# Induced and higher-dimensional stable independence

Michael Lieberman<EMAIL_ADDRESS>Institute of Mathematics, Faculty of
Mechanical Engineering, Brno University of Technology, Brno, Czech Republic  ,
Jiří Rosický<EMAIL_ADDRESS>http://www.math.muni.cz/~rosicky/
Department of Mathematics and Statistics, Faculty of Science, Masaryk
University, Brno, Czech Republic  and  Sebastien Vasey

###### Abstract.

We provide several crucial technical extensions of the theory of stable
independence notions in accessible categories. In particular, we describe
circumstances under which a stable independence notion can be transferred from
a subcategory to a category as a whole, and examine a number of applications
to categories of groups and modules, extending results of [MA21a]. We prove,
too, that under the hypotheses of [LRV], a stable independence notion
immediately yields higher-dimensional independence as in [SV].

###### Key words and phrases:

Stable independence, accessible categories, classification theory, categories
of modules

###### 1991 Mathematics Subject Classification:

Primary: 03C45. Secondary: 03C48, 03C52, 13C60, 18C35.

The second author is supported by the Grant agency of the Czech Republic under
the grant 19-00902S

##  1\. Introduction

We here concern ourselves with stable independence in the context of
accessible categories. This notion has its origins in the model-theoretic
concept of stable nonforking, which can be thought of on one hand as a
freeness property of type extensions and, on the other, as a notion of
freeness or independence for amalgams of models. The latter perspective, taken
to its logical conclusion, leads to a formulation of stable independence as a
property of commutative squares in a general category, described by a family
of purely category-theoretic axioms, cf. [LRV19a]. This generalization is of
practical value: often in mathematics we begin with a nice category of objects
$\mathcal{K}$, then restrict to a particular family of desirable morphisms
$\mathcal{M}$, obtaining a subcategory, $\mathcal{K}_{\mathcal{M}}$, which
loses much of the useful structure of $\mathcal{K}$. Even if $\mathcal{K}$ is
locally finitely presentable (say, $\mathcal{K}=\mathbf{Ab}$, the category of
abelian groups and homomorphisms), if we take $\mathcal{M}$ to be a family of
monomorphisms (say, $\mathcal{M}$ consists of the pure monomorphisms in
$\mathbf{Ab}$), the category $\mathcal{K}_{\mathcal{M}}$ will no longer have
the pushouts available to us in $\mathcal{K}$. A central theme of [LRV] is
that stable independence—or, rather, stably independent squares—provide a
workable alternative to the missing pushouts, sufficient for many
applications.

Of greater importance, perhaps, is [LRV, 3.1], which asserts, roughly
speaking, that given a pair $(\mathcal{K},\mathcal{M})$, $\mathcal{K}$
cocomplete, $\mathcal{K}_{\mathcal{M}}$ admits a stable independence notion
just in case the family of morphisms $\mathcal{M}$ is cofibrantly generated;
that is, generated by pushouts, transfinite compositions and retracts from a
_set_ (as opposed to a proper class) of morphisms. In this case, the stable
independence notion must be precisely the one given by $\mathcal{M}$-effective
squares, generalizing the effective unions of [Bar88]. This provides a useful
link between stable independence and, for example, (combinatorial, i.e.
cofibrantly generated) cellular categories, which occur naturally in topology
and homological algebra. The translation it offers has already proved
fruitful: arguing by way of stable independence, one can give a very brief
proof of (a special case of) the fact that combinatorial structures are left-
induced, [LRV, 3.11]; the original proof of this fact, in [MR14], requires a
great deal of heavy category-theoretic machinery, including the good colimits
of Lurie (see [MRV14]). On the other hand, an analysis of questions involving
stable independence by way of cofibrant generation often leads to more
efficient proofs, and useful new results, particularly in algebra. For
example, [LRV, 4.3] resolves an open question concerning the circumstances
under which Ext-orthogonality classes of modules admit a stable independence
notion, and, while the proof of the cofibrant generation of pure monomorphisms
in locally presentable additive categories in [LPRV20] does not use stable
independence, it would not have been evident were it not for this connection.

We pursue further applications here, first deriving stable independence for a
host of algebraic categories. For this, we require an essential technical
lemma. A crucial part of the definition of a stable independence relation on a
category $\mathcal{K}$ is that it be accessible (Definition 2.2(2)), meaning,
roughly speaking, that the independent squares form an accessible subcategory
of the arrow category, $\mathcal{K}^{2}$: without this property, we say the
independence relation is weakly stable. Accessibility, which corresponds
loosely to the local character of nonforking, is neither particularly natural
nor easy to verify in practice. For a continuous independence notion, on the
other hand, we require only that the appropriate subcategory of
$\mathcal{K}^{2}$ be closed under directed colimits—this is far more common,
as we will see in Section 3. The essential result of Section 2 is that, in
many cases, we can infer the existence of a stable independence notion from
that of a continuous weakly stable independence notion (Theorem 2.7). In
particular, if an accessible category $\mathcal{K}$ has a continuous weakly
stable independence notion, and has a stable independence notion on a
sufficiently nicely embedded subcategory, then in fact the independence notion
on $\mathcal{K}$ must be stable. Thanks to recent work on stability—in the
sense of Galois types—in categories of groups and modules (e.g. [KMA20],
[MA21a]), we obtain continuous weakly stable independence notions for free in,
for example, Abelian $p$-groups with pure embeddings, or torsion $R$-modules
with pure embeddings. A model-theoretic argument shows that these weakly
stable notions restrict to stable independence on subcategories of
sufficiently saturated (that is, universal injective) objects, and Theorem 2.7
allows us to lift this stability to the categories themselves. The results are
summarised in Theorems 3.8 and 3.9.

Finally, we show that in the setting of [LRV], the existence of a stable
independence notion—essentially, a family of nice commutative squares—implies
the existence of higher-dimensional stable independence: nice families of
cubes, hypercubes, etc. We note that higher-dimensional amalgams of this form
have played an important role in the analysis of categoricity in
$L_{\omega_{1},\omega}$ ([She83a] and [She83b], where they were first
introduced) and, more recently, in connection with the categoricity conjecture
for abstract elementary classes ([SV]). In the aforementioned cases, it is a
significant technical challenge to ensure that the existence of such amalgams
in low dimensions can be pushed to higher dimensions ([SV] needs the weak
generalized continuum hypothesis, for example). As our underlying category is
locally presentable—hence has arbitrary pushouts—existence is more or less
automatic.

We assume a familiarity with accessible categories ([AR94], [MP89]). A passing
acquaintance with the category-theoretic formulation of stable independence
relations ([LRV19a], [LRV]) would be very useful in providing motivation for
the discussion that follows.

We are grateful to Marcos Mazari-Armida for his detailed feedback on an early
draft of this paper. We are grateful, too, to the anonymous referee for
comments that have led to several important clarifications.

##  2\. Lifting stable independence

While a much fuller picture can be given of the transfer of stable
independence notions along adjunctions, we here concern ourselves with a very
limited special case. In particular, we show that if an accessible category
$\mathcal{K}$ has a _$\aleph_{0}$ -continuous_ weakly stable independence
notion, and has a stable independence notion on a sufficiently nice
subcategory, then in fact $\mathcal{K}$ has a stable independence notion. This
allows us to extend certain stability results concerning classes of modules in
[MA21a]: while that paper establishes that a number of such classes are stable
in the sense of Galois types, we show that they possess stable independence
notions. This is stronger: for example, the category of $\aleph_{1}$-free
groups and pure embeddings is Galois-stable (see [MA, 5.9]), but does not have
a stable independence notion, albeit for a rather trivial reason—this category
does not have the amalgamation property.

For the sake of completeness, we briefly recall the definitions of the
essential notions here. More detailed treatments can be found in [LRV19a] and
[LRV].

######  Definition 2.1.

Let $\mathcal{K}$ be a category.

  1. (1)

We define an _independence notion_ (or _independence relation_) on
$\mathcal{K}$ as a class $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ of commutatives squares (called
_$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ -independent_, or simply
_independent_) such that, for any commutative diagram

$\textstyle{E}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the square spanning $A$, $B$, $C$, and $D$ is independent if and only if the
square spanning $A$, $B$, $C$, and $E$ is independent.

  2. (2)

We say that an independence notion on $\mathcal{K}$ is _weakly stable_ if it
satisfies the following conditions:

    1. (a)

Symmetry: $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is closed under reflection across the
diagonal from bottom left to top right.

    2. (b)

Transitivity: $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is closed under vertical and
horizontal composition of squares.

    3. (c)

Existence: Any span $B\leftarrow A\to C$ can be completed to a
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$-independent square.

    4. (d)

Uniqueness: Any two completions of a span $B\leftarrow A\to C$ to
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$-independent squares are equivalent up
to amalgamation, in the sense of the diagram in (1) above.

In this section, we will concern ourselves largely with the tension between
_weakly stable_ and _stable_ independence, where the latter builds in a
crucial accessibility condition.

######  Definition 2.2.

Let $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ be an independence notion on category
$\mathcal{K}$, satisfying the existence and transitivity properties.

  1. (1)

We denote by $\mathcal{K}_{\downarrow}$ the subcategory of the arrow category
$\mathcal{K}^{2}$ whose objects are $\mathcal{K}$-morphisms, and whose
morphisms are $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$-independent squares. (Our assumption
on $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ in the preamble of this definition is
needed only to ensure that $\mathcal{K}_{\downarrow}$ does indeed form a
category.)

  2. (2)

We say that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is _$\lambda$ -accessible_, $\lambda$
an infinite regular cardinal, if the category $\mathcal{K}_{\downarrow}$ is
$\lambda$-accessible. We say that $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is _accessible_ if it is
$\lambda$-accessible for some $\lambda$.

  3. (3)

We say that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is a _stable independence notion_ it
if is weakly stable and accessible.

We note that in the model-theoretic context, weakenings of stability—for
example, simplicity—are obtained by weakening existence and/or uniqueness,
these being the more difficult property to verify, model-theoretically. Here
accessibility (which corresponds roughly to what model theorists might refer
to as _local and finite character_) is far thornier: by and large, it is
easier to detect when a weakly stable independence notion satisfies the
(weaker) condition of $\lambda$-continuity:

######  Definition 2.3.

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 an independence notion on category
$\mathcal{K}$, satisfying the existence and transitivity properties. We say
that $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is $\lambda$-continuous if
$\mathcal{K}_{\downarrow}$ is closed in $\mathcal{K}^{2}$ under
$\lambda$-directed colimits.

That $\lambda$-continuity follows from $\lambda$-accessibility, incidentally,
is [LRV19a, 3.26].

We now turn to the central result of this section, which ensures that if an
accessible category $\mathcal{K}$ with all morphisms monomorphisms has a
continuous, weakly stable independence notion and a stable independence notion
on a sufficiently nice subcategory $\mathcal{L}$, then in fact there is a
stable independence notion on $\mathcal{K}$ itself.

In this context, “sufficiently nice” will mean, precisely, that $\mathcal{L}$
is a _cofinal_ subcategory of $\mathcal{K}$:

######  Definition 2.4.

We say that a functor $F:\mathcal{L}\to\mathcal{K}$ is _cofinal_ if for any
$K\in\mathcal{K}$, and any finite sequence
$\left(FL_{i}\xrightarrow{f_{i}}K\right)_{i\in I}$, there exists
$L\in\mathcal{L}$, $K\xrightarrow{g}FL$ and
$\left(FL_{i}\xrightarrow{Fg_{i}}FL\right)_{i\in I}$ such that $Fg_{i}=g\circ
f_{i}$ for all $i\in I$.

We say that a subcategory $\mathcal{L}$ of a category $\mathcal{K}$ is
_cofinal_ if the inclusion is cofinal.

######  Example 2.5.

  1. (1)

If $\mathcal{K}_{\ast}$ is a full subcategory of $\mathcal{K}$ and for every
$M\in\mathcal{K}$ there exists $M\to N$ with $N\in\mathcal{K}_{\ast}$, then
$\mathcal{K}_{\ast}$ is a cofinal subcategory of $\mathcal{K}$. Of course, by
choosing $I=\emptyset$, every cofinal subcategory has the latter property.

  2. (2)

The category of $\lambda$-saturated models of an elementary class is a cofinal
subcategory. This generalizes to $\mu$-AECs, [LRV19a, 7.7].

######  Remark 2.6.

We note that this notion of cofinality is somewhat weaker than the one found
elsewhere in the category-theoretic literature, e.g. [AR94, 0.11]. One can
show, however, that the two definitions coincide if one assumes the
amalgamation property.

######  Theorem 2.7.

Let $\mathcal{K}$ be an accessible category and let $\mathcal{L}$ be an
accessibly embedded cofinal full subcategory of $\mathcal{K}$. If:

  1. (1)

All morphisms of $\mathcal{K}$ are monos.

  2. (2)

$\mathcal{K}$ has an $\aleph_{0}$-continuous weakly stable independence
notion.

  3. (3)

$\mathcal{L}$ has a stable independence notion.

Then $\mathcal{K}$ has a stable independence notion.

###### Proof.

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 an $\aleph_{0}$-continuous weakly
stable independence notion on $\mathcal{K}$. The $\aleph_{0}$-continuity 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$}}}\limits$ implies in particular that
$\mathcal{K}$ has directed colimits, so $\mathcal{L}$ has directed bounds. It
is easy to check 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$}}}\limits$ restricted to $\mathcal{L}$ is a
weakly stable independence notion, hence by the canonicity theorem ([LRV,
A.6]) it must be stable. We wish to show that $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is stable, so must show, in
particular, that $\mathcal{K}_{\downarrow}$ is accessible. 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$}}}\limits$ restricted to $\mathcal{L}$ is
accessible, $\mathcal{L}$ itself must be an accessible category. Taking $\mu$
bigger if needed, we may assume without loss of generality that $\mathcal{K}$,
$\mathcal{L}$, and $\mathcal{L}_{\downarrow}$ are $\mu$-accessible. Let
$\mathcal{K}_{\mu}$ denote the full subcategory of $\mathcal{K}$ consisting of
$\mu$-presentable objects. Let $\mathcal{K}^{\ast}$ consist of those morphisms
in $\mathcal{K}$ which are $\mu$-directed colimits in
$\mathcal{K}_{\downarrow}$ of morphisms in $\mathcal{K}_{\mu}^{2}$. Recalling
that the morphisms in $\mathcal{K}_{\mu}^{2}$ are precisely the
$\mu$-presentable objects of $\mathcal{K}^{2}$, it suffices to see that
$\mathcal{K}^{\ast}=\mathcal{K}^{2}$. Clearly,
$\mathcal{K}_{\mu}^{2}=\mathcal{K}_{\mu}^{\ast}$ and
$\mathcal{L}^{2}\subseteq\mathcal{K}^{\ast}$.

Concretely, we will prove the following:

  * (i)

$\mathcal{K}^{\ast}$ is closed under composition.

  * (ii)

$\mathcal{K}^{\ast}$ is left cancellable in $\mathcal{K}^{2}$.

  * (iii)

If $M\in\mathcal{K}$, there exists $M\to N$ in $\mathcal{K}^{\ast}$ with $N$
in $\mathcal{L}$.

To verify that this is sufficient, take $f:M\to N$ in $\mathcal{K}^{2}$. We
will show that, under assumptions (i) and (ii), if $M$ satisfies the condition
of (iii), then $f$ must be in $\mathcal{K}^{\ast}$. To begin, since $M$
satisfies (iii), there is $g:M\to M^{\prime}$ in $\mathcal{K}^{\ast}$ with
$M^{\prime}$ in $\mathcal{L}$. Without loss of generality,
$f\in\mathcal{K}^{2}_{\lambda}$ and $\mathcal{K}^{2}$ is $\lambda$-accessible
for some $\lambda>\mu$. Following [Ros97], there is a morphism
$h:M^{\prime}\to M^{\ast}$ where $M^{\ast}$ is $\lambda$-saturated in
$\mathcal{K}$. Thus there is $t:N\to M^{\ast}$ such that $tf=hg$. Following
2.5(1), there is $p:M^{\ast}\to L$ with $L$ in $\mathcal{L}$. Since
$ph\in\mathcal{L}^{2}\subseteq\mathcal{K}^{\ast}$, (i) implies that
$phg\in\mathcal{K}^{\ast}$. Hence, because $ptf=phg$, (ii) implies that
$f\in\mathcal{K}^{\ast}$, as desired. We now prove statements (i)-(iii):

(i) Consider $f:K\to L$ and $g:L\to M$ in $\mathcal{K}^{\ast}$. Let
$f=\operatorname{colim}_{i}f_{i}$ and $g=\operatorname{colim}_{j}g_{j}$ be
$\mu$-directed colimits in $\mathcal{K}_{\downarrow}$ of morphisms in
$\mathcal{K}_{\mu}^{2}$. We may assume that
$(k_{ii^{\prime}},l_{ii^{\prime}}^{1}):f_{i}\to f_{i^{\prime}}$ and
$(l_{jj^{\prime}}^{2},m_{jj^{\prime}}):g_{j}\to g_{j^{\prime}}$ are canonical
diagrams of $f$ and $g$ with respect to $\mathcal{K}^{2}_{\lambda}$ in
$\mathcal{K}_{\downarrow}$. Given $i_{0}\in I$ and $j_{0}\in J$, there is
$j_{1}\geq j_{0}$ in $J$ such that $l_{i_{0}}^{1}:L_{i_{0}}^{1}\to L$
factorizes through $l_{j_{1}}^{2}:L_{j_{1}}^{2}\to L$. Similarly, there is
$i_{1}\geq i_{0}$ in $I$ such that $l_{j_{1}}^{2}$ factorizes through
$l_{i_{1}}^{1}$. Continuing this procedure and taking a colimit, we get
$L_{i_{\omega}}^{1}\cong L_{j_{\omega}}^{2}$. Then $f_{i_{\omega}}$ and
$g_{j_{\omega}}$ are composable and $g_{i_{\omega}}f_{j_{\omega}}$ is in
$\mathcal{K}^{\ast}$. Continuing in this way, we obtain that
$gf\in\mathcal{K}^{\ast}$.

(ii) Consider $f:K\to L$ and $g:L\to M$ such that $gf\in\mathcal{K}^{\ast}$.
Let $\mathcal{K}^{\to\to}$ be the category of composable pairs of morphisms in
$\mathcal{K}$ and $(f,g)=\operatorname{colim}_{i}(f_{i},g_{i})$ be a
$\mu$-directed colimit in $\mathcal{K}^{\to\to}$ of morphisms in
$\mathcal{K}_{\mu}^{\to\to}$. Let $gf=\operatorname{colim}_{j}h_{j}$ be a
$\mu$-directed colimit in $\mathcal{K}_{\downarrow}$ of morphisms in
$\mathcal{K}_{\mu}^{2}$. Here, $f_{i}:K^{1}_{i}\to L_{i}$, $g_{i}:L_{i}\to
M^{1}_{i}$ and $h_{j}:K^{2}_{j}\to M^{2}_{j}$. As in the preceding argument,
for every $i_{0}\in I$ and $j_{0}\in J$ there are $i_{\omega}\geq i_{0}$ in
$I$ and $j_{\omega}\geq j_{0}$ in $J$ such that $K^{1}_{i_{\omega}}\cong
K^{2}_{j_{\omega}}$. In the same way, for every $i_{0}\in I$ and $j_{0}\in J$
there are $i_{\omega}\geq i_{0}$ in $I$ and $j_{\omega}\geq j_{0}$ in $J$ such
that $M^{1}_{i_{\omega}}\cong M^{2}_{j_{\omega}}$. By iterating both
procedures, we show that for every $i_{0}\in I$ and $j_{0}\in J$ there are
$i_{\omega}\geq i_{0}$ in $I$ and $j_{\omega}\geq j_{0}$ in $J$ such that
$M^{1}_{i_{\omega}}\cong M^{2}_{j_{\omega}}$ and $K^{1}_{i_{\omega}}\cong
K^{2}_{j_{\omega}}$. Then $h_{j_{\omega}}=g_{i_{\omega}}f_{i_{\omega}}$, which
implies that $f_{i_{\omega}}\to f$ is a morphism in
$\mathcal{K}_{\downarrow}$. Hence $f\in\mathcal{K}^{\ast}$.

(iii) Assume that the claim does not hold and let $M$ have the smallest
presentation rank $r$ among objects violating (iii). Following [BR12, 4.2],
$r=\lambda^{+}\geq\mu^{+}$. Under the hypothesis of the theorem, $\mathcal{K}$
is well $\lambda^{+}$-filtrable (see [LRV20, 8.8(2)], noting that, since the
morphisms in $\mathcal{K}$ are monos, filtrability and well-filtrability
coincide). In particular, $M=\operatorname{colim}M_{i}$ can be expressed as
the colimit of a smooth chain of $\lambda$-presentable objects where
$i\leq\lambda^{+}$. There is $h_{0}:M_{0}\to N_{0}$ in $\mathcal{K}^{\ast}$
with $N_{0}\in\mathcal{L}$. There is a $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$-independent square

$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{N_{1}}$$\textstyle{M_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{01}}$$\scriptstyle{h_{0}}$$\textstyle{N_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n_{01}}$

in $\mathcal{K}$ and, since $\mathcal{L}$ is cofinal in $\mathcal{K}$, we may
assume that $N_{1}$ is in $\mathcal{L}$. Since $M_{1}$ satisfies (iii),
$h_{1}$ is in $\mathcal{K}^{\ast}$. We iterate this procedure, proceeding as
above at successor stages. At limit stages, we take colimits, although this
requires somewhat more care: for short chains, we must make use of the
cofinality of $\mathcal{L}$ to ensure that the object in the upper right
corner is still in $\mathcal{L}$. Taking the colimit of the resulting
$\lambda^{+}$-chain, we have $h:M\to N$ in $\mathcal{K}^{\ast}$ with $N$ in
$\mathcal{L}$ (thanks to the $\lambda^{+}$-accessibility of $\mathcal{L}$),
which contradicts our initial assumption.

∎

##  3\. Stable independence in categories of groups and modules

That we are able to lift stable independence in the sense of Theorem 2.7
yields immediate benefits, namely the proof of stable independence in a host
of categories that arise naturally in algebra. This is a consequence not only
of the theorem, but the following recent developments:

  1. (1)

One of the essential ideas of [LRV19b] is that continuous weakly stable
independent relations are abundant and easily detectable in the algebraic
context, typically taking the form of _effective squares_. We briefly recall
some of the necessary terminology, as it will also be required in Section 4.

  2. (2)

In [KMA20] and [MA21a], a large number of algebraic categories are shown to
have precisely the model-theoretic properties required to ensure the existence
of a cofinal full subcategory equipped with a stable independence relation.

Taken together, this yields a host of algebraic categories with stable
independence.

We begin by recalling a few pieces of necessary terminology from [LRV], which
will allow us to give sufficient conditions for the existence of a
$\aleph_{0}$-continuous weakly stable independence relation.

######  Definition 3.1.

Let $\mathcal{K}$ be a category, and let $\mathcal{M}$ be a class of morphisms
in $\mathcal{K}$.

  1. (1)

We say that $\mathcal{M}$ is _almost nice_ if it is satisfies the following
conditions:

    1. (a)

$\mathcal{M}$ is _normal_ : it contains all isomorphisms in $\mathcal{K}$ and
is closed under composition.

    2. (b)

$\mathcal{M}$ is _coherent_ : whenever $f$ and $g$ are composable morphisms
with $gf\in\mathcal{M}$ and $g\in\mathcal{M}$, then $f\in\mathcal{M}$ as well.

    3. (c)

$\mathcal{M}$ is a _coclan_ : the pushout of any two morphisms, at least one
of which is in $\mathcal{M}$, exists, and $\mathcal{M}$ is closed under
pushouts.

Incidentally, we say that $\mathcal{M}$ is _nice_ if it is also closed under
retracts in $\mathcal{K}^{2}$. This means that if $(u,v):g\to f$ and
$(r,s):f\to g$ are morphisms in $\mathcal{K}^{2}$ such that
$(r,s)(u,v)=\operatorname{id}_{g}$ then $f\in\mathcal{M}$ implies that
$g\in\mathcal{M}$.

  2. (2)

We say that $\mathcal{M}$ is _$\lambda$ -continuous_, $\lambda$ an infinite
regular cardinal, if $\mathcal{K}$ has $\lambda$-directed colimits, and
$\mathcal{M}$ is closed under $\lambda$-directed colimits in $\mathcal{K}$.

  3. (3)

We say that $\mathcal{M}$ is $\lambda$-accessible if it is
$\lambda$-continuous and both $\mathcal{K}$ and $\mathcal{K}_{\mathcal{M}}$
are $\lambda$-accessible. $\mathcal{M}$ is accessible if it is
$\lambda$-accessible for some $\lambda$.

######  Notation 3.2.

Note that if $\mathcal{M}$ is normal, we can form a subcategory
$\mathcal{K}_{\mathcal{M}}$ of $\mathcal{K}$ whose objects are those of
$\mathcal{K}$ and whose morphisms are precisely those in $\mathcal{M}$.

We define a natural candidate for an independence relation on
$\mathcal{K}_{\mathcal{M}}$ in the form of _$\mathcal{M}$ -effective squares_,
following [LRV, 2.2,2.3].

######  Definition 3.3.

Let $\mathcal{K}$ be a category, and let $\mathcal{M}$ be a class of morphisms
in $\mathcal{K}$. An _$\mathcal{M}$ -effective square_ is a commutative square
of $\mathcal{K}$-morphisms

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{D}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$

such that the pushout $P$ of $f$ and $g$ exists, and the induced morphism
$P\to D$ is in $\mathcal{M}$.

We note that, in case $\mathcal{M}$ consists of the regular monomorphisms in
$\mathcal{K}$, $\mathcal{M}$-effective squares are precisely the _effective
unions_ of [Bar88].

######  Fact 3.4.

If $\mathcal{K}$ has pushouts and $\mathcal{M}$ is almost nice,
$\mathcal{M}$-effective squares form a weakly stable independence relation on
$\mathcal{K}_{\mathcal{M}}$ [LRV, 2.7]. If, moreover, $\mathcal{M}$ is
$\lambda$-continuous, this independence relation is $\lambda$-continuous [LRV,
2.11].

This will guarantee the existence of $\aleph_{0}$-continuous, weakly stable
independence relations in a number of familiar algebraic categories that have
recently been the subject of analyses using the tools of abstract model
theory. The essential fact we require, which allows us to isolate a cofinal,
full subcategory equipped with a stable independence relation—and thus apply
Theorem 2.7—is fundamentally model-theoretic.

We recall that in an abstract elementary class $\mathcal{K}$, the syntactic
types familiar from classical model theory are replaced by _Galois_ (or
_orbital_) _types_ : given a model $M\in\mathcal{K}$, Galois types over $M$
are typically identified with orbits of tuples in a large, strongly
homogeneous _monster model_ under automorphisms fixing $M$. A class is said to
be _$\lambda$ -Galois stable_ if there are at most $\lambda$ Galois (1-)types
over any $M\in\mathcal{K}$ of cardinality $\lambda$. Moreover, Galois types in
a class $\mathcal{K}$ are said to be _$ <\aleph_{0}$-short_ if, roughly
speaking, the type of any tuple is completely determined by the types of its
finite subtuples. Readers unfamiliar with these properties may wish to treat
the following as a black box:

######  Lemma 3.5.

Let $\mathcal{K}$ be an _abstract elementary class_. If

  1. (1)

$\mathcal{K}$ has the amalgamation property,

  2. (2)

$\mathcal{K}$ is Galois-stable, and

  3. (3)

types in $\mathcal{K}$ are $<\aleph_{0}$-short over models,

then there is a full, cofinal subcategory of $\mathcal{K}$—consisting of
sufficiently saturated models—on which there is a stable independence notion.

###### Proof sketch.

In essence, finite shortness puts us in the realm of homogeneous model theory,
where the desired result is already known. While we omit the full argument, we
hope that the following outline will be sufficient for the interested reader.
In the process, we will make free use of the technique of _Galois
Morleyization_ introduced in [Vas1]: in an $<\aleph_{0}$-short AEC one can
identify types of finite sequences over the empty set with finitary
quantifier-free formulas (formally, by expanding the language).

Following [HS00, §3], the assumptions of the lemma yield a relation _$p$ is
free over $M$_, for $p$ a Galois type over a model $N\leq M$, that satisfies
all the properties of stable independence provided $M$ and $N$ are
sufficiently saturated (in particular, in the terminology of that paper, $M$
and $N$ must be $a$-saturated, a consequence of $\lambda$-saturation in some
sufficiently big $\lambda$, see [HS00, 1.9.4]). Note that the properties of
independence verified in that paper are the model-theoretic analogues of the
category-theoretic definition we discuss here, but the two definitions are
equivalent, [LRV, 8.14]. There is the issue, too, that in [HS00], $p$ is
assumed to be the type of a finite sequence—as we are concerned with types of
infinite sequences, we must show that the existence/extension property of
[HS00] can be transferred to this context. This can be done relatively easily,
making use of the compactness theorem for homogeneous model theory: the
complete type $p$ of a sequence of arbitrary length is satisfiable just in
case its restrictions to finite subsequences are satisfiable (see [HS00, 1.1],
or, more explicitly, [ABV19, 3.8,3.9]). Given the type $p$ of a sequence of
arbitrary length over $M$, and $\bar{x}$ a finite subsequence, the restriction
of $p$ to $\bar{x}$ has a free extension over $N$. Consider the set of all
such free extensions, regarded as quantifier-free formulas (via Galois
Morleyization, if necessary). The resulting set is complete and, by the
extension property of freeness for types of _finite_ sequences, all of its
restrictions to finite sets of variables are consistent. By construction, the
resulting type is the free extension of $p$ over $N$. ∎

We now obtain stable independence relations on a wide array of algebraic
categories using Theorem 2.7, Fact 3.4, and Lemma 3.5—we note that [MA]
constructs stable independence relations in many of the same cases, by more
concrete means.

As a template for our approach, consider:

######  Theorem 3.6.

For any ring with unit $R$, the category of (left) $R$-modules and pure
monomorphisms, $\mathbf{RMod}_{pure}$, has a stable independence notion.

###### Proof.

By [KMA20], $\mathbf{RMod}_{pure}$ forms an AEC, has amalgamation, is stable,
and types are $(<\aleph_{0})$-short over models; that is, it satisfies all the
hypotheses of Lemma 3.5. Thus $\mathcal{K}$ must have a stable independence
relation on its sufficiently saturated models, which form a cofinal, full
subcategory. By Theorem 2.7 and Fact 3.4, then, $\mathbf{RMod}_{pure}$ has a
stable independence relation. ∎

As an aside, in light of Fact 4.3 below (originally appearing as [LRV, 3.1]),
it follows that:

######  Corollary 3.7.

Pure monomorphisms are cofibrantly generated (generated from a _set_ of
morphisms by pushouts, tranfinite composition, and retracts) in
$\mathbf{RMod}$.

Note that this is a special case of [LPRV20, 3.13], which holds not just for
$R$-modules but arbitrary locally finitely presentable additive categories. Of
greater interest are the other applications of this style of argument. In
particular,

######  Theorem 3.8.

Let $R$ be an integral domain. The following categories of modules have a
stable independence relation:

  1. (1)

Torsion $R$-modules with pure monomorphisms.

  2. (2)

$R$-divisible modules with pure monomorphisms (recall that a module $M$ is
$R$-divisible if for any nonzero $m\in M$ and nonzero $r\in R$, there is $n\in
M$ with $rn=m$).

###### Proof.

  1. (1)

The category of torsion $R$-modules and pure monomorphisms satisfies the
condition of Lemma 3.5, by [MA21a, 4.2(4)] and [MA21a, 4.8(2)]

  2. (2)

Similarly to (1), using [MA21a, 4.2(5)] in place of [MA21a, 4.2(4)].

∎

We obtain stable independence relations on an assortment of familiar
categories of groups, as well, again taking advantage of recent model-
theoretic results—again, [MA] actually obtains similar results, by different
means.

######  Theorem 3.9.

The following categories of groups all have a stable independence relation.

  1. (1)

Abelian groups with monomophisms (respectively, pure monomorphisms).

  2. (2)

Torsion-free abelian groups with pure monomorphisms. Similarly, reduced
torsion-free abelian groups with pure monomorphisms.

  3. (3)

Abelian $p$-groups with monomorphisms (respectively, pure monomorphisms), $p$
any prime.

  4. (4)

Torsion abelian groups with monomorphisms (respectively, pure monomorphisms).

  5. (5)

Divisible abelian groups with monomorphisms.

###### Proof.

As in the proof of Theorem 3.8, we satisfy ourselves with indicating the
model-theoretic sources that ensure the category satisfies the hypotheses of
Lemma 3.5.

  1. (1)

[BCG+] and [MA21b, 3.12], and [KMA20, 3.16] (respectively).

  2. (2)

The reduced case is [She17, 1.2(3)]; general torsion free groups are addressed
in, e.g. [KMA20, 3.14].

  3. (3)

By [MA21a, 4.8(3)] and [MA21a, 3.5].

  4. (4)

By [MA21a, 4.8(3)] and [MA21a, 4.8(1)], respectively.

  5. (5)

By [MA21a, 4.8(3)].

∎

We expect that considerably more applications of Theorem 2.7 of this form are
within easy reach: the forthcoming paper [MA], for example, provides clear
avenues for future work along these lines.

##  4\. Higher-dimensional independence

We turn now to a different, and perhaps more natural question: we know that if
we have a stable independence relation on a category $\mathcal{K}$, we obtain
a well behaved subcategory $\mathcal{K}_{\downarrow}$ of the category of
morphisms $\mathcal{K}^{2}$ consisting of the independent squares. Is it the
case, too, that there is a stable independence notion on
$\mathcal{K}_{\downarrow}$—consisting now of commutative _cubes_ in
$\mathcal{K}$—and under what conditions? Is there, in turn, a stable
independence relation on these cubes?

We wish to examine, in short, the existence and behavior of _higher-
dimensional_ stable independence relations.

It should be noted that this is not an exercise in abstraction: higher-
dimensional independence relations have played a significant role in recent
advances in model theory. The idea, introduced by Shelah, is vital in his
analysis of the classification theory of $L_{\omega_{1},\omega}$ in [She83a]
and [She83b], and in his proof of the first-order Main Gap, cf. [She90, Ch.
XII]. More recently, these notions have made crucial appearances in a number
of categoricity transfer arguments, most notably in [Zil05], for quasiminimal
pregeometry classes, and [SV], for abstract elementary classes with
amalgamation, assuming weak GCH. Of particular interest are _excellent_
classes, which possess independence notions in all finite dimensions.

To be precise, we propose the following notions of $n$-dimensional stable
independence and excellence, adapted to our context—these should specialize to
the standard ones in the model-theoretic examples mentioned above.

######  Definition 4.1.

Let $\mathcal{K}$ be a category. For $n\geq 1$, we define an _$n$ -dimensional
stable independence notion on $\mathcal{K}$_, $\Gamma$, and its induced
category, $\mathcal{K}^{\Gamma}$, proceeding by induction on $n$:

  * •

We say that $\Gamma$ is a $1$-dimensional stable independence notion on
$\mathcal{K}$ just in case it is the collection of all morphisms in
$\mathcal{K}$. In this case, we define $\mathcal{K}^{\Gamma}=\mathcal{K}$.

  * •

An $(n+1)$-dimensional stable independence notion on $\mathcal{K}$ consists of
a pair $(\Gamma_{n},\Gamma)$, where

    * –

$\Gamma_{n}$ is an $n$-dimensional stable independence notion on
$\mathcal{K}$, and

    * –

$\Gamma$ is a stable independence notion on $\mathcal{K}^{\Gamma_{n}}$, in the
sense of Definition 2.2(3).

  * •

Given an $(n+1)$-dimensional stable independence notion
$\Gamma_{n+1}=(\Gamma_{n},\Gamma)$ on $\mathcal{K}$, we define
$\mathcal{K}^{\Gamma_{n+1}}$ to be the category
$(\mathcal{K}^{\Gamma_{n}})^{\Gamma}$, whose objects are morphisms of
$\mathcal{K}^{\Gamma_{n}}$ and whose morphisms are the $\Gamma$-independent
squares. Note that this is precisely $(\mathcal{K}^{\Gamma_{n}})_{\downarrow}$
with $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits=\Gamma$.

Note that the stable independence notions considered in Sections 2 and 3 are
precisely $2$-dimensional independence notions—in the sense above—on the
appropriate categories. The best case scenario is the following:

######  Definition 4.2.

We say that a category $\mathcal{K}$ is _excellent_ if for all $n\geq 1$,
$\mathcal{K}$ has an $n$-dimensional stable independence notion $\Gamma_{n}$
so that $\mathcal{K}^{\Gamma_{n}}$ has directed colimits.

As noted in the introduction, excellence is far from the norm in the model-
theoretic context, as the existence property will typically fail for
sufficiently high-dimensional relations: [GKK15a] and [GKK15b] develop a
comprehensive theory of such obstructions. Here we restrict ourselves to the
setting of [LRV]; that is, with locally presentable ambient category
$\mathcal{K}$ and well behaved class of morphisms $\mathcal{M}$. With this
added structure, the obstructions disappear: if there is a stable independence
relation on $\mathcal{K}_{\mathcal{M}}$, it is excellent.

All of the difficulty lies in the inductive step: given an $n$-dimensional
stable independence notion $\Gamma$ on $\mathcal{K}$, how do we construct a
stable independence notion $\Gamma^{\prime}$ so that
$(\mathcal{K}^{\Gamma})^{\Gamma^{\prime}}$ has directed colimits?

In fact, we consider a simpler—but entirely equivalent—problem, whose
solution, Proposition 4.4 below, should be of independent interest. In
particular, we take advantage of the fact that, in this framework, existence
of stable independence notions of a particular dimension is equivalent to
cofibrant generation of a suitable family of morphisms, via the central result
of [LRV]. For the sake of completeness, we include that result here, phrased
in terms better suited to the current context:

######  Fact 4.3.

([LRV, 3.1]) Let $\mathcal{K}$ be a locally presentable category, and let
$\mathcal{M}$ be a nice, accessible, and $\aleph_{0}$-continuous class of
morphisms in $\mathcal{K}$. The following are equivalent:

  1. (1)

$\mathcal{K}_{\mathcal{M}}$ has a stable independence notion.

  2. (2)

$\mathcal{M}$-effective squares form a stable independence notion in
$\mathcal{K}_{\mathcal{M}}$.

  3. (3)

$\mathcal{M}$ is cofibrantly generated in $\mathcal{K}$.

Note that we have unpacked much of the terminology used in [LRV, 3.1], for the
benefit of the reader. Recall that a class of morphisms $\mathcal{M}$ is
_cofibrantly generated_ if it can be generated from a set—as opposed to a
proper class—of morphisms by pushouts, transfinite compositions, and retracts.

######  Proposition 4.4.

Let $\mathcal{K}$ be locally presentable and $\mathcal{M}$ be nice and
$\aleph_{0}$-continuous in $\mathcal{K}$. Let $\mathcal{M}!$ consist of
$\mathcal{M}$-effective morphisms in $\mathcal{K}^{2}$. Then $\mathcal{M}!$ is
nice and $\aleph_{0}$-continuous in $\mathcal{K}^{2}$.

###### Proof.

(a) $\mathcal{M}!$ is normal: Isomorphisms in $\mathcal{K}^{2}$ are
commutative squares whose horizontal arrows are isomorphisms. Such squares are
pushouts, hence $\mathcal{M}$-effective. The composition of two
$\mathcal{M}$-effective morphisms is $\mathcal{M}$-effective (see [LRV, 2.7]).

(b) $\mathcal{M}!$ is coherent: see [LRV, 2.10].

(c) $\mathcal{M}!$ is a coclan: We have to show that pushouts of
$\mathcal{M}$-effective squares are $\mathcal{M}$-effective. Let

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{2}}$$\textstyle{D}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{1}}$$\scriptstyle{m_{0}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{3}}$

be an $\mathcal{M}$-effective square considered as a morphism
$(m_{0},m_{2}):m_{1}\to m_{3}$ in $\mathcal{K}^{2}$. Let

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{C^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{1}}$$\scriptstyle{a}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{1}}$

be a commutative square considered as a morphism $(a,c):m_{1}\to
m^{\prime}_{1}$. Consider the pushout

$\textstyle{m_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(b,d)}$$\textstyle{m_{3}^{\prime}}$$\textstyle{m_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(m_{0},m_{2})}$$\scriptstyle{(a,c)}$$\textstyle{m_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(m^{\prime}_{0},m_{2}^{\prime})}$

in $\mathcal{K}^{2}$. This means that

$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{0}}$$\textstyle{B^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{m_{0}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$

and

$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{2}}$$\textstyle{D^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{m_{2}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$

are pushouts and $m^{\prime}_{3}$ is the induced morphism, i.e.,
$m^{\prime}_{3}m^{\prime}_{0}=m^{\prime}_{2}m^{\prime}_{1}$ and
$m^{\prime}_{3}b=dm_{3}$. It suffices to show that the square

$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{2}}$$\textstyle{D^{\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{1}}$$\scriptstyle{m^{\prime}_{0}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{3}}$

is $\mathcal{M}$-effective.

Consider a pushout

$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{0}}$$\textstyle{P^{\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}_{1}}$$\scriptstyle{m^{\prime}_{0}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{1}}$

We must show that the induced morphism $t^{\prime}:P^{\prime}\to D^{\prime}$
is in $\mathcal{M}$. Returning to the original square, we know that the
induced morphism $t:P\to D$ is in $\mathcal{M}$ where

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\textstyle{P}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{1}}$$\scriptstyle{m_{0}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$

is a pushout. For this, it suffices to show that

$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{D^{\prime}}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{t}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$

is a pushout where $p:P\to P^{\prime}$ is the induced morphism; that is,
$pp_{0}=p^{\prime}_{0}c$ and $pp_{1}=p^{\prime}_{1}b$.

Consider morphisms $u:P^{\prime}\to X$ and $v:D\to X$ such that $up=vt$. Then

$up^{\prime}_{0}c=upp_{0}=vtp_{0}=vm_{2}.$

Thus there exists a unique $q:D^{\prime}\to X$ such that
$qm^{\prime}_{2}=up^{\prime}_{0}$ and $qd=v$. It remains to show that
$qt^{\prime}=u$. We have

$qt^{\prime}p^{\prime}_{0}=qm^{\prime}_{2}=up^{\prime}_{0}$

and $qt^{\prime}p^{\prime}_{1}=qm^{\prime}_{3}$. To finish the proof, we need
that $up^{\prime}_{1}=qm^{\prime}_{3}$ because then $qt^{\prime}=u$. We have

$qm^{\prime}_{3}m^{\prime}_{0}=qm^{\prime}_{2}m^{\prime}_{1}=up^{\prime}_{0}m^{\prime}_{1}=up^{\prime}_{1}m^{\prime}_{0}$

and

$qm^{\prime}_{3}b=qdm_{3}=vm_{3}=vtp_{1}=upp_{1}=up^{\prime}_{1}b.$

Hence $qm^{\prime}_{3}=up^{\prime}_{1}$.

(d) $\mathcal{M}!$ is nice: Since $\mathcal{M}$ is closed under retracts,
$\mathcal{M}!$ is closed under retracts.

(e) $\mathcal{M}!$ is $\aleph_{0}$-continuous: see [LRV, 2.11]. ∎

######  Theorem 4.5.

Let $\mathcal{K}$ be a locally presentable category, and let $\mathcal{M}$ be
a nice, accessible, and $\aleph_{0}$-continuous class of morphisms in
$\mathcal{K}$. If $\mathcal{K}_{\mathcal{M}}$ has a stable independence
notion, it is excellent.

###### Proof.

We have the obvious one-dimensional stable independence notion on
$\mathcal{K}_{\mathcal{M}}$, with
$(\mathcal{K}_{\mathcal{M}})^{\Gamma_{1}}=\mathcal{K}_{\mathcal{M}}$ and
$\Gamma_{1}$ consisting precisely of $\mathcal{M}$, the class of morphisms in
$\mathcal{K}_{\mathcal{M}}$. Following Fact 4.3, $\mathcal{M}$-effective
squares form a stable, $\aleph_{0}$-continuous independence notion $\Gamma$ on
$\mathcal{K}_{\mathcal{M}}$. Take $\Gamma_{2}=(\Gamma_{1},\Gamma)$. Then

$(\mathcal{K}_{\mathcal{M}})^{\Gamma_{2}}=(\mathcal{K}_{\mathcal{M}})_{\downarrow}\subseteq(\mathcal{K}_{\mathcal{M}})^{2}\subseteq\mathcal{K}^{2}$

is an accessible category closed under directed colimits in $\mathcal{K}^{2}$.

Assume that we have an $n$-dimensional stable independence notion
$\Gamma_{n}$, $n>1$, on $\mathcal{K}_{\mathcal{M}}$ where

$(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}}\subseteq(\mathcal{K}_{\mathcal{M}})^{2^{n-1}}\subseteq\mathcal{K}^{2^{n-1}}$

is an accessible category closed under directed colimits in
$\mathcal{K}^{2^{n-1}}$ and a nice and $\aleph_{0}$-continuous class
$\mathcal{M}_{n-1}$ of morphisms in $\mathcal{K}^{2^{n-1}}$. Following
Proposition 4.4, $\mathcal{M}_{n}=(\mathcal{M}_{n-1})!$ is nice and
$\aleph_{0}$-continuous class of morphisms in $\mathcal{K}^{2^{n}}$. We have
to show that $\mathcal{M}_{n}$-effective squares yield a stable independence
notion on $(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}}$. Following Fact 3.4,
$\mathcal{M}_{n}$-effective squares form a weakly stable independence notion
on $(\mathcal{K}^{2^{n}})_{\mathcal{M}_{n}}$ which is, moreover,
$\aleph_{0}$-continuous. $(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}}$ is a full
subcategory of $(\mathcal{K}^{2^{n}})_{\mathcal{M}_{n}}$ consisting of those
$\mathcal{M}_{n}$-squares which are $\mathcal{M}_{n}$-effective. Since
$\mathcal{M}_{n}$ is nice, the proof of [LRV, 2.7] yields that
$\mathcal{M}_{n}$-effective squares yield a weakly stable independence notion
on $(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}}$. Similarly, the proof of [LRV,
2.11] yields that this weakly stable independence notion is
$\aleph_{0}$-continuous.

It remains to show that the category
$(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n+1}}=((\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}})_{\downarrow}$
is accessible.

We note that an $\mathcal{M}_{n}$-square

$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f^{\prime}_{0},f^{\prime}_{1})}$$\textstyle{d}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g_{0},g_{1})}$$\scriptstyle{(f_{0},f_{1})}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(h_{0},h_{1})}$

is in fact a cube of the following form:

---
$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{D_{2}}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{f_{0}^{\prime}}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{\prime}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{0}}$$\scriptstyle{b}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{a}$$\scriptstyle{g_{0}}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{g_{1}}$

where the top, bottom, front and rear squares are all
$\mathcal{M}_{n-1}$-effective.

Such an $\mathcal{M}_{n}$-square is $\mathcal{M}_{n}$-effective if and only if
the derived square induced by pushing out on the right and left hand sides of
the cube is $\mathcal{M}_{n-1}$-effective. That is, if the following squares
are pushouts

$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\textstyle{P}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{0}}$$\textstyle{P^{\prime}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\scriptstyle{f_{0}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{f_{1}}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{1}}$

and $q:P\to D_{1}$, $q^{\prime}:P^{\prime}\to D_{2}$ and $p:P\to P^{\prime}$
are the obvious induced morphisms, the derived square

$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}}$$\textstyle{D_{2}}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$

must be $\mathcal{M}_{n-1}$-effective.

The category $(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n+1}}$ thus consists of the
full subcategory of $((\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}})^{2}$ on the
$\mathcal{M}_{n}$-effective squares whose derived squares are
$\mathcal{M}_{n-1}$-effective. In particular, the following is a pullback of
categories:

$\textstyle{((\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{(\mathcal{K}_{\mathcal{M}})^{2^{n-1}}}$$\textstyle{\mathcal{K}^{\Gamma_{n+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{G}}$$\scriptstyle{\bar{F}}$$\textstyle{(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$

where $F$ sends an $\mathcal{M}_{n}$-square to its derived square and $G$ is
the inclusion. The functor $G$ is transportable because
$\mathcal{M}_{n-1}$-effective squares are closed under isomorphisms of
$\mathcal{M}_{n-1}$-squares. Following [MP89, 5.1.1], the pullback above is in
fact a _Pullback_ (or _bipullback_) of the corresponding categories. Hence
$(\mathcal{K}_{\mathcal{M}})^{\Gamma_{n+1}}$ is an accessible category (see
[MP89, 5.1.6]). ∎

######  Remark 4.6.

The category $\mathbf{RMod}_{pure}$ from Theorem 3.6 is excellent. Similarly,
the category $\mathbf{RMod}_{emb}$ of $R$-modules and regular monomorphisms.
The latter follows from the fact that $\mathbf{RMod}$ has effective unions,
hence regular monomorphisms are cofibrantly generated. In fact, the same is
true in any Grothendieck abelian category or Grothendieck topos ([Bar88]).

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