# A note on stable Kim-forking

Yvon Bossut

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Yvon Bossut. A note on stable Kim-forking. 2025. hal-04767527v3

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A NOTE ON WEAKLY STABLE KIM-FORKING

Yvon BOSSUT

Abstract : We define weak stable Kim-forking, a notion that generalizes stable forking to the context of NSOP$_1$ theories. We adapt some of the known results on stable forking to this context.

# 1 Introduction

NSOP$_1$ theories have recently been studied as a generalization of simple theories. The notion of Kim-forking plays a similar role in NSOP$_1$ theories than forking in simple theories. Kim and Pillay [14] have shown that forking independence in simple theories is characterized by some of its properties (see [17] for example). Chernikov and Ramsey have shown similar results for Kim-independence in NSOP$_1$ theories, [8, Theorem 5.8].

Here Kim-forking is considered at the level of formulas. The stable forking conjecture is the conjecture that in a simple theory, if a complete type forks over a set of parameter, then there is a stable formula inside of this type which forks. In short, that forking is witnessed by stable formulas. A generalization of the notion of stable forking to the context of NSOP$_1$ theories, namely weak stable Kim-forking, is given here. We try to connect this notion to the results known about stable forking, which can be found in [6]. For example it is known that some theory $T$ has stable forking then the imaginary extension $T^{eq}$ also does. However I could not prove the equivalent result here for some reasons discussed in Section 5.

We can also cite on this subject [1] a recent article by Baldwin, Freitag and Mutchnik in which they demonstrate, among other things, that Kim-forking in an NSOP$_1$ theory is always witnessed by a simple formula, that is, one that does not have the tree property.

# 2 Preliminaries : Independence relations and local types

## 2.1 Without the existence axiom

Let $M \models T$ be a monster model of some complete theory. We introduce the relations of independence that will appear here.

**Definition 2.1.1.** Let $b_0, e$ be tuples of $M$ and $\varphi(x, b_0)$ be a formula.

1. $\varphi(x, b_0)$ divides over $e$ if there is an $e$-indiscernible sequence $I = (b_i : i < \omega)$ such that :
$$ \{\varphi(x, b_i) : i < \omega\} \text{ is inconsistent.} $$

In that case we say that $\varphi(x, b_0)$ divides over $e$ with respect to $I$.

2. A partial type $p(x, b)$ forks over $e$ if it implies a finite disjunction of formulas each of which divides over $e$.

3. Let $a \in M$. We write $a \searrow_e^f b$ to denote the assertion that $tp(a/eb_0)$ does not fork over $e$ and $a \searrow_e^d b_0$ to denote the assertion that $tp(a/eb_0)$ does not divide over $e$.

4. A sequence of tuples $(a_i : i \in I)$ is called e-Morley if it is e-indiscernible and if $a_i \searrow_e^f a_{<i}$ for every $i \in I$.
**Definition 2.1.2.** A set $e \subseteq \mathbb{M}$ is an extension basis if $a \downarrow_e^f e$ for every $a \in \mathbb{M}$. A theory $T$ has existence if every set is an extension basis. This is equivalent to saying that for every $e \in \mathbb{M}$ and every $p(x) \in S(e)$ there is an $e$-Morley sequence in $p$.

**Remark 2.1.3.** Every model is an extension basis, simple theories satisfy existence, and it was a conjecture that NSOP₁ theories does as well before a counterexample was found by Mutchnik in [16].

**Definition 2.1.4.** Consider a small model $M \prec \mathbb{M}$. A sequence of tuples $(a_i : i \in I)$ is called Coheir Morley over $M$ if it is $M$-indiscernible and if $\text{tp}(a_i/Ma_{<i})$ is finitely satisfiable in $M$ for every $i \in I$. Such a sequence is always $M$-Morley.

There are a lot of simplicity-like results characterizing NSOP₁-theories in terms of properties of Kim-forking, for now let us set the following definition :

**Definition 2.1.5.** Let $T$ be a complete theory. $T$ is NSOP₁ if and only if Kim-independence is symmetric over models, meaning that $a \downarrow_M^K b$ iff $b \downarrow_M^K a$ for every $a, b \in \mathbb{M}$ and $M \prec \mathbb{M}$.

Without the hypothesis of existence we define Kim-forking over models, this notion has been studied in [10] and [12] for example.

**Definition 2.1.6.** Consider a small model $M \prec \mathbb{M}$. Let $b_0 \in \mathbb{M}$ and let $\varphi(x, b_0)$ be a partial type over $Mb$.

1. $\varphi(x, b_0)$ *Kim-divides over M* if there is an $M$-Coheir-Morley sequence $(b_i : i < \omega)$ such that :

$$ \{\varphi(x, b_i) : i < \omega\} \text{ is inconsistent.} $$

We will say that $\varphi(x, b_0)$ *Kim-divides over M* with respect to $(b_i : i < \omega)$.

2. $p(x, b)$ *Kim-forks over M* if it implies a finite disjunction of formulas each of which Kim-divides over $M$.

3. Let $a \in \mathbb{M}$. We write $a \downarrow_M^K b_0$ to denote the assertion that $\text{tp}(a/Mb)$ does not Kim-fork over $M$ and $a \downarrow_M^{Kd} b$ to denote the assertion that $\text{tp}(a/Mb)$ does not Kim-divide over $M$.

This is the notion of Kim-forking that we will use in Section 3.

## 2.2 Assuming existence

If we assume that the theory we are working in satisfies existence we can define Kim-forking over arbitrary sets :

**Definition 2.2.1.** Consider a tuple $e \in \mathbb{M}$. Let $b_0 \in \mathbb{M}$ and $\varphi(x, b_0)$ be a partial type over $eb$. $\varphi(x, b_0)$ *Kim-divides over e* if there is an $e$-Morley sequence $(b_i : i < \omega)$ such that $\{\varphi(x, b_i) : i < \omega\}$ is inconsistent. We will say that $\varphi(x, b_0)$ *Kim-divides over e* with respect to $(b_i : i < \omega)$. Kim-forking over $e$ is defined similarly.

This is the notion of Kim-forking that we will use in Section 4, its properties in NSOP₁ theories have been studied in [11] and [9] for example. To give some example the following notion will also appear in this section :

**Definition 2.2.2.** We define *algebraic independence* as the relation $A \downarrow_C^a B := acl(AC) \cap acl(BC) = acl(C)$.
In this section we consider Kim-forking at the level of formulas. The stable forking conjecture is the conjecture that in a simple theory, if a complete type forks over a set of parameter, then there is a stable formula inside of this type which forks. In short, that forking is witnessed by stable formulas.

A generalization of the notion of stable forking to the context of $\text{NSOP}_1$ theories, namely weak stable Kim-forking, is given here. We connect this notion to some results known about stable forking which can be found in [6]. It is known that if a theory $T$ has stable forking then the imaginary extension $T^{\text{eq}}$ also does. However I could not prove the equivalent result here for some reasons discussed in Section 5.

## 2.3 Local types

We recall some basic notions around local types. Let $\varphi(x, y)$ be a formula.

**Definition 2.3.1.** A $\varphi$-formula over $A$ is a boolean combination of formulas of the form $\varphi(x, a)$ for $a \in A$. A generalized $\varphi$-formula over $A$ is a formula over $A$ that is equivalent to a boolean combination of $\varphi$-formulas, possibly with parameters outside of $A$.

A complete $\varphi$-type over $A$ (resp. complete generalized $\varphi$-type over $A$) $p$ is a consistent set of $\varphi$-formulas (resp. generalized $\varphi$-formulas) over $A$ such that given any $\psi(x)$ $\varphi$-formula (resp. generalized $\varphi$-formula) over $A$ either $\psi(x) \in p$ or $\neg\psi(x) \in p$.

The set of complete $\varphi$-types (resp. generalized $\varphi$-types) over $A$ is written $S_{\varphi}(A)$ (resp. $S_{\varphi^*}(A)$) and the $\varphi$-type (resp. generalized $\varphi$-type) of $a$ over $A$ is written $\text{tp}_{\varphi}(a/A)$ (resp. $\text{tp}_{\varphi^*}(a/A)$).

**Definition 2.3.2.** A $\varphi$-type $p$ is said to be definable over some set of parameters $A$ if there is a formula $d_p x \varphi(x, y)$ with parameters in $A$ such that $\varphi(x, b) \in p$ if and only if $\models d_p x \varphi(x, b)$.

**Fact 2.3.3.** [14, Chapter 2] If $\varphi(x, y, e)$ is a stable formula and $M$ is a model such that $e \in M$, then every $\varphi(x, y, e)$-type $p$ over $M$ is definable over $M$ and has a unique non-forking extension, which has the same definition. We write $d_p x \varphi(x, y, e)$ its definition and $\text{Cb}(d_p x \varphi(x, y, e))$ the canonical parameter of this definition.

The previous result holds more generally for types over algebraically closed sets in $T^{\text{eq}}$.

**Fact 2.3.4.** [4, Proposition 6.13]: Let $\varphi(x, y)$ be a stable formula and $E = \text{acl}^{\text{eq}}(E) \subseteq B$. Then any generalized $\varphi$-type $p_{\varphi} \in S_{\varphi^*}(E)$ is definable over $E$, and it has a unique extension $q \in S_{\varphi^*}(B)$ which is also definable over $E$. The formula $d_p x \varphi(x, y)$ defining $p$ then admits a canonical parameter in $E$, which we write $\text{Cb}(d_p x \varphi(x, y))$.

## 3 Weak stable Kim-forking over models

In this section we work in any complete theory $T$ in a language $\mathcal{L}$, all sets of parameters and elements are assumed to be in a monster model $\mathbb{M}$ of $T$. We show some basic results about what can and can not happen when Kim-dividing is witnessed by stable formulas, and isolate the notion of 'weak stable Kim-forking' as the correct generalization of stable forking in simple theories to the $\text{NSOP}_1$ context. In this section we only consider types over models and Kim-forking over models.

**Definition 3.1.** A theory $T$ has *stable Kim-forking over models* (resp. *stable forking over models*) if whenever a $\mathcal{L} \not\subsetneq_M B$ for $M \subseteq B$ and $M$ a model (resp. $\mathcal{L} \not\subsetneq_B M$) there is a stable formula $\varphi(x, y) \in \mathcal{L}$ and $b \in B$ such that $a \models \varphi(x, b)$ and $\varphi(x, b)$ Kim-forks (resp. forks) over $M$.
**Definition 3.2.** A theory $T$ has weak stable Kim-forking over models (resp. weak stable forking over models) if whenever $a \llcorner_M^K B$ (resp. $a \llcorner_M^f B$) for $M \subseteq B$ and $M$ a model there is a stable formula $\varphi(x, y, m) \in \mathcal{L}_m$ (i.e. the formula with fixed $m \in M$ is stable) and $b \in B$ such that $|\varphi(a, b, m)| = |\varphi(x, b, m)$ Kim-forks (resp. forks) over $M$.

It is known that a theory with stable forking is simple [6, Proposition 1.3]. We show that a theory with stable Kim-forking over models is simple and that a theory with weak stable forking over models is simple. We also show that a theory with weak stable Kim-forking over models is NSOP$_1$ and not necessarily simple.

**Lemma 3.3.** Let $M \subseteq N$ be models of a theory $T$ with weak stable forking over models. For any type $p = \text{tp}(a/N)$, $a \downarrow_M^f N$ if and only if for every stable formula $\varphi(x, y, m)$ with $m \in M$ the canonical parameter $\text{Cb}(d_p x \varphi(x, y, m))$ is in $M$.

*Proof.* [⇒] By Fact 2.3.3 we know that the $\varphi(x, y, m)$-type of $a$ over $M$ is definable over $M$ and admits a unique non-forking extension to $N$, which has the same definition over $M$, so in particular the same canonical parameter $\text{Cb}(d_p x \varphi(x, y, m)) \in M$.

[ ⇐ ] Conversely assume that $p$ forks over $M$ and that $M$ contains the canonical parameters. Then there is a stable formula $\varphi(x, y, m)$ with $m \in M$ and $n_0 \in N$ such that $\varphi(x, n_0, m) \in p$ and $\varphi(x, n_0, m)$ forks over $M$.

So there is an $M$-indiscernible sequence $(n_i)_{i<\omega}$ such that $\{\varphi(x, n_i, m) : i < \omega\}$ is inconsistent. Now since $\text{Cb}(d_p x \varphi(x, y, m)) \in \text{acl}^{\text{eq}}(M)$ by indiscernibility $|\varphi(x, n_i, m)| = |\varphi(x, n_i, m)|$ for all $i < \omega$, so $\{\varphi(x, n_i, m) : i < \omega\}$ is satisfied by any realization of a non-forking extension of $p$ to $N(n_i)_{i<\omega}$, a contradiction. □

**Proposition 3.4.** Let $T$ be a theory with weak stable forking over models, then $T$ is simple.

*Proof.* We show that $\downarrow^f$ satisfies local character. Let $a$ be a finite tuple, $M$ a model, $p = \text{tp}(a/M)$. We consider the set $C_0 = \{\text{Cb}(d_p x \varphi(x, y)): \varphi(x, y) \in \mathcal{L} \text{ a stable formula}\}$. We take a model $M_0 \subseteq M$ such that $|M_0| = |T|$ and $M_0$ contains a representative of every element of $C_0$.

We iterate the process taking $C_{i+1} = \{\text{Cb}(d_p x \varphi(x, y, e)): \varphi(x, y, e) \in \mathcal{L}_e \text{ a stable formula with } e \in M_i\}$ and $M_i \subseteq M_{i+1} \subseteq M$ such that $|M_{i+1}| = |T|$ and $M_{i+1}$ contains a representative of every element of $C_{i+1}$. Let $M' := \bigcup_{i<\omega} M_i$. By Lemma 3.3 $a \downarrow_{M'}^f M$ and $M'$ has size $|T|$. □

**Lemma 3.5.** Let $\mathfrak{M}$ be a monster model of a theory $T$ with weak stable Kim-forking over models. Let $\mathfrak{M} \subseteq \mathfrak{B} \subseteq \mathfrak{M}$ with $\mathfrak{M}$ a model and $p := \text{tp}(a/B)$ a type. Then $a \downarrow_M^K B$ if for every stable formula $\varphi(x, y, m)$ with $m \in M$ the canonical parameter $\text{Cb}(d_p x \varphi(x, y, m))$ is in $M$.

*Proof.* Assume that $p$ Kim-forks over $M$ and that $M$ contains all of the canonical parameters. There is a stable formula $\varphi(x, y, m)$ with $m \in M$ and $n_0 \in N$ such that $\varphi(x, n_0, m) \in p$ and $\varphi(x, n_0, m)$ Kim-forks over $M$. So there is an $M$-Morley sequence $(n_i)_{i<\omega}$ such that $\{\varphi(x, n_i, m): i < \omega\}$ is inconsistent.

By assumption $\text{Cb}(d_p x \varphi(x, y, m)) \in M$, so by indiscernibility $|\varphi(x, n_i, m)| = |\varphi(x, n_i, m)|$ for all $i < \omega$, and $\{\varphi(x, n_i, m): i < \omega\}$ is satisfied by any realization of a non-forking extension of $p$ to $N(n_i)_{i<\omega}$, which contradicts the fact that $\varphi(x, n_0, m)$ Kim-forks over $M$. □

**Proposition 3.6.** Let $T$ be a theory with weak stable Kim-forking over models, then $T$ is NSOP$_1$.

*Proof.* We show that $\downarrow^K$ satisfies the local character criterion of [7, Proposition 2.6] to show that $T$ is NSOP$_1$. Assume that $T$ has weak stable Kim-forking and that there exists a continuous increasing
sequence of models of size $|T|$ ($M_i$)$_{i<\kappa}$ for $\kappa = |T|^{+}$ and a finite tuple $a$ such that $a \not\ll_M^K M_{i+1}$ for all $i < \kappa$. Let $M := \bigcup_{i<\kappa} M_i$ and $p := \text{tp}(a/M)$.

For every $i < \kappa$ the number of stable formulas with parameters in $M_i$ is at most $|T|$, since $\kappa > |T|$ is regular there is a smallest index $\alpha_i \ge i$ such that for every stable formula $\varphi(x, y, e)$ with $e \in M_i$, $\text{Cb}(d_p x \varphi(x, y, e)) \in M_{\alpha_i}$. Now let us consider $(\beta_i)_{i<\omega}$ defined by induction by $\beta_0 = 0$ and $\beta_{i+1} = \alpha_{\beta_i}$. Take $\beta = \bigcup_{i<\omega} \beta_i$, by continuity $\alpha_\beta = \beta$, so $a \downarrow_M^K M$ by Lemma 3.5, which contradicts $a \not\ll_M^K M_{\beta+1}$. $\square$

**Corollary 3.7.** Let $T$ be a theory with stable Kim-forking over models, then $T$ is simple.

*Proof.* By Proposition 3.6 we know that $T$ is NSOP$_1$. If $T$ is non-simple we can build an arbitrary long sequence of Kim-forking extensions with an instance of $\text{TP}_2$ (as in [5, Proposition 3.6]). Now we can run the same proof as in [6, Proposition 1.3]: There is a stable formula with a dividing chain of infinite length and hence with the tree property, which is not possible. $\square$

**Lemma 3.8.** Let $T$ be an theory. If $p = \text{tp}(a/M)$ is a complete type over a model $M$, $M \subseteq N$ is a model, and $\varphi(x, y, m)$ is a stable formula for some fixed $m \in M$, then there is a unique complete $\varphi$-type extending $p$ that does not Kim-fork over $M$. The unique global $\varphi(x, y, e)$-type $q_\varphi$ extending $p$ without Kim-forking is definable over $M$, we write $d_p^K x\varphi(x, y, e)$ its definition and $\text{Cb}(d_p^K x\varphi(x, y, e))$ the canonical parameter of this definition.

*Proof.* Assume that there are $a', a'' \models p$ such that $a' \downarrow_M^K N$, $a'' \downarrow_M^K N$ and that for some $b \in N$, $\models \varphi(a', b, m)$ and $\models \neg\varphi(a'', b, m)$. Consider an $M$-Coheir Morley sequence $(N_i)_{i<\omega}$ such that $N = N_0$. Let $(b_i)_{i<\omega}$ be the corresponding sequence with $b_0 = b$, and let $a'_i, a''_i$ be such that $a'_i N_i \equiv_M a' N_0$ and $a''_i N_i \equiv_M a'' N_0$, so $\models \varphi(a'_i, b_i, m)$ and $\models \neg\varphi(a''_i, b_i, m)$ for every $i < \omega$.

By applying the amalgamation theorem for Kim-forking over $M$ along the sequence $N_i$ we can construct a sequence $(a_i)_{i<\omega}$ such that $\models \varphi(a_i, b_j, m)$ iff $i < j$, contradicting the stability of $\varphi(x, y, m)$. So $\varphi(x, b, e) \in q_\varphi$ iff $p \cup \{\varphi(x, b, e)\}$ does not Kim-fork over $M$, and the complementary of this set is $\{b : p \cup \{\varphi(x, b, e)\} \text{ Kim forks over } M\}$. Both of these sets are $M$-type definable, so they are definable over $M$. $\square$

**Remark 3.9.** By Fact 2.3.3 we know that there is a unique non-forking extension of $p \in S_\varphi(M)$ to any set of parameters, and by the above result we know that there is a unique non Kim-forking extension. So these two extension coincide, and we get that the Shelah definition $d_p x\varphi(x, y, e)$ coincides with the definition $\text{Cb}(d_p^K x\varphi(x, y, e))$.

**Lemma 3.10.** Let $M \subseteq N$ be models in a theory $T$ with weak stable Kim-forking over models. Let $q = \text{tp}(a/N)$ and $p = \text{tp}(a/M)$. Then $a \downarrow_M^K N$ if and only if for every stable formula $\varphi(x, y, m)$ with $m \in M$ the canonical parameter $\text{Cb}(d_q^K x\varphi(x, y, m)) \in N$ is equal to $\text{Cb}(d_q^K x\varphi(x, y, m)) \in M$. These two canonical parameters are equal if $\text{Cb}(d_q^K x\varphi(x, y, m)) \in M$.

*Proof.* $[\implies]$ Assume that $a \downarrow_M^K N$ and that $\text{Cb}(d_p x \varphi(x, y, e)) \notin M$ for some stable formula $\varphi(x, y, m)$ with $m \in M$. Let $p_\varphi := \text{tp}_\varphi(a/M)$. Lemma 3.8, which we can apply by Proposition 3.6, yields that there is a unique complete $\varphi(x, y, m)$-type $q_\varphi$ over $N$ such that $p \cup q_\varphi$ does not Kim-fork over $M$. In particular since a non-forking extension always exist we get that $p \cup q_\varphi$ does not fork over $M$, so $q_\varphi$ coincides with the unique non-forking extension of $p_\varphi$ to $N$, so $q_\varphi$ and $p_\varphi$ have the same definition and the same canonical parameter.

$[\Leftarrow]$ Is just Lemma 3.5. $\square$
# 4 Over algebraically closed sets

In this section we work in a monster model $\mathbb{M}$ of a theory $T$ with existence. We can define weak stable Kim-forking over algebraically closed sets in a similar fashion. We will make a small use of hyperimaginaries. As previously we will consider stable formulas $\varphi(x, y, e)$ for some fixed parameters, for convenience we will sometimes shorten to $\varphi$ and write $\text{tp}_{\varphi}(a/A)$ instead of $\text{tp}_{\varphi(x,y,e)}(a/A)$ for example.

**Definition 4.1.** A theory $T$ has weak stable Kim-forking if whenever $a \not\perp_E^K B$ for $E = \text{acl}(E) \subseteq B$ there is a stable formula $\varphi(x, y, e) \in \mathcal{L}_e$ (i.e. the formula with fixed $e \in E$ is stable) and $b \in B$ such that $a \models \varphi(x, b, e)$ and $\varphi(x, b, e)$ Kim-forks over $E$.

**Remark 4.2.** By Fact 2.3.4 we know that if $\varphi(x, y)$ is a stable formula and $E = \text{acl}^{\text{eq}}(E) \subseteq B$ then any generalized $\varphi$-type $p_{\varphi} \in S_{\varphi^*}(E)$ has a unique extension $q \in S_{\varphi^*}(B)$ which is also definable over $E$. It is easy to check that this extension does not Kim-fork over $E$:

*Write $p$ as $p(x, B)$. If $(B_i)_{i<\omega}$ is an $E$-Morley sequence then by indiscernibility the unique extension $q \in S_{\varphi^*}((B_i)_{i<\omega})$ which is also definable over $E$ implies $\cup_{i<\omega} p(x, B_i)$, so this type is consistent.*

**Remark 4.3.** [15, Remark 1.3] If $\varphi(x, a) \equiv \psi(x, b)$ and $\psi(x, z)$ is stable, then for some $\mu(y) \in \text{tp}(a)$, $\varphi(x, y) \wedge \mu(y)$ is stable. In particular if $\psi(x, b)$ is a generalized $\varphi$-formula for some stable formula $\varphi(x, y)$, $\psi(x, y)$ might not be stable, but adding a fragment of $\text{tp}(b)$ will give a stable formula.

**Definition 4.4.** Let $E \subseteq \mathbb{M}$ be a tuple and $p(x)$ be a type over $E$. We call $p$ a *Kim-amalgamation basis* if whenever $p_1$ and $p_2$ are non-Kim-forking extensions of $p$ over some sets of parameters $A$ and $B$ respectively, with $A, B \subseteq \mathbb{M}^{\text{eq}}$, $E \subseteq A \cap B$ and $A \downarrow_E^K B$, then the type $p_1 \cup p_2$ does not Kim-fork over $E$. Any complete type over a model or a boundedly closed set is a Kim-amalgamation base: [2, Theorem 2.5.8].

The proof of the following result works similarly as the one of Lemma 3.8. Here we need $p$ to be a Kim-amalgamation basis in order to apply the independence theorem, and Remark 4.3 in order to have stable formulas.

**Lemma 4.5.** *Assume that $T$ is NSOP$_1$. If $p = \text{tp}(a/E)$ is a Kim-amalgamation basis, $E \subseteq B$, and $\varphi(x, y, e)$ is a stable formula for some fixed $e \in E$, then there is a unique complete generalized $\varphi$-type $p_{\varphi^*}$ extending $p$ that does not Kim-fork over $E$ and it is definable over $E$. In particular, for every generalized $\varphi$-formula $\psi(x, b)$ with $b \in B$, exactly one of $p \cup \{\psi(x, b)\}$ and $p \cup \{\neg\psi(x, b)\}$ Kim-forks over $E$.*

*Proof.* Assume that there are $a', a'' \models p$ such that $a' \downarrow_E^K B$, $a'' \downarrow_E^K B$ and that for some generalized $\varphi$-formula $\psi(x, b)$ with $b \in B$, $\models \psi(a', b)$ and $\models \neg\psi(a'', b)$. By Remark 4.3 there is a formula $\mu(y) \in \text{tp}(b/E)$ such that $\psi(x, y) \wedge \mu(y)$ is stable. Consider an $E$-Morley sequence $(b_i)_{i<\omega}$ such that $b = b_0$.

Let $a'_i, a''_i$ be such that $a'_i b_i \equiv_E a'b_0$ and $a''_i b_i \equiv_E a''b_0$. Then $\models \psi(a'_i, b_i) \wedge \mu(b_i)$ and $\models \neg\psi(a''_i, b_i) \wedge \mu(b_i)$ for every $i < \omega$. By applying the amalgamation theorem for Kim-forking over $E$ along the sequence $b_i$ we can construct a sequence $(a_i)_{i<\omega}$ such that $\models \psi(a_i, b_j) \wedge \mu(b_j)$ iff $i < j$, contradicting the stability of $\psi(x, y) \wedge \mu(y)$.

Consider some generalized $\varphi$-formula $\psi(x, b)$ with $b \in B$. $\psi(x, b) \in p_{\varphi^*}$ if and only if $p \cup \{\psi(x, b)\}$ does not Kim-fork over $E$, and by completeness $\psi(x, b) \notin p_{\varphi^*}$ if and only if $p \cup \{\neg\psi(x, b)\}$ does not Kim-fork over $E$. Both of these sets are $E$-type definable, so they are definable over $E$. $\square$

Using both results of uniqueness we get that Lemma 4.5 holds more generally for generalized $\varphi$-types over algebraically closed sets in $T^{\text{eq}}$.
**Corollary 4.6.** Assume that $T$ is $\text{NSOP}_1$. If $p = \text{tp}_{\varphi^*}(a/E)$ with $E = \text{acl}^{\text{eq}}(E)$, $E \subseteq B$, and $\varphi(x, y, e)$ is a stable formula for some fixed $e \in E$, then there is a unique complete generalized $\varphi$-type extending $p$ that does not $\text{Kim-fork}$ over $E$, and such a type does not $\text{Kim-fork}$ over $E$ iff it is definable over $E$.

*Proof.* Fix some completion $p' \in S(\text{bdd}(E))$, by Lemma 4.5 there is a unique $q' \in S_{\varphi^*}(B)$ such that $p' \cup q'$ does not $\text{Kim-fork}$ over $E$ (we recall that a type $\text{Kim-forks}$ over $E$ if and only if it $\text{Kim-forks}$ over $\text{bdd}(E)$). So it coincides with the unique extension of $p$ over $B$ that is definable over $E$ from Remark 4.2. $\square$

**Remark 4.7.** Corollary 4.6 implies that in any $\text{NSOP}_1$ theory $\text{Kim-forking}$ is transitive for complete generalized $\varphi$-type over algebraically closed sets in the imaginaries: Let $E = \text{acl}^{\text{eq}}(E) \subseteq B = \text{acl}^{\text{eq}}(B) \subseteq D$, $p = \text{tp}_{\varphi^*}(a/E)$, $p' = \text{tp}_{\varphi^*}(a/B)$ and $p'' = \text{tp}_{\varphi^*}(a/D)$. If $p''$ does not $\text{Kim-fork}$ over $B$ and $p'$ does not $\text{Kim-fork}$ over $E$ then $p''$ does not $\text{Kim-fork}$ over $E$.

**Definition 4.8.** If $p = \text{tp}_{\varphi^*}(a/E)$ for $E = \text{acl}^{\text{eq}}(E)$, $E \subseteq B$, and $\varphi(x, y, e)$ is a stable formula for some fixed $e \in E$, then the unique global $\varphi(x, y, e)$-type $q_{\varphi}$ extending $p_{\varphi} := \text{tp}_{\varphi_e}(a/E)$ without forking over $E$ is definable over $E$, we write $d_p^K x\varphi(x, y, e)$ its definition and $\text{Cb}(d_p^K x\varphi(x, y, e))$ the canonical parameter of this definition.

**Remark 4.9.** By uniqueness we get that when both the Shelah definition $d_p x\varphi(x, y, e)$ and the previous one exist they coincide. By considering a non-$\text{Kim-forking}$ extension of $\text{tp}(a/E)$ to a model $M$ we get that the definition $d_p^K x\psi(x, y, e)$, which is a formula with parameters in $\text{acl}^{\text{eq}}(E)$, is equivalent to a positive boolean combination of formulas of the form $\psi(a', y)$ with $a' \in M$.

**Corollary 4.10.** Assume that $T$ has weak stable $\text{Kim-forking}$. If $p = \text{tp}(a/B)$ with $B = \text{acl}^{\text{eq}}(B)$ and $E = \text{acl}(E) \subseteq B$, then a $\searrow_E^K B$ if and only if $\text{Cb}(d_p x\varphi(x, y, e)) \in \text{acl}^{\text{eq}}(E)$ for every stable formula $\varphi(x, y, e)$ with $e \in E$.

*Proof.* $[\iff]$ Follows from a similar proof as the one of Lemma 3.5.

$[\implies]$ Let $p' := \text{tp}(a/\text{acl}^{\text{eq}}(E))$. Assume that a $\searrow_E^K B$ and consider some stable formula $\varphi(x, y, e)$ with $e \in E$. By Corollary 4.6 we know that there is a unique complete $\varphi(x, y, e)$-type $q_{\varphi}$ over $B$ such that $p' \cup q_{\varphi}$ does not $\text{Kim-fork}$ over $E$, so this type coincides with the $\varphi(x, y, e)$-type of $a$ over $B$ since a $\searrow_E^K B$, so they have the same definition and the same canonical parameter, which lies in $\text{acl}^{\text{eq}}(E)$. $\square$

**Remark 4.11.** [6, Remark 1.2] Let $\varphi(x, y, e)$ be a stable formula (with some parameters $e$). If $\theta(v, x, e) \vdash \exists^{-n}x\theta(v, x, e)$ then $\psi(v, y, e) = \exists x(\theta(v, x, e) \wedge \varphi(x, y, e))$ is also stable.

We will use this fact to get a stable formula from another one by replacing the variables by some 'larger variables' in the sense of the algebraic closure.

**Remark 4.12.** Corollary 4.10 implies that if $T$ is an theories with existence and weak stable $\text{Kim-forking}$, given some tuples $E = \text{acl}^{\text{eq}}(E) \subseteq B = \text{acl}^{\text{eq}}(B)$ and a tuple $a$ there is a smaller subset $E \subseteq B_0 = \text{acl}^{\text{eq}}(B_0) \subseteq B$ such that a $\searrow_{B_0}^K B$, in the sense that any $E \subseteq B' = \text{acl}^{\text{eq}}(B') \subseteq B$ such that a $\searrow_{B'}^K B$ must contain $B_0$. So $T^{\text{eq}}$ has weak canonical bases in the sense of [13, Definition 4.1].

**Lemma 4.13.** Assume that $T$ is $\text{NSOP}_1$ and has existence. Let $E \subseteq B \subseteq D$ be sets of parameters, $a$ be a tuple such that a $\searrow_B^K D$, $\varphi(x, y, e)$ be a stable formula with $e \in E$ and $d \in D$ be such that $\varphi(x, d, e)$ $\text{Kim-forks}$ over $E$ and $\models \varphi(a, d, e)$. Then there is a stable formula $\psi(x, y, e)$ and some $b \in B$ such that $\psi(x, b, e)$ $\text{Kim-forks}$ over $E$ and $\models \psi(a, b, e)$.
*Proof.* Let $p = \text{tp}_{\varphi^*}(a/\text{acl}^{\text{eq}}(E))$, $p' := \text{tp}_{\varphi^*}(a/\text{acl}^{\text{eq}}(B))$ and $p'' = \text{tp}_{\varphi^*}(a/D)$.

Then $\text{Cb}(d_{p''}^K x \psi(x, y, e)) \in \text{acl}^{\text{eq}}(B) \setminus \text{acl}^{\text{eq}}(E)$. In fact since $\varphi(x, d, e)$ Kim-forks over $E$ the definition of a $\varphi(x, y, e)$-type containing $\varphi(x, d, e)$ cannot lie in $\text{acl}^{\text{eq}}(E)$. On the other hand $\text{tp}_{\varphi^*}(a/D)$ does not Kim-forks over $B$, so by uniqueness and definition 4.8 we get that the definition of $\text{tp}_{\varphi^*}(a/B)$ and the one of $\text{tp}_{\varphi^*}(a/D)$ coincide, so it lies in $\text{acl}^{\text{eq}}(B)$.

$p'$ Kim-forks over $E$ by Remark 4.7, so there is a generalized $\varphi$-formula $\psi(x, b')$ with $b' \in \text{acl}^{\text{eq}}(B)$ such that $\models \psi(a, b')$ and $\psi(a, b')$ Kim-forks over $E$. By Remark 4.3 we can assume that $\psi(x, y)$ is stable. Take $b \in B$ such that $b'$ is algebraic over $b$, $\theta(z, y, e)$ a formula witnessing this. By considering a strengthening of $\theta$ we can assume that whenever $\models \theta(\beta', b)$ then $\varphi(x, \beta')$ Kim-divides over $E$. By Remark 4.11 the formula $\exists y(\theta(z, y, e) \wedge \psi(x, y, e))$ is stable, is satisfied by $a, b$ and $\exists y(\theta(b, y, e) \wedge \psi(x, y, e))$ Kim-forks over $E$. $\square$

It is enough to show weak stable Kim forking for types over models:

**Lemma 4.14.** Let $T$ be a theory with existence such that whenever $a \searrow_E^K N$ for $N$ a model and $E = \text{acl}(E) \subseteq N$ there is $e \in E$, $n \in N$, $\varphi(x, y, e)$ a stable formula such that $\models \varphi(a, n, e)$ and $\varphi(x, n, e)$ Kim-forks over $E$. Then $T$ has weak stable Kim-forking.

*Proof.* Let $a, E \subseteq B$ such that $a \searrow_E^K B$. Let $N$ be a model containing $B$ such that $a \downarrow_B^f N$. So $a \searrow_E^K N$ and by assumption there is a stable formula $\psi(x, y, e)$ such that $\psi(x, n, e)$ Kim-forks over $E$ for some $n \in N$. By Lemma 4.13 there is a stable formula $\varphi(x, y, e)$ and some $b \in B$ such that $\varphi(x, b, e)$ Kim-forks over $E$ and $\models \varphi(x, y, e)$. The fact that $T$ is, which we need to use Lemma 4.13, is a consequence of the fact that the hypothesis of this lemma implies weak stable Kim-forking over models, which implies by Proposition 3.6. $\square$

**Examples of theories with weak stable Kim-forking:**

1. The first example we give here is the theory $T_{eq,P}$ of parameterized equivalence relations. This theory has two sorts $O$ for objects and $P$ for parameters. The language is $\mathcal{L} = \{=, \equiv\}$, with $\equiv \subseteq P \times O^2$. We write $x \equiv_p y$ to mean that $(p, x, y) \in \equiv$. $T_{eq,P}$ expresses that for any parameter $p \in P$ the relation $\equiv_p$ is an equivalence relation on $O$ with an infinity of classes all of which are infinite, and that for different parameters these relations interact randomly (see [8, Section 6.3] and [3] for more details on this theory).

$T_{eq,P}$ has existence, and for arbitrary sets $E \subseteq A, B$:

1. $A \downarrow_E^K B$ iff $A \cap B = E$ and $O(A) \downarrow_{O(E)}^p O(B)$ for every $p \in P(E)$ where $\downarrow^p$ is forking independence in the theory $T_\infty$ of an equivalence relation with an infinite number of classes all infinite.

2. $A \downarrow_E^f B$ iff $A \cap B = E$ and $O(A) \downarrow_{O(E)}^p O(B)$ for every $p \in P(B)$.

The first condition is witnessed by stable formulas (namely equalities). For the second one, the difference between Kim-forking and forking here is that in the first case the parameter is inside of the basis.

For a given parameter $p$ the formula $\varphi(x, y) = (x \equiv_p y)$ is stable since it defines an equivalence relation, so Kim-forking is always witnessed by stable formulas in this theory. However if we do not fix $p$, the formula $\psi(x, yz) := (x \equiv_z y)$ is unstable, so forking in not always witnessed by stable formulas in this theory (which we already knew from Proposition 3.4).
**2** - The theory of vector spaces of infinite dimension over a field with weak stable Kim-forking (so in particular over a stable field) with a generic bilinear form $sT_{\infty}^K$ (which are studied in the Chapter 3) is an other example. A difference with the previous example is that in this case there is a non-trivial algebraic closure, and that Kim-independence is characterized only for algebraically closed structures: If $E = \mathrm{acl}(E) \subseteq A = \mathrm{acl}(A)$, $B = \mathrm{acl}(B)$ then $A \downarrow_E^K B$ if and only if $\langle A \rangle \cap \langle B \rangle = \langle E \rangle$ and $K(A) \downarrow_{K(E)}^K K(B)$.

The failure of one of these two conditions is witnessed by some stable formula, but for $sT_{\infty}^K$ to have weak stable Kim-forking we need this witness to exist even when $A$ and $B$ are not algebraically closed. This is what we prove now.

For this we use Remark 4.11 and the fact that in an NSOP$_1$ theory with existence $A \downarrow_E^K B$ if and only if $\mathrm{acl}(EA) \downarrow_{\mathrm{acl}(E)}^K \mathrm{acl}(EB)$ for any sets $E, A, B$.

Assume that $a \downarrow_{\mathrm{acl}(E)}^K b$, then $\mathrm{acl}(Ea) \downarrow_{\mathrm{acl}(E)}^K \mathrm{acl}(Eb)$, so there is a stable formula $\varphi(x, y, e)$ with $e \in \mathrm{acl}(E)$ and $\bar{a} \in \mathrm{acl}(Ea)$, $\bar{b} \in \mathrm{acl}(Eb)$ such that $\models \varphi(\bar{a}, \bar{b}, e)$ and $\varphi(x, \bar{b}, e)$ Kim-divides over $E$. So we can assume that $\bar{a} \in \mathrm{acl}(ea)$ and $\bar{b} \in \mathrm{acl}(eb)$.

Let $\theta_a(\bar{x}, a, e)$ be a formula witnessing the fact that $\bar{a} \in \mathrm{acl}(Ea)$. Let $\theta_b(\bar{y}, b, e)$ be a formula witnessing the fact that $\bar{b} \in \mathrm{acl}(Eb)$, by strengthening $\theta_b$ we can assume that $\varphi(\bar{x}, \bar{b}', e)$ Kim-divides over $E$ whenever $\models \theta_b(\bar{b}', b, e)$. For this we eventually have to enlarge the tuple $e$.

By Remark 4.11 the formula $\psi(x, y, e) := \theta_a(\bar{x}, x, e) \wedge \theta_b(\bar{y}, y, e) \wedge \varphi(\bar{x}, \bar{y}, e)$ is stable, by construction $\models \psi(a, b, e)$ and the formula $\psi(x, b, e)$ Kim-forks over $E$.

**3** - With a similar proof we can show that if an NSOP$_1$ theory satisfies that $\downarrow^K = \downarrow^a$ over arbitrary sets then it has weakly stable Kim-forking. In fact in that case $a \downarrow_E^K b$ is witnessed by $\exists x \exists y (\varphi(x, a, e) \wedge \psi(y, b, e) \wedge x = y)$ with $\varphi(x, a, e)$ and $\psi(y, b, e)$ algebraic formulas, and the stable formula we apply Remark 4.11 to is simply ($x = y$). Examples of such theories are given in the Chapter 2 of this manuscript.

# 5 The question of imaginaries

It is shown in section 2 of [6] that if a theory $T$ has stable forking so does $T^{eq}$. We can ask the same question about weak stable Kim-forking. Here we adapt [6, Proposition 2.2] and then discuss the reasons why we can not adapt the proof of Casanovas in its entirety. In this section we work with a theory $T$ with existence and weak stable Kim-forking.

**Proposition 5.1.** If $T$ has weak stable Kim-forking and existence, then $T^{eq}$ has weak stable Kim-forking over real parameters: Whenever $a_F \downarrow_E^K B$ for $E$ an algebraically closed set of real elements, $B \supseteq E$ possibly containing imaginaries and $a_F \in \mathbb{M}^{eq}$ there is a stable formula $\varphi(x, y, e)$ with $e \in E$ and $b \in B$ such that $a_F \models \varphi(x, b, e)$ and $\varphi(x, b, e)$ Kim-forks over $E$.

*Proof.* Let $a_F, E, B$ be as in the statement. We choose a saturated model $M \supseteq B$ such that $a_F \in \mathrm{dcl}^{eq}(M)$ and a representative $a$ of $a_F$ such that $a \downarrow_{Ea_F}^f M$ (using existence). Then $a \downarrow_E^K M$. By assumption there is a stable formula $\varphi(x, y, e)$ for $e \in E$, and a tuple $m \in M$ such that $\models \varphi(a, m, e)$ and $\varphi(x, m, e)$ Kim-forks over $E$.

Let $p = \mathrm{tp}_{\varphi^*}(a/M)$. By Lemma 4.13 there is a stable formula $\psi(x, y, e)$ and $b \in B$ such that $\models \psi(a, b, e)$ and $\psi(x, b, e)$ Kim-forks over $E$. Since $a \downarrow_{Ea_F}^K B$ the $\psi(x, y, e)$-type of $a$ over $B$ is definable over $Ea_F$.
Let $d \in acl^{eq}(Ea_F)$ be its canonical base and $\chi(d, y)$ be its definition, by Remark 4.3 we can assume that $\chi(w, y)$ is a stable formula.

**Claim:** If $q(w) := tp(d/E)$ then $q(w) \cup \{\chi(w, b)\}$ Kim-forks over $E$.

*Proof:* Consider an $E$-Morley sequence $(b_i)_{i<\omega}$ with $b_0 = b$. If $d' \models q(w) \cup \{\chi(w, b_i) : i < \omega\}$ by saturation there is some $(b'_i)_{i<\omega} \in M$ such that $d'(b_i)_{i<\omega} \equiv_E d(b'_i)_{i<\omega}$, so $\models \chi(d, b'_i)$ for all $i < \omega$, so $\psi(x, b'_i, e) \in tp_\psi(a/M)$, so $\{\psi(x, b'_i, e) : i < \omega\}$ is consistent, which contradicts the fact that $\psi(x, b, e)$ Kim-divides over $E$. $\square$

There is some $\mu(w, e) \in q(w, e)$ such that $\chi'(w, b, e) := \chi(w, b) \wedge \mu(w, e)$ Kim-forks over $E$, also $\chi'(w, y, e)$ is stable. We know that $d \in acl^{eq}(Ea_F)$, we can assume that $d \in acl^{eq}(ea_F)$, take $\theta(w, a_F, e)$ some formula witnessing this, i.e. $\models \theta(d, a_F, e)$ and $\models \exists^n w \theta(w, x, z)$.

Let $\alpha(v, y, e) := \exists w(\theta(v, w, e) \wedge \chi(w, y, e))$. By Remark 4.11 $\alpha(v, y, e)$ is stable, $\models \alpha(a_F, b, e)$ and $\alpha(v, b, e)$ Kim-forks over $E$. For the last point if there were $a'_F \models \alpha(v, b, e)$ such that $a' \searrow_E^K b$, then there is $d' \in acl(ea')$ such that $\models \chi'(d', b, e)$, which contradicts $d' \searrow_E^b b$. $\square$

We would like to show that $T^{eq}$ satisfies weak stable forking. Consider $e_F$ a single imaginary, a tuple $a$ and a model $M$ containing $e_F$ such that $a \searrow_M^K M$. We want to find some stable formula $\varphi(x, y, e_F)$ and $m \in M$ such that $\models \varphi(a, m, e_F)$ and $\varphi(x, m, e_F)$ Kim-forks over $e_F$.

There is some representative $e$ of $e_F$ such that $a \searrow_{e_F}^f Me$, so $a \searrow_M eM$. By assumption we can find a stable formula $\varphi(x, y, e)$ and $m \in M$ such that $\models \varphi(a, m, e)$ and $\varphi(x, m, e)$ Kim-forks over $e$. The problem now is that there is no obvious way of preserving these conditions when reducing $e$ to $e_F$. The same difficulty appears if we want to deduce weak stable Kim-forking from weak stable Kim-forking over models.

# 6 Some questions

We conclude this chapter with the following questions:

1. Does existence and weak stable Kim-forking over models imply weak stable Kim-forking over arbitrary sets?

2. If $T^{eq}$ has existence and weak stable Kim-forking, does $T^{eq}$ have weak stable Kim-forking?

3. More specifically, can we prove that if a simple theory $T$ has weak stable forking then $T^{eq}$ also have weak stable forking?

4. Is there an theory that does not have weak stable Kim-forking? Either over models or algebraically closed sets if it has existence.

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