Canonical forking in AECs
AMS 2010 Subject Classification: Primary: 03C48, 03C45 and 03C52. Secondary: 03C55, 03C75, 03C85 and 03E55.
Abstract.
Boney and Grossberg [BG] proved that every nice AEC has an independence relation. We prove that this relation is unique: in any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally the properties of independence relations for AECs and also prove a canonicity result for Shelah’s good frames. The usual tools of firstorder logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.
Key words and phrases:
Abstract Elementary Classes; forking; Classification Theory; stability; good framesContents
1. Introduction
Let be an abstract elementary class (AEC) which satisfies amalgamation, joint embedding, and which does not have maximal models. These assumptions allow us to work inside its monster model . The main results of this paper are:
To understand the relevance of the results, some history is necessary.
In 1970, Shelah discovered the notion “ forks over ” (for ), a generalization of Morley’s rank in stable theories. Its basic properties were published in [She78].
In 1974, Lascar [Las76, Theorem 4.9] established that for superstable theories, any relation between , , satisfying the basic properties of forking is Shelah’s forking relation. In 1984, Harnik and Harrington [HH84, Theorem 5.8] extended Lascar’s abstract characterization to stable theories. Their main device was the finite equivalence relation theorem. In 1997, Kim and Pillay [KP97, Theorem 4.2] published an extension to simple theories, using the independence theorem (also known as the typeamalgamation theorem).
This paper deals with the characterization of independence relations in various nonelementary classes. An early attempt on this problem can be found in Kolesnikov’s [Kol05], which focuses on some important particular cases (e.g. homogeneous model theory and classes of atomic models). We work in a more general context, and only rely on the abstract properties of independence. We cannot assume that types are sets of formulas, so work only with Galois (i.e. orbital) types.
In [She87, Chapter II] (which later appeared as [She09b, Chapter V.B]), Shelah gave the first axiomatic definition of independence in AECs, and showed that it generalized firstorder forking. In [She09a, Chapter II], Shelah gave a similar definition, localized to models of a particular size (the socalled “good frame”). Shelah proved that a good frame existed, under very strong assumptions (typically, the class is required to be categorical in two consecutive cardinals).
Recently, working with a different set of assumptions (the existence of a monster model and tameness), Boney and Grossberg [BG] gave conditions (namely a form of Galois stability and the extension property for coheir) under which an AEC has a global independence relation. This showed that one could study independence in a broad family of AECs. Our paper is strongly motivated by both [She09a, Chapter II] and [BG].
The paper is structured as follows. In Section 2, we fix our notation, and review some of the basic concepts in the theory of AECs. In Section 3, we introduce independence relations, the main object of study of this paper, as well as some important properties they could satisfy, such as extension and uniqueness. We consider two examples: coheir and nonsplitting.
In Section 4, we prove a weaker version of (1) (Corollary 4.14) that has some extra assumptions. This is the core of the paper.
In Section 5, we go back to the properties listed in Section 3 and investigate relations between them. We show that some of the hypotheses in Corollary 4.14 are redundant. For example, we show that the symmetry and transitivity properties follow from existence, extension, uniqueness, and local character. We conclude by proving (1). Finally, in Section 6, we apply our methods to the coheir relation considered in [BG] and to Shelah’s good frames, proving (2) and (3).
While we work in a more general framework, the basic results of Sections 23 often have proofs that are very similar to their firstorder analogs. Readers feeling confident in their knowledge of firstorder nonforking can start reading directly from Section 4 and refer back to Sections 23 as needed.
This paper was written while the first and fourth authors were working on a Ph.D. under the direction of Rami Grossberg at Carnegie Mellon University. They would like to thank Professor Grossberg for his guidance and assistance in their research in general and in this work specifically.
An early version of this paper was circulated already in early 2014. Since that time, Theorem 5.14 has been used by the fourth author to build a good frame from amalgamation, tameness, and categoricity in a suitable cardinal [Vasa]. With VanDieren, the fourth author has also used it to deduce a certain symmetry property for nonsplitting in classes with amalgamation categorical in a highenough cardinal [VV], with consequences on the uniqueness of limit models. The question of canonicity of forking in more local setups (e.g. when the independence relation is only defined for certain types over models of a certain size) is pursued further in [Vasb]. The latter preprint addresses Questions 5.5, 6.14, 7.1, and 7.2 posed in this paper.
2. Notation and prerequisites
We assume the reader is familiar with abstract elementary classes and the basic related concepts. We briefly review what we need in this paper, and set up some notation.
Hypothesis 2.1.
We work in a fixed abstract elementary class which satisfies amalgamation and joint embedding, and has no maximal models.
2.1. The monster model
Definition 2.2.
Let be a cardinal. For models , we say is a universal extension of if for any , with , can be embedded inside over , i.e. there exists a embedding fixing pointwise. We say is a universal extension of if it is a universal extension of .
Definition 2.3.
Let be a cardinal. We say a model is model homogeneous if for any , is a universal extension of . We say is saturated if it is model homogeneous (this is equivalent to the classical definition by [She01, Lemma 0.26]).
Definition 2.4 (Monster model).
Since has amalgamation and joint embedding properties and has no maximal models, we can build a strictly increasing continuous chain , where for all , is universal over . We call the union the monster model^{1}^{1}1Since is a proper class, it is strictly speaking not an element of . We ignore this detail, since we could always replace OR in the definition of by a cardinal much bigger than the size of the models under discussion. of .
Any model of can be embedded inside the monster model, so we will adopt the convention that any set or model we consider is a subset or a substructure of .
We write for the set of automorphisms of fixing pointwise. When , we omit it.
We will use the following without comments.
Remark 2.5.
Let , be models. By our convention, and , thus by the coherence axiom, implies .
Definition 2.6.
Let be an index set. Let , be sequences of sets, and let be a set. We write to mean that , and for all , . We write to mean that for some . When is empty, we omit it.
We will most often use this notation when has a single element, or when all the sets are singletons. In the later case, we identify a set with the corresponding singleton, i.e. if and are sequences, we write instead of , with , . We write for the equivalence class of . This corresponds to the usual notion of Galois types first defined in [She01, Definition 0.17].
Note that for sets , we have precisely when there are enumerations , of and respectively such that .
2.2. Tameness and stability
Although we will make no serious use of it in this paper, we briefly review the notion of tameness. While it appears implicitly in [She99], tameness was first introduced in [GV06b] and used in [GV06a] to prove an upward categoricity transfer. Our definition follows [Bon14b, Definition 3.1].
Definition 2.7 (Tameness).
Let . Let be a cardinal. We say is tame for length types if for any tuples of length , and any , if , there exists of size such that .
We say is tame for length types if for any tuples of length , and any , if , there exists of size such that .
We say is tame if it is tame for length types. We say is fully tame if it is tame for all lengths. Similarly for tame.
The following dual of tameness is introduced in [Bon14b, Definition 3.3]:
Definition 2.8 (Type shortness).
Let . Let be a cardinal. We say is type short over sized models if for any index set , any enumerations , of type , and any , if , there is of size such that . Here .
We define type short over sized models similarly.
We say is fully type short if it is type short over sized models for all . Similarly for type short.
We also recall that we can define a notion of stability:
Definition 2.9 (Stability).
Let and be cardinals. We say is stable in if for any , has cardinality . Equivalently, given any collection , where for all , , there exists such that .
We say is stable in if it is stable in .
We say is stable if it is stable in for some . We say is stable if it is stable in for some . We write “unstable” instead of “not stable”.
Remark 2.10.
If , and is stable in , then is stable in .
The following follows from [Bon, Theorem 3.1].
Fact 2.11.
Let . Let be a cardinal. Assume is stable in and . Then is stable in .
3. Independence relations
In this section, we define independence relations, the main object of study of this paper. We then consider two examples: coheir and nonsplitting.
3.1. Basic definitions
Definition 3.1 (Independence relation).
An independence relation is a set of triples of the form where is a set, are models (i.e. ), . Write for . When , we may write for . We require that satisfies the following properties:

(I) Invariance: Assume . Then if and only if .

(M) Left and right monotonicity: If , , , then .

(B) Base monotonicity: If , and , then .
We write for restricted to the base set , and similarly for e.g. .
In what follows, always denotes an independence relation.
Remark 3.2.
To avoid relying on a monster model, we could introduce an ambient model as a fourth parameter in the above definition (i.e. we would write ). This would match the approach in [She09b, Chapter V.B] and [She09a, Chapter II] where the existence of a monster model is not assumed. We would require that contains the other parameters , and . To avoid cluttering the notation, we will not adopt this approach, but generalizing most of our results to this context should cause no major difficulty. Some simple cases will be treated in the discussion of good frames in Section 6. In [Vasb], many of the results of this paper are stated in a “monsterless” framework.
We will consider the following properties of independence^{2}^{2}2Continuity, transitivity, uniqueness, existence and extension are adapted from [MS90]. Symmetry comes from [She09a, Chapter II].:

Continuity: If , then there exists , of size strictly less than such that for all containing , .

Left transitivity: If , and , with , then .

Right transitivity: If , and , with , then .

Symmetry: If , then there is with such that . If is a model extending , one can take ^{3}^{3}3This second part actually follows from monotonicity and the first part..

Uniqueness: If , , and , then for some so that .

The following properties hold:

Existence: for all sets and models , .

Extension: Given a set , and , if , then there is such that .


Local character: for all , where for all , all increasing, continuous chains and all sets of size , there is some so .

Strong extension: A technical property used in the proof of canonicity. See Definition 4.4.
For a property that is not local character, and a model, when we say has , we mean has (i.e. has when the base is restricted to be ). If is either or , means we assume in the definition.
Whenever we are considering two independence relations and , we write as a shorthand for “ has ”, and similarly for .
Notice the following important consequence of :
Remark 3.3.
Assume has . Then for any , and , there is such that (use to see , and then use ).
Assuming , this last statement is actually equivalent to .
The property will be introduced and motivated later in the paper. For now, we note that there is an asymmetry in our definition of an independence relation: the parameter on the left is allowed to be an arbitrary set, while the parameter on the right must be a model extending the base. This is because we have in mind the analogy “ if and only if does not fork over ”, and in AECs, types over models are much better behaved than types over sets.
The price to pay is that the statement of symmetry is not easy to work with. Assume for example we know an independence relation satisfies and . Should it satisfy ? Surprisingly, this is not easy to show. We prove it in Lemma 5.9, assuming . For now, we prepare the ground by showing how to extend an independence relation to take arbitrary sets on the right hand side.
Definition 3.4 (Closure of an independence relation).
We call a closure of if is a relation defined on all triples of the form , where is a model (but maybe ). We require it satisfies the following properties:

For all , and all , if and only if .

Invariance: If , then if and only if .

Left and right monotonicity: If and , , then .

Base monotonicity: If , and , then .
The minimal closure of is the relation defined by if and only if there exists , with , so that .
It is straightforward to check that the minimal closure of is the smallest closure of but there might be others (and they also sometimes turn out to be useful), see the coheir and explicit nonsplitting examples below.
We can adapt the list of properties to a closure .
Definition 3.5.

We say has if for all sets , if and only if .

We say that has if whenever , there exists , of size strictly less than such that .

We say that has if whenever , and , there exists such that .

We say that has if whenever , , and , there is with .

We say that has if whenever , , and , we have .

The statements of , , are unchanged. We will not need to use on a closure.
For an arbitrary closure, we cannot say much about the relationship between the properties satisfied by and those satisfied by . The situation is different for the minimal closure, but we defer our analysis to section 5.
Remark 3.6.
Shelah’s notion of a good frame introduced in [She09a, Chapter II] is another axiomatic approach to independence in AECs. There are several key differences with our framework. In particular, good frames only operate on sized models and singleton sets. On the other hand, the theory of good frames is very developed; see e.g. [She09a, JS12, JS13].
An earlier framework which is closer to our own is the “Existential framework” (see [She09b, Definition V.B.1.9]). The key differences are that only defines when , , (essentially) assumes , while we seldom need continuity, and local character (a property crucial to our canonicity proof) is absent from the axioms of .
3.2. Examples
Though so far developed abstractly, this framework includes many previously studied independence relations.
Definition 3.7 (Coheir, [Bg]).
Fix a cardinal . We call a set small if it is of size less than . For , define
for every small and ,  
there is such that . 
One can readily check that satisfies the properties of an independence relation. was first studied in [BG], based on results of [MS90] and [Bon14b], and generalizes the firstorder notion of coheir. An alternative name for this notion is satisfiability. Sufficient conditions for this relation to be wellbehaved (i.e. to have most of the properties listed above) are given in [BG, Theorem 5.1], reproduced here as Fact 3.16.
Definition 3.8.
We define a natural closure for :
for every small and ,  
there is such that . 
It is straightforward to check that is indeed a closure of , but it is not clear at all that this is the minimal one. This closure will be useful in the proof of local character (Theorem 6.4) Note that differs from the notion of coheir given in [MS90]; there, types are consistent sets of formulas from a fragment of for strongly compact and the notion there (see [MS90, Definition 4.5]) allows parameters from and .
Definition 3.9 (nonsplitting, [She99]).
Let . For , we say if and only if for for all with , , if , then there is such that .
There is also a definition of nonsplitting that does not depend on a cardinal .
Definition 3.10 (Nonsplitting).
For ,
An equivalent definition of nonsplitting is given by the following.
Proposition 3.11.
if and only if for all with , , if , then for some with (equivalently, for all enumerations of ).
The analog statement also holds for nonsplitting.
Proof.
Assume , and is such that . Let . Then , and fixes . In other words, is as needed. Conversely, assume . Find such that . Then is the desired witness that . ∎
Using Proposition 3.11 to check base monotonicity, it is easy to see that both and are independence relations. These notions of splitting in AECs were first explored in [She99], but have seen a wide array of uses; see [SV99, Van06, Van13, GVV, Vasa] for examples. nonsplitting is more common in the literature, but we focus on nonsplitting here. Using tameness, there is a correspondence between the two:
Proposition 3.12.
Let and . If is tame for length types and , then
Proof.
We use the equivalence given by Proposition 3.11. Let , and suppose . Then there are so for and , but for some enumeration of . By tameness, there is so that . Without loss of generality, . Let . Then and witness that . ∎
A variant is explicit nonsplitting, which allows the ’s to be sets instead of requiring models; this is based on explicit nonstrong splitting from [She99, Definition 4.11.2].
Definition 3.13 (Explicit Nonsplitting).
For , we say if and only if for for all , if , then there is such that .
From the definition, we see immediately that . Of course, the corresponding version of Proposition 3.11 also holds for , so it is again straightforward to check that is an independence relation. One advantage of using over is that it has a natural closure:
Definition 3.14.
We say if and only if for for all , if , then there is such that .
Again, it is not clear this is the minimal closure. We will have no use for this closure, so for most of the paper we will stick with regular nonsplitting.
Nonsplitting will be used mostly as a technical tool to state and prove intermediate lemmas, while coheir will be relevant only in Section 6.
3.3. Properties of coheir and nonsplitting
We now investigate the properties satisfied by coheir and nonsplitting. Here is what holds in general:
Proposition 3.15.
Let .

and have , and .

If is saturated, and have .

, , and have .
Proof.
Just check the definitions. ∎
While extension and uniqueness are usually considered very strong assumptions, it is worth noting that nonsplitting satisfies a weak version of them, see [Van06, Theorems I.4.10, I.4.12]. It is also well known that nonsplitting has local character assuming tameness and stability (see e.g. [GV06b, Fact 4.6]). This will not be used.
Regarding coheir, the following^{4}^{4}4Since this paper was first submitted, a stronger result has been proven (for example one need not assume ). See [Vasc, Theorem 5.15]. appears in [BG] :
Fact 3.16.
Let be regular. Assume is fully tame, fully type short, has no weak order property^{5}^{5}5See [BG, Definition 4.2]. and has ^{6}^{6}6All the properties mentioned in this Lemma are valid for models of size only..
Then has and .
Moreover, if is strongly compact, then the tameness and typeshortness hypotheses hold for free, has , and “no weak order property” is implied by “ so .”
4. Comparing two independence relations
In this section, we prove the main result of this paper (canonicity of forking), modulo some extra hypotheses that will be eliminated in Section 5. After discussing some preliminary lemmas, we introduce a strengthening of the extension property, , which plays a crucial role in the proof. We then prove canonicity using (Corollary 4.8). Finally, we show follows from some of the more classical properties that we had previously introduced (Corollary 4.13), obtaining the main result of this section (Corollary 4.14). We conclude by giving some examples showing our hypotheses are close to optimal.
For the rest of this section, we fix two independence relations and . Recall from Definition 3.1 that this means they satisfy , and . We aim to show that if and satisfy enough of the properties introduced in Section 3, then .
The first easy observation is that given some uniqueness, only one direction is necessary^{7}^{7}7Shelah states as an exercise a variation of this lemma in [She09a, Exercise II.6.6.(1)].:
Lemma 4.1.
Let be a model. Assume:
Then .
Proof.
Assume . By , find so that . By hypothesis (1), . By , . By , . ∎
With a similar idea, one can relate an arbitrary independence relation to nonsplitting^{8}^{8}8Shelah gives a variation of this lemma in [She09a, Claim III.2.20.(1)].:
Lemma 4.2.
Assume . Then .
Proof.
Assume . Let and . By monotonicity, for . By invariance, . By , there is with . By Proposition 3.11, . ∎
A similar result holds for , see Lemma 5.6.
The following consequence of invariance will be used repeatedly:
Lemma 4.3.
Assume satisfies . Assume , and . Then there is such that .
Proof.
By , there is , . Thus , so letting and applying invariance, we obtain . ∎
Even though we will not use it, we note that an analogous result holds for left extension, see Lemma 5.8.
We now would like to strengthen Lemma 4.3 as follows: suppose we are given , , and assume is “very big” (e.g. it is saturated), but does not contain . Can we find with , and ?
We give this property a name:
Definition 4.4 (Strong extension).
An independence relation has (strong extension) if for any and any set , there is such that for all , there is with and .
Intuitively, says that no matter which isomorphic copy of we pick, even if does not contain , is so big that we can still find inside with the right property. This is stronger than in the following sense:
Proposition 4.5.
If has , has . If in addition has , then has . Thus if has and , it has .
Proof.
Use monotonicity and Remark 3.3. ∎
Remark 4.6.
Example 4.15 shows does not follow from .
Strong extension allows us to prove canonicity:
Lemma 4.7.
Assume . Assume also that .
Then .
Proof.
Assume . We show . Fix as described by . By Lemma 4.3, we can find such that . By definition of , one can pick with and .
We have , , and , so by definition of nonsplitting, . By invariance, , as needed. ∎
Corollary 4.8 (Canonicity of forking from strong extension).
Assume:

.

.
Then .
We now proceed to show that follows from , , and . We will use the following important concept:
Definition 4.9 (Independent sequence).
Let be a linearly ordered set. A sequence of sets is independent over a model if there is a strictly increasing continuous chain of models such that for all :

and .

.
This generalizes the notion of independent sequence from the firstorder case. The most natural definition would only require (for some closure