is formalized by the following lemma:

######  Lemma 11.14.

Let $\mathfrak{i}$ be a very good multidimensional independence notion with
domain $\\{\lambda,\lambda^{+}\\}$. If $\mathfrak{i}^{\text{brim}}$ has
$(\lambda^{+},n)$-existence, then $\mathfrak{i}^{\text{brim}}$ has
$(\lambda,n+1)$-existence.

###### Proof.

By $(\lambda^{+},n)$-existence, pick $\mathbf{m}$ a brimmed independent
$(\lambda^{+},\mathcal{P}(n))$-system. By Lemma 11.9(4),
$\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i}^{\text{brim}},\mathcal{P}(n)}$
is resolvable, so there exists a strictly increasing continuous chain
$\langle\mathbf{m}_{i}:i<\lambda^{+}\rangle$ in
$\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\text{brim}},\mathcal{P}(n)}$
where for each $i<\lambda^{+}$, $\mathbf{m}_{i}$ is a brimmed independent
$(\lambda,\mathcal{P}(n))$-system, and
$\mathbf{m}=\bigcup_{i<\lambda^{+}}\mathbf{m}_{i}$. Now by definition of
$\leq_{\mathfrak{K}^{\text{proper}}_{\mathfrak{i}^{\text{brim}},\mathcal{P}(n)}}$,
we must have that $\mathbf{m}_{0}\ast\mathbf{m}_{1}$ is a brimmed independent
$(\lambda,\mathcal{P}(n+1))$-system, as desired. ∎

Thus we obtain the existence properties for free if the interval of cardinals
we are working in is closed under the successor operation.

######  Lemma 11.15.

Let $\mathfrak{i}$ be a very good independence notion with domain
$\Theta:=[\lambda_{1},\lambda_{2})$. If $\lambda_{2}$ is a limit cardinal,
then for any $\lambda\in\Theta$, $\mathfrak{i}^{\text{brim}}$ has
$(\lambda,<\omega)$-existence.

###### Proof.

We prove by induction on $n<\omega$ that for any $\lambda\in\Theta$,
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n)$-existence. When $n=0$, this is
part of Lemma 11.9. Assume now inductively that $\mathfrak{i}^{\text{brim}}$
has $(\lambda,n)$-existence for all $\lambda\in\Theta$. Let
$\lambda\in\Theta$. Since $\lambda_{2}$ is limit, $\lambda^{+}\in\Theta$. By
the induction hypothesis, $\mathfrak{i}^{\text{brim}}$ has
$(\lambda^{+},n)$-existence. By Lemma 11.14 (applied to
$\mathfrak{i}\upharpoonright\\{\lambda,\lambda^{+}\\}$),
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+1)$-existence, as desired. ∎

We now attack the harder question of obtain $(n+1)$-dimensional uniqueness
from the $n$-dimensional properties. This is done using the weak diamond:

######  Lemma 11.16 (The stepping up lemma).

Let $\lambda_{1}$ and $\lambda_{2}$ be infinite cardinals such that
$2^{\lambda_{1}}=2^{<\lambda_{2}}<2^{\lambda_{2}}$. Let $\mathfrak{i}$ be a
very good independence relation with domain
$\Theta:=[\lambda_{1},\lambda_{2}]$. Assume that for all $\lambda\in\Theta$,
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n)$-existence and
$(\lambda,n)$-uniqueness. Then:

  1. (1)

For all $\lambda\in[\lambda_{1},\lambda_{2})$, $\mathfrak{i}^{\text{brim}}$
has $(\lambda,n+1)$-existence.

  2. (2)

There exists $\lambda\in[\lambda_{1},\lambda_{2})$ such that
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+1)$-uniqueness.

  3. (3)

If $\mathfrak{i}$ is nice and $\mathfrak{i}^{\text{brim}}$ has
$(\lambda_{2},n+1)$-existence, then $\mathfrak{i}^{\text{brim}}$ has
$(\lambda,n+1)$-uniqueness for _all_ $\lambda\in[\lambda_{1},\lambda_{2})$.

###### Proof.

  1. (1)

By Lemma 11.14 applied to
$\mathfrak{i}\upharpoonright\\{\lambda,\lambda^{+}\\}$.

  2. (2)

By Lemma 11.10, for every $\lambda\in[\lambda_{1},\lambda_{2}]$,
$\mathfrak{i}^{\text{brim}}$ has strong $(\lambda,n)$-uniqueness and
$(\lambda,n)$-extension, and for every $\lambda\in[\lambda_{1},\lambda_{2})$,
$\mathfrak{i}^{\text{brim}}$ also has $(\lambda,n+1)$-extension. Write
$\mathfrak{K}^{\ast}:=\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i},\mathfrak{i}^{\text{brim}},\mathcal{P}(n)}$
(Definition 11.8).

Claim: There is $N\in\mathfrak{K}_{[\lambda_{1},\lambda_{2})}^{\ast}$ which is
a $\phi_{n}$-amalgamation base in $\mathfrak{K}_{\|N\|}^{\ast}$.

Proof of Claim: Let $\mathfrak{K}^{\ast\ast}$ be $|\mathfrak{K}^{\ast}|$
ordered by $\mathbf{m}\leq_{\mathfrak{K}^{\ast\ast}}\mathbf{m}^{\prime}$ if
and only if $\mathbf{m}\leq_{\mathfrak{K}}^{d}\mathbf{m}^{\prime}$ (Definition
8.6(2)). We apply Theorem 5.8, where $\mathfrak{K}$,
$\mathfrak{K}^{\ast},\phi$ there stand for $\mathfrak{K}^{\ast}$,
$\mathfrak{K}^{\ast\ast},\phi_{n}$ here. Note first that
$\mathfrak{K}_{[\lambda_{1},\lambda_{2})}^{\ast}$ is not empty since
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+1)$-existence for each
$\lambda\in[\lambda_{1},\lambda_{2})$. Thus if we can check that the
hypotheses of Theorem 5.8, we will obtain the conclusion of the claim. First
observe that $\mathfrak{K}^{\ast}$ is $[\lambda_{1},\lambda_{2})$-continuous
by Lemma 11.9(5). Next, there is a $\phi_{n}$-universal model in
$\mathfrak{K}_{\lambda_{2}}^{\ast\ast}$ by Lemma 10.15(3). Further,
$\mathfrak{K}_{\lambda_{2}}^{\ast\ast}$ is $\phi_{n}$-categorical by Lemma
10.15(2). Moreover, for any $\lambda\in[\lambda_{1},\lambda_{2})$,
$\mathfrak{K}_{\lambda}^{\ast}$ has $\phi$-uniqueness (Lemma 10.15(4)).
Finally, for any $\lambda\in[\lambda_{1},\lambda_{2})$, if we can
$\phi_{n}$-amalgamate in $\mathfrak{K}^{\ast\ast}$, then we can
$\phi_{n}$-amalgamate in $\mathfrak{K}^{\ast}$ by Lemma 10.15(5).
$\dagger_{\text{Claim}}$

Now let $\lambda:=\|N\|$, and apply Theorem 10.16, where
$\mathfrak{i},\mathfrak{i}^{\ast}$ there stand for
$\mathfrak{i}_{\lambda},(\mathfrak{i}^{\text{brim}})_{\lambda}$ here. We get
that $\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+1)$-uniqueness, as desired.

  3. (3)

Assume that $\mathfrak{i}$ is nice and $\mathfrak{i}^{\text{brim}}$ has
$(\lambda_{2},n+1)$-existence. It is enough to show that
$\mathfrak{i}^{\text{brim}}$ has $(\lambda_{1},n+1)$-uniqueness. Then given an
arbitrary $\lambda\in[\lambda_{1},\lambda_{2})$, we can run the same argument
with $\Theta=[\lambda,\lambda_{2}]$. By the previous part, there is
$\lambda\in[\lambda_{1},\lambda_{2})$ such that $\mathfrak{i}^{\text{brim}}$
has $(\lambda,n+1)$-uniqueness. Let $\mathbf{m}_{0}$ be a brimmed independent
$\mathcal{P}^{-}(n+1)$ system in $\lambda_{1}$ and let
$\mathbf{m}_{1},\mathbf{m}_{2}$ be brimmed independent
$\mathcal{P}(n+1)$-systems in $\lambda_{1}$ such that
$\mathbf{m}_{1}\upharpoonright\mathcal{P}^{-}(n+1)=\mathbf{m}_{2}\upharpoonright\mathcal{P}^{-}(n+1)=\mathbf{m}_{0}$.
We build $\langle\mathbf{m}_{\ell}^{i}:i\leq\lambda\rangle$ strictly
increasing continuous in $\mathfrak{K}_{\mathfrak{i},\mathcal{P}(n+1)}$, such
that for all $i<\lambda$:

    1. (a)

$\mathbf{m}_{\ell}^{0}=\mathbf{m}_{\ell}$, $\ell=1,2$.

    2. (b)

All the models in $\mathbf{m}_{\ell}^{i}$ have cardinality $\lambda_{1}+|i|$.

    3. (c)

$\mathbf{m}_{\ell}^{i+1}$ is brimmed over $\mathbf{m}_{\ell}^{i}$, for
$\ell=1,2$.

    4. (d)

$\mathbf{m}_{1}^{i}\upharpoonright\mathcal{P}^{-}(n+1)=\mathbf{m}_{2}^{i}\upharpoonright\mathcal{P}^{-}(n+1)$.

This is possible by definition of niceness (using that the extensions are
brimmed, we can take unions at limits). Now let
$\mathbf{m}^{\ast}:=\mathbf{m}_{1}^{\lambda}\upharpoonright\mathcal{P}^{-}(n+1)=\mathbf{m}_{2}^{\lambda}\upharpoonright\mathcal{P}^{-}(n+1)$.
We claim that $\mathbf{m}^{\ast}$ is a uniqueness base. From that it will
directly follow that $\mathbf{m}_{0}$ is also a uniqueness base, as desired.
We would like to use that $\mathfrak{i}^{\text{brim}}$ has
$(\lambda,n+1)$-uniqueness, but we can’t do so directly since
$\mathbf{m}^{\ast}$ may not be brimmed (all the models in the system are
brimmed, but the extensions between the models may not be). We want to apply
Lemma 9.14, where $\mathfrak{i},\mathfrak{i}^{\ast},J$ there stand for
$\mathfrak{i}_{\lambda}\upharpoonright\mathcal{I}_{n+2}$,
$(\mathfrak{i}^{\text{brim}})_{\lambda}\upharpoonright\mathcal{I}_{n+2},\mathcal{P}(n+1)$
here. To apply it, we need to check that $\mathfrak{i}$ has
$(\lambda,n+2)$-extension.

By Lemma 9.12 and categoricity, it is enough to check that
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+2)$-extension. We know already
that $\mathfrak{i}^{\text{brim}}$ has $(\lambda^{\prime},n+1)$-existence for
all $\lambda^{\prime}\in[\lambda_{1},\lambda_{2}]$ (when
$\lambda^{\prime}=\lambda_{2}$, this is a hypothesis, and when
$\lambda^{\prime}\in[\lambda_{1},\lambda_{2})$, this was derived in the first
part of this lemma). In particular, $\mathfrak{i}^{\text{brim}}$ has
$(\lambda^{+},n+1)$-existence. By Lemma 11.14, $\mathfrak{i}^{\text{brim}}$
has $(\lambda,n+2)$-existence. Since $\mathfrak{i}^{\text{brim}}$ has
$(\lambda,n+1)$-uniqueness, we get by Lemma 11.10 that
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+2)$-extension, as desired.

∎

We obtain two sufficient conditions to have all the $n$-dimensional
properties: weak GCH or niceness.

######  Theorem 11.17 (The brimmed excellence theorem).

Let $\langle\lambda_{n}:n<\omega\rangle$ be an increasing sequence of infinite
cardinals such that $2^{\lambda_{n}}=2^{<\lambda_{n+1}}<2^{\lambda_{n+1}}$ for
all $n<\omega$. Let $\lambda_{\omega}:=\sup_{n<\omega}\lambda_{n}$ and let
$\Theta:=[\lambda_{0},\lambda_{\omega})$. Let $\mathfrak{i}$ be a very good
independence notion with domain $\Theta$. Assume that at least one of the
following conditions holds:

  1. (1)

$\mathfrak{i}$ is nice.

  2. (2)

$\lambda_{n+1}=\lambda_{n}^{+}$ for all $n<\omega$.

Then for all $\lambda\in\Theta$, $\mathfrak{i}^{\text{brim}}$ has
$(\lambda,<\omega)$-existence and $(\lambda,<\omega)$-uniqueness.

###### Proof.

By Lemma 11.15 (where $\lambda_{1},\lambda_{2}$ there stand for
$\lambda_{0},\lambda_{\omega}$ here), for any $\lambda\in\Theta$,
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,<\omega)$-existence. Next, we prove
by induction on $n<\omega$ that for all $\lambda\in\Theta$,
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n)$-uniqueness. When $n=0$, this is
part of Lemma 11.9. Assume now that for all $\lambda\in\Theta$, $\mathfrak{i}$
has $(\lambda,n)$-uniqueness. Fix $k<\omega$. By Lemma 11.16 (where
$\lambda_{1},\lambda_{2}$ there stand for $\lambda_{k},\lambda_{k+1}$ here),
$\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+1)$-uniqueness for all
$\lambda\in[\lambda_{k},\lambda_{k+1})$. Since $k$ was arbitrary, this shows
that $\mathfrak{i}^{\text{brim}}$ has $(\lambda,n+1)$-uniqueness for all
$\lambda\in\Theta$. ∎

##  12\. Building primes

In this section, we show that prime models over independent systems can be
built, provided that we have the $n$-dimensional extension and uniqueness
properties. This is the _prime extension theorem_ , Theorem 12.5.

To build primes, we need the following technical concept, which says that a
system can in a sense only be extended in an independent way. This is similar
to the definition of a _reduced tower_ [SV99, 3.1.11]. The differences are
that we work with systems and are slightly more local (we work inside a given
system).

######  Definition 12.1.

Let $\mathfrak{i}=(\mathfrak{K},\mathcal{I},\operatorname{NF})$ be a
multidimensional independence relation and let $\mathbf{m}$ be an independent
$I$-system, with $I=J\cup\\{v\\}$, $J<v$. Let $\mathbf{m}_{\ast}$ be an
independent $J$-system. We say that $\mathbf{m}$ is _reduced inside
$\mathbf{m}_{\ast}$_ if:

  1. (1)

$\mathbf{m}\upharpoonright J\leq_{\mathfrak{i}}\mathbf{m}_{\ast}$.

  2. (2)

Whenever $\mathbf{m}^{\prime}$ is an independent $I$-system and
$f:\mathbf{m}\rightarrow\mathbf{m}^{\prime}$ is a systems embedding, if:

    1. (a)

$f$ is the identity on $J$.

    2. (b)

$\mathbf{m}\upharpoonright
J\leq_{\mathfrak{i}}\mathbf{m}^{\prime}\upharpoonright J$.

    3. (c)

$\mathbf{m}^{\prime}\upharpoonright J\leq_{\mathfrak{i}}\mathbf{m}_{\ast}$.

Then $f(\mathbf{m})\leq_{\mathfrak{i}}\mathbf{m}^{\prime}$.

Reduced systems exist:

######  Lemma 12.2 (Existence of reduced systems).

Let $\mathfrak{i}$ be a very good multidimensional independence notion. Let
$\Theta:=\operatorname{dom}(\mathfrak{K}_{\mathfrak{i}})$. Let
$I=J\cup\\{v\\}$ be in $\mathcal{I}_{\mathfrak{i}}$, $J<v$. Assume that $I$ is
an initial segment of $\mathcal{P}(n)$. Let $\lambda\in\Theta$ be such that
$\lambda^{+}\in\Theta$ and $\mathfrak{i}$ has $(\lambda,n+1)$-extension.

For any independent $J$-system $\mathbf{m}$ in $\mathfrak{i}_{\lambda^{+}}$,
there exists a $\leq_{\mathfrak{i}}$-increasing chain of $I$-systems
$\langle\mathbf{m}_{i}:i<\lambda^{+}\rangle$ in $\mathfrak{i}_{\lambda}$ such
that each $\mathbf{m}_{i}$ is reduced in $\mathbf{m}$ and
$\bigcup_{i<\lambda^{+}}\mathbf{m}_{i}\upharpoonright J=\mathbf{m}$.

###### Proof.

Without loss of generality, $\lambda=\min(\Theta)$. Let
$\mathfrak{i}^{\prime}$ be as given by Lemma 11.5. Let $\langle
a_{i}:i<\lambda^{+}\rangle$ be an enumeration of $\mathbf{m}$. We build
$\langle\mathbf{m}_{i}^{0}:i<\lambda^{+}\rangle$ a
$\leq_{\mathfrak{i}}$-increasing continuous chain of independent $J$-systems
in $\mathfrak{i}_{\lambda}$, $\langle N_{i}:i<\lambda^{+}\rangle$ in
$\mathfrak{K}_{\mathfrak{i}_{\lambda}}$, and $\langle f_{i,j}:i\leq
j<\lambda^{+}\rangle$ such that for all $i<\lambda^{+}$:

  1. (1)

$f_{i,j}:N_{i}\rightarrow N_{j}$ form a continuous directed system.

  2. (2)

$\mathbf{m}_{i}^{0}\smallfrown N_{i}$ is an independent $I$-system.

  3. (3)

$f_{i,j}$ fixes $\mathbf{m}_{i}^{0}$.

  4. (4)

$\mathbf{m}_{i+1}^{0}$ contains $a_{i}$.

  5. (5)

If $\mathbf{m}_{i}^{0}\smallfrown N_{i}$ is _not_ reduced in $\mathbf{m}$,
then $\mathbf{m}_{i+1}^{0}$, $N_{i+1}$, and $f_{i,i+1}$ witness it. That is,
$\mathbf{m}_{i}^{0}\smallfrown
f_{i,i+1}[N_{i}]\not\leq_{\mathfrak{i}}\mathbf{m}_{i+1}^{0}\smallfrown
N_{i+1}$.

This is possible using the extension property and taking direct limits at
limits (we use strong continuity to see that the direct limit is still an
independent system). This is enough: let $\mathbf{m}_{\lambda^{+}}^{0}$,
$f_{i,\lambda^{+}}$, $N_{\lambda^{+}}$ be the direct limit of the system. Note
that $\mathbf{m}_{\lambda^{+}}^{0}\smallfrown N_{\lambda^{+}}$ is an
independent $I$-system in $\mathfrak{i}^{\prime}$, and since
$\mathfrak{K}_{\mathfrak{i}^{\prime},I}$ is an AEC, this system can be
$\leq_{\mathfrak{i}}$-resolved into independent systems of cardinality
$\lambda$. By Lemma 5.3, this implies that there is a club
$C\subseteq\lambda^{+}$ such that for any $i\in C$,
$\mathbf{m}_{i}^{0}\smallfrown
f_{i,\lambda^{+}}[N_{i}]\leq_{\mathfrak{i}}\mathbf{m}_{\lambda^{+}}^{0}\smallfrown
N_{\lambda^{+}}$. Now if $i\in C$ and $\mathbf{m}_{i}^{0}\smallfrown N_{i}$ is
_not_ reduced in $\mathbf{m}$, then by property (5),
$\mathbf{m}_{i}^{0}\smallfrown
f_{i,i+1}[N_{i}]\not\leq_{\mathfrak{i}}\mathbf{m}_{i+1}^{0}\smallfrown
N_{i+1}$. Taking the image of this statement under $f_{i+1,\lambda^{+}}$, we
get that $\mathbf{m}_{i}^{0}\smallfrown
f_{i,\lambda^{+}}[N_{i}]\not\leq_{\mathfrak{i}}\mathbf{m}_{i+1}^{0}\smallfrown
f_{i+1,\lambda^{+}}[N_{i+1}]$. By monotonicity 2, this means that
$\mathbf{m}_{i}^{0}\smallfrown
f_{i,\lambda^{+}}[N_{i}]\not\leq_{\mathfrak{i}}\mathbf{m}_{i+1}^{0}\smallfrown
N_{\lambda^{+}}$, a contradiction. Thus for any $i\in C$,
$\mathbf{m}_{i}^{0}\smallfrown N_{i}$ _is_ reduced in $\mathbf{m}$. Let
$\langle\alpha_{i}:i<\lambda^{+}\rangle$ be a normal enumeration of $C$ and
let $\mathbf{m}_{i}:=\mathbf{m}_{\alpha_{i}}^{0}\smallfrown N_{\alpha_{i}}$. ∎

We use reduced systems to build prime ones:

######  Definition 12.3.

Let $\mathfrak{i}$ be a multidimensional independence relation.

  1. (1)

Let $I=J\cup\\{v\\}$ be in $\mathcal{I}_{\mathfrak{i}}$ with $J<v$, and let
$\mathbf{m}$ be an independent $I$-system. We say that $\mathbf{m}$ is _prime_
if whenever $\mathbf{m}^{\prime}$ is an independent $J$-system with
$\mathbf{m}\upharpoonright J=\mathbf{m}^{\prime}\upharpoonright J$, there
exists $f:\mathbf{m}\rightarrow\mathbf{m}^{\prime}$ a systems embedding fixing
$\mathbf{m}\upharpoonright J$.

  2. (2)

We say that $\mathfrak{i}$ has _prime extension_ if for any
$I=J\cup\\{v\\}\in\mathcal{I}_{\mathfrak{i}}$ with $J<v$ and any independent
$J$-system $\mathbf{m}$, there is a _prime_ $I$-system $\mathbf{m}^{\prime}$
such that $\mathbf{m}\cong\mathbf{m}^{\prime}\upharpoonright J$. We define
variations such as $(\lambda,n)$-prime extension as in Definition 10.2.

######  Lemma 12.4.

Let $\mathfrak{i}$ be a very good multidimensional independence notion, let
$\Theta:=\operatorname{dom}(\mathfrak{K}_{\mathfrak{i}})$. Let
$I=J\cup\\{v\\}$ be in $\mathcal{I}_{\mathfrak{i}}$, $J<v$. Assume that $I$ is
an initial segment of $\mathcal{P}(n)$. Let $\lambda\in\Theta$ be such that
$\lambda^{+}\in\Theta$, $\mathfrak{i}$ has $(\lambda,n+1)$-extension and
$(\lambda,n+1)$-uniqueness. If $\mathfrak{K}_{\mathfrak{i}}$ is categorical in
$\lambda^{+}$, then any independent $J$-system $\mathbf{m}$ in
$\mathfrak{i}_{\lambda^{+}}$ can be extended to a _prime_ independent
$I$-system in $\mathfrak{i}_{\lambda^{+}}$.

###### Proof.

Similar to [She09a, III.4.9] (or [Vas17a, 3.6]), using Lemma 12.2 to get the
resolution into reduced triples. ∎

######  Theorem 12.5 (The prime extension theorem).

Let $\mathfrak{i}$ be a very good multidimensional independence notion, let
$\Theta:=\operatorname{dom}(\mathfrak{K}_{\mathfrak{i}})$. Let
$\lambda\in\Theta$ be such that $\lambda^{+}\in\Theta$ and
$\mathfrak{K}_{\mathfrak{i}}$ is categorical in $\lambda^{+}$. If
$\mathfrak{i}$ has $(\lambda,n)$-extension and $(\lambda,n)$-uniqueness, then
$\mathfrak{i}$ has $(\lambda^{+},<n)$-prime extension.

###### Proof.

Immediate from Lemma 12.4. ∎

##  13\. Excellent classes

In this section, we introduce the definition of an _excellent_ AEC (this
generalizes the definition in [She83a, She83b]). We show how to obtain
excellent AECs from the setups considered earlier (Lemma 13.3) and explain why
excellent classes are local and have primes (Theorem 13.9), hence (as we will
see in the next section) admit categoricity transfers. We also show how to get
excellence in compact AECs (Theorem 13.7) and in $(<\omega)$-extendible good
frames, using the weak diamond (Theorem 13.6).

######  Definition 13.1.

Let $\mathfrak{i}$ be a multidimensional independence relation. We call
$\mathfrak{i}$ _excellent_ if:

  1. (1)

$\mathfrak{K}_{\mathfrak{i}}$ is an AEC.

  2. (2)

$\mathfrak{i}$ is very good.

  3. (3)

$\mathfrak{i}$ has extension and uniqueness.

We say that an AEC $\mathfrak{K}$ is _excellent_ if there exists an excellent
multidimensional independence relation $\mathfrak{i}$ such that
$\mathfrak{K}=\mathfrak{K}_{\mathfrak{i}}$.

Reasonable multidimensional independence relations will be local:

######  Lemma 13.2.

Let $\mathfrak{i}$ be a multidimensional independence relation. If
$\mathfrak{K}_{\mathfrak{i}}$ is an AEC, $\mathfrak{i}$ has extension and
uniqueness, and $\mathfrak{i}$ satisfies the definition of being very good,
except possibly for $\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i}))$ being a good frame,
then $\mathfrak{K}_{\mathfrak{i}}$ is
$\text{LS}(\mathfrak{K}_{\mathfrak{i}})$-local.

###### Proof.

As in [She09a, III.1.10] (see also [GK]). ∎

The next two lemmas tell us how to derive excellence from the conclusion of
the brimmed excellence theorem (Theorem 11.17).

######  Lemma 13.3.

Let $\mathfrak{i}$ be a very good multidimensional independence relation. Let
$\lambda:=\min(\operatorname{dom}(\mathfrak{i}))$. If $\mathfrak{i}$ has
$(\lambda,<\omega)$-extension and $(\lambda,<\omega)$-uniqueness, then there
exists a unique excellent multidimensional independence relation
$\mathfrak{i}^{\prime}$ such that
$(\mathfrak{i}^{\prime})_{\lambda}=\mathfrak{i}_{\lambda}$. In particular, the
AEC generated by $(\mathfrak{K}_{\mathfrak{i}})_{\lambda}$ is excellent.

###### Proof.

Let $\mathfrak{K}$ be the AEC generated by
$(\mathfrak{K}_{\mathfrak{i}})_{\lambda}$. Let $\mathfrak{i}^{\prime}$ be as
described by Lemma 11.5. It satisfies all the conditions in the definition of
being very good, except perhaps for existence, 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$}}}\limits(\mathfrak{i}^{\prime}))$ being a good
frame. By Lemma 11.6, $\mathfrak{i}^{\prime}$ has extension and uniqueness. By
Lemma 13.2, $\mathfrak{K}$ is $\text{LS}(\mathfrak{K})$-local. Since
$\mathfrak{i}^{\prime}$ has extension, $\mathfrak{K}$ has in particular
amalgamation. By Fact 6.11, there is a good frame $\mathfrak{s}$ with
underlying class $\mathfrak{K}$. Let
$\mathfrak{t}:=\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i}^{\prime}))$. It is clear
from the definition that $\mathfrak{s}_{\lambda}=\mathfrak{t}_{\lambda}$.
Moreover, $\mathfrak{t}$ satisfies most of the axioms of a good frame (Fact
7.5). The only axiom that can fail is local character. However since
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i}^{\prime})$ is very good,
it has local character, so every type does not $\mathfrak{t}$-fork over a
model of size $\lambda$. By Lemma 6.5 and Remark 6.6, this implies that
$\mathfrak{s}=\mathfrak{t}$, so $\mathfrak{t}$ is a good frame. Thus
$\mathfrak{i}^{\prime}$ is very good, and hence excellent. ∎

######  Lemma 13.4.

Let $\mathfrak{i}$ be a very good multidimensional independence notion and let
$\Theta:=\operatorname{dom}(\mathfrak{i})$. Let $\lambda:=\min(\Theta)$ and
let $\mathfrak{K}$ be the AEC generated by
$\left(\mathfrak{K}_{\mathfrak{i}}\right)_{\lambda}$. If:

  1. (1)

$\mathfrak{K}$ is categorical in $\lambda$.

  2. (2)

$\mathfrak{i}^{\text{brim}}$ has $(\lambda,<\omega)$-existence and
$(\lambda,<\omega)$-uniqueness.

Then $\mathfrak{K}$ is excellent.

###### Proof.

By Lemma 11.10, $\mathfrak{i}^{\ast}$ also has strong
$(\lambda,<\omega)$-uniqueness. By Theorems 9.12 and 9.15, we obtain that
$\mathfrak{i}$ also has $(\lambda,<\omega)$-extension and
$(\lambda,<\omega)$-uniqueness. Now apply Lemma 13.3. ∎

Assuming a weak version of the generalized continuum hypothesis, we obtain
excellence in $(<\omega)$-extendible very good frames. We will use the
following notation:

######  Definition 13.5.

For $S$ a class of cardinals, we write $\operatorname{WGCH}(S)$ for the
statement “$2^{\lambda}<2^{\lambda^{+}}$ for all $\lambda\in S$”. We write
$\operatorname{WGCH}$ instead of $\operatorname{WGCH}(\operatorname{Card})$,
where Card is the class of all cardinals.

######  Theorem 13.6.

Let $\mathfrak{s}$ be a $(<\omega)$-extendible categorical very good
$\lambda$-frame. Let $\mathfrak{K}$ be the AEC generated by
$\mathfrak{K}_{\mathfrak{s}}$. If
$\operatorname{WGCH}([\lambda,\lambda^{+\omega}))$ holds, then $\mathfrak{K}$
is excellent.

###### Proof.

Let $\Theta:=[\lambda,\lambda^{+\omega})$. By Fact 7.20(4), there is a very
good categorical frame $\mathfrak{t}$ such that
$\operatorname{dom}(\mathfrak{t})=\Theta$ and
$\mathfrak{t}_{\lambda^{+n}}=\mathfrak{s}^{+n}$ for all $n<\omega$. By
definition of being very good, there is a very 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 $\mathfrak{K}_{\mathfrak{t}}$.

Fix $\mathfrak{i}$ and $\mathfrak{i}^{\ast}$ satisfying the conclusion of
Theorem 11.3. By Lemma 11.17, we have that $\mathfrak{i}^{\ast}$ has
$(\lambda,<\omega)$-existence and $(\lambda,<\omega)$-uniqueness. Now apply
Lemma 13.4. ∎

In a compact AEC, excellence follows from categoricity (without any cardinal
arithmetic hypothesis)

######  Theorem 13.7.

Let $\mathfrak{K}$ be a compact AEC. If $\mathfrak{K}$ is categorical in some
$\mu>\text{LS}(\mathfrak{K})$, then
${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+6}\text{-sat}}$ is an excellent
AEC.

###### Proof.

By Corollary 4.8, $\mathfrak{K}$ is categorical in a proper class of
cardinals, so we might as well assume that
$\mu\geq\beth_{\omega}(\text{LS}(\mathfrak{K}))$. Fix $\mathfrak{i}$
satisfying the conclusion of Theorem 11.13. Let
$\Theta:=[\text{LS}(\mathfrak{K})^{+6},\beth_{\omega}(\text{LS}(\mathfrak{K})))$.
By Lemma 11.17, we have that $\mathfrak{i}^{\text{brim}}$ has
$(\text{LS}(\mathfrak{K})^{+6},<\omega)$-existence and
$(\text{LS}(\mathfrak{K})^{+6},<\omega)$-uniqueness. The result now follows
from Lemma 13.4. ∎

An appropriate subclass of saturated models of an excellent class will always
have primes in the following sense:

######  Definition 13.8 ([She09a, III.3.2]).

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

  1. (1)

A _prime triple_ is a triple $(a,M,N)$ such that $M\leq_{\mathfrak{K}}N$,
$a\in|N|$, and whenever $\mathbf{tp}(b/M;N^{\prime})=\mathbf{tp}(a/M;N)$,
there exists a $\mathfrak{K}$-embedding $f:N\xrightarrow[M]{}N^{\prime}$ such
that $f(b)=a$.

  2. (2)

$\mathfrak{K}$ _has primes_ if for any $p\in\mathscr{S}(M)$ there exists a
prime triple $(a,M,N)$ such that $p=\mathbf{tp}(a/M;N)$.

######  Theorem 13.9 (Structure of excellent AECs).

If $\mathfrak{K}$ is an excellent AEC, then $\mathfrak{K}$ is not empty, has
amalgamation, joint embedding, no maximal models and is
$\text{LS}(\mathfrak{K})$-local. Moreover,
${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}}$ has primes.

###### Proof.

Let $\lambda:=\text{LS}(\mathfrak{K})$. Let $\mathfrak{i}$ be an excellent
multidimensional independence relation with
$\mathfrak{K}_{\mathfrak{i}}=\mathfrak{K}$. $\mathfrak{K}$ is not empty and
has no maximal models because $\mathfrak{i}$ has $1$-extension, and
$\mathfrak{K}$ has amalgamation and joint embedding because $\mathfrak{i}$ has
$1$-uniqueness (see Lemma 10.5). By Lemma 13.2, $\mathfrak{K}$ is
$\text{LS}(\mathfrak{K})$-local. Let $\mathfrak{i}^{\ast}$ be the restriction
of $\mathfrak{i}$ to the fragmented AEC $\mathfrak{K}^{\ast}$ such that
$\mathfrak{K}^{\ast}_{\leq\lambda}=\mathfrak{K}_{\lambda}$ and
$\mathfrak{K}^{\ast}_{>\lambda}={\mathfrak{K}}^{\lambda^{+}\text{-sat}}$. It
is easy to check that $\mathfrak{i}^{\ast}$ is still a very good
multidimensional independence relation with extension and uniqueness. By
Theorem 12.5, $\mathfrak{i}^{\ast}$ has $(\lambda^{+},<\omega)$-prime
extension. By standard arguments similar to those in [She83b, §5], one gets
that $\mathfrak{i}^{\ast}_{\geq\lambda^{+}}$ has prime extension.

Now by [She09a, III.4.9] (see also [Vas17a, 3.6]),
$\mathfrak{K}^{\ast}_{\lambda^{+}}$ has primes. It remains to check that
$\mathfrak{K}^{\ast}_{>\lambda^{+}}$ has primes. Let
$M\in\mathfrak{K}^{\ast}_{>\lambda^{+}}$ and let $p\in\mathscr{S}(M)$. Let
$N\in\mathfrak{K}^{\ast}$ be such that $M\leq_{\mathfrak{K}}N$ and $N$
realizes $p$, say with $a$. By local character for $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits:=\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits(\mathfrak{i}^{\ast})$, there exists
$M_{0},N_{0}\in\mathfrak{K}_{\lambda}^{\ast}$ such that $a\in|N_{0}|$ and
$M\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}N_{0}$. Since
$\mathfrak{K}_{\lambda^{+}}^{\ast}$ has primes, we may pick
$N_{0}^{\ast}\leq_{\mathfrak{K}}N_{0}$ such that $(a,M_{0},N_{0})$ is a prime
triple in $\mathfrak{K}^{\ast}$. By monotonicity, we still have
$M\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}N_{0}^{\ast}$, so by prime
extension (applied to the independent system $\mathbf{m}$ consisting of
$M_{0}$, $M$, and $N_{0}^{\ast}$), there exists a prime model $N^{\ast}$ over
$\mathbf{m}$, which can be taken to be contained inside $N$. Now check that
$(a,M,N^{\ast})$ is a prime triple. ∎

##  14\. Excellence and categoricity: the main theorems

In this section, we combine the results derived so far about excellence with
known categoricity transfers to obtain the main theorems of this paper. In
addition to Facts 3.25, 3.26, the facts we will use about categoricity are:

######  Fact 14.1.

Let $\mathfrak{K}$ be an $\text{LS}(\mathfrak{K})$-local AEC with
amalgamation, and arbitrarily large models. Assume that $\mathfrak{K}$ is
categorical in _some_ $\mu>\text{LS}(\mathfrak{K})$. Then:

  1. (1)

[Vas17b, 10.9]666The main ideas of the proof appear in [Vas17d, Vas18]. If
$\mathfrak{K}$ has primes then $\mathfrak{K}$ is categorical in all
$\mu^{\prime}\geq\mu$.

  2. (2)

[Vas17b, 10.3,10.4]777The upward part of the transfer (i.e. getting
categoricity in $\mu^{\prime}\geq\mu$) is due to Grossberg and VanDieren
[GV06c, GV06a]. If $\mathfrak{K}$ is categorical in $\text{LS}(\mathfrak{K})$
and $\mu$ is a successor, then $\mathfrak{K}$ is categorical in _all_
$\mu^{\prime}\geq\text{LS}(\mathfrak{K})$.

We obtain the following categoricity transfer for excellent classes:

######  Theorem 14.2.

Let $\mathfrak{K}$ be an excellent AEC. If $\mathfrak{K}$ is categorical in
_some_ $\mu>\text{LS}(\mathfrak{K})$, then there exists
$\chi<h(\text{LS}(\mathfrak{K}))$ such that $\mathfrak{K}$ is categorical in
_all_ $\mu^{\prime}\geq\min(\mu,\chi)$. If in addition $\mathfrak{K}$ is
categorical in $\text{LS}(\mathfrak{K})$, then $\mathfrak{K}$ is categorical
in all $\mu^{\prime}>\text{LS}(\mathfrak{K})$.

###### Proof.

By Fact 3.26, we may assume that $\mu>\text{LS}(\mathfrak{K})^{+}$. By Fact
14.1 and Theorem 13.9,
${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}}$ is categorical on a
tail of cardinals. By Fact 3.25, $\mathfrak{K}$ is categorical in all
$\mu^{\prime}\geq\min(\mu,\chi)$. If $\mathfrak{K}$ is categorical in
$\text{LS}(\mathfrak{K})$, then it will be categorical in a successor, hence
by Fact 14.1 again, $\mathfrak{K}$ will be categorical in all cardinals above
$\text{LS}(\mathfrak{K})$. ∎

When working inside a $(<\omega)$-extendible categorical good $\lambda$-frame,
it is known how to transfer categoricity across finite successors of
$\lambda$, so we can be more precise:

######  Lemma 14.3.

Let $\mathfrak{s}$ be a $(<\omega)$-extendible categorical good
$\lambda$-frame. Let $\mathfrak{K}$ be the AEC generated by
$\mathfrak{K}_{\mathfrak{s}}$. Let $n<\omega$ and let $\mathfrak{K}^{\ast}$ be
the AEC generated by $\mathfrak{K}_{\mathfrak{s}^{+n}}$. If
$\mathfrak{K}^{\ast}$ is excellent, then:

  1. (1)

$\mathfrak{K}$ has arbitrarily large models.

  2. (2)

If in addition $\mathfrak{K}$ is categorical in _some_ $\mu>\lambda$, then
$\mathfrak{K}$ is categorical in _all_ $\mu^{\prime}>\lambda$,
$\mathfrak{K}_{\geq\mu}$ has amalgamation, no maximal models, and is
$\mu$-local.

###### Proof.

If $\mu\leq\lambda^{+n}$, then $\mu$ is a successor cardinal, so by Fact 6.9
we get that $\mathfrak{K}$ is also categorical in $\lambda^{+(n+1)}$. Thus we
can assume that $\mu>\lambda^{+n}$. Note that $\mathfrak{K}^{\ast}$ is
categorical in $\lambda^{+n}=\text{LS}(\mathfrak{K}^{\ast})$. By Theorem 14.2,
$\mathfrak{K}^{\ast}$ is categorical in all $\mu^{\prime}\geq\lambda^{+n}$. By
Fact 6.9, this implies that $\mathfrak{K}$ is categorical in all
$\mu^{\prime}>\lambda$. That $\mathfrak{K}_{\geq\mu}$ has amalgamation, no
maximal models, and is $\mu$-local is by Theorem 13.9. ∎

Thus assuming some instances of the weak diamond, we obtain the following
eventual categoricity transfer for $(<\omega)$-extendible frames:

######  Corollary 14.4.

Let $\mathfrak{s}$ be a $(<\omega)$-extendible categorical good
$\lambda$-frame. Let $\mathfrak{K}$ be the AEC generated by
$\mathfrak{K}_{\mathfrak{s}}$. Assume
$\operatorname{WGCH}([\lambda^{+n},\lambda^{+\omega}))$ holds for some
$n<\omega$. Then $\mathfrak{K}$ has arbitrarily large models and if
$\mathfrak{K}$ is categorical in _some_ $\mu>\lambda$, then $\mathfrak{K}$ is
categorical in _all_ $\mu^{\prime}>\lambda$ and moreover
$\mathfrak{K}_{\geq\mu}$ has amalgamation, no maximal models, and is
$\mu$-local.

###### Proof.

Let $\mathfrak{t}:=\mathfrak{s}^{+(n+3)}$. By Fact 7.20(2), $\mathfrak{t}$ is
very good. By Theorem 13.6, the AEC $\mathfrak{K}^{\ast}$ generated by
$\mathfrak{K}_{\mathfrak{t}}$ is excellent. Now apply Lemma 14.3. ∎

Specializing to compact AECs, we get:

######  Theorem 14.5.

Let $\mathfrak{K}$ be a compact AEC. Let $\mu>\text{LS}(\mathfrak{K})$. If
$\mathfrak{K}$ is categorical in $\mu$, then there exists
$\chi<h(\text{LS}(\mathfrak{K}))$ such that $\mathfrak{K}$ is categorical in
all $\mu^{\prime}\geq\min(\mu,\chi)$.

###### Proof.

By Corollary 4.8, $\mathfrak{K}$ has amalgamation, no maximal models, and is
categorical in a proper class of cardinals. We will also use without comments
Fact 3.24, which says in particular that ${\mathfrak{K}}^{\lambda\text{-sat}}$
is an AEC for any $\lambda>\text{LS}(\mathfrak{K})$. By Theorem 13.7,
${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+6}\text{-sat}}$ is excellent. By
Theorem 14.2, ${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+6}\text{-sat}}$ is
categorical in all high-enough cardinals. Since $\mathfrak{K}$ itself is
categorical in a proper class of cardinals, this implies that
${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}}$ is also categorical
in all high-enough cardinals, and so in particular in a high-enough successor
cardinal. Now ${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}}$ is
categorical in
$\text{LS}(\mathfrak{K})^{+}=\text{LS}({\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}})$,
and hence by Fact 14.1 it is categorical in _all_
$\mu^{\prime}\geq\text{LS}(\mathfrak{K})^{+}$. Now apply Morley’s omitting
type theorem for AECs (Fact 3.25) to get the desired conclusion about
categoricity in $\mathfrak{K}$. ∎

######  Corollary 14.6.

Let $\mathfrak{K}$ be a compact AEC. If $\mathfrak{K}$ is categorical in
_some_ $\mu\geq h(\text{LS}(\mathfrak{K}))$, then $\mathfrak{K}$ is
categorical in _all_ $\mu^{\prime}\geq h(\text{LS}(\mathfrak{K}))$.

###### Proof.

This is a special case of Theorem 14.5. ∎

######  Corollary 14.7.

  1. (1)

Let $\mathfrak{K}$ be an AEC and let $\kappa>\text{LS}(\mathfrak{K})$ be a
strongly compact cardinal. If $\mathfrak{K}$ is categorical in _some_ $\mu\geq
h(\kappa)$, then $\mathfrak{K}$ is categorical in _all_ $\mu^{\prime}\geq
h(\kappa)$.

  2. (2)

Let $T$ be a theory in $\mathbb{L}_{\kappa,\omega}$, $\kappa$ a strongly
compact cardinal. If $T$ is categorical in _some_ $\mu\geq
h(|T|+|\tau(T)|+\kappa)$, then $T$ is categorical in _all_ $\mu^{\prime}\geq
h(|T|+|\tau(T)|+\kappa)$.

  3. (3)

Let $\psi$ be an $\mathbb{L}_{\kappa,\omega}$ sentence. If $\psi$ is
categorical in _some_ $\mu\geq h(\kappa)$, then $\psi$ is categorical in _all_
$\lambda^{\prime}\geq h(\kappa)$.

###### Proof.

The first two parts are by Fact 4.3 and Corollary 14.6. The third part follows
from the second because an $\mathbb{L}_{\kappa,\omega}$ sentence has a
vocabulary of size strictly less than $\kappa$. ∎

Regarding AECs with amalgamation, the following lemma says that (assuming
WGCH) with a little bit of locality, we can transfer categoricity. Note that
the proof only uses amalgamation below the categoricity cardinal, but
transfers it above.

######  Lemma 14.8.

Let $\mathfrak{K}$ be an AEC with arbitrarily large models. Assume
$\operatorname{WGCH}([\text{LS}(\mathfrak{K}),\text{LS}(\mathfrak{K})^{+\omega}))$.
Let $\mu>\text{LS}(\mathfrak{K})^{+}$ and assume that
$\mathfrak{K}_{<\max(\mu,\text{LS}(\mathfrak{K})^{+\omega})}$ has amalgamation
and no maximal models. If
$\mathfrak{K}^{\text{sat}}_{(\text{LS}(\mathfrak{K}),\text{LS}(\mathfrak{K})^{+\omega})}$
is $\text{LS}(\mathfrak{K})$-local and $\mathfrak{K}$ is categorical in $\mu$,
then there is $\chi<h(\text{LS}(\mathfrak{K}))$ such that $\mathfrak{K}$ is
categorical in all $\mu^{\prime}\geq\min(\mu,\chi)$.

###### Proof.

We use without further comments (Fact 3.24) that 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, the model of cardinality $\mu$ is saturated.
By Fact 6.2, there is a (categorical) good frame $\mathfrak{s}$ on
$\mathfrak{K}^{\text{sat}}_{[\text{LS}(\mathfrak{K})^{+},\text{LS}(\mathfrak{K})^{+\omega})}$.
By Fact 7.16, $\mathfrak{s}$ is $(<\omega)$-extendible. By Corollary 14.4,
${\mathfrak{K}}^{\text{LS}(\mathfrak{K})^{+}\text{-sat}}$ is categorical in
all $\mu^{\prime}\geq\text{LS}(\mathfrak{K})^{+}$ and has amalgamation above
$\mu$. In particular, $\mathfrak{K}$ is categorical in all
$\mu^{\prime}\geq\mu$ and $\mathfrak{K}_{\geq\mu}$ has amalgamation. Since we
knew that $\mathfrak{K}_{<\mu}$ had amalgamation, we have that $\mathfrak{K}$
has amalgamation. By Morley’s omitting type theorem for AECs (Fact 3.25),
there is $\chi<h(\text{LS}(\mathfrak{K}))$ such that $\mathfrak{K}$ is
categorical in all $\mu^{\prime}\geq\min(\mu,\chi)$. ∎

Since locality can be derived from high-enough categoricity, we obtain:

######  Theorem 14.9.

Let $\mathfrak{K}$ be an AEC with arbitrarily large models. Let
$\mu\geq\aleph_{\text{LS}(\mathfrak{K})^{+}}$. Assume there exists
unboundedly-many $\lambda<\aleph_{\text{LS}(\mathfrak{K})^{+}}$ such that
$\operatorname{WGCH}([\lambda,\lambda^{+\omega}))$ holds. If $\mathfrak{K}$ is
categorical in $\mu$ and $\mathfrak{K}_{<\mu}$ has amalgamation and no maximal
models, then there exists $\chi<\aleph_{\text{LS}(\mathfrak{K})^{+}}$ such
that $\mathfrak{K}$ is categorical in all $\mu^{\prime}\geq\min(h(\chi),\mu)$.

###### Proof.

By Fact 3.24(2) (where $\lambda$ there stands for
$\aleph_{\text{LS}(\mathfrak{K})^{+}}$ here), there is
$\chi_{0}<\aleph_{\text{LS}(\mathfrak{K})^{+}}$ such that
$\mathfrak{K}^{\text{sat}}_{(\chi_{0},\aleph_{\text{LS}(\mathfrak{K})^{+}})}$
is $\chi_{0}$-local. By Lemma 14.8 applied to $\mathfrak{K}_{\geq\chi_{0}}$,
we get the result. ∎

######  Corollary 14.10.

Assume $\operatorname{WGCH}$ (see Definition 13.5). Let $\mathfrak{K}$ be an
AEC with amalgamation. If $\mathfrak{K}$ is categorical in _some_ $\mu\geq
h(\aleph_{\text{LS}(\mathfrak{K})^{+}})$, then $\mathfrak{K}$ is categorical
in _all_ $\mu^{\prime}\geq h(\aleph_{\text{LS}(\mathfrak{K})^{+}})$.

###### Proof.

This is a special case of Theorem 14.9. It is well-known that AECs with a
model in $h(\text{LS}(\mathfrak{K}))$ have arbitrarily large models. Moreover,
by considering the equivalence relation “$M$ and $N$ embed into a common
model”, we can partition the AEC into disjoint classes, each of which have
joint embedding. One can then work inside the unique class with arbitrarily
large models, which will have joint embedding and no maximal models. It is
then easy to see that the original class must also be categorical in all high-
enough cardinals. See for example [Vas16a, 10.13] for the details of this
argument. ∎

We can also obtain results from completely local hypotheses about the number
of models in the $\lambda^{+n}$’s. This was already stated in [She09a,
III.12.43]. Below, $\mathbb{I}(\mathfrak{K},\lambda)$ denotes the number of
models (up to isomorphism) in $\mathfrak{K}_{\lambda}$. On
$\mu_{\text{unif}}$, see [She09b, VII.0.4] for the definition and [She09b,
VII.9.4] on what is known (morally,
$\mu_{\text{unif}}(\lambda^{+},2^{\lambda})=2^{\lambda^{+}}$ when
$2^{\lambda}<2^{\lambda^{+}}$).

######  Theorem 14.11.

Let $\mathfrak{K}$ be an AEC and let $\lambda\geq\text{LS}(\mathfrak{K})$ be
such that $\operatorname{WGCH}([\lambda,\lambda^{+\omega)})$ holds. Assume
that $\mathfrak{K}$ is categorical in $\lambda$, $\lambda^{+}$,
$\mathfrak{K}_{\lambda^{++}}\neq\emptyset$, and
$\mathbb{I}(\mathfrak{K},\lambda^{+n})<\mu_{\text{unif}}(\lambda^{+n},2^{\lambda^{+(n-1)}})$
for all $n\in[2,\omega)$. Then $\mathfrak{K}$ is categorical in all
$\mu>\lambda$.

###### Proof.

As in the proof of [She09a, II.9.2], we get that there is an
$\omega$-successful $\text{good}^{+}$ $\lambda$-frame $\mathfrak{s}$ on
$\mathfrak{K}_{\lambda}$. This implies (Fact 7.20(1)) that $\mathfrak{s}$ is
$(<\omega)$-extendible. Now apply Lemma 14.3. ∎

When $\lambda=\aleph_{0}$, the hypotheses can be weakened and we obtain the
following generalization of the main result of [She83a, She83b]:

######  Theorem 14.12.

Let $\mathfrak{K}$ be a $\operatorname{PC}_{\aleph_{0}}$ AEC (so
$\text{LS}(\mathfrak{K})=\aleph_{0}$). Assume
$\operatorname{WGCH}([\aleph_{0},\aleph_{\omega}))$. Assume that
$\mathfrak{K}$ is categorical in $\aleph_{0}$,
$1\leq\mathbb{I}(\mathfrak{K},\aleph_{1})<2^{\aleph_{1}}$, and
$\mathbb{I}(\mathfrak{K},\aleph_{n})<\mu_{\text{unif}}(\aleph_{n},2^{\aleph_{n-1}})$
for all $n\in[2,\omega)$. Then $\mathfrak{K}$ has arbitrarily large models.
Moreover, if $\mathfrak{K}$ is categorical in _some_ uncountable cardinal,
then $\mathfrak{K}$ is categorical in _all_ uncountable cardinals.

###### Proof.

As in the proof of Theorem 14.11, using [She09a, II.9.3] instead of [She09a,
II.9.2]. ∎

## Index

  * $(<\omega)$-extendible Definition 7.19
  * $(\lambda,n)$-$P$ item 4
  * $(\lambda,\delta)$-brimmed over (a model) item 2
  * $[A]^{<\mu}$ §2
  * $[A]^{\mu}$ §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$}}}\limits}$ item 1
  * $|\mathfrak{K}|$ §2, item 1
  * $|M|$ §2
  * $\|M\|$ §2
  * $\leq_{\mathfrak{i}}$ Definition 8.14
  * $\leq_{\mathfrak{K}}$ §2
  * $\leq_{\mathfrak{K}}^{d}$ item 2
  * $\bot$ §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$}}}\limits$, see 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$}}}\limits(\mathfrak{i})$ Definition 8.17
  * abstract class Definition 3.1
  * abstract elementary class item 5
  * AEC, see abstract elementary class
  * AEC generated by Definition 3.17
  * almost very good multidimensional independence relation Definition 11.4
  * amalgam, see $\phi$-amalgam
  * amalgamation §3
  * amalgamation base, see $\phi$-amalgamation base
  * amalgamation property, see $\phi$-amalgamation property
  * brimmed item 5
  * brimmed over (a model) item 3
  * brimmed over (a set) item 4
  * brimmed over (for two systems) Definition 11.11
  * brimmed system Definition 11.7
  * categorical, see $\phi$-categorical
  * categorical good frame Definition 6.1
  * chain axioms item 4c
  * closed class of semilattice Definition 8.9
  * coherent abstract class item 3
  * compact AEC Definition 4.1
  * continuous abstract class item 2
  * disjoint system item 7
  * does not fork over item 2, Definition 7.4
  * domain of a good frame Definition 6.1
  * domain of an abstract class Definition 3.6
  * $\operatorname{dom}{\mathfrak{K}}$ Definition 3.6
  * equal, see $\phi$-equal
  * excellent Definition 13.1
  * existence (for multidimensional systems) Definition 8.16
  * extendible Definition 7.15
  * extension (for multidimensional systems) Definition 8.16
  * extension base item 1
  * forking item 2, Definition 7.4
  * fragmented abstract elementary class item 4
  * fully disjoint system item 8
  * generated by, see AEC generated by
  * good frame Definition 6.1
  * good two-dimensional independence notion Definition 7.6
  * has primes Definition 13.8
  * $\mathfrak{i}$, see multidimensional independence relation
  * $\mathfrak{i}(\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower 2.0pt\hbox{$\textstyle\smile$}}}\limits)$ Definition 8.19
  * $\mathbb{I}(\mathfrak{K},\lambda)$ §14
  * $\mathfrak{i}\upharpoonright\mathcal{J}$ Definition 8.12
  * $\mathfrak{i}\upharpoonright\mathfrak{K}^{\ast}$ Definition 8.12
  * $I\times J$ §2
  * $\mathcal{I}_{<\omega}$ item 1
  * $\mathfrak{i}^{\text{brim}}$ Definition 11.7
  * $\mathcal{I}_{\mathfrak{i}}$, see multidimensional independence relation
  * $\mathcal{I}_{n}$ item 1
  * isomorphism of systems item 5
  * $\mathfrak{i}_{\lambda}$ Definition 8.12
  * $\mathfrak{K}$ §2
  * $\mathfrak{K}_{\leq\lambda}$ Notation 3.3
  * $\mathfrak{K}_{\geq\lambda}$ Notation 3.3
  * $\mathfrak{K}_{0}\subseteq\mathfrak{K}$ item 2
  * $\mathfrak{K}^{\text{brim}}$ item 3
  * $\mathfrak{K}^{I}$ Definition 8.8
  * $\mathfrak{K}_{\mathfrak{i}}$, see multidimensional independence relation
  * $\mathfrak{K}_{\mathfrak{i},I}$ Definition 8.14
  * $\mathfrak{K}^{\text{proper},\ast}_{\mathfrak{i},\mathfrak{i}^{\ast},I}$ Definition 11.8
  * $\mathfrak{K}^{\text{proper}}_{\mathfrak{i},I}$ Definition 8.15
  * $\mathfrak{K}^{\text{proper}}_{\mathfrak{i},\mathfrak{i}^{\ast},I}$ Definition 10.12
  * $\mathfrak{K}_{\mathfrak{s}}$ Definition 6.1
  * $\mathfrak{K}^{\text{sat}}$ item 2
  * $\mathfrak{K}_{\Theta}$ Notation 3.3
  * $\mathfrak{K}_{\lambda}$ Notation 3.3
  * ${\mathfrak{K}}^{\lambda\text{-sat}}$ item 1
  * lattice §2
  * local Definition 3.4
  * locality Definition 3.4
  * Löwenheim-Skolem-Tarski axiom item 4b
  * Löwenheim-Skolem-Tarski number item 1
  * $\text{LS}(\mathfrak{K})$ item 1
  * $\mathbf{m}\upharpoonright I$ item 1
  * $M\subseteq^{\ast}N$, see weak substructure
  * $\mathbf{m}_{1}\ast\mathbf{m}_{2}$ item 3
  * model-homogeneous item 6
  * multidimensional independence notion, see multidimensional independence relation
  * multidimensional independence relation Definition 8.11
  * $n$-extendible Definition 7.19
  * $n$-okay, see okay
  * $n$-$P$ item 3
  * $n(I,u)$ Definition 10.7
  * $n(I)$ Definition 10.7
  * $\operatorname{NF}_{\mathfrak{i}}$, see multidimensional independence relation
  * nice very good multidimensional independence relation Definition 11.12
  * okay item 2
  * orbital type §3
  * $\mathcal{P}^{-}(n)$ §2
  * $\mathcal{P}^{-}(u)$ §2
  * $\mathcal{P}(n)$ §2
  * $\mathcal{P}(u)$ §2
  * prime extension item 2
  * prime system item 1
  * prime triple Definition 13.8
  * proper Definition 8.2
  * reduced inside Definition 12.1
  * reflects down Definition 7.7
  * resolvable Definition 3.13
  * respects $\mathfrak{s}$ Definition 7.2
  * restricted chain axioms item 4c
  * $\mathfrak{s}$, see good frame
  * $\mathfrak{s}(\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower 2.0pt\hbox{$\textstyle\smile$}}}\limits)$ item 3
  * $\mathfrak{s}^{+}$ Definition 7.18
  * $\mathfrak{s}^{+n}$ Definition 7.19
  * $s\land t$ §2
  * saturated item 7
  * semilattice §2
  * short Definition 4.4
  * skeleton of a multidimensional independence relation Definition 9.5
  * skeleton of an abstract class Definition 3.8
  * smoothness axiom item 4(c)ii
  * span, see $\phi$-span
  * stability §3
  * strong continuity (for a multidimensional independence relation) Definition 11.1
  * strong continuity (for a two-dimensional independence relation) Definition 7.11
  * strong uniqueness (for multidimensional systems) Definition 8.16
  * strong uniqueness base item 2
  * sub-abstract class Definition 3.7
  * subsemilattice §2
  * $\mathfrak{s}^{up}$ Definition 6.10
  * system Definition 8.1
  * system embedding item 4
  * $\mathbf{tp}_{\mathfrak{K}}(\bar{b}/A;N)$ §3
  * two-dimensional independence notion, see two-dimensional independence relation
  * two-dimensional independence relation Definition 7.1
  * uniqueness (for multidimensional systems) Definition 8.16
  * uniqueness base item 3
  * universal, see $\phi$-universal
  * universal over item 1
  * very good multidimensional independence relation Definition 11.2
  * very good two-dimensional independence notion Definition 7.11
  * vocabulary of systems Definition 8.8
  * weak AEC item 6
  * weak substructure Definition 5.2
  * $\operatorname{WGCH}$, see $\operatorname{WGCH}(S)$
  * $\operatorname{WGCH}(S)$ Definition 13.5
  * $\mu_{\text{unif}}$ §14
  * $\pi(\mathbf{m})$ item 6
  * $\tau^{I}$ Definition 8.8
  * $\phi$-amalgam item 3
  * $\phi$-amalgamation base item 4
  * $\phi$-amalgamation property item 5
  * $\phi$-categorical item 8
  * $\phi$-equal item 1
  * $\phi$-span item 2
  * $\phi$-uniqueness item 7
  * $\phi$-universal item 6
  * $\phi_{n}$ Definition 10.14

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