1 Introduction
Stable (nonforking) independence is a central notion of model theory. In the first-order context, it was introduced by Shelah in [Reference Shelah24], and constitutes an essential tool both in that book and in decades of subsequent work in first-order model theory. In the now-dominant anchor notation introduced by Makkai [Reference Makkai18], this is rendered as a relation on quadruples of sets,
, understood to mean that B and C are independent—in a precise syntactic sense—over A in D. Generalizing (and serving many of the same purposes as) linear or algebraic independence, this independence notion can also be directly axiomatized as the canonical quaternary relation satisfying a list of essential properties, including, but not limited to invariance, monotonicity, uniqueness, and local character. This notion was subsequently extended—again in syntactic form—to abstract elementary classes in [Reference Shelah26], with corresponding direct axiomatization presented in [Reference Bican, El Bashir and Enochs9]. In earlier work of the authors [Reference Lieberman, Rosický and Vasey16], it was shown that the latter axiomatization, in particular, leads naturally to the formulation of nonforking in an abstract category as a calculus of special commutative squares,
identified as “independent.” That this specializes to stable independence in abstract elementary classes is proven in [Reference Lieberman, Rosický and Vasey16, Section 8]; connections with the classical notion are examined in, e.g., [Reference Vasey27, Example 5.7(6)]. It is shown in [Reference Lieberman, Rosický and Vasey16], moreover, that this axiomatization is canonical in accessible categories with monomorphisms, assuming that they have chain bounds, thereby extending the canonicity theorem for abstract elementary classes of [Reference Bican, El Bashir and Enochs9]. The present paper hinges on a pair of observations implicit in the earlier one. First, the aforementioned commutative squares can be thought as a replacement for pushouts in situations where they—an often indispensable tool in category-theoretic constructions—are not available: for example, when all morphisms are monomorphisms, which is typical for abstract elementary classes. Second, we need not, in fact, assume that all morphisms are monomorphisms: we conclude in Section 5, for example, that canonicity holds in still greater generality, in arbitrary accessible categories with chain bounds. More broadly, this shifts stable independence away from the model-theoretic framework, and means that the benefits of the first observation are applicable across the broad swathe of category theory. The chief aim of the present paper is to highlight one particularly fruitful connection that arises as a result, between stable independence and the important homotopy-theoretic concepts of cellular and cofibrant generation.
Cellular categories were introduced in [Reference Makkai and Rosický19] as cocomplete categories equipped with a class of morphisms (called cellular) containing all isomorphisms and closed under pushouts and transfinite compositions. These categories are abundant in homotopy theory because any Quillen model category carries two cellular structures given by cofibrations and trivial cofibrations respectively. These cellular categories are, in addition, retract-closed (in the category of morphisms). A retract-closed cellular category is cofibrantly generated if it is generated by a set of morphisms using pushouts, transfinite compositions, and retracts. In locally presentable categories, this implies that cellular morphisms form a left part of a weak factorization system. In [Reference Makkai and Rosický19], retract-closed cofibrantly generated cellular locally presentable categories were called combinatorial. The main result of [Reference Makkai and Rosický19] is that combinatorial categories are closed under 2-limits, in particular under pseudopullbacks. A consequence is that combinatorial categories are left-induced in a sense that, given a colimit preserving functor
$F:\mathcal {K}\to \mathcal {L}$
from a locally presentable category
$\mathcal {K}$
to a combinatorial category
$\mathcal {L}$
then preimages of cellular morphisms form a combinatorial structure on
$\mathcal {K}$
. This was later used, e.g., in [Reference Hess, Kȩdziorek, Riehl and Shipley14]. The proof is quite delicate and depends on Lurie’s concept of a good colimit (see [Reference Makkai, Rosický and Vokřínek20]).
The main result of the present paper is that, in the special case when cellular morphisms are coherent and
$\aleph _0$
-continuous, a retract-closed cellular category is combinatorial if and only if it carries a stable independence notion (Theorem 3.1). Independent squares coincide with cellular squares; that is, squares of cellular morphisms such that the unique morphism from the pushout is cellular. These squares are also used in [Reference Henry13]. Since a pre-image of an accessible category is accessible, this yields a simple proof that coherent and
$\aleph _0$
-continuous combinatorial categories are left-induced (see Corollary 3.11). While coherence is quite common, especially for trivial cofibrations,
$\aleph _0$
-continuity is more limiting. Nevertheless, our theorem covers many situations. In particular, we will show (Theorem 4.3) that the abstract elementary classes of “roots of Ext” studied in [Reference Beke and Rosický4] (for example the AEC of flat modules with flat monomorphisms) have a stable independence notion. Note, too, that since pure monomorphisms in a locally finitely presentable category are coherent and
$\aleph _0$
-continuous, the result of [Reference Lieberman, Positselski, Rosický and Vasey15], the proof of which relies on [Reference Makkai and Rosický19], actually falls within the framework of this paper.
In Section 5, we prove a strengthening of the canonicity result of [Reference Lieberman, Rosický and Vasey16]: if a category
$\mathcal {K}$
has chain bounds, it has at most one weakly stable independence relation (in fact, we prove something stronger still; cf. Theorem 6.6). This new canonicity theorem eliminates the requirement—present in [Reference Lieberman, Rosický and Vasey16]—that all morphisms in
$\mathcal {K}$
are monomorphisms, and is central to several of the results of this paper. Both Theorem 6.6 and its proof should be of independent interest: in connection with the latter, we show that it is possible to define, and to work with, independent sequences in an abstract category, that is, without reference to elements.
Concerning terminology, we will refer freely to [Reference Adámek and Rosický2, Reference Lieberman, Rosický and Vasey16, Reference Makkai and Rosický19] (concerning accessible categories, cellular categories, and stable independence respectively). A more comprehensive version of the present paper, with added background, can be found at https://arxiv.org/abs/1904.05691v2.
2 Cellular categories
Recall that a cocomplete category
$\mathcal {K}$
is called cellular if it is equipped with a class
$\mathcal {M}$
of morphisms containing all isomorphisms and closed under pushouts and transfinite compositions (see [Reference Makkai and Rosický19]).
Remark 2.1. A composition of two morphisms is a special case of a transfinite composition. Thus a cellular category
$(\mathcal {K},\mathcal {M})$
induces a subcategory
$\mathcal {K}_{\mathcal {M}}$
of the category
$\mathcal {K}$
whose objects are those in
$\mathcal {K}$
and whose morphisms are precisely those of
$\mathcal {M}$
. Since
$\mathcal {M}$
contains all isomorphisms, the subcategory
$\mathcal {K}_{\mathcal {M}}$
is isomorphism-closed. Still,
$\mathcal {K}_{\mathcal {M}}$
need not have pushouts.
In order to explain this, recall that
$\mathcal {M}$
is closed under pushouts whenever, given a pushout square
in
$\mathcal {K}$
with
$f\in \mathcal {M}$
, then
$h\in \mathcal {M}$
. But this does not mean that, if also
$g\in \mathcal {M}$
, this square is a pushout square in
$\mathcal {K}_{\mathcal {M}}$
. The latter means that given another commutative square in
$\mathcal {K}_{\mathcal {M}}$
, as below, with
$u,v\in \mathcal {M}$
,
then the induced morphism t is in
$\mathcal {M}$
.
Similarly, although
$\mathcal {M}$
is closed under transfinite compositions, these compositions do not to be colimits in
$\mathcal {K}_{\mathcal {M}}$
. In the latter case,
$\mathcal {K}_{\mathcal {M}}$
would be closed under colimits of smooth chains, which implies closure under all directed colimits (see [Reference Adámek and Rosický2, Corollary 1.7]).
Definition 2.2. Let
$(\mathcal {K},\mathcal {M})$
be a cellular category. A commutative square
is called cellular if the induced morphism
$t:P\to D$
from the pushout (see above) belongs to
$\mathcal {M}$
.
Remark 2.3. Cellular squares could also be called
$\mathcal {M}$
-effective. In the special case in which
$\mathcal {M}$
is the class of regular monomorphisms, this corresponds precisely to the effective squares considered in [Reference Lieberman, Rosický and Vasey16], and originating in [Reference Baldwin, Eklof and Trlifaj5].
Definition 2.4. A cellular category
$(\mathcal {K},\mathcal {M})$
will be called:
-
(1) coherent if whenever f and g are composable morphisms,
$gf \in \mathcal {M}$
, and
$g \in \mathcal {M}$
, then
$f \in \mathcal {M}$
, -
(2) left cancellable if
$gf \in \mathcal {M}$
implies
$f \in \mathcal {M}$
, -
(3)
$\lambda $
-continuous if
$\mathcal {K}_{\mathcal {M}}$
is closed under
$\lambda $
-directed colimits in
$\mathcal {K}$
, -
(4)
$\lambda $
-accessible if it is
$\lambda $
-continuous and both
$\mathcal {K}$
and
$\mathcal {K}_{\mathcal {M}}$
are
$\lambda $
-accessible, -
(5) accessible if it is
$\lambda $
-accessible for some
$\lambda $
.
Remark 2.5.
-
(1) Since a cellular category is cocomplete, an accessible cellular category has
$\mathcal {K}$
locally presentable. -
(2) It is easy to see that
$(\mathcal {K},\mathcal {M})$
is
$\lambda $
-continuous provided that
$\mathcal {M}$
is closed under
$\lambda $
-directed colimits in
$\mathcal {K}^2$
. In fact, given a
$\lambda $
-directed diagram
$D:I\to \mathcal {K}_{\mathcal {M}}$
and its colimit
$\delta _i:Di\to K$
in
$\mathcal {K}$
, then
$\delta _i=\operatorname {colim}_{i\leq j\in I}D_{i,j}$
, where the
$D_{i,j}:Di\to Dj$
are the appropriate diagram maps. Similarly, given a cocone
$\gamma _i:Di\to L$
in
$\mathcal {K}_{\mathcal {M}}$
then the induced morphism
$g:K\to L$
is precisely
$\operatorname {colim}_i\gamma _i$
.
Remark 2.6.
-
(1) In [Reference Lieberman, Rosický and Vasey16], we defined an independence relation (or independence notion) in a category
$\mathcal {K}$
as a class
of commutative square (called
-independent, or just independent, squares) such that, for any commutative diagram the square spanning
$A, B, C,$
and D is independent if and only if the square spanning
$A, B, C,$
and E is independent. A subcategory of
$\mathcal {K}^2$
(the category of morphisms in
$\mathcal {K}$
, whose objects are morphisms and morphisms are commutative squares) whose objects are morphisms and morphisms are independent squares was denoted as
$\mathcal {K}_{\operatorname {NF}}$
. Here, we will denote it as
$\mathcal {K}_{\downarrow }$
.
-
(2) In [Reference Lieberman, Rosický and Vasey16], as independence relation
was defined to be stable if it is symmetric, transitive, accessible, has existence, and has uniqueness. In case
satisfies all of the above conditions except accessibility, we say that it is weakly stable. -
(3) Accessibility of
means that the category
$\mathcal {K}_{\downarrow }$
is accessible, which implies, in particular, that it is closed in
$\mathcal {K}^2$
under
$\lambda $
-directed colimits for some
$\lambda $
(see [Reference Lieberman, Rosický and Vasey16, Remark 3.26]). If
satisfies the latter closure condition, we say that it is
$\lambda $
-continuous. -
(4) Accessibility of
also implies that
$\mathcal {K}$
is accessible (see [Reference Lieberman, Rosický and Vasey16, Lemma 3.27]).
Theorem 2.7. If
$(\mathcal {K},\mathcal {M})$
is a cellular category, then cellular squares form a weakly stable independence relation in
$\mathcal {K}_{\mathcal {M}}$
.
Proof. We first check that cellular squares form an independence notion. Assume that
$(A, B, C, D)$
is a commutative squareFootnote
1
in
$\mathcal {K}_{\mathcal {M}}$
and we are given a morphism
$D \to E$
in
$\mathcal {M}$
. If
$(A, B, C, D)$
is cellular, then closure of
$\mathcal {M}$
under composition yields that
$(A, B, C, E)$
is cellular. Conversely, if
$(A, B, C, E)$
is cellular, then the map
$P \to E$
from the pushout is in
$\mathcal {M}$
by assumption, and also
$D \to E$
is in
$\mathcal {M}$
, so by coherence also the map
$P \to D$
is in
$\mathcal {M}$
. Thus
$(A, B, C, D)$
is cellular.
This concludes the proof that cellular squares form an independence notion. Of course, the relation is also symmetric. Existence follows from closure under pushouts (and the fact that the identity map is an isomorphism, hence in
$\mathcal {M}$
). In order to prove the uniqueness property, consider cellular squares
$(A, B, C, D^1)$
and
$(A,B,C,D^2)$
with the same span
$B \leftarrow A\to C$
. Form the pushout
and take the induced morphisms
$P\to D^1$
and
$P\to D^2$
. They are in
$\mathcal {M}$
by cellularity. Then the pushout
amalgamates the starting diagram.
To prove transitivity, consider:
where both squares are cellular. We have to show that the outer rectangle is cellular. Thus we have to show that the induced morphism
$p:P\to F$
from the pushout
is in
$\mathcal {M}$
. This pushout is a composition of pushouts
Recalling the left square of the starting diagram, we have an induced morphism
$q:Q\to D$
. Consider the pushout
Since the left square of the starting diagram is cellular, q is in
$\mathcal {M}$
and thus
$\bar {q}$
is in
$\mathcal {M}$
. Composing this pushout with the right pushout square in the diagram above it, we obtain the pushout
The right square in the starting diagram is cellular, so the induced morphism
$p':P'\to F$
is in
$\mathcal {M}$
. Thus
$p=p'\bar {q}$
is in
$\mathcal {M}$
.
Remark 2.8. In the proof, we have not used the full strength of the assumption that
$\mathcal {M}$
is closed under transfinite compositions: here finite compositions suffice. Coherence is used only once, in the proof that cellular squares form an independence notion (specifically, in the proof that the top right corner can be made “smaller”). Instead of coherence, we could also have assumed the dual property, cocoherence: indeed, we know in the proof that the maps
$C \to D$
and
$C \to P$
are in
$\mathcal {M}$
, so cocoherence would give us immediately that
$P \to D$
is in
$\mathcal {M}$
. Note, however, that if
$\mathcal {M}$
is a class of monomorphisms, cocoherence is too strong an assumption: if a section
$i: A \to B$
is in
$\mathcal {M}$
, cocoherence would imply that the corresponding retract
$r: B \to A$
is in
$\mathcal {M}$
, and so r would have to be an isomorphism.
Notation 2.9. For a cellular category
$(\mathcal {K},\mathcal {M})$
, we write
$\mathcal {K}_{\mathcal {M},\downarrow }$
for
$\left (\mathcal {K}_{\mathcal {M}}\right )_{\downarrow }$
.
Remark 2.10. In a cellular category, cellular squares form a class of morphisms in
$\mathcal {K}^2$
. Following Theorem 2.7 this class is closed under composition, by transitivity of the associated weakly stable independence notion. Using [Reference Lieberman, Rosický and Vasey16, Lemma 3.18], it is isomorphism-closed. Using [Reference Lieberman, Rosický and Vasey16, Lemmas 3.20 and 3.21], cellular squares are left-cancellable.
Lemma 2.11. If
$(\mathcal {K},\mathcal {M})$
is a
$\lambda $
-continuous cellular category, then the independence relation given by cellular squares is
$\lambda $
-continuous.
Proof. Let
$(\mathcal {K},\mathcal {M})$
be
$\lambda $
-continuous. Let
$D:I\to \mathcal {K}_{\mathcal {M}, \downarrow }$
be a
$\lambda $
-directed diagram where
$Di$
is
$f_i:A_i\to B_i$
. Let
$f:A\to B$
be a colimit of D in
$\left (\mathcal {K}_{\mathcal {M}}\right )^2$
. For each
$i\in I$
, the pushout of the colimit coprojection
$A_i\to A$
along
$f_i$
, i.e.,
is a
$\lambda $
-directed colimit of pushouts
Thus the induced morphism
$p:P\to B$
is a
$\lambda $
-directed colimit of induced morphisms
$p_{i'}:P_{i'}\to B_{i'}$
. Since
$\mathcal {M}$
is
$\lambda $
-continuous, it follows that
$p \in \mathcal {M}$
. This shows that all the maps of the cocone
$(f_i \to g)_{i \in I}$
are independent squares. Similarly, one can check that this is a colimit cocone in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
. Thus
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is closed under
$\lambda $
-directed colimits in
$\left (\mathcal {K}_{\mathcal {M}}\right )^2$
.⊣
3 Combinatorial categories
A cellular category
$(\mathcal {K},\mathcal {M})$
is said to be retract-closed if
$\mathcal {M}$
is closed under retracts in the category
$\mathcal {K}^2$
. A retract-closed cellular category is called combinatorial if it is cofibrantly generated, i.e., if
$\mathcal {M}$
is the closure of a set
$\mathcal {X}$
of morphisms under pushouts, transfinite compositions, and retracts. In particular,
$\mathcal {M}=\operatorname {cof}(\mathcal {X})$
, where
$$ \begin{align*}\operatorname{cof}(\mathcal {X})=\operatorname{Rt}(\operatorname{Tc}(\operatorname{Po}(\mathcal {X})))= \operatorname{Rt}(\operatorname{cell}(\mathcal {X})), \end{align*} $$
where
$\operatorname {Po}$
denotes the closure under pushouts,
$\operatorname {Tc}$
under transfinite compositions, and
$\operatorname {Rt}$
under retracts (see [Reference Makkai and Rosický19]).
For
$\lambda $
a regular cardinal, we write
$\mathcal {K}_{\lambda }$
for the full subcategory of
$\mathcal {K}$
consisting of
$\lambda $
-presentable objects. We similarly denote by
$\mathcal {K}_{\lambda }^2$
the full subcategory of
$\mathcal {K}^2$
consisting of morphisms with
$\lambda $
-presentable domains and codomains. We will also write, for example,
$\mathcal {M}_{\lambda } := \mathcal {M} \cap \mathcal {K}_{\lambda }^2$
.
The next result, the main theorem of this paper, characterizes when cellular squares form a stable independence notion in terms of cofibrant generation of the corresponding class of morphisms.
To go from stable independence to cofibrant generation, we require a technical result from [Reference Lieberman, Rosický and Vasey17, Section 8] concerning the existence of filtrations. Recall that the presentability rank of an object A is the least regular cardinal
$\lambda $
such that A is
$\lambda $
-presentable. We say that A is filtrable if it can be written as the directed colimit of a chain of objects with lower presentability rank than A. We say that A is almost filtrable if it is a retract of such a chain. The chain is smooth if directed colimits are taken at every limit ordinal. By [Reference Lieberman, Rosický and Vasey17, Corollary 8.9], in any accessible category with directed colimits, there exists a regular cardinal
$\lambda $
such that any object with presentability rank at least
$\lambda $
is almost filtrable (and, moreover, the chain in the filtration can be chosen to be smooth). We say that a category satisfying the latter condition is almost well
$\lambda $
-filtrable.
Theorem 3.1 (Main theorem).
Let
$(\mathcal {K},\mathcal {M})$
be an accessible cellular category which is retract-closed, coherent, and
$\aleph _0$
-continuous. The following are equivalent:
-
(1)
$\mathcal {K}_{\mathcal {M}}$
has a stable independence notion. -
(2) Cellular squares form a stable independence notion in
$\mathcal {K}_{\mathcal {M}}$
. -
(3)
$(\mathcal {K},\mathcal {M})$
is combinatorial.
Proof. (1) implies (2): If
$\mathcal {K}_{\mathcal {M}}$
has a stable independence notion, then canonicity (Theorem 6.6—note that
$\mathcal {K}_{\mathcal {M}}$
has directed colimits, since
$\mathcal {M}$
is
$\aleph _0$
-continuous) together with Theorem 2.7 ensures that it is given by cellular squares. Note that if we know that all morphisms in
$\mathcal {M}$
are monos, then we do not need Theorem 6.6 and can use [Reference Lieberman, Rosický and Vasey16, Theorem 9.1] instead.
(2) implies (3): Assume that
$\mathcal {K}_{\mathcal {M}}$
has a stable independence
given by cellular squares. Thus
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is accessible and has directed colimits (by Lemma 2.11). By Remark 2.6(4),
$\mathcal {K}_{\mathcal {M}}$
is accessible, so
$\mathcal {M}$
is accessible. Using the preceding discussion, pick a regular uncountable cardinal
$\lambda $
such that both
$\mathcal {K}$
and
$\mathcal {K}_{\mathcal {M}, \downarrow }$
are
$\lambda $
-accessible and almost well
$\lambda $
-filtrable. Let
$\mathcal {M}_{\lambda }$
be the collection of morphisms in
$\mathcal {M}$
whose domains and codomains are
$\lambda $
-presentable (in
$\mathcal {K}$
). We will show that for each infinite cardinal
$\mu $
,
$\mathcal {M}_{\mu ^+} \subseteq \operatorname {cof} (\mathcal {M}_{\lambda })$
. We proceed by induction on
$\mu $
. When
$\mu < \lambda $
, this is trivial, so assume that
$\mu \ge \lambda $
. Note that, playing with pushouts, it is straightforward to check that the
$\mu ^+$
-presentable objects in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
are exactly the morphisms of
$\mathcal {M}_{\mu ^+}$
.
Every morphism h in
$\mathcal {M}_{\mu ^+}$
must be a retract of a filtrable object in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
. Now, retracts in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
are retracts in
$\mathcal {K}^2$
, so since we are looking at
$\operatorname {cof} (\mathcal {M}_{\lambda })$
it suffices to show that any morphism h in
$\mathcal {M}_{\mu ^+}$
which is filtrable in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is in
$\operatorname {cof} (\mathcal {M}_{\lambda })$
. So take such a morphism. Write
$h = h_0:K_0\to L$
. We will show that
$h_0\in \operatorname {cof} (\mathcal {M}_{\lambda })$
. Express
$h_0$
as a colimit of a smooth chain of morphisms
$t_{0i}\in \operatorname {cof} (\mathcal {M}_{\lambda })$
,
$i<\text {cf} (\mu )$
, between
$(<\mu ^+)$
-presentable objects in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
.
Form a pushout
and take the induced morphism
$h_1:K_1\to L$
. Since the starting square is cellular,
$h_1$
is in
$\mathcal {M}$
. Note also that
$K_1$
is
$\mu ^+$
-presentable. We have a commutative square
because
$h_1 h_{01} k_{01} = h_0 k_{01} = l_{01} t_{01}$
. We can express
$h_1$
as a colimit of a smooth chain of morphisms
$t_{1i}\in \operatorname {cof} (\mathcal {M}_{\lambda })$
,
$1\leq i<\text {cf} (\mu )$
, between
$<\mu ^+$
-presentable objects in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
which are above
$t_{01}$
Form a pushout
and take the induced morphisms
$h_2:K_2\to L$
. Again, by cellularity,
$h_2$
is in
$\mathcal {M}$
. In
$$ \begin{align*}K_0 \xrightarrow{\ h_{01}\ } K_1\xrightarrow{\ h_{12}\ } K_2\xrightarrow{\ h_2} L, \end{align*} $$
we put
$h_{02}=h_{12}h_{01}$
and continue transfinitely. This means that for
$i<\text {cf} (\mu )$
we express
$h_i$
as a colimit of a smooth chain of morphisms
$t_{ij}\in \operatorname {cof} (\mathcal {M}_{\lambda })$
,
$i\leq j<\text {cf} (\mu )$
, between
$(<\mu ^+)$
-presentable objects in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
which are above
$t_{0i}$
Form a pushout
and take the induced morphisms
$h_{i+1}:K_{i+1}\to L$
. By cellularity,
$h_{i+1}$
is in
$\mathcal {M}$
. We put
$h_{k,i+1}=h_{i,i+1}h_{ik}$
. At limit steps we take colimits. Then by construction
$L=K_{\text {cf} (\mu )}$
and
$h_0$
is the transfinite composition of
$(h_{ij})_{i<j<\text {cf} (\mu )}$
. We have just observed that each
$h_{ij}$
is in
$\operatorname {cof} (\mathcal {M}_{\lambda })$
, so
$h_0$
also is.
(3) implies (1): Assume that
$\mathcal {M}$
is accessible and cofibrantly generated in
$\mathcal {K}$
. Let
$\mathcal {X}$
be a subset of
$\mathcal {M}$
so that
$\mathcal {M} = \operatorname {cof} (\mathcal {X})$
. Let
$\lambda $
be a big-enough uncountable regular cardinal such that
$\mathcal {K}$
and
$\mathcal {K}_{\mathcal {M}}$
are
$\lambda $
-accessible, and all the morphisms in
$\mathcal {X}$
have
$\lambda $
-presentable domain and codomain. Note that, by coherence, for any regular
$\mu \ge \lambda $
, an object which is
$\mu $
-presentable in
$\mathcal {K}$
is
$\mu $
-presentable in
$\mathcal {K}_{\mathcal {M}}$
. We claim that
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is
$\lambda $
-accessible. First,
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is closed under directed colimits in
$\mathcal {K}_{\mathcal {M}}$
by Lemma 2.11. Now let
$\mathcal {M}_{\lambda }$
be the class of morphisms in
$\mathcal {M}$
with
$\lambda $
-presentable domain and codomain and let
$\mathcal {M}^{\ast }$
be the class of morphisms in
$\mathcal {M}$
that are
$\lambda $
-directed colimit (in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
) of morphisms in
$\mathcal {M}_{\lambda }$
. It suffices to see that
$\mathcal {M}^{\ast } = \mathcal {M}$
.
First, any pushout of a morphism in
$\mathcal {M}_{\lambda }$
is in
$\mathcal {M}^{\ast }$
. Consider such a pushout
where
$K_0$
and
$L_0$
are
$\lambda $
-presentable. Then K is a
$\lambda $
-directed colimits of
$\lambda $
-presentable objects
$K_i$
above
$K_0$
in
$\mathcal {K}_{\mathcal {M}}$
. Consider pushouts
It is easy to check that the
$L_i$
’s are also
$\lambda $
-presentable and that
$h=\operatorname {colim} h_i$
in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
. Thus
$h \in \mathcal {M}^{\ast }$
.
Second,
$\mathcal {M}^{\ast }$
is closed under compositions of morphisms from
$\operatorname {Po}_{\lambda }$
where
$\operatorname {Po}_{\lambda }$
consists of pushouts of morphisms from
$\mathcal {M}_{\lambda }$
. Let
$f:K\to L$
and
$g:L\to M$
belong to
$\operatorname {Po}_{\lambda }$
. As above, f is a
$\lambda $
-directed colimit (in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
),
$(k_i,l_i):f_i\to f$
of
$f_i\in \mathcal {M}_{\lambda }$
,
$f_i : K_i \to L_i$
. Moreover, g is a pushout of
$g_0:L_0\to M_0$
having
$L_0$
and
$M_0$
both
$\lambda $
-presentable. Without loss of generality, we can assume that
$L_0\to L$
factors through the
$L_i$
. We then take pushouts as above
This shows that
$gf$
is a
$\lambda $
-directed colimit of the
$g_if_i$
’s in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
.
Third,
$\mathcal {M}^{\ast }$
is closed under transfinite compositions of morphisms from
$\operatorname {Po}_{\lambda }$
. Let
$(f_{ij})_{i,j\leq \alpha }$
be such a transfinite composition. At limit steps,
$f_{0i}$
is the following directed colimit in
$\mathcal {K}_{\mathcal {M}, \downarrow }$
:
This shows that
$f_{0,i}$
is in
$\mathcal {M}^{\ast }$
(we used [Reference Lieberman, Rosický and Vasey16, Lemma 3.12]).
We have shown that any transfinite composition of pushouts from
$\mathcal {M}_{\lambda }$
is in
$\mathcal {M}^{\ast }$
. That is,
$\operatorname {cell} (\mathcal {M}_{\lambda }) = \operatorname {Tc} (\operatorname {Po} (\mathcal {M}_{\lambda })) \subseteq \mathcal {M}^{\ast }$
. Since
$\mathcal {M}$
is closed under pushouts, retracts, and transfinite compositions,
$\operatorname {cof} (\mathcal {X}) \cap \mathcal {K}_{\lambda }^2 \subseteq \mathcal {M}_{\lambda }$
. By [Reference Makkai, Rosický and Vokřínek20, Theorem B.1], it follows that
$\mathcal {M} = \operatorname {cof} (\mathcal {X}) = \operatorname {cell} (\mathcal {M}_{\lambda })$
. We deduce that
$\mathcal {M} = \mathcal {M}^{\ast }$
, as desired.⊣
Example 3.2.
-
(1) On any locally presentable category
$\mathcal {K}$
, there are two trivial cellular structures—the discrete
$(\mathcal {K},\operatorname {Iso})$
and the indiscrete
$(\mathcal {K},\mathcal {K}^2)$
. They are both combinatorial (see [Reference Makkai and Rosický19]), coherent, and
$\aleph _0$
-continuous. The first one is not accessible because
$\mathcal {K}_{\operatorname {Iso}}$
is not accessible (as long as
$\mathcal {K}$
is not small, in any case). The second is accessible and yields a stable independence relation where every commutative square is independent. -
(2) On every locally presentable category
$\mathcal {K}$
, there is a cellular structure where
$\mathcal {M}$
consists of regular monomorphisms. This cellular category is accessible, retract-closed, and coherent. If
$\mathcal {K}$
is locally finitely presentable, it is
$\aleph _0$
-continuous. Concrete examples include graphs with induced subgraph embeddings, groups, Banach spaces, Boolean algebras, Hilbert spaces, and any Grothendieck topos. The last two are combinatorial, and hence have a stable independence notion. See [Reference Lieberman, Rosický and Vasey16] for more details. -
(3) On every locally finitely presentable category
$\mathcal {K}$
, there is a cellular structure where
$\mathcal {M}$
consists of pure monomorphisms. This cellular category is accessible, retract-closed, coherent, and
$\aleph _0$
-continuous. The conditions under which this cellular structure is combinatorial are discussed in [Reference Borceux and Rosický10, Reference Lieberman, Positselski, Rosický and Vasey15]. For example, the latter shows that
$(\mathcal {K}, \mathcal {M})$
is combinatorial for any additive category
$\mathcal {K}$
.
Often, it is natural to look not at all objects, but just those objects A so that
$0 \to A$
is in
$\mathcal {M}$
(where
$0$
is an initial object):
Definition 3.3. Let
$(\mathcal {K},\mathcal {M})$
be a cellular category. An object A is called cellular if
$0\to A$
is cellular. Let
$\mathcal {C}$
denote the full subcategory of
$\mathcal {K}$
consisting of cellular objects.
Remark 3.4. Let
$\mathcal {M}_0$
be the class of cellular morphisms with a cellular domain (then the codomain is cellular too). Then
$(\mathcal {C},\mathcal {M}_0)$
satisfies all properties of a cellular category up to cocompleteness of
$\mathcal {C}$
. Thus it induces a subcategory
$\mathcal {C}_{\mathcal {M}_0}$
of
$\mathcal {C}$
consisting of cellular objects and cellular morphisms.
If
$\mathcal {M}$
is coherent, then every cellular morphism
$A\to B$
with
$B\in \mathcal {C}$
has
$A\in \mathcal {C}$
.
We have the following version of Theorem 3.1 for cofibrant objects. Its advantage is that we do not need to assume that
$(\mathcal {K},\mathcal {M})$
itself is accessible: it suffices to have
$\mathcal {K}$
accessible.
Theorem 3.5. Let
$\mathcal {K}$
be a retract-closed, coherent, and
$\aleph _0$
-continuous cellular category such that
$\mathcal {K}$
is accessible. The following are equivalent:
-
(1)
$\mathcal {C}_{\mathcal {M}_0}$
has a stable independence notion. -
(2)
$\mathcal {M}_0$
-effective squares form a stable independence notion in
$\mathcal {C}_{\mathcal {M}_0}$
. -
(3)
$\mathcal {M}_0$
is cofibrantly generated in
$\mathcal {C}$
.
Proof. Similar to the proof of Theorem 3.1. Following [Reference Makkai, Rosický and Vokřínek20, Corollary 5.2], (3) implies that
$\mathcal {C}_{\mathcal {M}_0}$
is accessible.
In many cases, the cellular squares will be pullback squares:
Fact 3.6 [Reference Ringel21], [Reference Adámek, Herrlich and Strecker1, Lemma 11.15].
Let
$(\mathcal {K},\mathcal {M})$
be a cellular category where every cellular morphism is a monomorphism. If:
-
(1) A pullback of two morphisms in
$\mathcal {M}$
is again in
$\mathcal {M}$
. -
(2) Every epimorphism in
$\mathcal {M}$
is an isomorphism.
Then every cellular square is a pullback square.
Conversely, it is natural to ask whether every pullback square is cellular. When
$\mathcal {M}$
is the class of regular monomorphisms, categories with this property are said to have effective unions, a condition isolated by Barr [Reference Baldwin, Eklof and Trlifaj5]. The connections of this special case with stable independence were investigated in [Reference Lieberman, Rosický and Vasey16, Section 5], where it was shown that having effective unions implies that effective squares form a stable independence notion. We show that the definition can be naturally parameterized by
$\mathcal {M}$
(this was done already for pure morphisms in [Reference Boney, Grossberg, Kolesnikov and Vasey10, Definition 2.2]), and the corresponding results generalized.
Definition 3.7. We say that a cellular category
$(\mathcal {K},\mathcal {M})$
has effective unions if:
-
(1) The pullback of any two morphisms in
$\mathcal {M}$
with common codomain exists and the projections are again in
$\mathcal {M}$
. -
(2) Any pullback square with morphisms in
$\mathcal {M}$
is cellular.
Theorem 3.8. Let
$(\mathcal {K},\mathcal {M})$
be a cellular category which is coherent, has effective unions, and with
$\mathcal {K}$
accessible. Then
$(\mathcal {K}, \mathcal {M})$
is accessible if and only if cellular squares form a stable independence notion in
$\mathcal {K}_{\mathcal {M}}$
.
Proof. If there is a stable independence notion in
$\mathcal {K}_{\mathcal {M}}$
, then by Remark 2.6(4),
$(\mathcal {K}, \mathcal {M})$
is accessible. Let us prove the converse. Pick a regular cardinal
$\lambda $
such that
$(\mathcal {K}, \mathcal {M})$
is
$\lambda $
-accessible. By Theorem 2.7, cellular squares form a weakly stable independence notion and by Lemma 2.11 this independence notion is
$\lambda $
-continuous. It remains to see that
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is accessible. Consider an object
$C \to D$
of
$\mathcal {K}_{\mathcal {M},\downarrow }$
. Since
$\mathcal {M}$
is
$\lambda $
-accessible, D can be written as a
$\lambda $
-directed colimit
$\langle D_i : i \in I \rangle $
of
$\lambda $
-presentable objects. Let
$C_i$
be the pullback of C and
$D_i$
over D. Then the resulting maps
$C_i \to D_i$
form a
$\lambda $
-directed system. Since
$\lambda $
-directed colimits commute with finite limits (see [Reference Adámek and Rosický2, Proposition 1.59], the pullback functor is accessible, so it must preserve arbitrarily large presentability ranks. Thus there is a bound on the presentability rank of
$C_i$
that depends only on
$\lambda $
. This shows that
$\mathcal {K}_{\mathcal {M}, \downarrow }$
is accessible.⊣
Note that, as opposed to Theorem 3.1, we did not need to assume that
$(\mathcal {K}, \mathcal {M})$
was
$\aleph _0$
-continuous (nor that
$(\mathcal {K}, \mathcal {M})$
was retract-closed). However, a category may fail to have effective unions even if the effective squares form a stable independence notion (this is the case, for example, in locally finite graphs with regular monos; see [Reference Lieberman, Rosický and Vasey16, Example 5.7]).
As a corollary, we obtain a quick proof that having effective unions implies cofibrant generation. This had been done “by hand” before for several special classes of morphisms [Reference Beke6, Proposition 1.12], [Reference Boney, Grossberg, Kolesnikov and Vasey10, Theorem 2.4].
Corollary 3.9. If
$(\mathcal {K},\mathcal {M})$
is an accessible cellular category which is coherent,
$\aleph _0$
-continuous, and has effective unions, then it is combinatorial.
Proof. By Theorem 3.8 cellular squares form a stable independence notion, so by Theorem 3.1 (noting that retract-closedness is not used for this direction)
$(\mathcal {K}, \mathcal {M})$
is cofibrantly generated.
Remark 3.10. Let
$F:\mathcal {K}\to \mathcal {L}$
be a colimit-preserving functor from a locally presentable category
$\mathcal {K}$
to a combinatorial category
$\mathcal {L}$
. We get a cellular structure on
$\mathcal {K}$
where f is cellular if and only if
$Ff$
is cellular. This cellular structure is called left-induced (see [Reference Makkai and Rosický19, Remark 3.8]). It was shown in [Reference Makkai and Rosický19], using a great deal of heavy machinery, that such left-induced cellular structures are combinatorial. With the aid of Theorem 3.1, we obtain a special case of this result without any effort.
Corollary 3.11. Let
$F:\mathcal {K}\to \mathcal {L}$
be a colimit preserving functor from a locally presentable category to a combinatorial category. If
$\mathcal {L}$
is coherent and
$\aleph _0$
-continuous, then
$\mathcal {K}$
is combinatorial.
Proof. Preimages of cellular squares are cellular and the left-induced cellular category
$\mathcal {K}$
is clearly retract-closed, coherent, and
$\aleph _0$
-continuous. We have a pseudopullback
Since a pseudopullback of accessible categories is accessible (see [Reference Adámek and Rosický2, Example 2n]),
in
$\mathcal {K}$
is accessible. The result now follows from Theorem 3.1.⊣
4 Abstract elementary classes of roots of Ext
Abstract elementary classes (or AECs) are a framework for abstract model theory introduced by Shelah [Reference Shelah and Baldwin25]. We will use the category-theoretic characterization of Beke–Rosický [Reference Borceux and Rosický7]: they are accessible categories with directed colimits and with all morphisms monomorphisms which embed “nicely” into finitely accessible categories.
Lemma 4.1. Let
$(\mathcal {K},\mathcal {M})$
be an accessible cellular category which is coherent and
$\aleph _0$
-continuous. Assume that
$\mathcal {K}$
is finitely accessible and all morphisms in
$\mathcal {M}$
are monomorphisms.
Proof. It is easy to verify that
$\mathcal {K}_{\mathcal {M}}$
satisfies the conditions in [Reference Borceux and Rosický7, Corollary 5.7]. When
$(\mathcal {K}, \mathcal {M})$
is combinatorial, one can use [Reference Makkai, Rosický and Vokřínek20, Corollary 5.2] to see that
$\mathcal {C}_{\mathcal {M}_0}$
is an AEC as well.⊣
In what follows, we will apply our main theorem to the AECs studied in [Reference Beke and Rosický4]. For a fixed (associative and unital) ring R, let
$R\text {-}\operatorname {\mathbf {Mod}}$
denote the category of (left) R-modules with homomorphisms. It is a locally finitely presentable category.
Definition 4.2. Given a class
$\mathcal {B}$
of R-modules, we define its Ext-orthogonality class,
${}^{\perp _{\infty }}\mathcal {B}$
, as follows:
$$ \begin{align*} {}^{\perp_{\infty}}\mathcal {B}=\{A\,:\,\mbox{Ext}^i(A,N)=0\mbox{ for all } 1 \le i < \omega \text{ and all } N\in \mathcal {B}\}. \end{align*} $$
Roughly speaking,
${}^{\perp _{\infty }}\mathcal {B}$
is the collection of R-modules that do not admit nontrivial extensions by modules in
$\mathcal {B}$
. For example, when
$\mathcal {B}$
is the class of all pure injective modules, then
${}^{\perp _{\infty }}\mathcal {B}$
is exactly the class of flat modules (see [Reference Eklof and Trlifaj12, Definition 5.3.22 and Lemma 7.1.4]).
From now on, we assume that
$\mathcal {B}$
is a class of pure injective modules. Let
$\mathcal {K} := R\text {-}\operatorname {\mathbf {Mod}}$
, and let
$\mathcal {C}$
be the full subcategory of
$R\text {-}\operatorname {\mathbf {Mod}}$
with objects from
${}^{\perp _{\infty }}\mathcal {B}$
. Let
$\mathcal {M}$
be the class of monomorphisms (in
$R\text {-}\operatorname {\mathbf {Mod}}$
) whose cokernel is in
${}^{\perp _{\infty }}\mathcal {B}$
. That is, a monomorphism
$A \xrightarrow {f} B$
is in
$\mathcal {M}$
if and only if
$B / f[A]$
is in
${}^{\perp _{\infty }}\mathcal {B}$
. Let
$\mathcal {M}_0$
be the class of elements in
$\mathcal {M}$
with domain and codomain in
$\mathcal {C}$
. Note that this coincides with the notation from Definition 3.3 and Remark 3.4.
The category
$\mathcal {C}_{\mathcal {M}_0}$
is studied from the point of view of model theory by Baldwin–Eklof–Trlifaj [Reference Beke and Rosický4], where they prove it is an AEC. They ask (see [Reference Beke and Rosický4, Question 4.1(1)]) what one can say about tameness and stability in
$\mathcal {C}_{\mathcal {M}_0}$
(see, for example, [Reference Baldwin3] for the relevant definitions). We now show, using our main theorem and known facts, that
$\mathcal {C}_{\mathcal {M}_0}$
has a stable independence notion; hence (by [Reference Lieberman, Rosický and Vasey16, Corollary 8.16]) it will always be stable and tame.
Theorem 4.3.
$(\mathcal {K}, \mathcal {M})$
is a coherent,
$\aleph _0$
-continuous, and retract-closed cellular category. Moreover,
$\mathcal {C}_{\mathcal {M}_0}$
is cofibrantly generated in
$\mathcal {C}$
. In particular,
$\mathcal {C}_{\mathcal {M}_0}$
is an AEC with a stable independence notion.
Proof. The “in particular” part of the statement follows from Theorem 3.5 and Lemma 4.1. For the first sentence, following [Reference Rosický23, Lemma 4.2],
$(\mathcal {K}, \mathcal {M})$
is a retract-closed cellular category. The coherence was observed in [Reference Beke and Rosický4, Lemma 1.14] and
$\aleph _0$
-continuity in [Reference Beke and Rosický4, Remark 1.6]. In fact, the latter follows from Remark 2.5(2) because
$\mathcal {K}_{\mathcal {M}}$
is closed under directed colimits in
$R\text {-}\operatorname {\mathbf {Mod}}$
(as outlined in, for example, [Reference Beke and Rosický4, Section 1]). It remains to see that
$\mathcal {C}_{\mathcal {M}_0}$
is cofibrantly generated in
$\mathcal {C}$
.
By [Reference Barr and Borceux8, Proposition 2] and [Reference Enochs and Jenda11, Theorem 8],
$\mathcal {C}_{\mathcal {M}_0}$
has refinements. This means there exists a regular cardinal
$\theta $
so that any object of
$\mathcal {C}$
can be written as the union of an increasing smooth chain
$\langle A_i : i < \alpha \rangle $
of submodules, with
$A_0$
the zero module and for all
$i < \alpha $
,
$A_{i + 1} / A_i$
in
$\mathcal {C}$
and
$\theta $
-presentable.
By the proof of [Reference Rosický23, Theorem 4.5],
$\mathcal {M}$
is cofibrantly generated by a set of maps f so that
$0 \to A \xrightarrow {f} F \to B \to 0$
is a short exact sequence, F is a free module, and B is a
$\theta $
-presentable object of
$\mathcal {L}$
. Since F is free,
$F \in \mathcal {C}$
as well, and hence
$A \in \mathcal {C}$
. Thus
$f \in \mathcal {M}_0$
. Thus
$\mathcal {M}$
is cofibrantly generated in
$\mathcal {K}$
by a subset of
$\mathcal {M}_0$
, showing in particular that
$\mathcal {M}_0$
is cofibrantly generated in
$\mathcal {C}$
.⊣
In this case, the cellular squares can be given a very concrete description:
Proposition 4.4. A square
is cellular if and only if the pushout
has
$E\in {}^{\perp _{\infty }}\mathcal {B}$
.
Proof. The first square above is cellular if and only if the unique morphism
$t:P\to D$
from the pushout
is in
$\mathcal {M}$
. It suffices to show that
$$ \begin{align*}\overline{v}\operatorname{coker} g=\operatorname{coker} t. \end{align*} $$
We have
$$ \begin{align*}\overline{v}\operatorname{coker} g\cdot t\cdot f'= \overline{v}\operatorname{coker} g\cdot g=0 \end{align*} $$
and
$$ \begin{align*}\overline{v}\operatorname{coker} g\cdot t\cdot u'= \overline{v}\operatorname{coker} g\cdot v =\overline{g}\operatorname{coker} v\cdot v=0. \end{align*} $$
Hence
$\overline {v}\operatorname {coker} g\cdot t=0$
.
Conversely, let
$ht=0$
where
$h:D\to X$
. Then
$hg=htf'=0$
and
$hv=htu'=0$
. Thus there are unique
$h':C_0\to X$
and
$h":B_0\to X$
such that
$$ \begin{align*} h'\operatorname{coker} g=h"\operatorname{coker} v=h. \end{align*} $$
Thus there is a unique
$p:E\to X$
such that
$p\overline {v}=h'$
and
$p\overline g=h"$
. Hence
$$ \begin{align*} p\overline{v}\operatorname{coker} g=h'\operatorname{coker} g=h. \end{align*} $$
Since
$\overline {v}\operatorname {coker} g$
is an epimorphism (
$\overline {v}$
is an epimorphism because pushouts preserve epimorphisms),
$\overline {v}\operatorname {coker} g=\operatorname {coker} t$
.
5 Appendix. Canonicity of stable independence
We prove here canonicity of stable independence without the hypothesis, present in [Reference Lieberman, Rosický and Vasey16, Theorem 9.1], that all morphisms are monomorphisms. This does not depend on the rest of the paper. Our proof is a category-theoretic version of the argument in [Reference Bican, El Bashir and Enochs9] which shows somewhat more transparently what is going on there. The key notion is that of an independent sequence:
Definition 5.1. Let
$\mathcal {K}$
be a category and let
be an independence notion on
$\mathcal {K}$
. Let
$f : M_0 \to M$
be a morphism in
$\mathcal {K}$
. An
-independent sequence for f consists of a nonzero ordinal
$\alpha $
and morphisms
$(f_i)_{i \le \alpha }$
and
$(g_{i, j})_{i \le j \le \alpha }$
such that for
$i \le j \le k \le \alpha $
:
-
•
$f = f_0$
and
$N_0=M$
. -
•
$f_i : M \to N_i$
for
$0<i$
. -
•
$g_{i, j} : N_i \to N_j$
. -
•
$g_{j,k} g_{i, j} = g_{i, k}$
,
$g_{i, i} = \operatorname {id}_{N_i}$
. -
• When
$i < j$
, the following square commutes and, when
$j < \alpha $
, is
-independent:
We call
$\alpha $
the length of the sequence. For a regular cardinal
$\lambda $
, we say the independent sequence is
$\lambda $
-smooth if whenever
$\text {cf} (i) \ge \lambda $
,
$N_i$
is the colimit of the system
$(g_{j, k})_{j \le k < i}$
. We say it is smooth if it is
$\aleph _0$
-smooth.
For example, an independent sequence of length 1 for
$f: M_0 \to M$
consists of
$f_0 = f$
,
$f_1 : M \to N_1$
, and
$g_{0, 1} : M = N_0 \to N_1$
such that
$f_1 f_0 = g_{0, 1} f_0$
. Since there are no independence requirements, it is essentially just the morphism
$f_0$
(the additional data are only relevant when
$\alpha $
is limit; we could have taken
$N_1 = N_0 = M$
and
$f_1 = \operatorname {id}_{M}$
). More interestingly, an independent sequence of length 2 consists essentially (because
$N_0 = M$
and
$g_{0,0} f_0 = f_0$
) of an independent square:
Thus it consists of two “independent copies” of M.
An independent sequence of length 3 will look like:
where the inner diamond
$(M_0, M, M, N_1)$
and the outer diamond
$(M_0, M, N_1, N_2)$
are independent (in fact, if
is monotonic, all commutative subsquares of the diagram will be independent). Essentially, the leftmost “copy” of M is independent of the two rightmost copies (in fact it is independent of
$N_1$
).
Existence allows us to build independent sequences. Recall that a category
$\mathcal {K}$
has chain bounds if any chain has a compatible cocone.
Lemma 5.2. If
$\mathcal {K}$
has
$\lambda $
-directed colimits, chain bounds, and
is a monotonic independence notion with existence, then for any morphism
$f: M_0 \to M$
and any ordinal
$\alpha $
, there exists a
$\lambda $
-smooth independent sequence for f of length
$\alpha $
. More generally, any independent sequence of length
$\alpha _0 < \alpha $
extends to one of length
$\alpha $
(in the natural sense).
Proof. By repeated use of existence.
The following local character lemma will be handy:
Lemma 5.3. Let
$\mathcal {K}$
be a category, and
an independence relation such that
$\mathcal {K}_{\downarrow }$
is a
$\lambda $
-accessible category. Let
$(M_i \to N_i)_{i < \lambda ^+}$
be a system of
$\lambda ^+$
-presentable objects in
$\mathcal {K}^2$
with colimit
$M \to N$
. Then there exists
$i < \lambda ^+$
such that the square
is independent.
Proof. Write I for
$\lambda ^+$
with the usual ordering. By taking colimits at ordinals of cofinality
$\lambda $
and adding them to the system, we can assume without loss of generality that the system is
$\lambda $
-smooth: for any
$i \in I$
of cofinality
$\lambda $
,
$M_i$
is the colimit of
$(M_{i_0})_{i_0 < i}$
.
Let
$(M_j' \to N_j')_{j \in J}$
be a
$\lambda ^+$
-directed system of
$\lambda ^+$
-presentable objects whose colimit in
$\mathcal {K}_{\downarrow }$
is
$M \to N$
; we know that
$\mathcal {K}_{\downarrow }$
is
$\lambda ^+$
-accessible. We build
$(i_{\alpha }, j_{\alpha })_{\alpha < \lambda }$
such that for all
$\alpha < \lambda $
:
-
(1)
$i_{\alpha } \in I$
,
$j_{\alpha } \in J$
. -
(2)
$i_{\alpha } < i_{\alpha + 1}$
. -
(3) The map from
$M_{i_{\alpha }} \to N_{i_{\alpha }}$
to
$M \to N$
factors through
$M_{j_{\alpha }}' \to N_{j_{\alpha }}'$
. -
(4) The map from
$M_{j_{\alpha }}' \to N_{j_{\alpha }}'$
to
$M \to N$
factors through
$M_{i_{\alpha + 1}} \to N_{i_{\alpha + 1}}$
.
This is possible since I and J are
$\lambda ^+$
-directed and
$M_i \to N_i$
and
$M_j' \to N_j'$
are
$\lambda ^+$
-presentable. Now, let
$i := \sup _{\alpha < \lambda } i_{\alpha }$
. The colimit in
$\mathcal {K}^2$
of
$(M_{i_{\alpha }} \to N_{i_{\alpha }})_{\alpha < \lambda }$
and
$(M_{j_{\alpha }}' \to N_{j_{\alpha }}')_{\alpha < \lambda }$
coincide and by
$\lambda $
-smoothness must be
$M_i \to N_i$
. By assumption, for all
$\alpha < \lambda $
, the square
is independent. Since
$\mathcal {K}_{\downarrow }$
has
$\lambda $
-directed colimits, this means that the square
is also independent.
A much simpler result than the canonicity theorem is:
Lemma 5.4. Assume
$\mathcal {K}$
is a category, and
,
are independence notions such that
,
has existence, and
has uniqueness. Then
.
Proof. Given a square
$M_0, M_1, M_2, M_3$
that is
-independent, use existence for
to
-amalgamate the span
$M_0 \to M_1$
,
$M_0 \to M_2$
, giving maps
$M_1 \to M_3'$
,
$M_2 \to M_3'$
. Now by uniqueness for
, the amalgam involving
$M_3$
and the one involving
$M_3'$
must be equivalent, and hence
$M_0, M_1, M_2, M_3$
are also
-independent.⊣
We can now prove the canonicity theorem. The idea is to use a generalization of the fact that, in a vector space, if I is linearly independent and a is a vector, there exists a finite subset
$I_0 \subseteq I$
such that
$(I - I_0) \cup \{a\}$
is independent. Thus we can remove a small subset of I and get something independent.
Lemma 5.5. Assume
$\mathcal {K}$
has chain bounds, and
,
are independence notions with existence such that:
-
(1)
is right monotonic. -
(2)
is transitive, left monotonic, and right accessible.
Then any span has an amalgam that is both
-independent and
-independent. In particular, if
has uniqueness then
.
Proof. Consider a span
$M_0 \xrightarrow [f_0]{} M$
,
$M_0 \xrightarrow [f_0']{} M'$
. Fix a regular cardinal
$\lambda $
such that
$\mathcal {K}_{\downarrow ^2}$
(the arrow category induced by
) is
$\lambda $
-accessible and
$M_0, M, M', f_0, f_0'$
are
$\lambda $
-presentable in all relevant categories.
Using Lemma 5.2, build an
-independent sequence for
$f_0$
,
$(f_i : M \to N_i)_{i \le \lambda ^+}$
,
$(g_{i, j}: N_i \to N_j)_{i \le j \le \lambda ^+}$
, where
$N_{\lambda ^+}$
is the colimit of
$(N_i)_{i < \lambda ^+}$
. Observe that
$$ \begin{align*}f_{\lambda^+}f_0=g_{0,\lambda^+}f_0. \end{align*} $$
Along the way, we ensure that
$N_i$
is
$\lambda ^+$
-presentable for
$i < \lambda ^+$
. Now
-amalgamate the span
$M_0 \to N_{\lambda ^+}$
,
$M_0 \to M'$
, giving an
-independent square:
with
$N_{\lambda ^+}'$
a
$\lambda ^{++}$
-presentable object. Reworking the proof of [Reference Rosický22, Lemma 1]—which requires directed colimits—to use the chain bounds available to us here, we can write
$N_{\lambda ^+}'$
as a colimit of
$\lambda ^+$
-presentables
$(g_{i,j}':N_i'\to N_{j}')_{i\leq j < \lambda ^+}$
, where:
-
(1) There is an arrow
$h_i:N_i \to N_i'$
for each
$i < \lambda ^+$
. -
(2) The
$N_i'$
lie above
$M'$
, in the sense that
$h':M'\to N_{\lambda ^+}'$
factors as and, moreover, that the morphisms
$$ \begin{align*} M'\stackrel{u_i}{\longrightarrow}N_i'\stackrel{g_{i,\lambda^+}'}{\longrightarrow}N_{\lambda^+}' \end{align*} $$
$h'f_0'=hg_{0,\lambda ^+}f_0:M_0\to N_{\lambda ^+}'$
factor identically through
$g_{i,\lambda ^+}'$
, i.e., Here we use
$$ \begin{align*} h_if_if_0=u_if_0'. \end{align*} $$
$\lambda $
-presentability of
$M_0$
,
$M'$
, and
$\lambda ^+$
-directedness of the chain.
Then h is a colimit of the
$h_i$
in
$\mathcal {K}^2$
and by Lemma 5.3, there exists
$i < \lambda ^+$
such that the square
is
-independent. By definition of an
-independent sequence, the square
is
-independent. By left transitivity, we obtain that the following is
-independent.
A chase through the diagrams above reveals that
$$ \begin{align*}g_{i,\lambda^+}'h_if_if_0=h'f_0'=hg_{0,\lambda^+}f_0=hf_{\lambda^+}f_0, \end{align*} $$
meaning that the outer square and the large upper triangle in the following diagram commute:
Thus the square
is
-independent.
By left monotonicity for
, then, the following is also
-independent:
Note, however, that the morphism from M to
$N_{\lambda ^+}'$
in the diagram above is not the same as the one in the
-amalgam of
$M_0 \to N_{\lambda ^+}$
,
$M_0 \to M'$
. In fact, we have a diagram of the form:
where the upper rectangle is
-independent and the outer “square”
$(f_0', f_0, h f_{\lambda ^+}, h')$
is
-independent. By right monotonicity for
, we get that
$(f_0', f_0, h f_{\lambda ^+}, h')$
is also
-independent. Thus it is the desired amalgam of
$f_0', f_0$
.⊣
Theorem 5.6 (The canonicity theorem).
Assume
$\mathcal {K}$
has chain bounds, and
,
are independence notions with existence and uniqueness such that:
-
(1)
is right monotonic. -
(2)
is transitive and right accessible.
Then
. In particular,
$\mathcal {K}$
has at most one stable independence notion.
Proof. Combine Lemmas 5.4 and 5.5. Note that right monotonicity for
follows from existence, uniqueness, and transitivity [Reference Lieberman, Rosický and Vasey16, Lemma 3.20].⊣
Corollary 5.7. Assume
$\mathcal {K}$
has chain bounds. If
is a transitive and right accessible independence notion with existence and uniqueness, then
is a stable independence notion. In particular, it is symmetric.
Proof. It suffices to see that
. For this, apply Theorem 5.6 with
and
(again,
is right monotonic by [Reference Lieberman, Rosický and Vasey16, Lemma 3.20]).⊣
Remark 5.8. Instead of chain bounds, it suffices to be able to build the appropriate independent sequences. See [Reference Lieberman, Rosický and Vasey16, Remark 9.6].
Acknowledgments
We thank Jan Trlifaj for helpful conversations about roots of Ext and Simon Henry for sharing [Reference Henry13] with us. We are also indebted to John Baldwin, Marcos Mazari-Armida, Misha Gavrilovich, and the anonymous referee for useful feedback. The second author is supported by the Grant agency of the Czech Republic under the grant 19-00902S.
