Convention 2.1. Throughout this paper we are only interested in regular cardinals, which we usually denote by $\kappa $ , $\lambda $ , and $\mu $ .

Fact 2.6 [Reference Kamsma18, Remark 2.8].

For any AECat $(\mathcal {C}, \mathcal {M})$ there are arbitrarily large $\kappa $ such that $(\mathcal {C}, \mathcal {M})$ is a $\kappa $ -AECat.

Proof The claim about $\lambda $ -accessibility is exactly [Reference Beke and Rosický2, Proposition 4.1]. Preservation and reflection of $\lambda $ -presentable objects follows using the same proofs as [Reference Beke and Rosický2, Proposition 4.3] and [Reference Beke and Rosický2, Lemma 3.6] respectively, where in the latter we use the former and the fact that all arrows in $\mathcal {C}$ are monomorphisms, so the inclusion functor reflects split epimorphisms (which are isomorphisms).

Convention 2.9. Usually, there will be only one relevant extension of models. So to prevent cluttering of notation we will not give such an extension a name. Given such an extension $M \to N$ and some arrow $a: A \to M$ we will then denote the arrow $A \xrightarrow {a} M \to N$ by a as well.

Fact 2.11 [Reference Kamsma18, Proposition 3.8].

If $\operatorname {\mathrm {gtp}}((a_i)_{i \in I}; M) = \operatorname {\mathrm {gtp}}((a_i')_{i \in I}; M')$ then $:$

1. (i) $($ restriction $)$ we have $\operatorname {\mathrm {gtp}}((a_i)_{i \in I_0}; M) = \operatorname {\mathrm {gtp}}((a_i')_{i \in I_0}; M')$ for any $I_0 \subseteq I;$

2. (ii) $($ monotonicity $)$ given an arrow $b_i: B_i \to \operatorname {\mathrm {dom}}(a_i)$ for each $i \in I$ , then

   $$\begin{align*}\operatorname{\mathrm{gtp}}((a_i)_{i \in I}, (a_i b_i)_{i \in I}; M) = \operatorname{\mathrm{gtp}}((a_i')_{i \in I}, (a_i' b_i)_{i \in I}; M') \end{align*}$$

   and thus $\operatorname {\mathrm {gtp}}((a_i b_i)_{i \in I}; M) = \operatorname {\mathrm {gtp}}((a_i' b_i)_{i \in I}; M')$ ;

3. (iii) $($ extension $)$ for any $(b; M)$ there is an extension $M' \to N$ and some $(b'; N)$ such that $\operatorname {\mathrm {gtp}}(b, (a_i)_{i \in I}; M) = \operatorname {\mathrm {gtp}}(b', (a_i')_{i \in I}; N)$ .

Fact 2.12 [Reference Kamsma18, Proposition 3.9].

If $\operatorname {\mathrm {gtp}}(a, b; M) = \operatorname {\mathrm {gtp}}(a', b'; M')$ and a factors through b, say as $a = bi$ , then $a'$ factors through $b'$ in the same way, so as $a' = b' i$ .

Fact 2.14 [Reference Kamsma18, Proposition 4.6].

$\mathrm {S}_{\mathrm {gtp}}((A_i)_{i \in I})$ is really just a set.

Convention 2.16. For a chain of initial segments $(M_i)_{i < \kappa }$ for some sequence $(a_i)_{i < \kappa }$ in M we will abuse notation and view $a_i$ as an arrow into $M_j$ for $i < j$ . Similarly, if c embeds in $(M_i)_{i < \kappa }$ , we view c as an arrow into $M_i$ for all $i < \kappa $ .

Proof We build the chain of initial segments $(M_i')_{i < \kappa }$ by induction. For the base case we note that we can write $M_0$ as a $\kappa $ -directed colimit of $\kappa $ -presentable models (using Proposition 2.7). As $\operatorname {\mathrm {dom}}(c)$ is $\kappa $ -presentable, c factors through some $M_0'$ in this diagram. The successor step is similar, using that $M_i'$ and $\operatorname {\mathrm {dom}}(a_i)$ are both $\kappa $ -presentable and must thus factor through some $\kappa $ -presentable $M_{i+1}' \leq M_{i+1}$ . In the limit step we just take the colimit $M_\ell ' = \operatorname {\mathrm {colim}}_{i < \ell } M_i'$ and the universal property then yields an arrow $M_\ell ' \to M_\ell $ .

Proof It suffices to prove that $(a_1 / b; M) \sim _{\operatorname {\mathrm {Lgtp}}} (a_2 / b; M)$ implies $(a_1' / b'; M') \sim _{\operatorname {\mathrm {Lgtp}}} (a_2' / b'; M')$ . Let $M \to N$ with $m_0: M_0 \to N$ witness $(a_1 / b; M) \sim _{\operatorname {\mathrm {Lgtp}}} (a_2 / b; M)$ . So we have that $b = m_0 b^*$ for some $b^*: B \to M_0$ , and $\operatorname {\mathrm {gtp}}(a_1, m_0; N) = \operatorname {\mathrm {gtp}}(a_2, m_0; N)$ . Let $N \to N' \leftarrow M'$ witness $\operatorname {\mathrm {gtp}}(a_1, a_2, b; N) = \operatorname {\mathrm {gtp}}(a_1, a_2, b; M) = \operatorname {\mathrm {gtp}}(a_1', a_2', b'; M')$ . Then we get the following commuting diagram:

So we have $m_0: M_0 \to N'$ and $b'$ factors though $m_0$ . This follows from the fact that the above diagram commutes, so $b': B \to M' \to N'$ and $B \to M_0 \to N \to N'$ are the same arrow. Furthermore, we have that $\operatorname {\mathrm {gtp}}(a_1, m_0; N') = \operatorname {\mathrm {gtp}}(a_2, m_0; N')$ and because $a_1$ and $a_2$ are the same arrows into $N'$ as $a_1'$ and $a_2'$ respectively, we get $\operatorname {\mathrm {gtp}}(a_1', m_0; N') = \operatorname {\mathrm {gtp}}(a_2', m_0; N')$ . So we conclude $(a_1' / b'; M') \sim _{\operatorname {\mathrm {Lgtp}}} (a_2' / b'; M')$ , as required.

Proof We will first prove the following claim: for any $b: B \to M$ there is $\lambda _b$ such that for any $b': B \to M'$ with $\operatorname {\mathrm {gtp}}(b'; M') = \operatorname {\mathrm {gtp}}(b; M)$ there are at most $\lambda _b$ many equivalence classes of Lascar strong Galois types over $b'$ in $\operatorname {\mathrm {Hom}}(A, M')$ . By Fact 2.14 the collection $\mathrm {S}_{\mathrm {gtp}}(A, M)$ is a set. We pick $\lambda _b = |\mathrm {S}_{\mathrm {gtp}}(A, M)|$ . Now let $M \to N \leftarrow M'$ witness $\operatorname {\mathrm {gtp}}(b'; M') = \operatorname {\mathrm {gtp}}(b; M)$ . For any two arrows $a, a': A \to M'$ we have that $\operatorname {\mathrm {gtp}}(a, m; N) = \operatorname {\mathrm {gtp}}(a', m; N)$ implies that $\operatorname {\mathrm {Lgtp}}(a / b'; M') = \operatorname {\mathrm {Lgtp}}(a' / b'; M')$ , by definition of $\operatorname {\mathrm {Lgtp}}$ . The claim then follows by choice of $\lambda _b$ .

By the claim we can take $\lambda $ to be the supremum of $\lambda _b$ , where b ranges over the representatives of the Galois types in $\mathrm {S}_{\mathrm {gtp}}(B)$ .

Proof This is essentially the same Fact 2.11, but then for Lascar strong Galois types. To prove it, apply the definition of Lascar strong Galois types to reduce to some equality of Galois types and then apply Fact 2.11.

Convention 4.1. We extend Convention 2.9 to subobjects: given an extension $M \to N$ and a subobject $A \leq M$ , we will view A as a subobject of N.

Convention 4.4. For a base class $\mathcal {B}$ and some subobject $C \leq M$ we will also write $C \in \mathcal {B}$ to mean that the domain of C is in $\mathcal {B}$ , and similarly for $C \not \in \mathcal {B}$ .

Fact 4.8. The following two facts are standard.

1. (i) The intersection of two club sets on $\operatorname {\mathrm {Sub}}^\kappa _{\mathcal {M}}(M)$ is again a club set.

2. (ii) If $M = \operatorname {\mathrm {colim}}_{i < \kappa } M_i$ , where $(M_i)_{i < \kappa }$ is a continuous chain of $\kappa $ -presentable models, then $\{M_i : i < \kappa \}$ is a club set on $\operatorname {\mathrm {Sub}}^\kappa _{\mathcal {M}}(M)$ .

Proof Fact (i) is standard, see for example [Reference Jech17, Theorem 8.2]. We just apply the argument to the poset $\operatorname {\mathrm {Sub}}^\kappa _{\mathcal {M}}(M)$ instead of to a cardinal considered as a poset. Fact (ii) is just unfolding definitions. The chain $(M_i)_{i < \kappa }$ is unbounded because it is $\kappa $ -directed, so any $\kappa $ -presentable $M' \leq M$ will factor through the chain, and continuity is precisely saying that the chain is a closed set.

Proof We first apply Extension to find $M \to N_1$ with $m': M \to N_1$ such that and $\operatorname {\mathrm {gtp}}(m', b, c; N_1) = \operatorname {\mathrm {gtp}}(m, b, c; N_1)$ . In particular this means that b and c factor through $m'$ by Fact 2.12. We apply Extension again to find an extension $n_1: N_1 \to N$ and $n_1': N_1 \to N$ with and $\operatorname {\mathrm {gtp}}(n_1', m'; N) = \operatorname {\mathrm {gtp}}(n_1, m'; N)$ . We define $d'$ to be the composition $D \xrightarrow {d} M \to N_1 \xrightarrow {n_1'} N$ . By Monotonicity we then have . We also have $\operatorname {\mathrm {gtp}}(d', m'; N) = \operatorname {\mathrm {gtp}}(d, m'; N)$ , so since b and c factor through $m'$ , and $m'$ has a model as domain, we indeed get $\operatorname {\mathrm {Lgtp}}(d'/b,c; N) = \operatorname {\mathrm {Lgtp}}(d/b,c; N)$ .

Proof Apply Proposition 4.12 to find $M \to N'$ with $(d'; N')$ such that and $\operatorname {\mathrm {Lgtp}}(d'/b,c; N') = \operatorname {\mathrm {Lgtp}}(d/b,c; N')$ . Then just pick $(a'; N)$ in an extension $N' \to N$ such that $\operatorname {\mathrm {Lgtp}}(a',d/b,c; N) = \operatorname {\mathrm {Lgtp}}(a,d'/b,c; N)$ .

Convention 4.14. We call the class function $\Upsilon $ for Club Local Character a local character function. For an object A we write $\Upsilon (A)$ for $\Upsilon (\lambda )$ where $\lambda $ is the least regular cardinal such that A is $\lambda $ -presentable.

Proof Let $\mathcal {F} \subseteq \operatorname {\mathrm {Sub}}^\kappa _{\mathcal {M}}(M)$ be the club set from Club Local Character. By Fact 4.8(ii) the chain $(M_i)_{i < \kappa }$ forms a club set on $\operatorname {\mathrm {Sub}}^\kappa _{\mathcal {M}}(M)$ . So by Fact 4.8(i) $\{ M_i : i < \kappa \} \cap \mathcal {F}$ is nonempty.

Proof We construct the witnesses of independence $(M_i)_{i < \kappa }$ and sequence $(a_i)_{i < \kappa }$ by induction. At stage i we will construct $a_i$ and $M_{i+1}$ . By Existence we have , and so we will have for all $i < \kappa $ . At every stage we will apply Extension to the latter.

For the base case we set $M_0 = M$ and use Extension to find $a_0$ and $M \to M_1$ with $\operatorname {\mathrm {gtp}}(a_0, c; M_1) = \operatorname {\mathrm {gtp}}(a, c; M)$ and . In the successor step we use Extension to find $M_{i+1} \to M_{i+2}$ and $a_{i+1}$ such that and $\operatorname {\mathrm {gtp}}(a_{i+1}, c; M_{i+2}) = \operatorname {\mathrm {gtp}}(a, c; M)$ . Finally, for limit $\ell < \kappa $ let $M_\ell = \operatorname {\mathrm {colim}}_{i < \ell } M_i$ . We use Extension to find $M_\ell \to M_{\ell +1}$ and $a_\ell $ with $\operatorname {\mathrm {gtp}}(a_\ell , c; M_{\ell +1}) = \operatorname {\mathrm {gtp}}(a, c; M)$ and . We finish the construction by taking $N = \operatorname {\mathrm {colim}}_{i < \kappa } M_i$ .

Proof This is just [Reference Kamsma18, Lemma 6.14]. The differences in terminology are discussed after Definition 4.10. In particular, the monotonicity assumption there is only necessary to get what we call Extension. Finally, there is no notion of base class there, but this is only relevant in the application of Existence, which is only applied with c in the base. Hence the assumption .

Proof We first prove that Independence Theorem implies 3-amalgamation. Let the set up be as in Definition 5.1. In the diagram below we find the dashed arrows by using $\operatorname {\mathrm {gtp}}(c, m; N_2) = \operatorname {\mathrm {gtp}}(c, m; N_3)$ and $\operatorname {\mathrm {gtp}}(b, m; N_1) = \operatorname {\mathrm {gtp}}(b, m; N_2)$ .

We write $a_1$ for the arrow $A \to N_1 \to N'$ and $a_3$ for the arrow $A \to N_3 \to N'$ . Then we have $\operatorname {\mathrm {Lgtp}}(a_1 / m; N') = \operatorname {\mathrm {Lgtp}}(a_3 / m; N')$ . We can thus apply Independence Theorem to find some extension $N' \to N^*$ with some $a^*: A \to N^*$ such that $\operatorname {\mathrm {Lgtp}}(a^*,b/m; N^*) = \operatorname {\mathrm {Lgtp}}(a_1,b/m; N^*)$ , $\operatorname {\mathrm {Lgtp}}(a^*,c/m; N^*) = \operatorname {\mathrm {Lgtp}}(a_3,c/m; N^*)$ and

. So in particular we have

by Monotonicity. Using $\operatorname {\mathrm {gtp}}(a, b, m; N_1) = \operatorname {\mathrm {gtp}}(a_1, b, m; N^*) = \operatorname {\mathrm {gtp}}(a^*, b, m; N^*)$ and $\operatorname {\mathrm {gtp}}(a, c, m; N_3) = \operatorname {\mathrm {gtp}}(a_3, c, m; N^*) = \operatorname {\mathrm {gtp}}(a^*, c, m; N^*)$ after each other we find an extension $N^* \to N$ together with extensions from $N_1$ and $N_3$ to N and we just forget about the previous extensions from $N_1$ and $N_3$ to $N^*$ . These two new extensions, together with $N_2 \to N^* \to N$ , then form the solution to our 3-amalgamation problem.

Now we prove the converse. So we assume 3-amalgamation and we prove Independence Theorem over models. So suppose that

,

and

and $\operatorname {\mathrm {Lgtp}}(a / m; N) = \operatorname {\mathrm {Lgtp}}(a' / m; N)$ . Then we can form the commuting diagram as below, where we find the dashed arrows by 3-amalgamation.

We take the extension $N \to N^*$ to be h and write $a^*$ for $fa = ga'$ . The application of 3-amalgamation yields . Furthermore, $\operatorname {\mathrm {gtp}}(a^*, b, m; N^*) = \operatorname {\mathrm {gtp}}(a, b, m; N) = \operatorname {\mathrm {gtp}}(a, b, m; N^*)$ , so $\operatorname {\mathrm {Lgtp}}(a^*, b / m; N^*) = \operatorname {\mathrm {Lgtp}}(a, b / m; N^*)$ because the domain of m is a model. Similarly, we also find $\operatorname {\mathrm {Lgtp}}(a^*, c / m; N^*) = \operatorname {\mathrm {Lgtp}}(a', c / m; N^*)$ , which concludes the proof.

Fact 5.3 [Reference Kamsma18, Proposition 6.16].

Let be a basic independence relation satisfying Stationarity then it also satisfies 3-amalgamation.

Proof This comes down to checking all the definitions, which is lengthy to do in detail. However, there is only one trick that we repeatedly use, and that is Fact 2.11(ii). To apply this trick we let $f, g, h$ be isomorphisms such that $a' = af$ , $b' = bg$ and $c' = ch$ . Then, using the above trick, we easily see that for any sequence $(b_i)_{i < \lambda }$ witnessing long dividing of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ we have that $(b_i g)_{i < \lambda }$ witnesses long dividing for $\operatorname {\mathrm {gtp}}(a, b g, c; M) = \operatorname {\mathrm {gtp}}(a, b', c; M)$ . Similarly we can replace c by $ch = c'$ and a by $af = a'$ . The same holds for isi-dividing, noting that any isi-sequence over c is also an isi-sequence over $ch = c'$ .

Proof Everything is direct from the definition, except for Right-Monotonicity for long dividing and Base-Monotonicity. For both we will prove the contraposition.

For Base-Monotonicity let $(a, b, c, c'; M)$ be such that $\operatorname {\mathrm {gtp}}(a, b, c'; M)$ long divides over $c'$ and $C \leq C' \leq B$ , where $C, C', B$ are the subobjects represented by $c, c', b$ respectively. Let $\mu $ be as in the definition of long dividing and let $\lambda \geq \mu $ . Then there is $(b_i)_{i < \lambda }$ in some N that witnesses long dividing of $\operatorname {\mathrm {gtp}}(a, b, c'; M)$ over $c'$ . We will prove that it also witnesses long dividing of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ . Indeed we have for all $i < \lambda $ that $\operatorname {\mathrm {gtp}}(b_i, c'; N) = \operatorname {\mathrm {gtp}}(b, c'; M)$ and thus $\operatorname {\mathrm {gtp}}(b_i, c; N) = \operatorname {\mathrm {gtp}}(b, c; M)$ , because $C \leq C'$ . Let $\kappa < \lambda $ be such that for $I \subseteq \lambda $ with $|I| = \kappa $ we have that $\operatorname {\mathrm {gtp}}(a, b, c'; M)$ is inconsistent for $(b_i)_{i \in I}$ . We claim that for such I we also have that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ is inconsistent for $(b_i)_{i \in I}$ . Suppose that there would be a realisation $(a'; N')$ for some extension $N \to N'$ , then $\operatorname {\mathrm {gtp}}(a', b_i, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; M)$ for all $i \in I$ . Since $c'$ factors through b and $b_i$ in the same way for all $i < \lambda $ , see Fact 2.12, we then have $\operatorname {\mathrm {gtp}}(a', b_i, c'; N') = \operatorname {\mathrm {gtp}}(a, b, c'; M)$ for all $i \in I$ , contradicting that $\operatorname {\mathrm {gtp}}(a, b, c'; M)$ is inconsistent for $(b_i)_{i \in I}$ . This proves Base-Monotonicity for long dividing. We have shown that the same sequences that witness long dividing of $\operatorname {\mathrm {gtp}}(a, b, c'; M)$ also witness long dividing of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ . As any isi-sequence over $c'$ is an isi-sequence over c, the same proof shows that isi-dividing has Base-Monotonicity.

Now we prove Right-Monotonicity for long dividing. Let $(a, b, b', c; M)$ be such that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ long divides over c and b factors through $b'$ . For any sequence $(b_i)_{i < \lambda }$ in some N witnessing long dividing we can form $(b_i')_{i < \lambda }$ by letting $b_i'$ be such that $\operatorname {\mathrm {gtp}}(b_i', b_i, c; N) = \operatorname {\mathrm {gtp}}(b', b, c; M)$ for all $i < \lambda $ (possibly replacing N by an extension in the process). Then for $I \subseteq \lambda $ a realisation of $\operatorname {\mathrm {gtp}}(a, b', c; M)$ for $(b_i')_{i \in I}$ would also be a realisation of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ for $(b_i)_{i \in I}$ . So if we let $\kappa < \lambda $ be such that for every $I \subseteq \lambda $ with $|I| = \kappa $ we have that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ is inconsistent for $(b_i)_{i \in I}$ , we also get that $\operatorname {\mathrm {gtp}}(a, b', c; M)$ is inconsistent for $(b_i')_{i \in I}$ for any such I. We conclude that $\operatorname {\mathrm {gtp}}(a, b', c; M)$ long divides over c.

Proof We use the same trick as we did in Proposition 6.4. Let f be the isomorphism such that $a' = af$ . Let $M \to N$ be an extension with $((a_j)_{j \in J}, (d_j)_{j \in J}; N)$ witnessing isi-forking of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ . Then using Proposition 6.4 we have that $\operatorname {\mathrm {gtp}}(a_j f, d_j, c'; N)$ isi-divides over $c'$ for all $j \in J$ . We claim that isi-forking of $\operatorname {\mathrm {gtp}}(a', b', c'; M)$ is witnessed by $((a_j f)_{j \in J}, (d_j)_{j \in J}; N)$ , for which we are now left to check (ii) from Definition 6.7.

Let g and h be isomorphisms such that $b' = bg$ and $c' = ch$ . Let $N \to N^*$ be an extension with some $(a^*; N^*)$ such that $\operatorname {\mathrm {gtp}}(a^*, b', c'; N^*) = \operatorname {\mathrm {gtp}}(a', b', c'; N)$ . Then

$$ \begin{align*} \operatorname{\mathrm{gtp}}(a^* f^{-1}, b, c; N^*) &= \operatorname{\mathrm{gtp}}(a^* f^{-1}, b' g^{-1}, c' h^{-1}; N^*) \\ &= \operatorname{\mathrm{gtp}}(a' f^{-1}, b' g^{-1}, c' h^{-1}; N) \\ &= \operatorname{\mathrm{gtp}}(a, b, c; N), \end{align*} $$

so there is $j \in J$ with $\operatorname {\mathrm {gtp}}(a^* f^{-1}, d_j, c; N^*) = \operatorname {\mathrm {gtp}}(a_j, d_j, c; N)$ . We conclude that $\operatorname {\mathrm {gtp}}(a^*, d_j, c'; N^*) = \operatorname {\mathrm {gtp}}(a^* f^{-1} f, d_j, c h; N^*) = \operatorname {\mathrm {gtp}}(a_j f, d_j, c'; N)$ .

Proof The properties Invariance and Right-Monotonicity are direct from the definition. We prove the contraposition of the remaining three.

For Left-Monotonicity suppose that $\operatorname {\mathrm {gtp}}(a', b, c; M)$ isi-forks over c and let $(a; M)$ be such that $a'$ factors through a. Let $((a^{\prime }_j)_{j \in J}, (d_j)_{j \in J}; N)$ in some extension $M \to N$ witness the isi-forking. Let f be such that $af = a'$ . The following is a set by Fact 2.14:

$$ \begin{align*} F = \{\operatorname{\mathrm{gtp}}(a^*, d_j, c; N^*) :\, &N^* \text{ is an extension of } N \text{ and } j \in J \text{ and} \\ &\operatorname{\mathrm{gtp}}(a^*, b, c; N^*) = \operatorname{\mathrm{gtp}}(a, b, c; M) \text{ and } \\ &\operatorname{\mathrm{gtp}}(a^* f, d_j, c; N^*) = \operatorname{\mathrm{gtp}}(a_j', d_j, c; N)\}. \end{align*} $$

By Left-Monotonicity of isi-dividing, every Galois type in F isi-divides over c. By inductively amalgamating things we find one extension $N \to N^*$ with $((a_k)_{k \in K}, (d_k)_{k \in K}; N^*)$ such that every Galois type in F is realised by $(a_k, d_k, c; N^*)$ for some $k \in K$ . This then witnesses isi-forking of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ over c.

For Extension let $(a, b, b', c; M)$ be such that b factors through $b'$ and for every $(a'; N)$ in some extension $M \to N$ with $\operatorname {\mathrm {gtp}}(a', b, c; N) = \operatorname {\mathrm {gtp}}(a, b, c; M)$ we have that $\operatorname {\mathrm {gtp}}(a', b', c; N)$ isi-forks over c. So the conclusion of the Extension property for $\operatorname {\mathrm {gtp}}(a, b, c; M)$ fails. We have to prove that then $\operatorname {\mathrm {gtp}}(a, b, c; M)$ isi-forks over c.

For each Galois type in $\mathrm {S}_{\mathrm {gtp}}(\operatorname {\mathrm {dom}}(a), \operatorname {\mathrm {dom}}(b'), \operatorname {\mathrm {dom}}(c))$ we fix some witnesses of isi-forking. By Fact 2.14 and the definition of isi-forking, the following is a set:

$$ \begin{align*} F = \{&\operatorname{\mathrm{gtp}}(a', d, c; N) : N \text{ is an extension of } M \text{ and}\\ &\quad \operatorname{\mathrm{gtp}}(a', b, c; N) = \operatorname{\mathrm{gtp}}(a, b, c; M) \text{ and } \\ &\quad \operatorname{\mathrm{gtp}}(a', d, c; N) \text{ is a fixed witness of isi-forking of } \operatorname{\mathrm{gtp}}(a', b', c; N)\}. \end{align*} $$

By inductively amalgamating things we find one extension $N \to N^*$ together with $((a_j)_{j \in J}, (d_j)_{j \in J}; N^*)$ such that every Galois type in F is realised by $(a_j, d_j, c; N^*)$ for some $j \in J$ . This then witnesses isi-forking of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ over c.

Finally, for Base-Monotonicity let $(a, b, c, c'; M)$ be such that $\operatorname {\mathrm {gtp}}(a, b, c'; M)$ isi-forks over $c'$ and $C \leq C' \leq B$ , where $C, C', B$ are the subobjects represented by $c, c', b$ respectively. Let $((a_j)_{j \in J}, (d_j)_{j \in J}; N)$ witness this in some extension $M \to N$ . We claim that this also witnesses isi-forking of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ over c. Indeed, let $a': A \to N'$ for some extension $N \to N'$ be such that $\operatorname {\mathrm {gtp}}(a', b, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; N)$ . We have $C' \leq B$ , so $\operatorname {\mathrm {gtp}}(a', b, c'; N') = \operatorname {\mathrm {gtp}}(a, b, c'; N)$ . So there must be some $j \in J$ such that $\operatorname {\mathrm {gtp}}(a', d_j, c'; N') = \operatorname {\mathrm {gtp}}(a_j, d_j, c'; N)$ . As $C \leq C'$ this restricts to $\operatorname {\mathrm {gtp}}(a', d_j, c; N') = \operatorname {\mathrm {gtp}}(a_j, d_j, c; N)$ , which concludes the proof.

Proof The first implication is just the contrapositive of the trivial statement that isi-dividing implies isi-forking. If the converse of this implication holds, then isi-dividing and isi-forking coincide and so isi-dividing satisfies Right-Monotonicity and Extension by Proposition 6.11.

We are left to prove that if isi-dividing satisfies Right-Monotonicity and Extension then isi-forking implies isi-dividing. Suppose for a contradiction that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ isi-forks over c but does not isi-divide over c. Let $((a_j)_{j \in J}, (d_j)_{j \in J}; N)$ for some extension $M \to N$ witness the isi-forking of $\operatorname {\mathrm {gtp}}(a, b, c; M)$ . By Extension for isi-dividing we find an extension $N \to N'$ with $(a'; N')$ such that $\operatorname {\mathrm {gtp}}(a', b, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; N)$ and $\operatorname {\mathrm {gtp}}(a', N, c; N')$ does not isi-divide. By isi-forking, there must be $j \in J$ such that $\operatorname {\mathrm {gtp}}(a', d_j, c; N')$ isi-divides over c contradicting Right-Monotonicity of isi-dividing.

Proof Combine Proposition 4.18 and Proposition 6.11.

Proof Suppose that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ does not isi-divide over c. Let $\kappa \geq \Upsilon (A)$ such that $(\mathcal {C}, \mathcal {M})$ is a $\kappa $ -AECat and $\operatorname {\mathrm {dom}}(a)$ and $\operatorname {\mathrm {dom}}(c)$ are $\kappa $ -presentable. By Proposition 4.19 we find a long enough

-independent isi-sequence $(b_i)_{i < \lambda }$ over c in some $M \to N$ with $\lambda> \kappa $ and $\operatorname {\mathrm {gtp}}(b_i, c; N) = \operatorname {\mathrm {gtp}}(b, c; M)$ for all $i < \lambda $ . Let $(M_i)_{i < \lambda }$ be witnesses of independence. Since $\operatorname {\mathrm {gtp}}(a, b, c; M)$ does not isi-divide over c there is $I \subseteq \lambda $ with $|I| = \kappa $ such that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ is consistent for $(b_i)_{i \in I}$ . Let $a'$ be a realisation for $(b_i)_{i \in I}$ , which for convenience we may assume to be in N. By possibly deleting an end segment from I we may assume that I has the order type of $\kappa $ . Using Lemma 2.17 we may assume that each object in the chain $(M_i)_{i \in I}$ is $\kappa $ -presentable, where Monotonicity implies that these are still witnesses of independence. Then by chain local character, Lemma 4.15, we find $i_0 \in I$ such that

where $M_I = \operatorname {\mathrm {colim}}_{i \in I} M_i$ . By Monotonicity and Symmetry we then have

We also have

So by Transitivity we have

and the result follows by Symmetry and because $\operatorname {\mathrm {gtp}}(a', b_{i_0}, c; N) = \operatorname {\mathrm {gtp}}(a, b, c; M)$ .

For the final claim we just note that if we do not have Union we can still apply Proposition 4.18 instead of Proposition 4.19 to get an arbitrarily long -independent sequence. It might just not be an isi-sequence. Then the rest of the proof goes through as written.

Proof Let $(M_i)_{i < \delta }$ be witnesses of independence for $(b_i)_{i < \delta }$ , which we may assume to be subobjects of N. We will add one more link $M_\delta $ to the chain. If $\delta $ is a limit ordinal we set $M_\delta = \operatorname {\mathrm {colim}}_{i < \delta } M_i$ . If $\delta $ is a successor ordinal we set $M_\delta = N$ .

We will by induction construct a chain $(N_i)_{i \leq \delta }$ with $N_0$ extending N, together with extensions $\{m_i': M_i \to N_i\}_{i \leq \delta }$ and an arrow $(a"; N_0)$ such that $m_0' = m_0$ and $\operatorname {\mathrm {Lgtp}}(a" / c; N_0) = \operatorname {\mathrm {Lgtp}}(a / c; N_0)$ while at stage i we have:

1. (i) the extensions $\{m_j': M_j \to N_j\}_{j \leq i}$ are natural in the sense that

   

   commutes for all $j \leq i$ ;

2. (ii) if i is a successor, say $i = j + 1$ , then $\operatorname {\mathrm {Lgtp}}(a", b_j' / c; N_i) = \operatorname {\mathrm {Lgtp}}(a, b / c; N_i)$ , where $b_j'$ is the composition $B \xrightarrow {b_j} M_i \xrightarrow {m_i'} N_i$ ;

3. (iii) .

Base case. By Existence we have , so we can apply strong extension (Corollary 4.13) to find $N \to N_0$ and $(a"; N_0)$ with and $\operatorname {\mathrm {Lgtp}}(a" / c; N_0) = \operatorname {\mathrm {Lgtp}}(a / c; N_0)$ .

Successor step. Suppose we have constructed $N_i$ and $m_i'$ . As $M_i$ is an amalgamation base we have $\operatorname {\mathrm {gtp}}(m_i; N_i) = \operatorname {\mathrm {gtp}}(m_i'; N_i)$ . By (i) we have that $m_0'$ factors through $m_i'$ in the same way that $m_0$ factors through $m_i$ , so $\operatorname {\mathrm {gtp}}(m_i', m_0'; N_i) = \operatorname {\mathrm {gtp}}(m_i, m_0; N_i)$ . Since $m_0' = m_0$ have $\operatorname {\mathrm {gtp}}(m_i', m_0; N_i) = \operatorname {\mathrm {gtp}}(m_i, m_0; N_i)$ , so $\operatorname {\mathrm {Lgtp}}(m_i' / c; N_i) = \operatorname {\mathrm {Lgtp}}(m_i / c; N_i)$ . We thus find $(a^*, m_{i+1}^*; N^*)$ for some $N_i \to N^*$ such that $\operatorname {\mathrm {Lgtp}}(m_{i+1}^*, m_i' / c; N^*) = \operatorname {\mathrm {Lgtp}}(m_{i+1}, m_i / c; N^*)$ and $\operatorname {\mathrm {Lgtp}}(a^*, b_i^* / c; N^*) = \operatorname {\mathrm {Lgtp}}(a, b / c; N^*)$ , where $b_i^*$ is given by $B \xrightarrow {b_i} M_{i+1} \xrightarrow {m_{i+1}^*} N^*$ . For this last construction we used that $\operatorname {\mathrm {Lgtp}}(b_i / c; N) = \operatorname {\mathrm {Lgtp}}(b / c; N)$ and that $b_i$ factors through $m_{i+1}$ . Then , and . So by Independence Theorem we find $N^* \to N_{i+1}$ and $(a^{**}; N_{i+1})$ with $\operatorname {\mathrm {Lgtp}}(a^{**}, b_i^* / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a^*, b_i^* / c; N_{i+1})$ , $\operatorname {\mathrm {Lgtp}}(a^{**}, m_i' / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a", m_i' / c; N_{i+1})$ and . By Monotonicity we get . Using $\operatorname {\mathrm {Lgtp}}(a^{**}, m_i' / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a", m_i' / c; N_{i+1})$ we find $m_{i+1}': M_{i+1} \to N_{i+1}$ (after replacing $N_{i+1}$ by an extension) such that $\operatorname {\mathrm {Lgtp}}(a^{**}, m_{i+1}^*, m_i' / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a", m_{i+1}', m_i' / c; N_{i+1})$ . We verify the induction hypothesis:

1. (i) we have by construction that $\operatorname {\mathrm {gtp}}(m_{i+1}', m_i'; N_{i+1}) = \operatorname {\mathrm {gtp}}(m_{i+1}^*, m_i'; N_{i+1}) = \operatorname {\mathrm {gtp}}(m_{i+1}, m_i: N_{i+1})$ , so $m_i'$ factors through $m_{i+1}'$ in the same way that $m_i$ factors through $m_{i+1}$ by Fact 2.12, and naturality follows;

2. (ii) using $\operatorname {\mathrm {Lgtp}}(a^{**}, m_{i+1}^* / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a", m_{i+1}' / c; N_{i+1})$ and that $b_i^*$ and $b_i'$ are $B \xrightarrow {b_i} M_{i+1} \xrightarrow {m_{i+1}^*} N_{i+1}$ and $B \xrightarrow {b_i} M_{i+1} \xrightarrow {m_{i+1}'} N_{i+1}$ respectively by definition, we have $\operatorname {\mathrm {Lgtp}}(a", b_i' / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a^{**}, b_i^*/ c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a^*, b_i^* / c; N_{i+1}) = \operatorname {\mathrm {Lgtp}}(a, b / c; N_{i+1})$ ;

3. (iii) by and Invariance.

Limit step. For limit $\ell $ let $N_\ell = \operatorname {\mathrm {colim}}_{i < \ell } N_i$ . By (i) from the induction hypothesis the arrows $m_i'$ composed with the coprojections $N_i \to N_\ell $ form a cocone on $(M_i)_{i < \ell }$ . By continuity $M_\ell = \operatorname {\mathrm {colim}}_{i < \ell } M_i$ , so there is a universal arrow $m_\ell ': M_\ell \to N_\ell $ . This directly establishes (i). Property (ii) is vacuous. Property (iii) follows from the induction hypothesis and Union.

Having finished the inductive construction, we have two arrows $M_\delta \to N_\delta $ , namely $m_\delta : M_\delta \to N \to N_\delta $ and the $m_\delta '$ we just constructed. By (i) from the induction hypothesis we have $\operatorname {\mathrm {gtp}}(m_\delta , m_0; N_\delta ) = \operatorname {\mathrm {gtp}}(m_\delta ', m_0; N_\delta )$ . So we find an extension $N_\delta \to N'$ and some $(a'; N')$ such that $\operatorname {\mathrm {gtp}}(a', m_\delta , m_0; N') = \operatorname {\mathrm {gtp}}(a", m_\delta ', m_0; N_\delta )$ . Using that c factors through $m_0$ and (ii) from the induction hypothesis, we find that for any $i < \delta $ we have $\operatorname {\mathrm {Lgtp}}(a', b_i / c; N') = \operatorname {\mathrm {Lgtp}}(a", b_i' / c; N') = \operatorname {\mathrm {Lgtp}}(a, b / c; N')$ . By (iii) from the induction hypothesis we also have . So if $\delta $ was a successor ordinal we had $M_\delta = N$ and we are done. Otherwise we can just apply Extension and relabel things to get .

The final claim follows because $a'$ is a realisation of $\operatorname {\mathrm {gtp}}(a, b, c; N)$ for $(b_i)_{i < \delta }$ .

Theorem 1.2, repeated. Let $(\mathcal {C}, \mathcal {M})$ be an AECat with the amalgamation property, and suppose that is a simple independence relation. Then over .

Proof The implication

is already given by Theorem 7.1. For the converse we will assume that

and we will prove that

. Pick some representatives $a, b, c$ of $A, B, C$ . Let $\mu $ be such that $(\mathcal {C}, \mathcal {M})$ is a $\mu $ -AECat and let $\lambda> \Upsilon (B) + \mu $ . Let $(b_i)_{i < \lambda }$ be an isi-sequence over c in some $M \to N$ , with chain of initial segments $(M_i)_{i < \lambda }$ and $\operatorname {\mathrm {gtp}}(b_i, c; N) = \operatorname {\mathrm {gtp}}(b, c; M)$ for all $i < \lambda $ . Let $\Upsilon (B) + \mu \leq \kappa < \lambda $ . By Lemma 2.17 we may assume that $M_i$ is $\kappa $ -presentable for all $i < \kappa $ . We can thus apply chain local character, Lemma 4.15, to find $i_0 < \kappa $ such that

. We will aim to show that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ is consistent for $(b_i)_{i_0 \leq i < \kappa }$ . We use $\operatorname {\mathrm {gtp}}(b_{i_0}, c; N) = \operatorname {\mathrm {gtp}}(b, c; M)$ to find a common extension $M \to N' \leftarrow N$ where $b = b_{i_0}$ as arrows into $N'$ . By applying Extension to the assumption

we then find $(a'; N')$ (possibly after replacing $N'$ by an extension) such that

and $\operatorname {\mathrm {gtp}}(a', b, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; M)$ . Then by Base-Monotonicity and Monotonicity we find

. For any $i_0 \leq i < \kappa $ we have $\operatorname {\mathrm {gtp}}(b_i, m_i, m_{i_0}; N') = \operatorname {\mathrm {gtp}}(b_\kappa , m_i, m_{i_0}; N')$ because $(b_i)_{i < \lambda }$ is an isi-sequence. So by Monotonicity and Invariance and the earlier fact that

, we find

for all $i_0 \leq i < \kappa $ . So $(b_i)_{i_0 \leq i < \kappa }$ is a

-independent sequence. We can thus apply the generalised independence theorem, Lemma 7.2, to conclude that $\operatorname {\mathrm {gtp}}(a', b, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; M)$ is indeed consistent for $(b_i)_{i_0 \leq i < \kappa }$ . As $\kappa $ was arbitrarily large below $\lambda $ , $\lambda $ itself was arbitarily large and $(b_i)_{i_0 \leq i < \kappa }$ is a subsequence of an arbitrary isi-sequence of length $\lambda $ we conclude that indeed

.

Finally, the claim follows from Proposition 6.12 because has Extension and Right-Monotonicity.

Theorem 1.1, repeated. Let $(\mathcal {C}, \mathcal {M})$ be an AECat with the amalgamation property and let $\mathcal {B}$ be some base class. Suppose that $(\mathcal {C}, \mathcal {M})$ satisfies the $\mathcal {B}$ -existence axiom and suppose that there is an NSOP $_1$ -like independence relation over $\mathcal {B}$ . Then over $\mathcal {B}$ .

Proof Suppose that with $C \in \mathcal {B}$ and pick some representatives $a, b, c$ of $A, B, C$ . There is a bound $\mu $ on the cardinality of the set of Lascar strong Galois types compatible with $(b, c; M)$ , see Proposition 3.4. Let $\lambda> \mu $ and let $(b_i)_{i < \lambda }$ be a -independent sequence in some $M \to N$ with $\operatorname {\mathrm {gtp}}(b_i, c; N) = \operatorname {\mathrm {gtp}}(b, c; M)$ for all $i < \lambda $ . Then $(b_i)_{i < \lambda }$ is also -independent, by Theorem 7.1 and Proposition 6.12. We have to show that for every $\kappa < \lambda $ there is $I \subseteq \lambda $ with $|I| = \kappa $ such that $\operatorname {\mathrm {gtp}}(a, b, c; M)$ is consistent for $(b_i)_{i \in I}$ . So let $\kappa < \lambda $ . Then by the choice of $\mu $ and $\lambda $ there must be some $I \subseteq \lambda $ with $|I| = \kappa $ such that $\operatorname {\mathrm {Lgtp}}(b_i / c; N) = \operatorname {\mathrm {Lgtp}}(b_j / c; N)$ for all $i,j \in I$ . Let $i_0$ be the least element of I. Let $(a'; N')$ for some extension $N \to N'$ be such that $\operatorname {\mathrm {gtp}}(a', b_{i_0}, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; M)$ . Then we can apply the generalised independence theorem, Lemma 7.2, to see that $\operatorname {\mathrm {gtp}}(a', b_{i_0}, c; N') = \operatorname {\mathrm {gtp}}(a, b, c; M)$ is consistent for $(b_i)_{i \in I}$ . We conclude that indeed .

For the other direction, suppose that

with $C \in \mathcal {B}$ . Let $\kappa \geq \Upsilon (A)$ be such that $(\mathcal {C}, \mathcal {M})$ is a $\kappa $ -AECat and A and C are $\kappa $ -presentable. Let $(b_i)_{i < \lambda }$ be a long enough

-independent sequence in some extension $M \to N$ , with $\lambda> \kappa $ , witnesses of independence $(M_i)_{i < \lambda }$ and $\operatorname {\mathrm {gtp}}(b_i, c; N) = \operatorname {\mathrm {gtp}}(b, c; M)$ for all $i < \lambda $ . Such a sequence exists by Corollary 6.15, because we assumed the $\mathcal {B}$ -existence axiom. By Theorem 7.1 and Proposition 6.12 this is also a

-independent sequence. By definition of long Kim-dividing there is $I \subseteq \lambda $ with $|I| = \kappa $ such that $\operatorname {\mathrm {gtp}}(a, b, c; N)$ is consistent for $(b_i)_{i \in I}$ . Let $a'$ be a realisation for this (we may assume $a'$ is an arrow into N). By possibly deleting an end segment from I we may assume that I has the order type of $\kappa $ . Using Lemma 2.17 we may assume that each object in the chain $(M_i)_{i \in I}$ is $\kappa $ -presentable, where Monotonicity guarantees that these are still witnesses of independence. Then by chain local character, Lemma 4.15, we find $i_0 \in I$ such that

where $M_I = \operatorname {\mathrm {colim}}_{i \in I} M_i$ . So by Monotonicity and Symmetry we have

Furthermore, we have

So by Transitivity we have

. The result then follows by Symmetry and the fact that $\operatorname {\mathrm {gtp}}(a', b_{i_0}, c; N) = \operatorname {\mathrm {gtp}}(a, b, c; M)$ .

Theorem 1.3, repeated. Let $(\mathcal {C}, \mathcal {M})$ be an AECat with the amalgamation property and suppose that

is a stable or a simple independence relation in $(\mathcal {C}, \mathcal {M})$ . Suppose furthermore that

is an NSOP $_1$ -like independence relation in $(\mathcal {C}, \mathcal {M})$ with

. Then

Proof This follows directly from Theorem 1.2 and Theorem 1.1. To apply the latter we need the -existence axiom. This is automatic, as over by Theorem 1.2 and we have Existence by assumption.

Fact 7.8 [Reference Kamsma19, Proposition 2.39].

In a thick positive theory T having the same Lascar strong type over some parameter set C is the transitive closure of having the same type over finitely $\lambda _T$ -saturated models containing C.

Convention 8.3. We will only deal with invariant equivalence relations. To further simplify the notation we will drop the M from the notation. So we write $a \equiv _c b$ instead of $a \equiv _{c, M} b$ . Because of invariance it does not matter if we consider a and b as arrows into M or as arrows into an extension of M.

Proof The first implication follows because any strong 2- -Morley sequence is in particular strongly 2-c-indiscernible. The final implication follows because having the same Lascar strong Galois type is a bounded invariant equivalence relation, see Example 8.4(iv). We prove the middle implication. So let $\equiv $ be a bounded invariant equivalence relation over C. It is enough to prove that $a \sim ^2_c b$ implies $a \equiv _c b$ . Let $\kappa $ be the bound of $\equiv $ . Since $a \sim ^2_c b$ we find a 2-c-indiscernible sequence $(a_i)_{i < \kappa ^+}$ in some extension N of M with a and b on it. Without loss of generality we may assume $a_0 = a$ and $a_1 = b$ . By boundedness we find $i < j < \kappa ^+$ such that $a_i \equiv _c a_j$ . By 2-c-indiscernibility we have $\operatorname {\mathrm {gtp}}(a, b, c; N) = \operatorname {\mathrm {gtp}}(a_i, a_j, c; N)$ . So $a \equiv _c b$ follows from invariance.

Proof By Remark 7.3 we can apply Lemma 7.2, the generalised independence theorem, while avoiding referring to Lascar strong Galois types. We can thus inductively apply Lemma 7.2 to elongate $(a_i)_{i < \delta }$ to any length we want. We do this by letting $a_0$ , $a_1$ and m play the roles of b, a and c respectively. The final claim follows because in that case $(a, b)$ is a 2- -Morley sequence of length two.

Proof By Existence we have , so we can apply Extension to find $N \to N'$ and some $(a'; N')$ such that while $\operatorname {\mathrm {gtp}}(a', m; N') = \operatorname {\mathrm {gtp}}(a, m; N')$ . By Monotonicity we then have and . So by Lemma 8.10 we have , and we are done.