Forking independence from the categorical point of view
Michael Lieberman Jiří Rosický Sebastien Vasey
math.LO math.CT 03C45 (Primary), 18C35, 03C48, 03C52, 03C55, 03C75, 03E55
  (Secondary)



  
  
  Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in μ-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible—and useful—to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a category and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory.

  

  


ψ class intersections on Hassett Spaces for genus 0 with all weights 1/2
    Nand Sharma
    December 30, 2023
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§ INTRODUCTION




 §.§ Background


Forking is a model-theoretic notion generalizing linear independence in vector spaces and algebraic independence in fields. A central notion of modern model theory, it was developed by Saharon Shelah <cit.> for classes of models axiomatized by a stable first-order theory[While there are generalizations of the theory of stable forking to certain unstable first-order classes, most notably those axiomatized by NIP and simple theories, the present paper focuses exclusively on the stable case.]. Recall that a first-order theory T has the order property if there exists a model M of T, a formula ϕ (; ) and a sequence _i : i < ω of finite tuples in M such that for i, j < ω, M ϕ[_i; _j] if and only if i < j. A theory T is stable precisely when it does not have the order property. For example, both the theory of vector spaces over ℚ and the theory of algebraically closed fields of characteristic zero are stable, but the theory of linear orders and that of graphs are not (a formula witnessing the order property inside the random graph, for example, is ϕ (x_1 x_2, y_1 y_2) := x_1 E y_2  (x_2 E y_1)).

Roughly, Shelah defines forking so that a type p over a set C (i.e. a set of formulas with parameters from C) does not fork over a subset A of C if it is a “generic” extension of p  A. In particular, p is essentially determined by p  A. Thus nonforking[It is somewhat unfortunate that the negation of forking, “nonforking” is the positive notion. Nevertheless, this terminology is now well established.] can be seen as a notion of free extension. There is another way to see nonforking: in his survey on stability theory, Makkai <cit.> introduces the anchor symbol  and defines ABC (when working inside a monster model ) to mean that for any finite tuples  of elements from B, ( / A C) does not fork over A. This notation ABC, which can be read as “B is independent from C over A,” simplifies the statement of some of the properties of forking. For example, symmetry can be written as “ABC if and only if ACB”.

Forking is especially well-behaved when the base set A above is a model of the theory[To study forking over sets and not just models, one can consider a certain category of sufficiently algebraically closed sets (Example <ref>(<ref>)) in which the nonforking amalgams determine the behavior of forking over any set.]. In fact, to understand it category-theoretically, it is useful to consider the case in which all the sets under consideration are models, specifically M_0M_1M_2M_3, with M_0 ≼ M_ℓ≼ M_3, ℓ = 1,2. Here, we write M_3 for an ambient model inside which all the types are computed. One can also view such a quadruple as a commutative diagram of embeddings, also known as an amalgam.  In this sense, we may identify a nonforking notion over models with a particular choice of such amalgams. Note that consideration of nonforking amalgams plays an important role in model theory, as it leads to definitions such as that of an independent system of models (see <cit.> or <cit.>). This is for example a key concept in both the statement and proof of Shelah's main gap theorem <cit.>, a celebrated achievement of first-order classification theory.




 §.§ Main questions and earlier work


The present paper seeks to answer the following questions:



  * What are the basic category-theoretic properties of nonforking amalgams in the category of models of a stable first-order theory? More precisely, we would like a list of properties that are:

  
  
  * Invariant under equivalence of categories, e.g. they should not depend on what underlying concrete functor we use to represent the category.
  
  * Canonical: in any reasonable category, there should be at most one notion satisfying the properties of nonforking amalgamation.
  

  * In what other categories is there a notion satisfying these properties?   


There has already been a substantial amount of work on more purely model-theoretic versions of these questions. Consider, in particular, Harnik and Harrington <cit.>, which characterizes forking in a stable first-order theory by a list of four axioms on a relation of inclusion between types. These axioms are, however, not suitably category-theoretic, as they depend on seeing types as sets of formulas. Recently, Boney, Grossberg, Kolesnikov, and the third author <cit.> characterized stable forking in the general framework of abstract elementary classes (AECs), encompassing first-order theories but also classes axiomatized by infinitary logics such as _∞, ω. The characterization of <cit.> is phrased in terms of an anchor relation , but still uses the underlying set representations of the objects in the category.

The second question was considered early on by Shelah, with partial results in homogeneous model theory <cit.>, _ω_1, ω <cit.>, universal classes <cit.>, and AECs (e.g. in his two-volume book <cit.>). In fact, there is a growing body of literature on forking-like notions in AECs. We refer the reader to the survey of Boney and the third author <cit.>, but let us mention in particular the work of Boney and Grossberg <cit.>, which generalizes work of Makkai and Shelah <cit.> from _κ, ω (κ a strongly compact cardinal) to AECs, and builds a global forking-like independence notion on a subclass of sufficiently saturated models of the AEC. Interestingly, the subclass is not itself an AEC, but is still closed under κ-directed unions (as opposed to arbitrary directed unions). This was one motivation for developing κ-AECs <cit.>. This is a very broad framework for model theory. In fact, it has a category-theoretic equivalent: per Fact <ref>, a κ-AEC is exactly (up to equivalence of category) an accessible category whose morphisms are monomorphisms (see Makkai-Paré <cit.> or Adámek-Rosický <cit.> on accessible categories and their broader relationship to model theory).



 §.§ Main results


It therefore seems natural to investigate forking in arbitrary accessible categories (perhaps with all morphisms monomorphisms). The present paper makes the following contributions:



  * We define when a category has what we call a stable independence notion (Definition <ref>). Roughly, this is a class of distinguished squares that satisfies, in particular, an existence property, a uniqueness property (a weakening of the definition of a pushout), as well as a transitivity property (corresponding to being closed under composition in a double categorical sense). The transitivity property makes the class of independent squares into a category, and we require that this category be accessible.

  * We show that this is the desired purely category-theoretic axiomatization of forking: in a μ-AEC , being stable (and, specifically, the accessibility of the category of independent squares) corresponds to having certain local character properties well known to model theorists (Theorem <ref>). Moreover this axiomatization is canonical, assuming that the class has chain bounds (that is, any increasing chain of models has an upper bound, see Definition <ref>).  This result, Theorem <ref>, generalizes <cit.> to μ-AECs.

  * Working purely abstractly, we exhibit a connection between forking and effective unions (an exactness property introduced by Barr <cit.>). Specifically, we show that, if we start with a locally presentable and coregular category  that has effective unions then _reg, the subcategory of  containing just the regular monomorphisms of , has a stable independence notion, see Theorem <ref>. In particular, this covers both Grothendieck toposes and Grothendieck abelian categories. That forking occurs in these contexts seems not to have been recognized before (although it has long been known that forking occurs in classes axiomatized by first-order theories of modules, see <cit.>).

  * Assuming a large cardinal axiom, we characterize precisely when a stable independence notion exists in any μ-AEC with chain  bounds (and hence in any accessible category with chain bounds whose morphisms are monomorphisms). This is Corollary <ref>: such a μ-AEC has a cofinal subclass with a stable independence notion if and only if it does not have a certain order property. This implies that the usual “syntactic” definition of stability (note that the usual definition in terms of counting types is too weak in this context, see Example <ref>) is equivalent to a purely category-theoretic statement. Thus model-theoretic stability is invariant under equivalence of category[For the category of models of a first-order theory, this can also be seen using Shelah's saturation spectrum theorem <cit.>, see also <cit.>. However no such saturation spectrum theorem is known in arbitrary μ-AECs.]. As a philosophical remark, Shelah <cit.> argues that classification-theoretic dividing lines should have both an “internal” and an “external” characterization. If we interpret “external” as “invariant under equivalence of category” and “internal” as “a property satisfied by a fixed model in the class”, then we see here this principle in action and obtain evidence that there is a “stability-like” dividing line in the general framework of accessible categories.




 §.§ Notes


In a sense, this paper falls naturally into two parts: after a brief review of some of the basic concepts that will be used (Section <ref>), we devote Sections <ref> through <ref> to the development of an analogue of stable (or nonforking) independence suited to a general category and, working purely abstractly, give conditions on a category under which such a relation is (a) guaranteed to exist, and (b) to take a particular recognizable form, e.g. the independent squares are precisely the pullback squares. This first half is intended to be congenial to a broad mathematical audience, and may be of particular use to those who have previously been reluctant to wade into the details of model-theoretic nonforking.

In the second half, we shift our attention to μ-AECs, which, being concrete, come with a great deal more machinery, and thus allow the development of a richer theory—the results and proofs here are more recognizably model-theoretic, and significantly more technical.  We note, however, that μ-AECs are accessible categories with all morphisms monomorphisms, and vice versa—so, in fact, we are working in a vastly more general context than the coregular locally presentable categories of, say, Section <ref>. Section <ref> gives some tools to move from an arbitrary category to a μ-AEC by restricting its class of morphisms: this provides a bridge between the paper's two halves. In Section <ref>, we give some basic model-theoretic tools for use in μ-AECs. In section <ref>, we consider what the properties of stable independence look like in a μ-AEC and show that they are equivalent to the more model-theoretic local character properties of forking in a stable first-order theory. In Section <ref>, we prove that stable independence notions are canonical, symmetric, and imply failure of a certain order property. In Section <ref>, we reverse this and show (assuming a large cardinal axiom) that failure of an order property implies there is a stable independence notion on a subclass of saturated models.

Throughout this paper, we assume the reader is familiar with basic category theory as presented e.g. in <cit.>. More particularly, we will spend much of our time in accessible, locally presentable, or locally multipresentable categories (see <cit.> for further details).  For connections between locally multipresentable categories—and locally polypresentable categories, which also make a brief appearance—and abstract model theory, we point readers to <cit.>.  We also assume some familiarity with μ-AECs and their relationship with accessible categories <cit.>.

We use the following notational conventions: we write K for a class of τ-structures and  (boldface) for a pair (K, ), where  is a partial order. We write  (script) for a category. We will abuse notation and write M ∈ instead of M ∈ K. For a structure M, we write U M for its universe, and |U M| for the cardinality of its universe. We write M ⊆ N to mean that M is a substructure of N. For α an ordinal, we let <αA (respectively, αA) denote the set of sequences of length less than (respectively, equal to) α with elements from the set A. We will abuse this notation slightly, writing <αM in place of <αU M.



 §.§ Acknowledgments


We thank Marcos Mazari Armida for very detailed feedback which greatly improved the paper. We also would like to thank the referee for helpful comments.



§ PRELIMINARIES


Intuitively, an accessible category is a category with all  sufficiently directed colimits and such that every object can be written as a highly directed colimit of “small” objects.  Here “small” is interpreted in terms of presentability, a notion of size that makes sense in an arbitrary (potentially non-concrete) category. It is important to note that this notion, which may well appear unfamiliar, restricts to precisely what it should in familiar cases: in the category of sets, a set is λ-presentable if and only if its cardinality is less than λ; in an AEC , the same is true for all λ>(). 


  Let  be a category and let λ be a regular cardinal.

  
  
  * An object M is λ-presentable if its hom-functor (M,-):→ preserves λ-directed colimits. Put another way, M is λ-presentable if for any morphism f:M→ N with N a λ-directed colimit ⟨ϕ_α:N_α→ N⟩, f factors essentially uniquely through one of the N_α, i.e. f=ϕ_α f_α for some f_α:M→ N_α.
  
  *  is λ-accessible if it has λ-directed colimits and  contains a set S of λ-presentable objects such that every object of  is a λ-directed colimit of objects in S.
  
  *  is accessible if it is λ'-accessible for some regular cardinal λ'.
  


We will often quote results on accessible categories from <cit.>.



Recall from <cit.> that a (μ-ary) abstract class is a pair = (K, ≤) such that K is a class of structures in a fixed μ-ary vocabulary τ = τ (), and ≤ is a partial order on K that refines the τ-substructure relation, with K and ≤ closed under τ-isomorphism. We say that such a  is coherent if M_0 ⊆ M_1  M_2 and M_0  M_2 implies M_0  M_1.

In any abstract class , there is a natural notion of morphism: we say that f: M → N is a -embedding if f is an isomorphism from M onto f[M] and f[M]  N. We can see an abstract class and its -embeddings as a category. In fact (see <cit.>), an abstract class is a replete and iso-full subcategory of the category of τ-structures with injective homomorphisms (insisting, as is customary in model theory, that relation symbols be reflected as well as preserved). Thus we also think of  as a (concrete) category. Saying that  has concrete μ-directed colimits amounts to saying that for any μ-directed system in , the union of the system is its colimit. That is,  satisfies the chain axioms of μ-AECs. In fact, let us now recall the definition of a μ-AEC from <cit.>:


  Let μ be a regular cardinal. An abstract class  is a μ-abstract elementary class (or μ-AEC for short) if it satisfies the following three axioms:

  
  
  * Coherence: for any M_0, M_1, M_2 ∈, if M_0 ⊆ M_1  M_2 and M_0  M_2, then M_0  M_1.
  
  * Chain axioms: if M_i : i ∈ I is a μ-directed system in , then:
    
    
  * M := ⋃_i ∈ I M_i is in .
    
  * M_i  M for all i ∈ I.
    
  * If M_i  N for all i ∈ I, then M  N.
    
  
  * Löwenheim-Skolem-Tarski (LST) axiom: there exists a cardinal λ = λ^<μ≥ |τ ()| + μ such that for any M ∈ and any A ⊆ U M, there exists M_0 ∈ with M_0  M, A ⊆ U M_0, and |U M_0| ≤ |A|^<μ + λ. We write () for the least such λ.
  


Note that when μ = ℵ_0, we recover Shelah's definition of an AEC from <cit.>. The connection between μ-AECs and accessible categories is given by <cit.>:


  If  is a μ-AEC, then it is an ()^+-accessible category with all μ-directed colimits whose morphisms are monomorphisms. Conversely, any μ-accessible category whose morphisms are monomorphisms is equivalent to a μ-AEC.


We end this section by briefly recalling that we can define a notion of Galois (orbital) type in any abstract class. This notion of type was introduced by Shelah for AECs but we use the notation and definitions from <cit.>. Loosely speaking, it is defined as the finest notion of type preserving -embeddings.  More precisely, the Galois type of a sequence  over a structure M∈ computed in an extension N M, denoted ( / M; N), is the equivalence class of the triple (M,,N) under the (transitive closure of) the relation ∼, where (M,,N)∼ (M,',N') if there is an amalgam 


    @=3pc
            N @.>[r]^g   N̅
    
            M  [u]  [r]    
            N' @.>[u]_g'
 
with g()=g'(') (see, e.g., <cit.>).  We denote by ^<∞ (M) the set of Galois types of any length over M; that is,


    ^<∞ (M) := { ( / M; N) | M  N, ∈<∞N}.


We will also use variations such as ^α (M) (the length is restricted to be α) or ^<∞ (B; N) (the base set is B and we only look at types of elements inside N).



§ STABLE INDEPENDENCE IN AN ARBITRARY CATEGORY


Throughout this section, we assume:


  We work inside a fixed category .


The goal of this section is to axiomatize stable amalgams as a particular category of commutative squares in  (see Definition <ref>). In fact, although we will not stress this perspective, we will identify this notion with a double subcategory of , the usual double category of commutative squares in .


  Let (f_1 : M_0 → M_1, f_2 : M_0 → M_2) be a span in .

  
    
  * An amalgam of (f_1, f_2) is a cospan (g_1 : M_1 → N, g_2 : M_2 → N) such that the following diagram commutes:

      
    @=3pc
            M_1 @.>[r]^g_1    N 
    
            M_0  [u]^f_1 [r]_f_2   
            M_2 @.>[u]_g_2


      An amalgamation diagram is a quadruple (f_1, f_2, g_1, g_2) such that (f_1, f_2) is a cospan and (g_1, g_2) is an amalgam thereof.
    
  * We say that  has the amalgamation property if every span has an amalgam.
    
  * Two amalgams g_1^a: M_1 → N^a, g_2^a : M_2 → N^a, g_1^b : M_1 → N^b, g_2^b : M_2 → N^b of (f_1, f_2) are equivalent (written (f_1, f_2, g_1^a, g_2^a) ∼^∗ (f_1, f_2, g_1^b, g_2^b) if there exists N and g^a, g^b making the following diagram commute:

        
    N^b @.>[r]^g^b    N_∗
    
        M_1 [ru]^g_1^b[rr]|>>>>>g_1^a       N^a @.>[u]_g^a
    
        M_0 [u]^f_1[r]_f_2    M_2 [uu]|>>>>>g_2^b[ur]_g_2^a

      
  * Let ∼ be the transitive closure of ∼^∗.
  



  If  has the amalgamation property, then ∼^∗ is already transitive <cit.>.


The idea of ∼ is to identify amalgams whose underlying spans look the same. From this perspective, an independence relation simply consists in the specification of a particular way to amalgamate each span.


  An independence relation (on ) is a set  of amalgamation diagrams that is closed under ∼. We write (f_1, f_2, g_1, g_2) instead of (f_1, f_2, g_1, g_2) ∈. If an amalgamation diagram (f_1, f_2, g_1, g_2) is in , we call it an -independent diagram (or just an independent diagram when  is clear from context).


  We will use the terms independence relation and independence notion completely interchangeably.




	We note a certain utility, too, in a diagrammatic representation analogous to that for pullbacks and pushouts; that is, one may wish to represent the assertion (f_1, f_2, g_1, g_2) by the annotated diagram
	
    @=3pc
            M_1 @[dr]|[r]^g_1    N 
    
            M_0  [u]^f_1 [r]_f_2   
            M_2 [u]_g_2

    This points to the fact that, in axiomatizing abstract independence, we will be delineating the properties of a (double) category of such squares—this is made precise in Definition <ref> below.


When we are working in a concrete class and the morphisms are simply inclusions of strong substructures, we will employ the obvious notational shortcut:


  For an independence relation  on an abstract class , we write M_0M_1M_2M_3 (or (M_0, M_1, M_2, M_3)) if M_0  M_ℓ M_3 for ℓ = 1,2 and (i_0, 1, i_0, 2, i_1, 3, i_2, 3), where i_l, k is the -embedding from 
  M_l to M_k.



  An independence relation  is invariant if it is invariant under isomorphisms of amalgamation diagrams (in the expected sense). 


One can always switch the left and right hand side of  and obtain another independence relation. Thus it is natural to define: 


  For  an independence relation on , we define the dual of , denoted ^d by ^d (f_1, f_2, g_1, g_2) if and only if (f_2, f_1, g_2, g_1). We say that  is symmetric if = ^d.


The following property is a strengthening of the amalgamation property: it asks that any span have an independent amalgam.


  We say that  has the existence property (or just has existence) if for any cospan (f_1, f_2), there is an amalgam (g_1, g_2) such that (f_1, f_2, g_1, g_2) (so in particular,  has the amalgamation property).


  If  has the existence property, then ^d has the existence property.


The existence property implies that the base is independent over itself. More generally:


  Assume that  has the existence property. Consider the commutative diagram

  
    @=3pc
        M_1 [r]^g_1    M_3 
    
        M_0  [u]^f_1 [r]_f_2   
        M_2 [u]_g_2


  If either f_1 or f_2 is an isomorphism, then the diagram is independent.


  In light of <ref>, it suffices to prove the result in case f_1 is an isomorphism.
  
  Assume that f_1 is an isomorphism and apply existence to the cospan (f_1, f_2). We obtain an amalgam (h_1: M_0 → M_3', h_2: M_2 → M_3') such that (f_1, f_2, h_1, h_2). Now because f_1 is an isomorphism, (g_1, g_2) is equivalent to (h_1, h_2). Indeed, any amalgam of (g_2, h_2) will witness the equivalence. Since  is (by definition of an independence relation) closed under ∼, we also have that (f_1, f_2, g_1, g_2).


The statement of the uniqueness property, below, may seem unusual. The idea is that we want every span to have an independent amalgam which is unique, not up to isomorphism (as we always want e.g. to be able to grow the ambient model) but up to equivalence of amalgams:


  We say that  has the uniqueness property if whenever (f_0, f_1, g_1, g_2) and (f_0, f_1, g_1', g_2'), we have that (f_0, f_1, g_1, g_2) ∼ (f_0, f_1, g_1', g_2').


  If  has the uniqueness property, then ^d has the uniqueness property.


In order to coherently compose independent squares, the following property is key:


  Let  be an independence relation. We say that  is right transitive if whenever we have:

  
    @=3pc
        M_1 [r]^f_1,3    M_3 [r]^f_3,5    M_5 
    
        M_0  [u]^f_0,1 [r]_f_0,2    M_2 [u]_f_2,3[r]_f_2, 4    M_4 [u]_f_4,5


  with (f_0, 1, f_0, 2, f_1, 3, f_2, 3) and (f_2,3, f_2,4 , f_3, 5, f_4, 5), then it is also the case that (f_0,1,  f_2, 4∘ f_0, 2, f_3, 5∘ f_1, 3, f_4, 5).

  We say that  is left transitive if ^d is right-transitive. We say that  is transitive if it is both left and right transitive.


Diagrammatically, right transitivity means precisely that the situation in the leftmost diagram below implies that on the right:

    @=3pc
        M_1 [r]^f_1,3@[dr]|    M_3@[dr]|[r]^f_3,5    M_5 
    
        M_0  [u]^f_0,1 [r]_f_0,2    M_2 [u]_f_2,3[r]_f_2, 4    M_4 [u]_f_4,5@=3pc
        M_1 [r]^f_3,5f_1,3@[dr]|    M_5 
    
        M_0  [u]^f_0,1 [r]_f_2,4f_0,2    M_4 [u]_f_2,3

In other words, right transitivity guarantees that the collection of independent squares is closed under horizontal composition. Dually, left transitivity gives closure under vertical composition. While this (in conjunction with the preceding properties) gives us the structure of double category, we opt instead, for practical reasons, to describe the category of independent squares in more straightforward 1-categorical terms:


  Given a right transitive independence relation  with existence, define  (where  stands for “nonforking”) to be the following category:

  
  
  * Its objects are morphisms f: M_1 → M_2 in .
  
  * A morphism in _ from f: M_1 → M_2 to g: N_1 → N_2 is a pair (h_1 : M_1 → N_1, h_2 : M_2 → N_2) such that the following is an independent diagram:
    
      
    @=3pc
            M_2 [r]^h_2    N_2 
    
            M_1  [u]^f [r]_h_1   
            N_1 [u]_g

      
    
  * Composition of morphisms is defined as expected. Note that right transitivity exactly gives that  is closed under composition and Lemma <ref> gives the existence of an identity morphism.
  


Clearly,  is a subcategory of the category ^2 of morphisms in  with the same objects as ^2.


  Let  be a right transitive independence relation with the existence property. Then (using Lemma <ref>)  is isomorphic to a full subcategory of .


We can use  to give a quick proof that invariance (Definition <ref>) follows from the properties defined so far.


  Let  be an independence relation. If  is right transitive and has existence[It suffices to assume the conclusion of Lemma <ref>.], then  is invariant. 


  Assume we are given two isomorphic amalgamation diagrams (f_1, f_2, g_1, g_2), (f_1', f_2', g_1', g_2'), where the first one is independent. To see that the second diagram is independent, it suffices to see that the morphism H = (f_2', g_1') from f_1' to g_2'  (a morphism of ^2) is in fact a morphism in . Now f_1' is (in ^2) isomorphic to f_1, and g_2' is isomorphic to g_2. Moreover, these isomorphisms are in  by Lemma <ref>. Composing these isomorphisms with (f_2, g_1) in , we obtain H, which therefore must also be in .


Before studying  further, we note that a nice-enough independence relation will be monotonic in the following sense:


  Let  be an independence relation. We say that  is right monotonic if whenever we have:

    
    @=3pc
        M_1 [rr]^f_1,4       M_4 
    
        M_0  [u]^f_0,1 [r]_f_0,2    M_2 [r]_f_2, 3    M_3 [u]_f_3,4


if (f_0,1, f_2, 3∘ f_0, 2, f_1, 4, f_3, 4) is an independent diagram, then (f_0, 1,f_0, 2,f_1, 4,f_3, 4∘ f_2, 3) is an independent diagram.

  We say that  is left monotonic if ^d is right monotonic. We say that  is monotonic if it is both left and right monotonic.



  Let  be an independence relation with existence and uniqueness. If  is right transitive, then  is right monotonic.


  We start with the diagram from Definition <ref> and assume that 
  
    (f_0,1, f_2, 3∘ f_0, 2, f_1, 4, f_3, 4)
 is an independent diagram. Using existence, pick g_1,4 : M_1 → M_4', g_2,4: M_2 → M_4' such that (f_0,1, f_0, 2, g_1, 4, g_2, 4). Now pick g_4,5 : M_4' → M_5', g_3,5: M_3 → M_5' such that (g_2, 4, f_2, 3, g_4,5, g_3, 4). By right transitivity, (f_0,1, f_2,3∘ f_0,2, g_4, 5∘ g_1, 4, g_3,5). By uniqueness, we obtain the following picture:

  
    @=3pc    M_4' [r]^g_4,5    M_5'  @.> [r]^h_5    N 
    
        M_1 [ur]^g_1,4[rr]|>>>>>>>>f_1,4       M_4  @.> [ur]_h_4
    
        M_0  [u]^f_0,1 [r]_f_0,2    M_2 [uu]|>>>>>>>g_2,4[r]_f_2, 3    M_3 [u]_f_3,4 @/_2pc/ [uu]_g_3,5


  This shows that the amalgams (f_0,1, f_0,2, g_1, 4, g_2, 4) and (f_0, 1, f_0, 2, f_1, 4, f_3,4∘ f_2,3) are equivalent. Since the first is an independent diagram, the second also is. 


The following variation will also be useful:


Let  be a right monotonic independence relation and consider

    @=3pc
        M_1 [r]^    M_3 [r]^    M_5 
    
        M_0  [u]^ [r]_    M_2 [u]_[r]_    M_4 [u]_

where the outer rectangle is independent. Then the left square is independent.


Since  is right monotonic, the square

    @=3pc
          M_1   [r]^    M_5 
    
            M_0  [u]^ [r]_   
            M_2 [u]_

is independent. Since this square is equivalent to the left square, the latter is independent. The equivalence is documented by     
the diagram

        
    M_3 @.>[r]^    M_5 
    
        M_1 [ru]^[rr]|>>>>>       M_5 @.>[u]_
    
        M_0 [u]^[r]_    M_2 [uu]|>>>>>[ur]_





We can also define a monotonicity property with respect to the base:


  Let  be an independence relation. We say that  is right base-monotonic if whenever we have:

  
    @=3pc
        M_1 [rr]^f_1,4       M_4 
    
        M_0  [u]^f_0,1 [r]_f_0,2    M_2 [r]_f_2, 3    M_3 [u]_f_3,4


  and the outer square (f_0,1, f_2,3∘ f_0,2, f_1,4, f_3,4) is independent, then there exists f_4,4': M_4 → M_4' and f_1,1': M_1 → M_1', f_2, 1': M_2 → M_1', f_1', 4' : M_1' → M_4' such that (f_2,1', f_2,3, f_1', 4', f_4, 4'∘ f_3,4) is independent and the diagram below commutes:

  
    @=3pc    M_1' @.>[r]_f_1', 4'    M_4' 
    
        M_1 @.>[ur]^f_1, 1'[rr]|>>>>>>>>>f_1,4       M_4 @.>[u]_f_4, 4'
    
        M_0  [u]^f_0,1 [r]_f_0,2    M_2 [r]_f_2, 3@.>[uu]|>>>>>>f_2, 1'    M_3 [u]_f_3,4


  As usual,  left base-monotonic means that ^d is right base-monotonic, and base-monotonic means both left and right.


Any nice-enough independence relation is base-monotonic. More precisely:


  If  is right transitive and has uniqueness and existence, then  is right base-monotonic.


  This is similar to (E)(b) in <cit.>. In detail, start with the setup of Definition <ref>. By Lemma <ref>, the square (f_0,1, f_0,2, f_1,4, f_3,4∘ f_2,3) is independent. By existence, find f_3,3^∗: M_3 → M_3^∗ and f_4,3^∗ : M_4 → M_3^∗ such that the square (f_3,4∘ f_2,3, f_2,3, f_4, 3', f_3, 3^∗) is independent. By right transitivity, the square (f_0, 1, f_2,3∘ f_0,2, f_4, 3^∗∘ f_1,4, f_3, 3^∗) is independent. By uniqueness, this square is equivalent to (f_0,1, f_2, 3∘ f_0,2, f_1,4, f_3,4). Amalgamate these two squares: let f_4, 4^∗: M_4 → M_4^∗ and f_3^∗, 4^∗: M_3^∗→ M_4^∗ be the amalgams. Finally, let M_1' := M_4, M_4' := M_4^∗, f_4, 4' := f_4, 4^∗ f_1, 1' := f_1, 4, f_2, 1' := f_3,4∘ f_2, 3, f_1', 4' := f_3^∗, 4^∗∘ f_4, 3^∗.


A transitive independence relation with existence and uniqueness still falls short of the forking-like notion we seek: we must require, too, that it satisfy a suitable local character property. We formulate this by requiring that the induced category be accessible:


  Let  be an independence relation.
  
  
  * We say that  is right accessible if it is right transitive and  is accessible (Definition <ref>). We say that  is left accessible if ^d is right accessible. We say that  is accessible if it is both left and right accessible.
  
  * We say that  is stable if it is symmetric, transitive, accessible, has existence, and has uniqueness.
  


  In several examples of interest,  will actually be an AEC. This is the case, for example, when  is the category of models of a stable first-order theory or when  has an almost fully good independence relation in the sense of <cit.>.



Let  be a right monotonic, right transitive, independence relation with existence. Assume that  is λ-accessible. Then the embedding of  into ^2 preserves λ-directed colimits. Indeed, consider a λ-directed diagram D:→. Let (u_1^d,u_2^d):Dd→ g be its colimit in ^2 and (u̅_1^d,u̅_2^d):Dd→g̅ its colimit in . 
We get a unique morphism (t_1,t_2):g→g̅ in ^2. Since the outer rectangle of 

    @=3pc
        N_i [r]^u_2^d    N [r]^t_2   N̅
    
        M_i  [u]^Dd [r]_u_1^d    M [u]_g[r]_t_1   M̅[u]_g̅

is independent for each d∈, <ref> implies that the left squares are independent for each d∈. It is easy to check that t_1 is an isomorphism, so by Lemma <ref>, the right square is also independent. Thus g is a colimit of D in .


While we have made no requirements on the underlying category , accessibility of  implies accessibility of :


  If  is right monotonic, right transitive, right accessible and has existence, then  is accessible.


Assume that  is λ-accessible. We begin by proving that  has λ-directed colimits. 
Let f_i, j : M_i → M_j | i, j ∈ I be a λ-directed diagram in . We know that  has 
λ-directed colimits so (identifying  with its copy in , see Remark <ref>), let 
(f_i^1, f_i^2) : (𝕀_M_i:M_i→ M_i) → (g: N_1 → N_2) | i ∈ I be a colimit in .

We first prove that g is an isomorphism. Indeed, (f^1_i,f^1_i): 𝕀_M_i→𝕀_N_1| i ∈ I is a cocone in . Thus there is a unique morphism (h_1,h_2): g →𝕀_N_1 such that for all i ∈ I, h_1f^1_i=f^1_i and h_2f^2_i=f^1_i. Since (𝕀_N_1,g): 𝕀_N_1→ g is a morphism in , for all i ∈ I, h_1f^1_i=f^1_i, and gh_2f^2_i=gf^1_i=f^2_i, we get that h_1=𝕀_N_1 and gh_2=𝕀_N_2. Since h_2g=h_1=𝕀_N_1, h_2 is the inverse of g, as desired.


We now claim that f_i^2 | i ∈ I is a colimit of the diagram in . Let g_i : M_i → N | i ∈ I be 
a cocone. Then by Lemma <ref>, (g_i, g_i):𝕀_M_i→𝕀_N | i ∈ I is a cocone in . Therefore there must exist a unique morphism (h_1, h_2) such that h_ℓ f_i^ℓ=g_i for ℓ = 1,2 and i ∈ I. In particular, h_2 f_i^2=g_i. Moreover, h_2 is unique (in ) with this property: if hf_i^2= g_i for all i ∈ I, then hgf_i^1=g_i and, by Lemma <ref> (recalling that, by the preceding paragraph, g is an isomorphism), (hg,h) is a morphism from g: N_1 → N_2 to 𝕀_N in . Hence h=h_2. This completes the proof that  has λ-directed colimits.

Since  has λ-directed colimits, so does ^2. By Remark <ref>, λ-directed colimits are the same in ^2 and .  It is easy to verify, then, that ^2, and hence also , are λ-accessible.



  We have just shown that the embedding of  into  is accessible. In the other direction, the proof above shows that the category  is accessibly embedded (see <cit.>) into .



Let us illustrate what a stable independence relation looks like by considering accessible categories with weak polycolimits:


  A polyinitial object is a set  of objects of a category  such that for every object M in :



  * There is a unique i ∈ having a morphism i → M.

  * For each i ∈, given f,g:i → M, there is a unique (isomorphism) h:i → i with fh=g.


The polycolimit of a diagram D in a category  is a polyinitial object in the category of cones on D, i.e. a set of cones such that for any cone on D, there will be an induced map from exactly one of the members of the set, and this map will be unique up to unique isomorphism, in the sense of <ref>(2). The weak polycolimit of D is defined similarly, except that we waive the uniqueness requirement on the map.


For example, the algebraic closures of the prime fields form a polyinitial object in the category of algebraically closed fields. 

We recall, too, the related concept of a multicolimit, again derived from the notion of a multiinitial set of objects: a multiinitial set of objects in a category  is given by replacing (2) above—uniqueness of the morphism i→ M up to isomorphism—by actual uniqueness.  That is, for each M∈, there is a unique i∈ℐ and a unique i→ M.

In the category of fields, for example, the prime fields form a multiinitial object.

We say, incidentally, that a category is locally λ-polypresentable (respectively, locally λ-multipresentable) if it is λ-accessible and has all polycolimits (respectively, multicolimits).  Note that locally λ-presentable implies locally λ-multipresentable implies locally λ-polypresentable implies λ-accessible.

   
    
  *  Let  be a stable independence relation in an accessible category  with weak polycolimits and all morphisms monomorphisms (for example in a μ-AEC admitting intersections, see <cit.>). Consider a span s=(f_1,f_2) and its weak polypushout (p_i,1,p_i,2), i∈ I_s (note that in general I_s may be empty but here it is not: the existence property of  implies amalgamation). This means that we have commutative squares

    @=3pc
    M_1 [r]^p_i,1    P_i 
    
    M_0  [u]^f_1 [r]_f_2   
    M_2 [u]_p_i,2

such that for any commutative square

    @=3pc
    M_1 [r]^g_1    M 
    
    M_0  [u]^f_1 [r]_f_2   
    M_2 [u]_g_2

there exist a unique i∈ I_s and (not necessarily unique) t:P_i→ M such that tp_i,1=g_1 and tp_i,2=g_2. By closure under ∼ and the uniqueness property of the weak polypushout, there is exactly one i∈ I_s such that M_0M_1M_2P_i.
By monotonicity, these choices are coherent in the sense that given a morphism of spans (𝕀_M_0,h_1,h_2):s→ s',
we get an induced morphism P_i→ P_i' where i is chosen from I_s and i' from I_s'. For a general object M, the characterization of the relation M_0M_1M_2M is by closure under ∼: the relation holds if and only if the induced map from the weak polypushout is P_i→ M for the unique P_i such that M_0M_1M_2P_i. 

On the other hand, a relation  given by a coherent choice of weak polypushouts satisfies all axioms of stable independence with the possible exception of accessibility (and existence, if  does not have amalgamation). We will see (Theorem <ref>) that in a category with chain bounds there is at most one coherent choice of weak polypushouts giving a stable independence relation.

  * We will see in the next two sections that for any coregular locally presentable category , pushouts of regular monomorphisms form a coherent choice of multipushouts in _reg (the subcategory of  with the same objects but only regular monomorphisms). These pushouts will in particular be pullback squares in _reg.

  * One cannot expect that an accessible category with weak polycolimits and all morphisms monomorphisms will have pushouts. In this case,
  we might try to get a stable independence relation by taking all commutative squares
  

    @=3pc
    M_1 [r]^    M_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

Without the existence of pushouts, this relation satisfies all axioms of a stable independence relation except the uniqueness axiom.

We note, too, that one must be careful in assuming pushouts, as this will often prove too strong of a restriction on the category .  For example, let  be an accessible category with multicolimits and all morphisms monomorphisms.  If  has pushouts, it must be small.  To see this, consider an object M,
the instance O of a multiinitial object with morphism O→ M and the pushout

    @=3pc
    M [r]^    M∐ M 
    
    O  [u]^ [r]_   
    M [u]_

Since the codiagonal M∐ M→ M is a split epimorphism, it is an isomorphism and thus the two coproduct injections
M→ M∐ M are equal. Consequently, for any N in , M has at most one morphism M→ N. Thus  is thin and, since it is an accessible category, it must therefore be small.



We end this section by giving several examples and non-examples:

 
  
  *  If  is the AEC of all sets (ordered with substructure), then M_0M_1M_2M_3 if and only if M_0 ⊆ M_ℓ⊆ M_3, ℓ = 1,2, and M_1 ∩ M_2 = M_0 is a stable independence relation.
  
  *  For F a field, if  is the AEC of all F-vector spaces (ordered with subspace), then M_0M_1M_2M_3 if and only if M_0 ⊆ M_ℓ⊆ M_3, ℓ = 1,2, and M_1 ∩ M_2 = M_0 is a stable independence relation. We will see that this example (along with (1) above) is an instance of Theorem <ref>, which gives a general condition on when pullback induces a stable independence notion.

  
  *  If  is the AEC of all algebraically closed fields of characteristic p (for p a fixed prime or 0) ordered by subfield, define  M_0M_1M_2M_3 if and only if M_0  M_ℓ M_3 and for any finite A ⊆ U M_1, and any a ∈ U M_1, the transcendence degree of a over A M_2 is the same as the transcendence degree of a over AM_0.  This is a stable independence notion. Note however that M_0  M_ℓ M_3 and M_1 ∩ M_2 = M_0 does not imply M_0M_1M_2M_3. This is because the pregeometry induced by algebraic closure is not modular. Nevertheless,  has weak polycolimits, hence we are in the setup described by Remark <ref>(<ref>).
  
  *  Let  be the AEC of all (not necessarily algebraically closed) fields of characteristic p (for p a fixed prime or 0) ordered by elementary substructure. Then it is well-known that  does not have any stable independence notion (indeed, ordered fields belong to , so T is not a stable theory). However, the subclass of ℵ_0-saturated models of  is nothing but the AEC of algebraically closed fields of characteristic p ordered by subfield (by the model-completeness of algebraically closed fields), which does have a stable independence notion. 
  
  *  Let T be a stable first-order theory. Write ABCM if and only if A, B, C ⊆ U M, M  T, and ( / AC; M) does not fork over A (in the original sense of Shelah, see <cit.>), where  is any enumeration of B and ( / AC; M) is the set of first-order formulas with parameters from AC satisfied by  in M. There is a certain expansion T^eq of T which “eliminates imaginaries” in the sense that for every definable equivalence relation E there is a definable function F_E sending two E-equivalent elements to the same object (i.e. the equivalence classes of E are also elements), see <cit.>.[We note that the passage from T to T^eq has a category-theoretic generalization, developed independently (from Shelah) by Makkai and Reyes; that is, the passage from a logical category 𝕋 to its pretopos completion, <cit.>.  That these ideas are connected is due to Makkai, see e.g. <cit.>.] Expanding to T^eq leads to an equivalent category of models, so without loss of generality T = T^eq. Then it is known that ABCM if and only if (A) (AB) (AC)M, where (A) denotes the set of elements b in M for which there is a formula ϕ (x) with parameters from A with only finitely many solutions, one of which is b. Thus it suffices to consider forking over algebraically closed sets. Let  be the AEC of algebraically closed sets in T, ordered by being a substructure (see <cit.>). Then  induces a stable independence relation on .
  
  *  Let  be the AEC of all (undirected) graphs, ordered by being a full subgraph. Define M_0M_1M_2M_3 to hold if and only if M_0 ⊆ M_ℓ⊆ M_3, ℓ = 1,2, M_1 ∩ M_2 = M_0, and there are no edges of M_3 going from M_1 \ M_0 to M_2 \ M_0. Then  satisfies all the axioms of a stable independence relation except accessibility. One can define a different independence notion (that will similarly satisfy all the axioms except accessibility) similarly by requiring instead that all possible cross edges are present between M_1 \ M_0 and M_2 \ M_0. This is <cit.>, see also <cit.>. We will see (Example <ref>) that when considering only graphs whose vertices have bounded degree,  is stable and becomes the only independence notion.
  
  *  Let  be the AEC of all graphs, as before. One can modify  so that it satisfies existence, uniqueness, and monotonicity but not transitivity: say M_0M_1M_2M_3 holds if and only if M_0 ⊆ M_ℓ⊆ M_3, ℓ = 1,2, and:
    
    
  * If M_0 is a finite graph, then no cross edges are present between M_1 \ M_0 and M_2 \ M_0.
    
  * If M_0 is infinite, all possible cross edges are present between M_1 \ M_0 and M_2 \ M_0.
    

    One can similarly construct an independence relation with existence and uniqueness that fails to be monotonic (use the isomorphism type of M_2 to decide whether all or no cross edges should be included). It is also easy to construct examples that have existence, but not uniqueness, or uniqueness but not existence, see <cit.>.
  
  *  In all the examples given so far, the morphisms were monomorphisms. This need not be the case, however: in any accessible category with pushouts, the class of all squares form a stable independence notion. In this sense, stable independence generalizes pushouts. We explore this theme further in the next two sections.






§ COREGULAR CATEGORIES AND EFFECTIVE UNIONS



Recall that a monomorphism (respectively, epimorphism) in a category is regular if it is an equalizer (respectively, coequalizer) of a pair of morphisms (see <cit.>). 

 Let  be a category.  We denote by _reg the full subcategory of  whose morphisms are precisely the regular morphisms of .


  In μ-AECs with disjoint amalgamation (and more generally in any category where spans can be completed to pullback squares), every monomorphism is regular.


A category  is called regular if it has finite limits, coequalizers of kernel pairs and regular epimorphisms are stable under pullbacks (see <cit.>). Coregularity is the dual of this notion.  So, in particular, a locally presentable category  is coregular if and only if regular monomorphisms are stable under pushouts. In this case, we have the factorization system (epimorphism, regular monomorphism) in . This implies that _reg is quite a well-behaved category:

 Let  be a coregular locally λ-presentable category. Then _reg is locally λ-multipresentable.

 Following <cit.>, _reg is closed under λ-directed
colimits in . Since λ-pure monomorphisms are regular (<cit.>), _reg is accessible by <cit.>.
We will prove that each object K of  is a λ-directed colimit of λ-presentable objects
in _reg. Let a_i:A_i→ K be a λ-directed colimit of λ-presentable objects A_i in . Since  has (epimorphism, regular monomorphism)-
factorizations, we take a_i=a_i”a_i' to be a (epimorphism, regular monomorphism) factorization of a_i. Then a_i”:B_i→ K is
a λ-directed colimit of regular monomorphisms. Analogously as in <cit.>(i), objects B_i are λ-presentable in _reg.
Thus _reg is λ-accessible.

_reg is clearly closed under equalizers in . Let p_i:P→ M_i be a wide pullback of regular monomorphisms
f_i:M_i→ M. Since regular monomorphisms form the right part of a factorization system, the composition f_ip_i is a regular monomorphism. For the same reason,
the p_i are regular monomorphisms. Thus _reg has connected colimits and, since it is λ-accessible, it is locally λ-multipresentable.





We will use the following important fact:

[<cit.>]
In a coregular locally presentable category, pushouts of regular monomorphisms are pullbacks.  

In fact, this is valid in any category where pushouts of regular monomorphisms are regular monomorphisms.


Note, however, that pushouts of regular monomorphisms in  need not be pushouts in _reg—the induced map from the pushout will not even be a monomorphism, in general, let alone a regular monomorphism.  To ask that this be the case is precisely to ask that  have effective unions. The existence of effective unions is an exactness property (introduced by Barr <cit.>) satisfied by both Grothendieck abelian categories and Grothendieck toposes and which is strong enough to ensure the existence of enough injective objects—see Remark <ref> below.


  A locally presentable category  has effective unions if whenever we have a pullback
  

    @=3pc
    M_1 [r]^    M_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_


in _reg, and the pushout


    @=3pc
    M_1 [r]^    P 
    
    M_0  [u]^ [r]_   
    M_2 [u]_


in , the induced morphism P→ M_3 is a regular monomorphism.





If  is a coregular locally presentable category then any morphism in _reg is a regular monomorphism. This follows from the fact that pushout squares in  will be pullbacks (Fact <ref>) and Remark <ref>.



  Effective unions are closely tied to the existence of enough injectives (see e.g. Chapter 4 in <cit.>): let  be a coregular locally presentable category such that regular monomorphisms are closed under directed colimits 
  (in the sense of <cit.>). Then _reg has directed colimits. Moreover, if  has effective unions then regular monomorphisms are cofibrantly generated (in the sense given at the beginning of <cit.>; the proof is the same as that of <cit.>). In particular, there is a set  of regular monomorphisms such that an object injective with respect to any s∈ is injective to all regular monomorphisms. Consequently, these injective objects form an accessible category (see <cit.>).


   
  
  *  Both Grothendieck toposes and Grothendieck abelian  categories are locally presentable coregular categories having effective unions and regular monomorphisms closed under directed colimits (see <cit.>). In particular, they include categories of R-modules and presheaf categories ^. The latter include the category of multigraphs.
  
  * The category  of graphs is locally presentable and coregular but it does not have effective unions. It suffices
to take the pullback of two vertices of an edge, where the pushout consists two vertices without any edge. Thus the embedding 
of this pushout to the edge is not regular (although, we note, it would be regular in multigraphs). See also Example <ref>(<ref>).
  
  * The category of groups and the category of Boolean algebras are locally presentable and coregular. Regular monomorphisms coincide with monomorphisms and are closed under directed colimits. But groups do not have enough injectives, and the injectives in Boolean algebras are complete Boolean algebras: these do not form an accessible category. Thus neither groups nor Boolean algebras have effective unions (see Remark <ref>).
  
  * The category  of Banach spaces (with linear contractions) is locally presentable and coregular. The regular monomorphisms are isometries and are closed under directed colimits. However  does not have effective unions: if it did, by Remark <ref>, regular monomorphisms would be cofibrantly generated. This contradicts <cit.>.  We provide an alternate proof of this fact in Remark <ref>, exploiting the connection between stable independence and the failure of the order property.
   
  * The category  of Hilbert spaces (with linear isometries) is accessible. Recall that Hilbert spaces are precisely the Banach spaces whose norm satisfies the parallelogram identity.  By the note above,  is a (full) subcategory of _reg.  Pullbacks in  exist and are calculated as
for Banach spaces.  If f:V→ W is a morphism 
in  then W≅ V⊕ V^⊥ where V^⊥ is the orthogonal complement of f(V) in W. Thus a typical span of maps consists of 
f:V→ V⊕ W_1 and g:V→ V⊕ W_2, with pushout V⊕ W_1⊕ W_2. Since a general pullback has the form

    @=3pc
    V⊕ W_1 [r]^    V⊕ W_1⊕ W_2⊕ W_3 
    
    V  [u]^ [r]_   
    V⊕ W_2 [u]_

and the induced map V⊕ W_1⊕ W_2→ V⊕ W_1⊕ W_2⊕ W_3 is clearly an equalizer,  has effective unions.





§ STABLE INDEPENDENCE AND EFFECTIVE UNIONS



Let  be a coregular locally presentable category. If  has effective unions, then _reg has a stable independence relation.


Put M_0M_1M_2M_3 if and only if the square

    @=3pc
    M_1 [r]^    M_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

is a pullback. We check that this satisfies all the axioms of stable independence. If M_0M_1M_2M_3 and M_3→ M'_3 then
M_0M_1M_2M'_3. Given a commutative square

    @=3pc
    M_1 [r]^    M”_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

and M”_3→ M_3 such that M_0M_1M_2M_3, then M_0M_1M_2M”_3. This shows that  is closed under equivalence of amalgams.  We have used here that if gf is a regular monomorphism, then f is a regular monomorphism (see <cit.> – recall that in any coregular locally presentable category we have the factorization system (epimorphism, regular monomorphism)). 

Given a span M_1← M_0→ M_2, we form the pushout

    @=3pc
    M_1 [r]^    M_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

Following Fact <ref>, this square is a pullback. Thus M_0M_1M_2M_3, which yields the existence property.

In order to prove the uniqueness property, consider M_0M_1M_2M'_3 and M_0M_1M_2M”_3 with the same
span M_1← M_0→ M_2. Form the pushout 

    @=3pc
    M_1 [r]^    P 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

and take the induced regular monomorphisms P→ M'_3 and P→ M”_3. Then the pushout

    @=3pc
    M'_3 [r]^    M_3 
    
    P  [u]^ [r]_   
    M”_3 [u]_


amalgamates the starting pullbacks.

The transitivity and symmetry properties are also easy to check. We can prove accessibility directly or use the soon to be proven Theorem <ref>: local character holds since the nonforking base is simply given by the pullback, and the witness property is also a straightforward consequence of the definition.
 

   
  	
  * We note that the assumption of local presentability is largely a matter of convenience, and the argument will go through under weaker assumptions.  In particular, the category need only be accessible with pullbacks and pushouts of monomorphisms.
    
  * Effective unions were only needed for the uniqueness property. On the other hand, if the independence relation defined by being a pullback square satisfies uniqueness, then the category has effective unions. Consider
M_0M_1M_2M_3 and form the pushout

    @=3pc
    M_1 [r]^    P 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

Since M_0M_1M_2P (by <ref>), we have an amalgam M_3→ M← P whose right leg is the composition
P→ M_3→ M. Thus P→ M_3 is a regular monomorphism.

  * The converse of Theorem <ref> fails: Example <ref> shows that the category  of graphs whose vertices have degree at most k (for fixed k < ω) is coregular, locally presentable, and has a stable independence relation in _reg, but does not have effective unions.

  * We show in Theorem <ref> that in any μ-AEC, independent squares over sufficiently saturated models are pullbacks.

  * Let  be a coregular locally presentable category having effective unions. Then in _reg, given a span (f_1,f_2),
 exactly one instance of a multipushout of f_1 and f_2 is a pullback. This follows from Remark <ref> and Theorem <ref>.
  



Both Grothendieck toposes and Grothendieck  abelian categories have stable independence relations. This follows from Theorem <ref>
and Example <ref>(<ref>) and subsumes <ref>(<ref>) and (<ref>). In fact, (<ref>) is valid for R-modules in general. This was known already for complete first-order theories of modules (see <cit.>): we have produced an alternate proof of this fact. Similarly, the category of Hilbert spaces (with linear isometries) has a stable independence notion—as  is accessible but not locally presentable, we must actually invoke the weakening mentioned in <ref>(1) above. This was also known from the model-theoretic analysis of the corresponding continuous first-order theory, see for example <cit.>.


In the rest of this section, we show that even without effective unions, we can still define an independence notion that satisfies all the axioms of stable independence except accessibility.


  Let  be a coregular locally presentable category.  Define ^∗ by taking commutative squares in _reg
  

    @=3pc
    M_1 [r]^    M_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

such that the induced morphism from the pushout

    @=3pc
    M_1 [r]^    P 
    
    M_0  [u]^ [r]_   
    M_2 [u]_

is a regular monomorphism.


These are precisely effective pullback squares


    @=3pc
    M_1 [r]^    M_3 
    
    M_0  [u]^ [r]_   
    M_2 [u]_


in the sense that the unique morphism P→ M_3 is a regular monomorphism. In , ^∗ is the relation  mentioned in Example <ref>(<ref>).


If  has effective unions then ^∗= where  is the relation from Theorem <ref>. This uses the fact that pushouts in  are pullbacks. 

Moreover, by the next result, together with Theorem <ref>, if _reg has chain bounds then whenever  is any stable independence relation on _reg we have that = ^∗.  


  Let  be a coregular locally presentable categories and let ^∗ be as in Definition <ref>.

  Then:
  
  
  * ^∗⊆ (where  is from Definition <ref>).
  
  * ^∗ is invariant, monotonic, symmetric, transitive, has existence and uniqueness.
  
  * If  is locally λ-presentable then the category  of ^∗-independent squares in _reg (from Definition <ref>) is closed under λ-directed colimits in ^2.
  
  * If  is locally λ-presentable, ^□ is the category of commutative squares in , and ^⊡ is the category of effective pullback squares in , then ^⊡ is λ-accessible with λ-presentable objects being squares of λ-presentable objects in .
  

   
  
  * Straightforward.
  
  * The properties of ^∗ are proven as in the proof of Theorem <ref>, except for transitivity. To prove transitivity, consider:
  

    @=3pc
        M_1 [r]^    M_3 [r]^    M_5 
    
        M_0  [u]^ [r]_    M_2 [u]_[r]_    M_4 [u]_

where both squares are effective pullbacks. We have to show that the outer rectangle is an effective pullback.
Thus we have to show that the induced morphism p:P→ M_5 from the pushout

    @=3pc
    M_1 [r]^    P 
    
    M_0  [u]^ [r]_   
    M_4 [u]_
  
is a regular monomorphism. This pushout is a composition of pushouts

    @=3pc
        M_1 [r]^    Q [r]^    P 
    
        M_0  [u]^ [r]_    M_2 [u]_[r]_    M_4 [u]_

Consider the pushout
 
    @=3pc
    M_3 [r]^    P' 
    
    Q  [u]^q [r]_   
    P [u]_q̅
  
where q:Q→ M_3 is the induced morphism. Since the left pullback above is effective, q is a regular monomorphism and thus
q̅ is a regular monomorphism. Since

    @=3pc
    M_3 [r]^    P' 
    
    M_2  [u]^ [r]_   
    M_4 [u]_
 
is a pushout and the right pullback above is effective, the induced morphism p':P'→ M_5 is a regular monomorphism. Thus
p=p'q̅ is a regular monomorphism. Conversely, if the outer rectangle and right square in the proof above are effective pullbacks then the left square is an effective pullback. This follows directly from monotonicity. In fact, the composition of q with M_3→ M_5 is a regular monomorphism as the composition of Q→ P with p. Thus q is a regular monomorphism (see <cit.>).

  * Let D:→ be a λ-directed diagram where Dd is f_d:M_d→ N_d. Let f:M→ N
be its colimit in ^2. For each d∈, the square

    @=3pc
    M [r]^f    N 
    
    M_d  [u]^ [r]_f_d   
    N_d [u]_

is a pullback because it is a λ-directed colimit of pullbacks

    @=3pc
    M_d' [r]^f_d'    N_d' 
    
    M_d  [u]^ [r]_f_d   
    N_d [u]_

and pullbacks commute with λ-directed colimits in  (see <cit.> 1.59). Analogously, the pushout

    @=3pc
    M [r]^g    P 
    
    M_d  [u]^ [r]_f_d   
    N_d [u]_

is a λ-directed colimit of pushouts

    @=3pc
    M_d' [r]^g_d'    P_d'
    
    M_d  [u]^ [r]_f_d   
    N_d [u]_

Thus the induced morphism p:P→ N is a λ-directed colimit of induced morphisms p_d':P_d'→ N_d'. Hence p is a regular monomorphism.

  * The category ^⊡ is accessible and accessibly embedded into the category ^□ because it is given by the following sketch (see <cit.> 2.60)  

    A [rr]^ [dd]_       B  [dl]_[dd]^u
       P [dr]^p   
    
    C  [rr]_v [ur]^       D

where the outer rectangle is a pullback, the inner quadrangle is a pushout and u,v,p are regular monomorphisms. There is a regular cardinal μ▹λ such that ^⊡ is μ-accessible and the inclusion G:^⊡→^□ preserves μ-directed colimits and μ-presentable objects.



We return to the relation ^∗ in Example <ref>, as it provides a useful example of the witness property (Definition <ref>).


  Let κ≥ 2 be a (possibly finite) cardinal. We call a graph κ-local if all its vertices have degree strictly less than κ. Let  be the category of κ-local graphs (seen as a full subcategory of the category  of graphs), and let θ := κ^+ + ℵ_0. Then _reg is closed in  under connected limits, pushouts and θ-directed colimits. Thus _reg is a locally θ-multipresentable category. Moreover if κ < ℵ_0, then  is coregular and locally finitely presentable. On the other hand, like ,  does not have effective unions. Nevertheless, we show that it has a stable independence relation. Let ^∗ be the independence relation on _reg given by Definition <ref> (in this case, this will be as given by Example <ref>(<ref>): two graphs are independent over the base if there are no cross edges except over the base). By the proof of Theorem <ref> (or by checking directly), ^∗ satisfies all the properties in the definition of a stable independence notion, except perhaps accessibility. However in this case (_reg)_ is accessible. Regular monomorphisms in  are induced by the relation of being a full subgraph, so let G ⊆ H be κ-local graphs with G a full subgraph of H. Given a set A ⊆ U H, there are full subgraphs G_0 ⊆ H_0 such that H_0 contains A, G_0 contains A ∩ G, |U H_0| ≤ |A| + θ, and the square  

    @=3pc
            G @[r]    H 
    
            G_0  [u]^ [r]_   
            H_0 [u]_

      
is an effective pullback (i.e. any cross edge between G and H_0 inside H is inside G_0). It suffices to add to A ∩ G all vertices of G connected with a vertex of A by an edge. Thus ^∗ is a stable independence relation. 

We consider the canonicity of this relation in Remark <ref>.
  




§ FROM ACCESSIBLE CATEGORY TO Μ-AEC


In Lemma <ref>, we examined the relationship between a certain accessible category  and the category _reg obtained by restricting to the regular monomorphisms.  In this section we prove several more results along these lines, both in connection with _reg and with _mono, where the morphisms are taken to be all monomorphisms in the original category . The latter is particularly useful in light of Fact <ref>: accessible categories with all morphisms monomorphisms are μ-AECs, and vice versa.  This provides the essential Rosetta stone that allows us to translate the uniqueness and canonicity results derived for μ-AECs below back to the abstract categorical framework. We deduce in particular (using the corresponding result of the third author for universal classes <cit.>) that the eventual categoricity conjecture holds for all locally ℵ_0-multipresentable categories (Corollary <ref>). 

 For a category , we denote by _mono the full subcategory of  whose morphisms are precisely the monomorphisms of .



















 

Let  be a λ-accessible category. Then _mono is accessible and has λ-directed colimits.


Let  be a (representative) full subcategory of  consisting of λ-presentable objects and

    E:→^^

be the canonical functor (see <cit.>). Then E is a full embedding preserving λ-directed colimits (see <cit.>). Clearly, E preserves monomorphisms too. Thus monomorphisms in  are stable under λ-directed colimits
because this is true in ^^ (see <cit.>). That is, given λ-directed diagrams D,D':→
and a natural monotransformation δ:D→ D', then δ: D→ D' is a monomorphism. Consequently,
_mono is closed under λ-directed colimits in . That is, given a λ-directed colimit of monomorphisms
in , (i) the colimit cocone consists of monomorphisms, and (ii) for every cocone of monomorphisms the factoring morphism 
is a monomorphism.

Following <cit.>, there is a regular cardinal μ▹λ such that each object of  is a μ-directed
colimit of μ-presentable λ-pure subobjects of . Thus _mono is μ-accessible.



Let μ be a regular cardinal such that λ⊴μ and ||<μ where || denotes the cardinality of the set
of morphisms of . Following <cit.>, _mono is (μ^<μ)^+-accessible.


When the starting class is locally multipresentable, the index of accessibility is preserved:


Let  be a locally λ-multipresentable category. Then _mono is locally λ-multipresentable.


Following <ref>, we have to prove that each object K of  is a λ-directed colimit of λ-presentable objects 
in _mono. Let a_i:A_i→ K be a λ-directed colimit of λ-presentable objects A_i in . Following <cit.>,  has (strong epimorphism, monomorphism)-factorizations;  is well-powered by
<cit.>. Let a_i=a_i”a_i' be a (strong epimorphism, monomorphism) factorization of a_i. Then a_i”:B_i→ K is 
a λ-directed colimit of monomorphisms. Following <cit.>, objects B_i are λ-presentable in _mono.
Thus _mono is λ-accessible.

It remains to show that _mono has connected limits. Since equalizers are monomorphisms, we only need that wide pullbacks 
of monomorphisms are monomorphisms. But this is evident.



If K is λ-presentable in  then it is λ-presentable in _mono. One cannot expect the converse: in a general locally λ-presentable category , for example, the objects of  that are λ-presentable in _mono (the λ-generated objects of ) are guaranteed only to be strong quotients of λ-presentables (see 
<cit.>).
 
 The eventual categoricity conjecture in the sense of internal sizes (see <cit.>) holds for locally ℵ_0-multipresentable categories.
 By contrast with the remark above, whenever  is locally ℵ_0-multipresentable, the embedding _mono→ preserves μ-presentable objects 
for all sufficiently large μ (see <cit.>). Recalling that any locally ℵ_0-multipresentable category is a universal AEC (<cit.>), the eventual categoricity conjecture for locally ℵ_0-multipresentable categories follows from the corresponding result for universal AECs in <cit.>. 


In order to be able to restrict to regular monomorphisms, we will assume existence of pushouts:


Let  be a λ-accessible category with pushouts. Then _reg is accessible and has λ-directed colimits.


We follow the proof of <ref>. We note, first, that regular monomorphisms are stable under λ-directed colimits 
in  (see <cit.> Proposition 2).





For the rest, it suffices to note that λ-pure morphisms are regular monomorphisms (see <cit.>).




§ SOME MODEL THEORY OF Μ-AECS


We now shift more firmly to the model-theoretic context, establishing certain technical results on μ-AECs that we will require for the proofs related to canonicity of stable independence relations, and their relationship to a suitable formulation of the order property.  As mentioned already, Lemma <ref> and Fact <ref> will allow us to translate such results back to the broader category-theoretic framework that we have considered up to this point. 

Roughly speaking, the following lemma says that under reasonable assumptions, presentability and cardinality very often coincide quite often in reasonable classes of structures. Parts of the proof are similar to (for example) <cit.>. The lemma will be used in the proof of Theorem <ref> and the proof of Theorem <ref>. Note that we do not assume that the class  is coherent, nor that M is a τ ()-substructure of N when M  N.


    Let = (K, ) be such that:
    
    
  * K is a class of structures in a fixed μ-ary vocabulary τ ().
    
  *  is a partial order on K such that M  N implies U M ⊆ U N.
    
  *  is closed under isomorphisms.
    

    Let μ≤θ be regular cardinals such that  is θ-accessible and has concrete μ-directed colimits. Let C be the class of cardinals λ such that for any M ∈, |U M| < λ if and only if M is λ_0-presentable for some λ_0 < λ. Then:

  
  
  * C is closed unbounded.
  
  *  If λ is such that λ = λ^<μ and θ≤λ^+, then for any M ∈ there exists M_i : i ∈ℐ increasing and λ^+-directed such that:
    
    
  * M = ⋃_i ∈ℐ M_i.
    
  * For all i ∈ℐ, M_i is a μ-directed union of at most λ-many θ-presentable objects. In particular, M_i is λ^+-presentable and if in addition (θ, λ^+] ∩ C ≠∅, then also |U M_i| ≤λ.
    
  * M_i = ⋃_j < i M_j for all i ∈ℐ such that {j ∈ℐ| j < i} is μ-directed.
    
  
  * If λ > θ is such that λ = λ^<μ, and C ∩ (θ, λ] ≠∅, then λ^+ ∈ C.
  

   
  
  * C is clearly closed. Now given a cardinal λ_0, build λ_i : i ≤ω increasing continuous such that for all i < ω, for any M ∈, if M is λ_i-presentable, then |U M| < λ_i + 1 and if |U M| ≤λ_i, then M is λ_i + 1-presentable. This is possible since there is up to isomorphism only a set of objects of cardinality at most λ_i and a set of λ_i-presentable objects. Now λ_ω is in C, as desired.
  
  * Let M ∈. We know that M can be written as a θ-directed union of θ-presentable objects M = ⋃_i ∈ I N_i. Now by the cardinal arithmetic assumption, any subset of I of cardinality at most λ can be completed to a μ-directed subset of I of cardinality at most λ. Thus letting ℐ be the set of μ-directed subsets of I of cardinality at most λ, we have that M = ⋃_J ∈ℐ N_J. This is a λ^+-directed system, thus letting for i ∈ℐ M_i := N_i, the M_i's are as desired. As for the “in particular” part, it is routine to check that the M_i's are λ^+-presentable (see <cit.>) and if (θ, λ^+] ∩ C ≠∅ this means that M_i is a union of at most λ-many objects of cardinality at most λ, so |U M_i| ≤λ.
  
  * Let M ∈. Let M_i : i ∈ℐ be as given by the previous part. We show that |U M| ≤λ if and only if M is λ^+-presentable. First, if |U M| ≤λ then as ℐ is λ^+-directed M = M_i for some i ∈ℐ and since M_i is λ^+-presentable we are done. Conversely if M is λ^+-presentable then again as ℐ is λ^+-directed we must have that M = M_i for some i ∈ℐ, hence |U M| = |U M_i| ≤λ, as desired.
  


The next result is similar to <cit.>, but we do not require anything on how presentability ranks relate to cardinalities. Rather, what we need is derived via Lemma <ref>. This also improves <cit.>, which is the case μ = ℵ_0.


  Let  be a coherent abstract class in a μ-ary vocabulary with concrete μ-directed colimits. Then  is accessible if and only if  is a μ-AEC.


  If  is a μ-AEC, then by Fact <ref>,  is an accessible category. We show the converse. Assume that  is accessible and fix θ≥μ such that  is θ-accessible (such a θ exists by <cit.>). Let C be as given by Lemma <ref>. We have to show that () exists. Let λ_0 := (min (C ∩ [θ^+, ∞)))^<μ. Note that θ < λ_0 = λ_0^<μ. We claim that () ≤λ_0. Let M ∈ and let A ⊆ |M|. Let λ := |A|^<μ + λ_0. By Lemma <ref>(<ref>), there exists a λ^+-directed resolution M_i : i ∈ℐ of M such that each M_i has cardinality at most λ. Since the resolution is λ^+-directed and |A| ≤λ, there exists i ∈ℐ such that A ⊆ U M_i. Since |U M_i| ≤λ = λ_0 + |A|^<μ, we are done.




 §.§ Model-homogeneous models


The following notions are very close to “κ-closed” and “κ-saturated” from <cit.>. The only difference is that we use cardinalities instead of presentability ranks. Most of the proofs there carry through with little change, however.


  Let  be a μ-AEC. For κ >  (), M  N, we say that M is κ-model-homogeneous in N if whenever M_0, N_0 ∈ are such that:

  
  
  * M_0  M.
  
  * M_0  N_0  N.
  
  * N_0 ∈_<κ
  

  Then there exists f: N_0 → M fixing M_0.

  We say that M is locally κ-model-homogeneous if it is κ-model-homogeneous in N for every N ∈ with M  N. We say that M is κ-model-homogeneous if whenever M_0  M, M_0  N_0 are such that N_0 ∈_<κ, there exists f: N_0 → M fixing M_0.



  If  has amalgamation, being locally κ-model-homogeneous is equivalent to being κ-model-homogeneous (see <cit.>). Further (still assuming amalgamation), if M is κ-model-homogeneous in N and N is κ-model-homogeneous, then M is κ-model-homogeneous. 



  Let  be a μ-AEC and let κ >  (). We let  be the abstract class of locally κ-model-homogeneous models in , ordered with the appropriate restriction of .


Given an increasing chain M_i : i < δ in a μ-AEC , the union of the chain may not be inside . The following conditions are a very useful weakening: we require that this chain have an upper bound. This is already used in <cit.>. Classes of saturated models in AECs, as well as μ-CAECs (e.g. metric classes) are examples satisfying this property, see <cit.>.


  A μ-AEC  has directed bounds if whenever M_i : i ∈ I is a directed system, there exists M ∈ such that M_i  M for all i ∈ I.

  We say that  has chain bounds if this holds whenever I is an ordinal.


In practice, we will use chain bounds. It allows us to build locally κ-model-homogeneous models. Moreover a large cardinal axiom (Vopěnka's Principle, see Chapter 6 of <cit.>) ensures that the class of all such models is well-behaved. In particular, it has the amalgamation property.


  Let  be a μ-AEC and let κ >  () be regular.

  
  
  * Assume that  has chain bounds. For any M ∈, there exists N ∈ such that M  N and N is locally κ-model-homogeneous. Moreover, we can take N so that | U N| ≤(|U M| +  ())^<κ.
  
  *  For any N ∈ and any A ⊆ U N, there exists M ∈ with M  N such that M contains A, M is κ-model-homogeneous in N, and |U M| ≤(|A| +  ())^<κ.
  
  * Assume that  has chain bounds. If either  has amalgamation or Vopěnka's principle holds, then  is a κ-AEC with chain bounds.
  
  * If κ is strongly compact, then any locally κ-model-homogeneous model M is a global amalgamation base: whenever M  M_ℓ, ℓ = 1,2, there exists N ∈ and f_ℓ : M_ℓ→ N fixing M.
  

   
  
  * Similar to <cit.>.
  
  * Similar to the above.
  
  * All the axioms are straightforward, except for the Löwenheim-Skolem-Tarski axiom. If  has amalgamation, this follows from Theorem <ref> and Remark <ref>. If Vopěnka's principle holds, then by <cit.>, the Löwenheim-Skolem-Tarski axiom holds (though we do not have a bound for it).
  
  * As in <cit.>, using the fact that one can take ultraproducts in μ-AECs <cit.>
  


Now that we have tools to get amalgamation, we investigate to what extent we can work inside a big homogeneous model as in the first-order case (and, indeed, as in AECs with amalgamation):


  For  a μ-AEC and κ >  (), we say that M ∈ is κ-universal if any M_0 ∈_<κ embeds into M.



  We say that a μ-AEC  has monster models if for any κ >  () there exists M ∈ which is both κ-universal and κ-model-homogeneous.


It turns out that building monster models requires (at least when μ = ℵ_1) chain bounds:


  Let  be a μ-AEC. If  is non-empty, has amalgamation, joint embedding, and chain bounds, then  has monster models. Conversely, if  has monster models, then  has amalgamation and joint embedding; if μ≤ℵ_1, moreover,  has chain bounds.


  If  has amalgamation, joint embedding, and has chain bounds, then the construction of a κ-universal κ-model-homogeneous model is standard.

  Conversely, if  has monster models, then it clearly has amalgamation and joint embedding. To see it has chain bounds when μ = ω_1, fix M_i : i < δ increasing. Without loss of generality, δ = δ. If δ≥ω_1, we can use the chain axioms of ℵ_1-AECs, so assume without loss of generality that δ = ω. Let M be κ-universal and κ-model-homogeneous, where κ := (∑_i < δ |U M_i|^<μ +  ())^+. We build f_i: M_i → M | i < ω increasing as follows:

  
  
  * For i = 0, use universality of M.
  
  * For i successor, use model-homogeneity of M.
  

  Now let N  M contain ⋃_i < ω f_i[M_i] and rename to obtain an upper bound to M_i : i < ω.


Assuming a large cardinal axiom, then, there is a well-behaved sub-μ-AEC of the original class:


  Let  be a μ-AEC, let κ >  () be strongly compact and assume Vopěnka's principle. If  has chain bounds, then there exists a subclass ^∗ of  such that:

  
  
  * ^∗ is a κ-AEC.
  
  * ^∗ has amalgamation, joint embedding, and chain bounds.
  
  * If  is not empty, ^∗ is not empty and can be chosen to have arbitrarily large models if  has arbitrarily large models. In this case, ^∗ will have monster models.
  
  * If  has joint embedding, then ^∗ is cofinal in , i.e. any M ∈ is contained inside an N ∈^∗.
  


  By Theorem <ref>,  is a κ-AEC with amalgamation and chain bounds. If  had joint embedding already, then  also had joint embedding, and hence one can take ^∗ :=. Otherwise, using the equivalence relation induced by “embedding in a common model,” one can partition  into disjoint κ-AECs _i^∗ : i ∈ I such that each has amalgamation, joint embedding, and chain bounds. If  has arbitrarily large models,  has arbitrarily large models and hence (since I is a set) we can pick i ∈ I such that _i^∗ has arbitrarily large models. Let ^∗ = _i^∗.



  Can one remove Vopěnka's principle from Corollary <ref>?


One approach would be to consider a stronger ordering on , e.g. M  N if and only if M  N and M __κ, κ N. In this case, however, we do not know whether having chain bounds is preserved.



§ STABLE AMALGAMATION INSIDE A Μ-AEC


In this section, we consider independence relations on μ-AECs. The main result is Theorem <ref>, characterizing stable independence in terms of model-theoretic local character properties of forking. All throughout, we assume:


  We work inside a fixed μ-AEC . We fix an invariant (Definition <ref>) independence relation  on .


We can see  as a relation on Galois types if we introduce some notation. A similar idea is already investigated in <cit.>, but there the left hand side of  is already assumed to be an arbitrary set. Thus the situation here requires slightly more caution.


  Write N_0ABN_3 if N_0  N_3, A ∪ B ⊆ U N_3, and there exists M_1, M_2, M_3 with A ⊆ U M_1, B ⊆ U M_2, N_3  M_3, and N_0M_1M_2M_3.

  We say that ( / B; N_3) does not fork over N_0 if N_0BN_3 (it is easy to see that this does not depend on the choice of representatives, see also Fact <ref>(<ref>)).


One can think of  as the “closure” of . The point is that we allow sets on the left and right hand side.


  It is tempting to try to define  in an arbitrary category as the class of diagrams

  
    @=3pc
        M_1 [r]     M_3 
    
        M_0  [ur]    
        M_2 [u]


  that can be extended to an independent diagram consisting of M_0, M_1', M_2' and M_3' such that the following commutes:

  
    @=3pc    M_1' @.>[r]     M_3' 
    
        M_1 [r] @.>[ru]     M_3 @.>[ru]     M_2' @.>[u]
    
        M_0 @.>[uur] @.>[urr]  [ur]     M_2 [u] @.>[ru]


  However, when there is already a morphism e.g. from M_0 to M_2, there is no reason to believe that the resulting diagram will also commute with that morphism. This is problematic when one tries to prove, for example, that  is transitive (see Theorem <ref>). Thus it is unclear to us how to define  in a non-concrete category.


Properties of  generalize to  as follows:


  Assume that  is monotonic.
  
  
  
  * Let M_0  M_ℓ M_3 for ℓ = 1, 2. Then M_0M_1M_2M_3 if and only if M_0M_1M_2M_3.
  
  *  (Preservation under -embeddings) Given M_0  M_3, A, B ⊆ U M_3, and f: M_3 → N_3, we have that M_0ABM_3 if and only if f[M_0]f[A]f[B]N_3.
  
  * (Monotonicity) If M_0ABM_3 and A_0 ⊆ A, B_0 ⊆ B, then M_0A_0B_0M_3.
  
  * (Normality) M_0ABM_3 if and only if M_0A M_0B M_0M_3.
  
  * (Base monotonicity) Assume that  is right base-monotonic. If M_0ABM_3, M_0  M_2  M_3, and U M_2 ⊆ B, then M_2ABM_3.
  
  * (Extension) Assume that  has existence. Whenever M  N and p ∈^<∞ (M), there exists q ∈^<∞ (N) extending p such that q does not fork over M.
  
  * (Symmetry) Assume that  is symmetric. Then  is symmetric: MABN holds if and only if MBAN holds.
  


  This is essentially given by the arguments in <cit.>. For the convenience of the reader, we sketch some details:

  
  
  * If M_0M_1M_2M_3, then directly from the definition M_0M_1M_2M_3. For the converse, use monotonicity, invariance, closure under ∼, the coherence axiom of μ-AECs, and the definition of .
  
  * Directly from the definitions, invariance, and closure under ∼.
  
  * Clear from the definition of .
  
  * The right to left direction is by monotonicity. For the left to right direction, suppose that we have M_1, M_2, M_3' witnessing M_0ABM_3, i.e. M_0  M_ℓ M_3' for ℓ = 1,2, M_3  M_3', A ⊆ U M_1, B ⊆ U M_2, and M_0M_1M_2M_3'. By the first part, M_0M_1M_2M_3'. By monotonicity, M_0A M_0BM_0M_3'. Since  preserves -embeddings, this implies that M_0AM_0BM_0M_3, as desired.
  
  * Directly from the definition and the coherence axiom of μ-AECs.
  
  * Say p =  ( / M; M'). By existence, find N' ∈ and f: M' → N' such that N  N', f fixes M, and Mf[M']NN'. Let q :=  (f () / N; N').
  
  * Directly from the definition.
    


Uniqueness and transitivity also generalize, but the argument is more involved than the corresponding one in <cit.> (essentially because we are taking a closure with respect to both the left and right hand side of ). We sketch a full proof here. 


  Assume that  is transitive and has existence and uniqueness.
  
  
  
  * (Uniqueness) Given p, q ∈^<∞ (B; N) with M  N and U M ⊆ B ⊆ U N, if p  M = q  M and p, q do not fork over M, then p = q.
  
  * (Transitivity) If M_0  M_2  M_3, M_0AM_2M_3 and M_2ABM_3, then M_0ABM_3.
  


  We proceed via a series of claims.

  Claim 1: Assume that M_0M_1M_2M_3 and M_0M_1'M_2M_3'. Let , ' be enumerations of M_1, M_1' respectively. If ( / M_0; M_3) =  (' / M_0; M_3') (i.e. the enumerations induce an isomorphism between M_1 and M_1' fixing M_0), then ( / M_2; M_3) =  (' / M_2; M_3').

  Proof of Claim 1: This follows directly from the uniqueness property as in <cit.>. †_Claim 1.

  Claim 2: Assume that M_0M_2M_1M_3 and M_0M_2M_1'M_3. There exists M_1”, M_3', f: M_1' → M_1” such that f fixes M_0, M_1  M_1” M_3', M_3  M_3', and M_0M_2M_1”M_3'.

  Proof of Claim 2: Let  be an enumeration of M_2. Let p :=  ( / M_0; M_3). By extension, let q ∈^<∞ (M_3) be such that q extends p and q does not fork over M_0. Say q =  (' / M_3; M_3'^∗), and let M_2' be the model enumerated by '. We have that M_0M_2'M_3M_3^∗. By monotonicity, M_0M_2'M_1M_3^∗. By Claim 1 (where the role of M_1 and M_2 is reversed), ( / M_1; M_3) =  (' / M_1; M_3^∗). Let g: M_3^∗→ M_3' with M_3  M_3' be such that g (') = and g fixes M_1. Let M_1” := g[M_3] and let f := g  M_1'. This is as desired. †_Claim 2.

  Claim 3: Assume that M_0M_1M_2M_3 and _1, _2 ∈<∞M_1 are such that (_1 / M_0; M_3) =  (_2 / M_0; M_3). Then (_1 / M_2; M_3) =  (_2 / M_2; M_3).

  Proof of Claim 3: Let f: M_3 → M_3', M_3  M_3' be such that f fixes M_0 and f (_1) = _2. Using extension, find g: M_3' → M_3” that fixes M_1, such that M_3  M_3” and M_1g[M_3']M_3M_3”. By transitivity (for ), M_0g[M_3']M_2M_3. Let h := gf. We have in particular that M_0h[M_1]M_2M_3. Letting  be an enumeration of M_1 and using Claim 1, this means that ( / M_2; M_3) =  (h () / M_2; M_3). In particular, (_1 / M_2; M_3) =  (_2 / M_2; M_3). †_Claim 3.

  Claim 4 (uniqueness for types over models): Let M_0  M_2 and let p_1, p_2 ∈^<∞ (M_2) be given such that both do not fork over M_0 and p_1  M_0 = p_2  M_0. Then p_1 = p_2.

  Proof of Claim 4: Say p_ℓ =  (_ℓ / M_2; N_ℓ). Without loss of generality N := N_1 = N_2). By definition of nonforking of types, M_0_ℓM_2N. Expanding the definition of  and extending N if necessary, we have that for some M_1^ℓ containing _ℓ, ℓ = 1,2, M_0M_1^ℓM_2N. By Claim 2 applied to ^d, there exists M_1', N', and f: M_1^2 → M_1' such that f fixes M_0, M_1^1  M_1', N  N', and M_0M_1'M_2N'. Let _2' := f (_2). By Claim 3, (_1 / M_2; N') =  (_2' / M_2; N'). Moreover, Claim 1 implies that (_2' / M_2; N') =  (_2 / M_2; N'), so we get the desired result. †_Claim 4

  Claim 5 (transitivity): Let M_0  M_1  M_2 and let p ∈^<∞ (M_2). If p does not fork over M_1 and p  M_1 does not fork over M_0, then p does not fork over M_0 (note that this implies the transitivity statement of the theorem, since by definition of  we can always extend B so that it is a model extending M_2).

  Proof of Claim 5: By extension, let q ∈^<∞ (M_2) extend p  M_0 such that q does not fork over M_0. By monotonicity, q  M_1 does not fork over M_0. By Claim 4 (applied to q  M_1 and p  M_1), p  M_1 = q  M_1. By base monotonicity, q does not fork over M_1. By Claim 4 again, p = q. †_Claim 5

  Claim 6: Let p ∈^<∞ (B; N) with M  N and U M ⊆ B ⊆ U N. If p does not fork over M, then there exists q ∈^<∞ (N) such that q extends p and q does not fork over M.

  Proof of Claim 6: First note that this is not quite the same as the extension property, since we start of with a type that already does not fork over a smaller base. Write p =  ( / B; N). Fix M', N' such that M  M'  N', N  N', B ⊆ U M', and MM'N'. Let p' :=  ( / M'; N'). Note that p' extends p and p' does not fork over M. Find q' ∈^<∞ (N') such that q' extends p' and q' does not fork over M'. By Claim 5 (transitivity), q' does not fork over M. Let q := q'  N. By monotonicity, it is as desired. †_Claim 7
  
  Claim 7 (uniqueness):  Given p_1, p_2 ∈^<∞ (B; N) with M  N and U M ⊆ B ⊆ U N, if p_1  M = p_2  M and p_1, p_2 do not fork over M, then p_1 = p_2.

  Proof of Claim 7: Using Claim 6, find q_ℓ∈^<∞ (N) such that q_ℓ does not fork over M and extends p_ℓ for ℓ = 1,2. Thus in particular q_1  M = p_1  M = p_2  M = q_2  M. By Claim 4 (uniqueness for types over models), q_1 = q_2. In particular, p_1 = q_1  B = q_2  B = p_2. †_Claim 7


We can now state the local character property of forking: every type does not fork over a “small” set:


  We say that  has right local character if for each cardinal α, there exists a cardinal λ (depending on α) such that for any p ∈^α (M) there exists M_0  M with |U M_0| ≤λ and p not forking over M_0. Equivalently—avoiding any mention of Galois types—given M  N and N_1  N, if N_1 has cardinality at most α there exists M_0 of cardinality at most λ and N_1', N', such that N_1  N_1'  N', N  N', and M_0N_1'MN'.

  We say that  has left local character if ^d has right local character. We say that  has local character if it has both left and right local character.


The idea of the definition of local character is as follows: given M  N and N_1  N, we may want to write that N_0N_1MN, where N_0 is the pullback of N_1 and M over N. However pullbacks may not exist (and even when they do, the desired independence may not hold, see Example <ref>(<ref>)). Thus we say instead that we can close the intersection of N_1 and M to a small (meaning of cardinality depending only on α, the size of N_1) model M_0 so that N_1 and M do not interact over M_0. Importantly, we still require that M_0  M.

We will also study the following locality property, introduced as the “model-witness property” in <cit.>. When θ = ℵ_0 (as in the first-order case), it is often called—somewhat confusingly—the finite character property of forking.


  Let θ be an infinite cardinal. We say that  has the right (<θ)-witness property if M_0M_1M_2M_3 holds whenever M_0  M_ℓ M_3, ℓ = 1,2, and M_0M_1AM_3 for all A ⊆ U M_2 with |A| < θ.

  We say that  has the left (<θ)-witness property if ^d has it and we say that  has the (<θ)-witness property if it has both the left and right one. When θ is omitted, we mean that the witness property holds for some θ.


The witness property is known to follow from appropriate tameness assumptions. Recall from <cit.> that  is fully (<θ)-tame and short if Galois types are determined by the restrictions of their domain and variables of size less than θ. This holds in particular if θ≥κ, for some κ >  () strongly compact <cit.> or if θ≥μ and  is a universal μ-AEC (see the argument inside Remark <ref>). For the convenience of the reader, we replicate the precise statement and proof of the witness property from full tameness and shortness here:

[4.5 in <cit.>]
  Let θ be an infinite cardinal and assume that  is transitive and has existence and uniqueness. If  is fully (<θ)-tame and short, then  has the (<θ)-witness property.


  We prove the right (<θ)-witness property. The left version follows by applying the same argument to ^d. Assume that M_0  M_ℓ M_3, ℓ = 1,2, and M_0M_1AM_3 for all A ⊆ U M_2 with |A| < θ. Let  be an enumeration of M_1 and let p :=  ( / M_2;M_3). By extension for , let q ∈^<∞ (M_2) extend p  M_0 and not fork over M_0. We show that p = q, which is enough since  respects -embeddings. By full tameness and shortness, it is enough to see that p  A = q  A for any A of size less than θ. Fix such an A. We know that p  A does not fork over M_0 by assumption, and q  A also does not fork over M_0 by monotonicity. Therefore by uniqueness for  (Theorem <ref>), p  A = q  A, as desired.


  The argument does not need the full strength of tameness and shortness (in fact it only uses tameness, albeit for types of arbitrary length).



	Recall that for  a coregular locally presentable category, we can define an independence relation ^∗ on _reg consisting of effective pullback squares (see Definition <ref>).  We note that ^∗ has the witness property. To see this, use the fact that _reg is locally multipresentable, and that any locally multipresentable category whose morphisms are monomorphisms is fully tame and short (by the equivalence from <cit.> and the proof of Boney's theorem that universal classes are tame, see <cit.>), and hence Fact <ref> applies.


The witness property also makes the local character property more uniform: the cardinal λ from Definition <ref> becomes a simple function of α:


  Assume that  is right transitive, and satisfies existence, uniqueness, and the left (<θ)-witness property. If  has right local character, then there exists a cardinal λ_0 such that for each p ∈^α (M), there exists M_0 ∈ with M_0  M, |U M_0| ≤λ_0 + α^<(θ + μ) and p not forking over M_0.


  Let λ_0 be such that λ_0 = λ_0^<μ +  () and satisfy Definition <ref> with α, λ there standing for θ, λ_0 here. Now given any p ∈^α (M), for any I ⊆α with |I| < θ, pick M_I  M of cardinality at most λ_0 such that p^I does not fork over M_I. In the end, let A := ⋃_I ⊆α, |I| < θ M_I. Note that |A| ≤λ_0 + α^<θ, so one can pick M_0  M containing A of size at most |A|^<μ +  (). By base monotonicity and the left (<θ)-witness property, M_0 is as desired. 


We now turn to the main result of this section. If  is reasonable, accessibility of  is equivalent to the conjunction of the witness and local character properties. We will use the following auxiliary class:


  Assume that  is monotonic and right transitive. Let = (, ) be the obvious coding of  into an abstract class: the vocabulary is τ () = τ () ∪{P}, where P is a unary predicate, and we think of the members of  as pairs of τ ()-structures (M, N) satisfying M  N (so the elements of M are the ones satisfying the predicate). We order  by (M_0, M_1)  (M_2, M_3) if and only if M_0M_1M_2M_3. 


  Assume that  is monotonic and right transitive. Then  is isomorphic (as a category) to , so we need not distinguish between the two. We have that  is closed under isomorphisms, is a partial order, and  satisfies the coherence axiom: if (M_0, M_1) ⊆ (M_0', M_1')  (M, N) and (M_0, M_1)  (M, N), then by monotonicity (Lemma <ref>) (M_0, M_1)  (M_0', M_1'). However in ,  may not refine the τ ()-substructure relation: we could have (M_0, M_1)  (M, N) but U M_0 ⊊ U M ∩ U M_1.



  Let  be a μ-AEC and let  be an invariant independence relation on . Assume that  is transitive, and has existence and uniqueness. The following are equivalent:

  
  
  *   is accessible.
  
  *  For some λ,  satisfies all the axioms in the definition of a λ-AEC, except that  may not refine the τ ()-substructure relation (see Remark <ref>).
  
  *   has the witness and local character properties.
  


   
  
  * (<ref>) implies (<ref>): Assume that  is accessible. Pick λ_0 such that  is λ_0-accessible. By Remark <ref>, λ_0-directed colimits in ^2 and  coincide, so in particular  has concrete λ_0-directed colimits. By the proof of Theorem <ref> (here we are using that Lemma <ref> does not require  to extend substructure), we obtain the desired result.
    
  * (<ref>) implies (<ref>): Fix a regular cardinal λ such that  is a λ-AEC, except that  may not extend substructure. Fix M_0, M_1, M_2, M_3 such that M_0  M_ℓ M_3, ℓ = 1,2 and for any A ⊆ U M_1 with |A| ≤ (), M_0AM_2M_3 holds. We show that M_0M_1M_2M_3, which will establish the (< ()^+)-witness property. Since  is (almost) a λ-AEC, we can find a ()^+-directed system (M_0^i, M_1^i) : i ∈ I such that (M_0, M_1) = ⋃_i ∈ I (M_0^i, M_1^i) and |U M_1^i| ≤ () for all i ∈ I. This implies in particular that for any i ∈ I, M_0^iM_1^iM_0M_1. We also know that M_0M_1^iM_2M_3. By transitivity for  (Theorem <ref>), M_0^iM_1^iM_2M_3. In other words, (M_0^i, M_1^i)  (M_2, M_3) for all i ∈ I. By smoothness, this implies that (M_0, M_1)  (M_2, M_3), i.e. M_0M_1M_2M_3, as desired.

Similarly, local character holds: let M_0 ∈ and let p ∈^α (M_0). Without loss of generality α≤ () (if not use the witness property). Say p =  ( / M_0; M_1). As before, find a ()^+-directed system (M_0^i, M_1^i) : i ∈ I such that (M_0, M_1) = ⋃_i ∈ I (M_0^i, M_1^i) and |UM_1^i| ≤ () for all i ∈ I. Since I is ()^+-directed, there is i ∈ I such that ∈αM_1^i. Thus in particular p does not fork over M_0^i. Since |U M_0^i| ≤ (), we are done.
  
  * (<ref>) implies (<ref>): Fix a regular cardinal θ such that  has the (<θ)-witness property and let λ be as given by Lemma <ref>. We may assume without loss of generality that λ = λ^θ + μ +  () and λ is regular. By Remark <ref>, it is enough to prove that  satisfies the Löwenheim-Skolem-Tarski and chain axioms of λ-AECs.

    
    
  * LST axiom: Let (M, N) ∈ and let A ⊆ U N. Without loss of generality |A| is infinite. We build M_i, N_i : i < λ increasing in  such that for all i < λ:
      
      
  * A ⊆ U N_0.
      
  * M_i  M, M_i  N_i  N.
      
  * |U M_i| ≤ |U N_i| ≤ |A|^<λ.
      
  * M_i⋃_j < i U N_jMN.
      

      This is enough: Let M_λ := ⋃_i < λ M_i, N_λ := ⋃_i < λ N_i. We claim that M_λN_λMN. Indeed, recall that λ≥θ so for any B ⊆ U N_λ with |B| < θ, there exists i < λ such that B ⊆ U N_i. In particular by monotonicity M_i + 1BMN. By base monotonicity, M_λBMN. Since B was arbitrary, by the (<θ)-witness property we indeed have that M_λN_λMN. However we also have that M_λ N_λ, so M_λN_λMN, and hence (M_λ, N_λ)  (M, N), as needed.

      This is possible: given M_j : j < i, N_j : j < i, take any M_i  M such that |U M_i| ≤ |A|^<λ, M_j  M_i for all j < i, A ∩ U M ⊆ U M_i, and M_i⋃_j < i N_jMN (use local character, Lemma <ref>, and base monotonicity). Now given M_j : j ≤ i and N_j : j < i, pick any N_i  N such that N_j  N_i for all j < i, |U N_i| ≤ |A|^<λ, M_i N_i, and A ⊆ U N_i.

    
  * Chain axioms: Fix (M_i, N_i) : i ∈ I a λ-directed system in  and let (M, N) := ⋃_i ∈ I (M_i, N_i). Clearly, (M, N) ∈. We first want to see that M_iN_iMN. We use the witness property. Fix A ⊆ U M of size less than θ. Since θ≤λ, there exists j ∈ I such that i ≤ j and A ⊆ U M_j. Since M_iN_iM_jN_j by assumption, we must have by monotonicity that M_iN_iAN. Since A was arbitrary, this implies that M_iN_iMN. Similarly, smoothness also holds: fix (M', N') such that (M_i, N_i)  (M', N') for all i ∈ I. We want to show that MNM'N'. Fix A ⊆ U N of size less than θ and fix i ∈ I such that A ⊆ U N_i. By assumption, M_iN_iM'N', so M_iAM'N', and so by base monotonicity MAM'N'. Since A was arbitrary, the witness property implies that MNM'N', as desired.
    
      


  The proof goes through if we assume only that  is invariant and monotonic, and that  is transitive.


We deduce that having a stable independence notion implies stability and tameness:


  Let  be a μ-AEC with a stable independence relation. Then:

  
  
  * (Stability for Galois types) For any α, there exists a proper class S_α of cardinals such that for any λ∈ S_α and M ∈_λ, |^<α (M)| = λ.
  
  * (Tameness) For any α, there exists a cardinal λ such that for any M ∈, p, q ∈^<α (M), p = q if and only if p  A = q  A for all A ⊆ U M with |A| < λ.
  


  Use Theorem <ref> and imitate the argument in <cit.>, using local character and uniqueness.


We will see in the next section (Corollary <ref>) that if, in addition,  has chain bounds, it fails to have a certain order property.



§ CANONICITY, SYMMETRY, AND THE ORDER PROPERTY


In this section, we prove (assuming chain bounds) that stable independence is canonical: there is at most one stable independence relation in any class. In fact, the symmetry property is not necessary, and hence can be deduced from the others. We show further that symmetry implies failure of an order property. Combined with Corollary <ref>, this shows that the class has several features of stable first-order theories.

Note that the proof of symmetry and the order property here are different from those in <cit.>: since we are working in a more general context than AECs, we do not have Ehrenfeucht-Mostowski models (<cit.>) and hence cannot directly deduce that the order property implies instability as in <cit.>. In fact, this fails in general, see Example <ref>.


  Let  be a μ-AEC which has chain bounds (see Definition <ref>). Let  be an invariant, transitive independence notion on  with existence, uniqueness, and right local character.

  Then any other invariant transitive relation on  satisfying existence, and uniqueness must be .


  First note that the properties of  carry over to , by Fact <ref> and Theorem <ref>. By Theorem <ref>, we can work inside a monster model  which is as homogeneous as we need (if  does not have joint embedding, we can partition it into subclasses that each have joint embedding by looking at the equivalence classes of the relation “embedding into a common model”). We want to imitate the argument of <cit.>. It falls into two parts. The first part shows that  must have a property there called (E_+) (a strong existence/extension property, which we recall below). The second part shows that having (E_+) implies canonicity. This latter part is implemented in <cit.>, and does not use the fact that  is an AEC: the argument works in any μ-AEC (and, in fact, in any coherent abstract class).

  It remains to check that  has (E_+): for any M N_0 and set A, there is N_0  N such that for all N'≅_N_0 N, there is N_0'≅_M N_0 with MAN_0' and N_0' N' (see <cit.>). The proof of (E_+) uses independent sequences. These are sequences A_i : i < δ (inside a monster model) such that for some N_i : i < δ (called the witnesses), N_i : i < δ is an increasing chain, ⋃_j < i A_j ⊆ U N_i and N_0A_iN_i for all i < δ. We say that A_i : i < δ is independent over M when there is a witnessing sequence N_i : i < δ with M = N_0.

  The first step <cit.> is to show that we can always build independent sequences: given A, M, and δ, there exists A_i : i < δ that are independent over M and such that the type of each A_i over M is the same as the type of A over M (for some enumerations of A_i and A). The argument uses the extension property for  and goes through in the present setup too.

  The second and last ingredient in the proof of (E_+) is to show <cit.> a certain local character property of independent sequences. The proof uses symmetry, so we will instead use the following variation:

  Claim: Let A be a set and let λ≥μ be a regular cardinal such that any type of a sequence of length |A| does not fork over a model of cardinality strictly less than λ (this exists by right local character). Then whenever M_i : i < λ is a ^d-independent sequence (i.e. an independent sequence with respect to ^d) with M  M_i for all i < λ, then there is i < λ with MAM_i.

  Proof of Claim: This is the same proof as in <cit.>, but we give it for the convenience of the reader. Let N_i : i < λ witness the independence and let N_λ := ⋃_i < λ N_i. By right local character and base monotonicity, there exists i < λ such that N_iAN_λ. By monotonicity, N_iAM_i. Since the M_i's are ^d-independent, we also have that MN_iM_i. Using left transitivity, we get that MAM_i, as desired. †_Claim

  Now that the claim is proven, the argument of <cit.> goes through to show that  has (E_+), completing the proof.


  It is enough to assume that  has monster models instead of having chain bounds, see Theorem <ref>.


We deduce the symmetry property:


  If  is a μ-AEC which has chain bounds and  is an invariant, transitive independence notion with existence, uniqueness and right local character, then  is symmetric. Thus if  has, in addition, the right (or left) witness property, it is a stable independence relation.


  ^d is invariant, transitive, and has existence and uniqueness, so by Theorem <ref> = ^d. The last sentence follows from Theorem <ref>.



  Let  be a category which has chain bounds and whose morphisms are monomorphisms. Then there is at most one stable independence notion on .


  By Lemma <ref>, the equivalence between μ-AECs and accessible categories with all morphisms monomorphisms, Theorem <ref>, and Theorem <ref>.



  Can one prove an even more general canonicity result? What if the morphisms of the category are not all monomorphisms? What if the category does not have chain bounds?


The following remark shows that a weakening of chain bounds is enough for the proof of the canonicity theorem to go through.


  Let κ be an infinite cardinal and consider the class  of κ-local graphs studied in Example <ref>. Then  has a stable independence notion  but does not have chain bounds. However the proof of Theorem <ref> still goes through: chain bounds are only used to build independent sequences, and, at least in this case, we can achieve the same effect by other methods. Let A be a κ-local graph and let M be a fixed κ-local graph. For simplicity, assume that M is a full subgraph of A. We build A_i :i < δ a sequence of κ-local graphs with A_i ≅_M A for all i < δ, as well as N_i : i < δ an increasing continuous sequence of κ-local graphs such that N_0 = M, N_i + 1 = N_i ∪ A_i and MA_iN_iN_i + 1 for i < δ. Using the existence property, it is easy to implement the successor step. For the limit step, it is enough to check that for i limit, N_i = ⋃_j < i N_j is again κ-local. So let v be a vertex of N_i. Pick j < i such that v ∈ N_j. By construction, there are no edges from v to A_j' for j' ≥ j, and moreover N_i = ⋃_j' < i A_j', so any edge of v in N_i must be contained in N_j, which is κ-local, as desired.


We will use the following definition of the order property, introduced by Shelah for AECs <cit.>. In the first-order case, it is equivalent to the usual definition.


  A μ-AEC  has the α-order property of length θ if there exists M ∈ and a sequence _i : i < θ such that:
  
  
  
  * _i  ∈αU M for all i < θ.
  
  * For all i_0 < j_0 < θ and i_1 < j_1 < θ, (_i_0_j_0 / ∅; M) ≠ (_j_1_i_1 / ∅; M).
  

   has the order property if there exists α such that for all θ,  has the α-order property of length θ.



  Let  be a μ-AEC and  be an invariant, transitive independence notion with existence, uniqueness, and right local character. If  has chain bounds, or more generally if  has symmetry, then  does not have the order property.

 
  Recall from Corollary <ref> that chain bounds implies symmetry in the context of the theorem. Assume for a contradiction that  has the order property. Pick α and θ such that  has the α-order property of length θ^+, with θ = θ^<μ≥ () “sufficiently big” (the proof will tell us how big) such that |^α (M)| ≤θ for all M ∈_≤θ (exists by the proof of Corollary <ref>). Pick _i : i < θ^+ and M witnessing the α-order property of length θ^+. Let M_i : i < θ^+ be an increasing sequence of submodels of M such that {_j : j < i}⊆ U M_i for all i < θ^+ and |U M_i| ≤θ.   For each i < θ^+ of sufficiently high cofinality, there exists j < i such that (_i / M_i) does not fork over M_j. By Fodor's lemma, we can therefore assume without loss of generality that (_i / M_i; M) does not fork over M_0 for all i < θ^+. Since |^α (M_0)| ≤θ, we can further assume without loss of generality that (_i / M_0; M) =  (_i' / M_0; M) for all i < i' < θ^+. By uniqueness, this implies that (_i / M_i; M) =  (_i' / M_i; M).

  We are now in the following setup: _0 _1 is a two-element independent sequence over M_0 (in this case, this means that M_0_0_1M), and so by symmetry also M_0_1_0M, so _1 _0 is a two-element independent sequence over M_0. Now the proof of <cit.> (specifically the successor step) tells us that if we have two two-element independent sequences  and ' ' over a model M_0, so that the types of their individual elements are equal over M_0 (i.e.  and ' agree over M_0 and  and ' agree over M_0), then in fact  and ' ' have the same type over M_0. Applying this here, we get that (_0 _1 / M_0; M) =  (_1 _0/ M_0; M). This contradicts that the sequence _i : i < θ^+ witnessed the order property. 



  If  is a μ-AEC with a stable independence relation, then  does not have the order property. 


  This is a consequence of Theorem <ref>, using the equivalence given by Theorem <ref>.


 We may now give an alternate proof of the fact that , the category of Banach spaces and linear contractions, does not have effective unions (see Example <ref>(4)). If  had effective unions, the corresponding μ-AEC  of Banach spaces with isometries would have a stable independence notion by Theorem <ref>. By Corollary <ref>, this means that  does not have the order property (see Definition <ref>). Take however the Banach space c_0 of complex-valued sequences a_n : n < ω going to zero with the supremum norm. Let e_n be the sequence that is one at position n and 0 elsewhere. Let f_n := ∑_i ≤ n e_i. Then e_m + f_n = 2 if and only if m ≤ n (see <cit.>). Thus c_0 satisfies an instance of the order property of length ω. By the compactness theorem for continuous first-order logic (see e.g. <cit.>), this implies that  must have the order property.  Contradiction.

Shelah <cit.> examines several definitions of stability for _κ, κ, κ a strongly compact cardinal, and shows that, while there are natural implications, there are also several non-equivalences. The following is a simple example:


  Let  be the ℵ_1-AEC of well-orderings, ordered by being a suborder. Then  has the order property but for every M ∈, |^<ω (M)| = |U M| + ℵ_0. 


On the other hand it is known that failure of the order property implies stability in terms of counting Galois types. Roughly, this is because the proof of the corresponding first-order fact can be carried out inside a fixed model, see <cit.>, <cit.>, or the proof of <cit.>.



§ STABLE INDEPENDENCE ON SATURATED MODELS


Putting together <cit.>, one obtains the following converse to Corollary <ref>: Assuming large cardinals, any μ-AEC which has chain bounds and does not have the order property admits a stable independence relation on a subclass of model-homogeneous models. This was essentially observed for AECs with amalgamation by Boney and Grossberg <cit.>, although categoricity is used there to prove local character. We work from the result of <cit.>, which focuses on the stable case and avoids any use of categoricity. Note that it is in general necessary to pass to a subclass of model-homogeneous models, see Example <ref>(<ref>).


  Assume Vopěnka's principle. Let  be a μ-AEC which has chain bounds. Let κ >  () be strongly compact. If  does not have the order property, then the κ-AEC  of locally κ-model-homogeneous models of  (see Definition <ref>) has a stable independence relation.


  By Theorem <ref>,  has amalgamation and is indeed a κ-AEC. First observe that for any α < κ,  does not have the α-order property of length κ. If it did, then we would be able to take repeated ultraproducts of the universe to make the sequence longer and get that  has the α-order property of any length. Moreover, the strongly compact also gives that  is fully (<κ)-tame and short (see <cit.>).

  Now define M_0M_1M_2M_3 to hold if and only if M_0  M_ℓ M_3, ℓ = 1,2, all are in , and for any ∈<κM_1, ( / M_2; M_3) is a κ-coheir over M_0. That is, for any A ⊆ U M_2 of cardinality less than κ, ( / A; M_3) is realized inside M_0.

  By the proofs of <cit.> or <cit.>, we get all the required properties. More precisely, it is straightforward to check that  is invariant and monotonic. Transitivity, symmetry, and uniqueness are proven as in <cit.> or <cit.>. The (<κ)-witness property then follows from the definition. Local character is <cit.> and existence is <cit.>. Using Theorem <ref>, we get that  is stable, as desired.


  The assumption of Vopěnka's principle can be removed if  has amalgamation.


We summarize:


  Assume Vopěnka's principle. Let  be a μ-AEC which has chain bounds. The following are equivalent:

  
  
  *  does not have the order property.
  
  * For some sub-λ-AEC ^∗⊆ which is cofinal in  and has the same ordering as , ^∗ has a stable independence notion.
  


  Combine Theorems <ref> (noting that if  has the order property, then any of its cofinal subclasses must have it) and <ref>. 


Note that in general, one can always restrict a stable independence relation to a subclass of model-homogeneous models. In fact:


  Let  be a μ-AEC with a stable independence relation . Let ^∗ be a subclass of  (ordered by the appropriate restriction of ) satisfying the following conditions:

  
  
  * ^∗ is a λ-AEC, for some λ.
  
  * ^∗ is cofinal in  (that is any M ∈ extends to an N ∈^∗).
  

  Then the restriction ^∗ of  to ^∗ is a stable independence relation on ^∗.


  We use Theorem <ref>. It is straightforward to check that ^∗ is an invariant, monotonic, symmetric, and transitive independence relation with the witness property. Existence and uniqueness follow from the corresponding properties of  and the fact that ^∗ is cofinal in . To see local character, we use local character of  together with the fact that ^∗ is a λ-AEC, hence satisfies the Löwenheim-Skolem-Tarski axiom.


Restricting to a subclass of sufficiently homogeneous models, we can also get that any stable square is a pullback square (i.e. is disjoint over the base). The argument is similar to <cit.> but we give a self-contained version here.


  Let  be a stable independence relation on a μ-AEC . Let λ >  () and let M  N both be in  with M λ-model-homogeneous. Let p ∈ (N) and assume that p does not fork over M_0, with M_0  M and |U M_0|^<μ < λ. Then p is algebraic (i.e. realized inside its domain) if and only if p  M is algebraic. 


  If p  M is algebraic, then p is clearly algebraic. Conversely, assume that p is algebraic. Pick a ∈ N realizing p and let N_0 ∈_<λ be such that M_0  N_0  N and N_0 contains a. Since M is λ-model-homogeneous, there exists f: N_0 → M fixing M_0. Now by monotonicity p  N_0 does not fork over M_0, so f (p  N_0) does not fork over M_0. Since f fixes M_0, p  M_0 = f (p  N_0)  M_0, so by uniqueness f (p  N_0) = p  f[N_0]. Since p  N_0 is algebraic, this implies that p  f[N_0] also is. Since f[N_0]  M, p  M is algebraic, as desired.



  Let  be a stable independence relation on a μ-AEC . There exists a regular cardinal λ >  () such that if M_0M_1M_2M_3 and M_0 is λ-model-homogeneous, then M_1 ∩ M_2 = M_0. 


  By local character, we can pick λ >  () regular so that λ_0^<μ < λ for all λ_0 < λ and moreover any type of length one does not fork over a model of cardinality strictly less than λ. Now assume that M_0M_1M_2M_3 and M_0 is λ-model-homogeneous. Let a ∈ M_1 ∩ M_2. We show that a ∈ M_0. Let p :=  (a / M_2; M_3). Note that p is algebraic. By how λ was chosen, there exists M_0'  M_0 such that M_0' ∈_<λ and p  M_0 does not fork over M_0'. By transitivity, p does not fork over M_0'. By Lemma <ref> (where M_0, M, N there stand for M_0', M_0, M_2 here), p  M_0 is algebraic, so a ∈ M_0, as desired.


amsalpha


