In other words, ${\mathtt{T}}\models\Gamma(p_{1},\ldots,p_{n})$ if and only if
there are $M\models^{\mathrm{pc}}{\mathtt{T}}$ and $a_{1},\ldots,a_{n}$ in $M$
such that $M\models\Gamma(a_{1},\ldots,a_{n})$ and
${\mathrm{ltp}}^{M}_{+}(a_{i})=p_{i}$ for each $i$.

We say that ${\mathtt{T}}$ is _semi-Hausdorff_ if, for any pointed variable
$x$, the diagonal
$\Delta^{x}_{{\mathrm{loc}}+}({\mathtt{T}})\coloneqq\\{(p,q)\in{\mathbf{S}}^{x}_{{\mathrm{loc}}+}({\mathtt{T}}):p=q\\}$
is local positive $\bigwedge$-definable.

In other words, ${\mathtt{T}}$ is semi-Hausdorff if and only if, for any
pointed variable $x$, there is a partial local positive type
$\Delta(x,y)\subseteq{\mathrm{LFor}}^{x,y}_{+}({\mathtt{L}})$, where $y$ is
disjoint to $x$ and has the same sort, such that, for any
$M\models^{\mathrm{pc}}{\mathtt{T}}$ and any $a,b\in M^{x}$,
$M\models\Delta(a,b)$ if and only if
${\mathrm{ltp}}^{M}_{+}(a)={\mathrm{ltp}}^{M}_{+}(b)$.

######  Lemma 4.26.

If ${\mathtt{T}}$ is Hausdorff, then ${\mathtt{T}}$ is semi-Hausdorff.

###### Proof.

Pick $x,y$ disjoint pointed variables of the same sort. Take

$\Delta(x,y)=\\{\varphi(x)\vee\psi(y):\varphi,\psi\in{\mathrm{LFor}}^{x}_{+}({\mathtt{L}}),\
\varphi\mathrel{\top}\psi\\}.$

We claim that $\Delta(x,y)$ defines the diagonal of
${\mathbf{S}}^{x}_{{\mathrm{loc}}+}({\mathtt{T}})$. In other words, we claim
that for any $M\models^{\mathrm{pc}}{\mathtt{T}}$ and $a,b\in M^{s}$,
$M\models\Delta(a,b)$ if and only if
${\mathrm{ltp}}_{+}(a)={\mathrm{ltp}}_{+}(b)$.

Say ${\mathrm{ltp}}_{+}(a)={\mathrm{ltp}}_{+}(b)$. Take
$\varphi,\psi\in{\mathrm{LFor}}^{x}_{+}({\mathtt{L}})$ arbitrary with
$\varphi\mathrel{\top}\psi$. In particular, we have that
$M\models\varphi(a)\vee\psi(a)$ and $M\models\varphi(b)\vee\psi(b)$. Without
loss of generality, say $M\models\psi(a)$. As
${\mathrm{ltp}}_{+}(a)={\mathrm{ltp}}_{+}(b)$, we conclude that
$M\models\psi(b)$, so $M\models\varphi(a)\vee\psi(b)$. As $\varphi,\psi$ are
arbitrary, we conclude that $M\models\Delta(a,b)$.

On the other hand, suppose ${\mathrm{ltp}}_{+}(a)\neq{\mathrm{ltp}}_{+}(b)$.
Then, by Hausdorffness, there are
$\varphi,\psi\in{\mathrm{LFor}}^{x}_{+}({\mathtt{L}})$ such that
$\varphi\mathrel{\top}\psi$ and $\varphi\notin{\mathrm{ltp}}_{+}(a)$ and
$\psi\notin{\mathrm{ltp}}_{+}(b)$. Hence, $M\not\models\varphi(a)\vee\psi(b)$
with $\varphi(x)\vee\psi(y)\in\Delta(x,y)$, concluding
$M\not\models\Delta(a,b)$. ∎

######  Remark 4.27.

In classic first-order logic, also reducts are continuous between the spaces
of types. It is a well known fact that, in positive logic, unfortunately, that
is no longer the case. The issue is that the reduct of a positive type could
be only a partial positive type.

Let $M$ be a local ${\mathtt{L}}$-structure, $x$ a variable and $A$ a set. The
_space of local positive types_ ${\mathbf{S}}^{x}_{{\mathrm{loc}}+}(M/A)$ of
$M$ on $x$ over $A$ is the set of local positive types of $M$ on $x$ over $A$;
the _space of local positive types_ ${\mathbf{S}}_{{\mathrm{loc}}+}(M/A)$ is
the respective sorted set. We omit $M$ if it is clear from the context. We
never omit $A$.

Let $M$ be a local ${\mathtt{L}}$-structure and $A$ a subset. Let
$\Gamma\subseteq{\mathrm{LFor}}^{x}_{+}({\mathtt{L}}(A))$ be a subset of local
positive formulas. We define

$\langle\Gamma(x)\rangle_{{\mathbf{S}}_{{\mathrm{loc}}+}(M/A)}\coloneqq\\{p\in{\mathbf{S}}^{x}_{{\mathrm{loc}}+}(M/A):\Gamma\subseteq
p\\}.$

As usual, we omit $M$ and $x$ if they are clear from the context. Never omit
$A$.

The _local positive logic topology_ of ${\mathbf{S}}^{x}_{{\mathrm{loc}}+}(A)$
is the topology defined by taking $\langle\Gamma\rangle$ as closed for every
$\Gamma\subseteq{\mathrm{LFor}}^{x}_{+}({\mathtt{L}}(A))$ — we add $\emptyset$
as closed set if it is needed.

######  Remark 4.28.

Tautologically,
${\mathbf{S}}_{{\mathrm{loc}}+}(A)={\mathbf{S}}_{{\mathrm{loc}}+}({\mathrm{Th}}_{+}(M_{A}))$
as topological spaces.

Let $M$ be a local ${\mathtt{L}}$-structure and $A$ a subset. We say that $M$
is _Hausdorff over $A$_ if ${\mathbf{S}}_{{\mathrm{loc}}+}(A)$ is Hausdorff.
In other words, $M$ is Hausdorff over $A$ if ${\mathrm{Th}}_{-}(M/A)$ is
Hausdorff.

We say that $M$ is _semi-Hausdorff over $A$_ if the diagonal
$\Delta^{x}_{+}(A)\coloneqq\\{(p,q)\in{\mathbf{S}}^{x}_{{\mathrm{loc}}+}(A):p=q\\}$
is local positive $\bigwedge_{A}$-definable for any pointed variable $x$. In
other words, $M$ is semi-Hausdorff over $A$ if ${\mathrm{Th}}_{-}(M/A)$ is
semi-Hausdorff.

######  Remark 4.29.

Note that, as reducts are not continuous, it is very hard to see relations
between the spaces of types over different parameters.

######  Definition 4.30.

Let $M\models{\mathtt{T}}$ and $I$ be a first-order structure. Let
$a=(a_{i})_{i\in I}$ be a sequence in $M^{x}$ for some variable $x$. We say
that $(a_{i})_{i\in I}$ is _local positive indiscernible_ if
${\mathrm{ltp}}^{M}_{+}(a_{\bar{\imath}})={\mathrm{ltp}}^{M}_{+}(a_{\bar{\jmath}})$
whenever
${\mathrm{qftp}}^{I}(\bar{\imath})={\mathrm{qftp}}^{I}(\bar{\jmath})$.

######  Lemma 4.31.

Let $M\models{\mathtt{T}}$ and $I$ be a linearly ordered set222We can consider
other first-order structures as long as they satisfy a standard (Ramsey)
lemma.. Let $a=(a_{i})_{i\in I}$ be a sequence in $M^{x}$ for some variable
$x$. Assume that there is a bound $\mathrm{B}(x)$ of $x$ and a locality
relation ${\mathrm{d}}$ on $x$ such that $M\models\mathrm{B}(a_{i})$ and
$M\models{\mathrm{d}}(a_{i},a_{j})$ for any $i,j\in I$. Then, there is a local
model $N\models{\mathtt{T}}$ and a local positive indiscernible sequence
$b=(b_{i})_{i\in I}$ of $N$ such that, for any local positive formula
$\varphi\in{\mathrm{LFor}}_{+}({\mathtt{L}})$, we have
$N\models\varphi(b_{\bar{\jmath}})$ whenever
$M\models\varphi(a_{\bar{\imath}})$ for every $\bar{\imath}$ with
${\mathrm{qftp}}(\bar{\imath})={\mathrm{qftp}}(\bar{\jmath})$.

###### Proof.

Take $M\preceq\widetilde{N}$ into a (non-local) $\kappa$-saturated structure
with $\kappa>|{\mathtt{L}}|+|I|$ — note that $\widetilde{N}$ is not longer a
local structure in general. By the Ehrenfeucht-Mostowski’s Standard Lemma, we
can find an indiscernible sequence $b=(\widetilde{b}_{i})_{i\in I}$ in
$\widetilde{N}$ such that, for any formula $\varphi$,
$\widetilde{N}\mathrel{\mathord{\models}_{\star}}\varphi(b_{\bar{\jmath}})$
whenever $M\models\varphi(a_{\bar{\imath}})$ for every $\bar{\imath}$ with
${\mathrm{qftp}}(\bar{\imath})={\mathrm{qftp}}(\bar{\jmath})$. Pick elements
$o=(o_{s})$ for each single sort such that $o_{s}=b_{0}$ for the sort of $x$,
$o_{s}$ is a constant if there are constants of sort $s$ and $o_{s}$ is taken
arbitrarily in the rest of cases. Let $\mathtt{D}(o)$ be the local component
at $o$. By the choice of $o$ and the fact that $b_{0}$ satisfies a bound, by
Lemma 1.8, we conclude that $\mathtt{D}(o)\models{\mathtt{T}}$ and satisfies
every local formula with parameters in $\mathtt{D}(o)$ satisfied by
$\widetilde{N}$. As $\widetilde{N}\models{\mathrm{d}}(b_{0},b_{i})$ for each
$i\in I$, we know that $b$ lies in $\mathtt{D}(o)$. Therefore, $\mathtt{D}(o)$
contains a local positive indiscernible sequence $b$ such that, for any local
positive formula $\varphi\in{\mathrm{LFor}}_{+}({\mathtt{L}})$, we have
$\mathtt{D}(o)\models\varphi(b_{\bar{\jmath}})$ whenever
$M\models\varphi(a_{\bar{\imath}})$ for every $\bar{\imath}$ with
${\mathrm{qftp}}(\bar{\imath})={\mathrm{qftp}}(\bar{\jmath})$. ∎

We say that two subsets $\Gamma_{1}(x)$ and $\Gamma_{2}(y)$ of formulas in
${\mathtt{L}}$ are _(local) compatible_ in ${\mathtt{T}}$ if there is a local
model $M\models{\mathtt{T}}$ realising both of them, i.e. there are $a\in
M^{x}$ and $b\in M^{y}$ such that $M\models\Gamma_{1}(a)$ and
$M\models\Gamma_{2}(b)$.

By definition, ${\mathtt{T}}$ is irreducible if and only if any two locally
satisfiable quantifier free positive formulas are compatible. By Theorem 3.11,
weak completeness is equivalent to irreducibility in countable languages,
which gives a characterisation of weak completeness in terms of compatibility.
The following lemma provides a characterisation of completeness in terms of
compatibility.

######  Lemma 4.32.

${\mathtt{T}}$ is complete if and only if any local positive type $p$ of
${\mathtt{T}}$ on a finite variable $x$ and any quantifier free positive
formula $\varphi(y)$ with $\exists y\,\varphi(y)\in{\mathtt{T}}_{+}$ are
compatible in ${\mathtt{T}}$.

###### Proof.

One direction is obvious. By Lemma 4.3, $p$ is realised in some
$M\models^{\mathrm{pc}}{\mathtt{T}}$. Thus, if ${\mathtt{T}}$ is complete,
since ${\mathrm{Th}}_{-}(M)={\mathtt{T}}$, $M\models{\mathtt{T}}_{+}$, so
$M\models\exists y\,\varphi(y)$, concluding that it realises $\varphi$.

Conversely assume ${\mathtt{T}}$ is incomplete. Then, there is some
$M\models^{\mathrm{pc}}{\mathtt{T}}$ such that $M$ is not a model of
${\mathtt{T}}_{+}$. Let $\exists y\,\varphi(y)\in{\mathtt{T}}_{+}$, with
$\varphi$ quantifier free, such that $M\models\neg\exists y\,\varphi(y)$. Let
$s=(s(y_{i}))_{y_{i}\in y}$ be the sort of $y$. For each single sort
$s(y_{i})$ pick a single variable $x_{s(y_{i})}$ disjoint to $y$ of sort
$s(y_{i})$ and let $x=\\{x_{s(y_{i})}:y_{i}\in y\\}$. Let $z$ be a variable
disjoint to $x$ and $y$ with no repeated single sorts such that $xz$ is
pointed. Choose $a\in M^{x}$ and $b\in M^{z}$, and set
$p\coloneqq{\mathrm{ltp}}_{+}^{M}(a)$ and
$q\coloneqq{\mathrm{ltp}}_{+}^{M}(ab)$.

For each $y_{i}\in y$, choose a locality relation ${\mathrm{d}}_{y_{i}}$ of
sort $s(y_{i})$ and set
${\mathrm{d}}(x,y)=\bigwedge{\mathrm{d}}_{y_{i}}(x_{s(y_{i})},y_{i})$. Then,
the set of formulas

$\Sigma_{\mathrm{d}}(x,y,z)\coloneqq
q(x,z)\wedge\varphi(y)\wedge{\mathrm{d}}(x,y)$

is inconsistent with ${\mathtt{T}}$.

Indeed, for the sake of contradiction, suppose otherwise and take
$N\mathrel{\mathord{\models}_{\star}}{\mathtt{T}}$ with $c\in N^{x}$, $d\in
N^{y}$ and $e\in N^{z}$ such that
$N\mathrel{\mathord{\models}_{\star}}\Sigma_{\mathrm{d}}(c,d,e)$. Consider
$N^{\prime}\coloneqq\mathtt{D}(ce)$ (and note $d\in N^{\prime}$ as
$N\mathrel{\mathord{\models}_{\star}}{\mathrm{d}}(c,d)$). By Lemma 1.8, since
${\mathtt{T}}$ is a negative local theory and $\Sigma_{\mathrm{d}}$ is a set
of local positive formulas, $N^{\prime}\models\Sigma_{\mathrm{d}}(c,d,e)$ and
$N^{\prime}\models{\mathtt{T}}$. By Theorem 2.7, there is a homomorphism
$h\colon N^{\prime}\to P$ to $P\models^{\mathrm{pc}}{\mathtt{T}}$. By Remark
2.3, we get that $P\models\Sigma_{\mathrm{d}}(h(c,d,e))$. In particular,
${\mathrm{ltp}}^{P}_{+}(h(c,e))=q$ by maximality of $q$. Hence, as
${\mathrm{ltp}}^{P}_{+}(h(c,e))=q$, we get that $P_{ce}\models\psi$ for any
$\psi\in{\mathrm{LFor}}^{0}_{-}({\mathtt{L}}(ce))$ realised by $M_{ce}$, where
$P_{ce}$ is the ${\mathtt{L}}(ce)$-expansion of $P$ given by $h$. As $ce$ is
pointed, this implies that $P_{ce}\models{\mathrm{Th}}_{-}(M/ce)$. By Remark
2.10, we get that $P_{ce}\models^{\mathrm{pc}}{\mathrm{Th}}_{-}(M/ce)$ and,
similarly, $M_{ce}\models^{\mathrm{pc}}{\mathrm{Th}}_{-}(M/ce)$. Therefore, by
Theorem 3.5 applied to ${\mathrm{Th}}_{-}(M/ce)$, we get that
${\mathrm{Th}}_{-}(P_{ce})={\mathrm{Th}}_{-}(M_{ce})$. Therefore, since
$P\models\exists y\,\varphi(y)$, we conclude that $M$ satisfies $\exists
y\,\varphi(y)$ as well, getting a contradiction.

Thus, for any choice of locality relations ${\mathrm{d}}$, there is some
$\psi_{\mathrm{d}}(x,z)\in q$ such that $\neg\exists
x,y,z\,\psi_{\mathrm{d}}(x,z)\wedge\varphi(y)\wedge{\mathrm{d}}(x,y)\in{\mathtt{T}}$
— where the existential quantifier ranges only on the variables that actually
appears in the formula. Now, note that $\exists z\,\psi_{\mathrm{d}}(x,z)\in
p$. Thus, we conclude that $p$ contradicts $\exists
y\,{\mathrm{d}}(x,y)\wedge\varphi(y)$ for any choice of ${\mathrm{d}}$. In
other words, $p$ and $\exists y\,\varphi(y)$ are incompatible. ∎

On the other hand, we have the following characterisation of the local joint
property in terms of compatibility.

######  Lemma 4.33.

A local negative theory ${\mathtt{T}}$ has the local joint property if and
only if any two local positive types on pointed variables are compatible.

###### Proof.

One direction is obvious. Let $p(x)$ and $q(y)$ be local positive types on
pointed variables. By definition, $p$ and $q$ are realised in some
$A,B\models{\mathtt{T}}$. If ${\mathtt{T}}$ has the local joint property,
there are homomorphisms $f\colon A\to M$ and $g\colon B\to M$ to
$M\models{\mathtt{T}}$. Then, by Remark 2.3, $M$ realises $p$ and $q$.

On the other hand, suppose that any two partial local positive types on
pointed variables are compatible. Let $A,B\models{\mathtt{T}}$. Pick $a\in
A^{x}$ and $b\in B^{y}$ pointed tuples (with $x$ and $y$ disjoint). Then, by
assumption, $p(x)\coloneqq{\mathrm{ltp}}(a)$ and
$q(y)\coloneqq{\mathrm{ltp}}_{+}(b)$ are compatible, so $p(x)\wedge q(y)$ is
boundedly satisfiable. Pick a feasible bound $\mathrm{B}(x,y)$. Let
$M^{\ast}\mathrel{\mathord{\models}_{\star}}{\mathtt{T}}$ be (non-local)
$\kappa$-saturated. Then, $p(x)\wedge q(y)\wedge\mathrm{B}(x,y)$ is realised
in $M^{\ast}$ by some $a^{\prime}$ and $b^{\prime}$. By saturation, there are
homomorphism $f\colon A\to M^{\ast}$ and $g\colon B\to M^{\ast}$ with
$f(a)=a^{\prime}$ and $g(b)=b^{\prime}$. Consider the substructure
$M\coloneqq\mathtt{D}(a^{\prime})$. By Lemma 1.8, $M\models{\mathtt{T}}$. As
$M^{\ast}\models\mathrm{B}(a^{\prime},b^{\prime})$, we get that $b^{\prime}$
is in $M$. Since $A$ and $B$ are local, we get that $f\colon A\to M$ and
$g\colon B\to M$, concluding that ${\mathtt{T}}$ has the local joint property.
∎

######  Question 4.34.

On the light of Lemma 4.32, does compatibility of local positive types on
finite variables imply the local joint property?

##  5\. Retractors

Recall that, unless otherwise stated, we have fixed a local language
${\mathtt{L}}$ and a locally satisfiable negative local theory ${\mathtt{T}}$.

######  Definition 5.1.

Let $M\models{\mathtt{T}}$ and $\kappa$ a cardinal. We say that $M$ is a
_local positively $\kappa$-ultrahomogeneous_ model of ${\mathtt{T}}$ if, for
any positive embedding $f\colon A\to M$ and any homomorphism $g\colon A\to B$
to $B\models{\mathtt{T}}$ with $|A|,|B|<\kappa$, there is a homomorphism
$h\colon B\to M$ such that $f=h\circ g$. We say that $M$ is _local positively
absolutely ultrahomogeneous_ model of ${\mathtt{T}}$ if it is local positively
$\kappa$-ultrahomogeneous for every cardinal $\kappa$.

From now on, for the sake of simplicity, we will abbreviate the terminology by
writing “ultrahomogeneous” in place of “local positively ultrahomogeneous”.

######  Lemma 5.2.

Let $M\models{\mathtt{T}}$ and $\kappa$ a cardinal. If $M$ is
$\kappa$-ultrahomogeneous for ${\mathtt{T}}$, then $M_{A}$ is
$\kappa$-ultrahomogeneous for ${\mathrm{Th}}_{-}(M/A)$ for every subset $A$
with $|A|<\kappa$. Conversely, if $M\models^{\mathrm{pc}}{\mathtt{T}}$ is
$\kappa$-ultrahomogeneous for ${\mathrm{Th}}_{-}(M)$, then $M$ is
$\kappa$-ultrahomogeneous for ${\mathtt{T}}$.

###### Proof.

Let $f\colon B_{A}\to M_{A}$ be a positive embedding and $g\colon B_{A}\to
C_{A}$ a homomorphism to $C_{A}\models{\mathrm{Th}}_{-}(M/A)$ with
$|B|,|C|<\kappa$. In particular, $B,C\models{\mathtt{T}}$ so, as $M$ is
$\kappa$-ultrahomogeneous for ${\mathtt{T}}$, there is an
${\mathtt{L}}$-homomorphism $h\colon C\to M$ such that $f=h\circ g$. Now,
$h(a^{C_{A}})=h(g(a^{B_{A}}))=f(a^{B_{A}})=a$ for any $a\in A$, so $h$ is in
fact an ${\mathtt{L}}(A)$-homomorphism. Hence, we conclude that $M_{A}$ is
$\kappa$-ultrahomogeneous for ${\mathrm{Th}}_{-}(M/A)$.

Conversely, suppose $M\models^{\mathrm{pc}}{\mathtt{T}}$ and $M$ is
$\kappa$-ultrahomogeneous for ${\mathrm{Th}}_{-}(M)$. Let $f\colon A\to M$ be
a positive embedding and $g\colon A\to B$ a homomorphism with
$A,B\models{\mathtt{T}}$ and $|A|,|B|<\kappa$. By Corollary 2.13, as
$M\models^{\mathrm{pc}}{\mathtt{T}}$, we get
$A\models^{\mathrm{pc}}{\mathtt{T}}$, so in particular $g$ is a positive
embedding too. By Remark 2.3, as $f$ and $g$ are positive embeddings,
${\mathrm{Th}}_{-}(A)={\mathrm{Th}}_{-}(B)={\mathrm{Th}}_{-}(M)$. Since $M$ is
$\kappa$-ultrahomogeneous for ${\mathrm{Th}}_{-}(M)$, we conclude that there
is a homomorphism $h\colon B\to M$ with $f=hg$. Hence, we conclude that $M$ is
$\kappa$-ultrahomogeneous for ${\mathtt{T}}$. ∎

######  Theorem 5.3.

For any $N\models{\mathtt{T}}$, there is a homomorphism $h\colon N\to M$ to a
$\kappa$-ultrahomogeneous $M\models^{\mathrm{pc}}{\mathtt{T}}$.

###### Proof.

Without loss of generality assume $\kappa\geq|{\mathtt{L}}|$. By Theorem 2.7,
find a homomorphism $f_{-1}\colon N\to M_{0}$ with
$M_{0}\models^{\mathrm{pc}}{\mathtt{T}}$. Take $\lambda$ cardinal such that
$\kappa\leq{\mathrm{cf}}(\lambda)$. We recursively define a direct sequence
$(M_{i},f_{i,j})_{i,j\in\lambda,i\geq j}$ of local positively closed models of
${\mathtt{T}}$ such that, for any $i,j\in\lambda$ with $j\leq i$ and any
homomorphism $g\colon A\to B$ from $A\preceq^{+}M_{j}$ to
$B\models{\mathtt{T}}$ with $|A|,|B|<\kappa$, there is a homomorphism $h\colon
B\to M_{i}$ with ${f_{ji}}_{\mid A}=h\circ g$.

Let $\alpha\in\lambda$ limit and suppose $(M_{i},f_{ij})_{i,j\in\lambda,i\geq
j}$ is already defined. We take respective direct limits
$M_{\alpha}=\underrightarrow{\lim}_{i<\alpha}M_{i}$,
$f_{\alpha,j}=\underrightarrow{\lim}_{j\leq i<\alpha}f_{i,j}$ and
$f_{\alpha,\alpha}={\mathrm{id}}$. By Lemma 2.6, $M_{\alpha}$ is a local
positively closed model of ${\mathtt{T}}$. On the other hand, for any
$j\in\alpha$, given a homomorphism $g\colon A\to B$ from $A\preceq^{+}M_{j}$
to $B\models{\mathtt{T}}$ with $|A|,|B|<\kappa$, there is $h\colon B\to
M_{j+1}$ homomorphism such that ${f_{j+1,j}}_{\mid A}=h\circ g$. Thus,
${f_{\alpha,j}}_{\mid A}=h^{\prime}\circ g$ with
$h^{\prime}=f_{\alpha,j+1}\circ h\colon B\to M_{\alpha}$ homomorphism.

Suppose $M_{i}$ and $(f_{ij})_{i\geq j}$ are already defined. Let
$(A_{\eta},B_{\eta},g_{\eta})_{\eta\in\mu_{i}}$ be an enumeration of all the
possible homomorphisms $g_{\eta}\colon A_{\eta}\to B_{\eta}$ where
$A_{\eta}\preceq^{+}M_{i}$, $B_{\eta}\models{\mathtt{T}}$ and
$|A_{\eta}|<\kappa$ and $B_{\eta}\in\kappa$. We recursively define a directed
sequence $(M^{\eta}_{i},f^{\eta,\nu}_{i})_{\eta,\nu\in\mu,\eta\geq\nu}$ of
local positively closed models of ${\mathtt{T}}$ such that $M^{0}_{i}=M_{i}$
and, for any $\eta\in\mu_{i}$ and $\nu\in\eta$, there is a homomorphism
$h\colon B_{\nu}\to M^{\eta}_{i}$ such that ${f^{\eta,0}_{i}}_{\mid
A_{\nu}}=h\circ g_{\nu}$. Then, we take
$M_{i+1}=\underrightarrow{\lim}_{\eta\in\mu_{i}}M^{\eta}_{i}$,
$f_{i+1,i}=\underrightarrow{\lim}_{\eta\in\mu_{i}}f^{\eta,0}_{i}$,
$f_{i+1,j}=f_{i+1,i}\circ f_{i,j}$ for $j<i$ and $f_{i+1,i+1}={\mathrm{id}}$.
By Lemma 2.6, $M_{i+1}$ is a local positively closed model of ${\mathtt{T}}$.
Let $A\preceq^{+}M_{j}$ with $j\leq i$ and $g\colon A\to B$ to
$B\models{\mathtt{T}}$ with $|A|,|B|<\kappa$. Consider $A^{\prime}=f_{i,j}(A)$
and take $\iota\colon B\to B^{\prime}$ isomorphism with
$|B^{\prime}|\in\kappa$. Then, there is $\eta\in\mu_{i}$ such that
$A^{\prime}=A_{\eta}$, $B^{\prime}=B_{\eta}$ and $g_{\eta}=\iota\circ
g\circ({f_{i,j}}_{\mid A})^{-1}$. We have then a homomorphism $h\colon B\to
M^{\eta+1}_{i}$ such that $f^{\eta,0}_{i}\circ{f_{i,j}}_{\mid
A}=h\circ\iota\circ g$. Thus, ${f_{i+1,j}}_{\mid A}=h^{\prime}\circ g$ with
$h^{\prime}=f^{\eta}_{i}\circ h\circ\iota$, where
$f^{\eta}_{i}=\underrightarrow{\lim}_{\eta<\nu<\mu_{i}}f^{\nu,\eta}_{i}$.

Let $\alpha\in\mu_{i}$ limit and suppose
$(M^{\eta}_{i},f^{\eta,\nu}_{i})_{\eta,\nu\in\alpha,\eta\geq\nu}$ is already
defined. We take the respective direct limits
$M^{\alpha}_{i}=\underrightarrow{\lim}_{\eta<\alpha}M^{\eta}_{i}$,
$f^{\alpha,\nu}_{i}=\underrightarrow{\lim}_{\nu\leq\eta<\alpha}f^{\eta,\nu}_{i}$
and $f^{\alpha,\alpha}_{i}={\mathrm{id}}$. By Lemma 2.6, we know that
$M^{\alpha}_{i}$ is a local positively closed model of ${\mathtt{T}}$. On the
other hand, for any $\eta\in\alpha$, there is a homomorphism $h\colon
B_{\eta}\to M^{\eta+1}_{i}$ such that ${f^{\eta+1,0}_{i}}_{\mid
A_{\eta}}=h\circ g_{\eta}$. Hence, ${f^{\alpha,0}_{i}}_{\mid
A_{\eta}}=h^{\prime}\circ g_{\eta}$ with
$h^{\prime}=f^{\alpha,\eta+1}_{i}\circ h$ homomorphism.

Suppose $M^{\eta}_{i}$ and $(f^{\eta,\nu}_{i})_{\eta\geq\nu}$ are already
defined. Consider the negative local ${\mathtt{L}}(A_{\eta})$-theory
${\mathtt{T}}^{\eta}_{i}\coloneqq{\mathrm{Th}}_{-}(M^{\eta}_{i}/A_{\eta})$
where we interpret $a^{M^{\eta}_{i}}=f^{\eta,0}_{i}(a)$ for each $a\in
A_{\eta}$. As $A_{\eta}\precsim^{+}M^{\eta}_{i}$ via $f^{\eta,0}_{i}$, we have
that ${\mathrm{Th}}_{-}(A_{\eta}/A)={\mathtt{T}}^{\eta}_{i}$. By Corollary
2.13, as $M^{\eta}_{i}$ is local positively closed for ${\mathtt{T}}$, we have
that $A_{\eta}\models^{\mathrm{pc}}{\mathtt{T}}$. By Lemma 2.9, we conclude
that $B_{\eta}\models{\mathtt{T}}^{\eta}_{i}$ with the interpretation
$a^{B_{\eta}}=g_{\eta}(a)$ for $a\in A_{\eta}$. As $A_{\eta}$ is pointed, by
Theorem 3.5, we conclude that ${\mathtt{T}}^{\eta}_{i}$ has the local joint
property. Therefore, using Theorem 2.7, there is a local model
$M^{\eta+1}_{i}\models^{\mathrm{pc}}{\mathtt{T}}^{\eta}_{i}$ and
${\mathtt{L}}(A_{\eta})$-homomorphisms $f^{\eta+1,\eta}_{i}\colon
M^{\eta}_{i}\to M^{\eta+1}_{i}$ and $h\colon B_{\eta}\to M^{\eta+1}_{i}$. We
take $M^{\eta+1}_{i}$ to be the ${\mathtt{L}}$-reduct,
$f^{\eta+1,\nu}_{i}=f^{\eta+1,\eta}_{i}\circ f^{\eta,\nu}_{i}$ for
$\nu\leq\eta$ and $f^{\eta+1,\eta+1}_{i}={\mathrm{id}}$. As
$f^{\eta+1,\eta}_{i}$ and $h$ are ${\mathtt{L}}(A_{\eta})$-homomorphisms, we
have that $f^{\eta+1,0}_{i}(a)=f^{\eta+1,\eta}_{i}\circ
f^{\eta,0}_{i}(a)=h\circ g_{\eta}(a)$ for every $a\in A_{\eta}$. Finally, note
that, by Lemma 2.9, $M^{\eta+1}_{i}\models^{\mathrm{pc}}{\mathtt{T}}$.

Take $M=\underrightarrow{\lim}_{i\in\lambda}M_{i}$,
$f_{i}=\underrightarrow{\lim}_{i\leq j<\lambda}f_{j,i}$ and $f=f_{0}\circ
f_{-1}\colon N\to M$. By Lemma 2.6, $M$ is a local positively closed model of
${\mathtt{T}}$. Let $f\colon A\to M$ be a positive embedding and $g\colon A\to
B$ a homomorphism with $|A|,|B|<\kappa$. As $\kappa<{\mathrm{cf}}(\lambda)$,
we get ${\mathrm{Im}}\,f\subseteq{\mathrm{Im}}\,f_{i}$ for some $i\in\lambda$.
As $f$ and $f_{i}$ are positive embeddings, there is a positive embedding
$f^{\prime}\colon A\to M_{i}$ with $f=f_{i}\circ f^{\prime}$. By construction,
there is $h\colon B\to M_{i+1}$ such that $f^{\prime}=h\circ g$, so
$f=h^{\prime}\circ g$ with $h^{\prime}=f_{i}\circ h$. Hence, $M$ is
$\kappa$-ultrahomogeneous for ${\mathtt{T}}$. ∎

######  Remark 5.4.

Using Löwenheim-Skolem Theorem we can add a bound on the size of $M$ in
Theorem 5.3. Indeed, suppose $|N|\leq\kappa$, then, taking
$\lambda=2^{\kappa}$ and using Löwenheim-Skolem Theorem at each step, we can
add that $|M|\leq 2^{\kappa}$. Explicitly, we can take $|M_{i}|\leq
2^{\kappa}$, $\mu_{i}\leq 2^{\kappa}$ and $|M^{\eta}_{i}|\leq 2^{\kappa}$ for
each $i,\eta\in 2^{\kappa}$.

######  Definition 5.5.

Let $M$ be a local ${\mathtt{L}}$-structure and $\kappa$ a cardinal. We say
that $M$ is _local positively $\kappa$-saturated_ if it realises every local
positive type on $x$ over $A$ for any variable $|x|<\kappa$ and any subset $A$
with $|A|<\kappa$ such that $M$ satisfies the local joint property over $A$.
We say that $M$ is _local positively absolutely saturated_ if it is local
positively $\kappa$-saturated for every $\kappa$.

From now on, for the sake of simplicity, we will abbreviate the terminology by
writing “saturated” in place of “local positively saturated”.

######  Remark 5.6.

Note that, by Theorem 3.5, every structure has the local joint property over
any pointed set of parameters. Thus, every $\kappa$-saturated model realises
every local positive type over any pointed set of parameters of size less than
$\kappa$.

######  Lemma 5.7.

Assume $\kappa>|{\mathtt{L}}|$. Every $\kappa$-ultrahomogeneous model for
${\mathtt{T}}$ is $\kappa$-saturated. Conversely, every $\kappa$-saturated
local positively closed model of ${\mathtt{T}}$ is $\kappa$-ultrahomogeneous
for ${\mathtt{T}}$.

###### Proof.

Let $M$ be $\kappa$-ultrahomogeneous for ${\mathtt{T}}$ and
$p\in{\mathbf{S}}^{x}_{{\mathrm{loc}}+}(A)$ with $|x|<\kappa$ and $A$ a subset
with $|A|<\kappa$ such that $M$ satisfies the local joint property over $A$.
By downwards Löwenheim-Skolem, there is $B_{A}\preceq M_{A}$ with
$|B|\leq|{\mathtt{L}}|+|A|<\kappa$. As $B_{A}\preceq M_{A}$, we have that
${\mathrm{Th}}_{-}(B_{A})={\mathrm{Th}}_{-}(M/A)$. Then, by Lemma 4.13, there
is an ${\mathtt{L}}(A)$-homomorphism $g\colon B_{A}\to N_{A}$ to
$N_{A}\models^{\mathrm{pc}}{\mathrm{Th}}_{-}(M/A)$ such that $N_{A}\models
p(b)$ for some $b$. Furthermore, we can take
$|N|\leq|{\mathtt{L}}|+|B|+|x|<\kappa$ by using downwards Löwenheim-Skolem
Theorem. As $M$ is $\kappa$-ultrahomogeneous for ${\mathtt{T}}$, we find a
homomorphism $h\colon N\to M$ such that ${\mathrm{id}}=h\circ g\colon B\to M$
is the natural inclusion. In particular, $h(a^{N})=h(g(a))=a$, so $h$ is an
${\mathtt{L}}(A)$-homomorphism too. By Remark 2.3, we conclude that $h(b)$
realises $p$ in $M$.

Conversely, suppose $M\models^{\mathrm{pc}}{\mathtt{T}}$ is
$\kappa$-saturated. Let $f\colon A\to M$ be a positive embedding and $g\colon
A\to B$ a homomorphism with $A,B\models{\mathtt{T}}$ and $|A|,|B|<\kappa$.
Since $M$ is local positively closed for ${\mathtt{T}}$, so is $A$ by
Corollary 2.13. Hence, $g$ is a positive embedding. As $f$ and $g$ are
positive embeddings, by Remark 2.3, we conclude that
${\mathrm{Th}}_{-}(B_{A})={\mathrm{Th}}_{-}(A/A)={\mathrm{Th}}_{-}(M/A)$ where
$B_{A}$ and $M_{A}$ are the natural ${\mathtt{L}}(A)$-expansions given by $g$
and $f$ respectively. Let $b=(b_{i})_{i\in I}$ be an enumeration of $B$ and
let $p$ be the local positive type of $b$ in $B_{A}$. Then, $p$ is a local
positive type of ${\mathrm{Th}}_{-}(M/A)$, i.e. a local positive type of $M$
over $g(A)$. By saturation of $M$, there is $c=(c_{i})_{i\in I}$ in $M$
realising $p$. Consider the map $h\colon B\to M$ given by $h\colon
b_{i}\mapsto c_{i}$. Obviously, $f=h\circ g$. Also, as $b$ and $c$ have the
same local positive type, they have in particular the same quantifier free
positive type, so $h$ is a homomorphism. ∎

######  Definition 5.8.

Let $f\colon A\to B$ be a homomorphism. A _retraction_ of $f$ is a
homomorphism $g\colon B\to A$ such that $g\circ f={\mathrm{id}}$. A _retractor
of ${\mathtt{T}}$_ is a local model $\mathfrak{C}\models{\mathtt{T}}$ such
that every homomorphism $f\colon\mathfrak{C}\to N$ to $N\models{\mathtt{T}}$
has a retraction.

######  Theorem 5.9.

The following are equivalent:

  1. (i)

$\mathfrak{C}$ is an absolutely ultrahomogeneous model of ${\mathtt{T}}$.

  2. (ii)

$\mathfrak{C}$ is a retractor of ${\mathtt{T}}$.

  3. (iii)

$\mathfrak{C}$ is an absolutely saturated local positively closed model of
${\mathtt{T}}$.

###### Proof.

  1. (i)$\Rightarrow$(ii)

Let $f\colon\mathfrak{C}\to A$ be a homomorphism with $A\models{\mathtt{T}}$.
Applying ultrahomogeneity with
${\mathrm{id}}\colon\mathfrak{C}\to\mathfrak{C}$, there is a homomorphism
$g\colon A\to\mathfrak{C}$ such that ${\mathrm{id}}=g\circ f$, so $g$ is a
retraction of $f$.

  2. (ii)$\Rightarrow$(iii)

Take a homomorphism $h\colon\mathfrak{C}\to M$ to $M\models{\mathtt{T}}$.
Since $\mathfrak{C}$ is a retractor, there is $g\colon M\to\mathfrak{C}$ such
that ${\mathrm{id}}=g\circ h$. Take
$\varphi\in{\mathrm{For}}^{x}_{+}({\mathtt{L}})$ and $a\in\mathfrak{C}^{x}$.
Suppose $M\models\varphi(h(a))$. Then, by Remark 2.3,
$\mathfrak{C}\models\varphi(gh(a))$, so $\mathfrak{C}\models\varphi(a)$ as
$gh(a)=a$. Hence, we conclude that $\mathfrak{C}$ is local positively closed.

Let $A$ be a subset such that $\mathfrak{C}$ satisfies the local joint
property over $A$ and let $p(x)$ be a local positive type of $\mathfrak{C}$
over $A$. By Lemma 4.13, there is an ${\mathtt{L}}(A)$-homomorphism
$f\colon\mathfrak{C}_{A}\to N_{A}$ with
$N_{A}\models^{\mathrm{pc}}{\mathrm{Th}}_{-}(\mathfrak{C}/A)$ and $b\in N^{x}$
such that $N_{A}\models p(b)$. Since $\mathfrak{C}$ is a retractor, there is a
homomorphism $g\colon N\to\mathfrak{C}$ such that ${\mathrm{id}}=g\circ f$. As
$f$ is an ${\mathtt{L}}(A)$-homomorphism, we conclude that $g$ is an
${\mathtt{L}}(A)$-homomorphism. Thus, by Remark 2.3, $\mathfrak{C}_{A}\models
p(g(b))$, concluding that $\mathfrak{C}$ realises $p$.

  3. (iii)$\Rightarrow$(i)

By Lemma 5.7. ∎

The following corollary is [Hru22a, Proposition 2.1(3)]. We give the proof for
the sake of completeness.

######  Corollary 5.10.

Let $\mathfrak{C}$ be a retractor of ${\mathtt{T}}$. Then, every endomorphism
of $\mathfrak{C}$ is an automorphism.

###### Proof.

Let $f\colon\mathfrak{C}\to\mathfrak{C}$ be a homomorphism. Since
$\mathfrak{C}$ is a retractor, there is $g\colon\mathfrak{C}\to\mathfrak{C}$
such that ${\mathrm{id}}=g\circ f$. In particular, $g$ is surjective. By
Theorem 5.9, $\mathfrak{C}\models^{\mathrm{pc}}{\mathtt{T}}$, so $g$ is a
positive embedding. In particular, $g$ is injective. Hence, $g$ is an
automorphism and, _a posteriori_ , $f=g^{-1}$ is an automorphism too. ∎

The following corollary is [Hru22a, Proposition 2.1(4)]. We provide an
alternative proof of it.

######  Corollary 5.11.

Let $\mathfrak{C}$ be a retractor of ${\mathtt{T}}$. Let $A$ be a subset of
$\mathfrak{C}$ and $x$ a variable pointed over ${\mathtt{L}}(A)$. Take
$a,b\in\mathfrak{C}^{x}$. Then,
${\mathrm{ltp}}^{\mathfrak{C}}_{+}(a/A)={\mathrm{ltp}}^{\mathfrak{C}}_{+}(b/A)$
if and only if there is an automorphism
$\sigma\in{\mathrm{Aut}}(\mathfrak{C}/A)$ such that $\sigma(a)=b$. In
particular,
${\mathrm{ltp}}^{\mathfrak{C}}_{+}(a/A)={\mathrm{ltp}}^{\mathfrak{C}}_{+}(b/A)$
if and only if ${\mathrm{tp}}(a/A)={\mathrm{tp}}(b/A)$

###### Proof.

The backward direction is obvious. On the other hand, consider the family
$\mathcal{K}$ of partial homomorphisms $f\colon D\subseteq\mathfrak{C}$ fixing
$A$ and mapping $f(a)=b$. Assuming
${\mathrm{tp}}^{\mathfrak{C}}_{{\mathrm{loc}}+}(a/A)={\mathrm{tp}}^{\mathfrak{C}}_{{\mathrm{loc}}+}(b/A)$,
we have that $\mathcal{K}\neq\emptyset$. Obviously, $\mathcal{K}$ is partially
ordered by $\subseteq$. Also, for any chain $\Omega\subseteq\mathcal{K}$, we
have that $\bigcup\Omega\in\mathcal{K}$. Therefore, by Zorn’s Lemma, there is
a maximal element $\sigma\in\mathcal{K}$. Write
$D\coloneqq\mathrm{Dom}\,\sigma$ and take an arbitrary single element
$c\in\mathfrak{C}$. As $a,A$ is pointed, $D$ is pointed. By saturation of
$\mathfrak{C}$, we conclude that there is an element $d$ such that
${\mathrm{ltp}}^{\mathfrak{C}}_{+}(c,D)={\mathrm{ltp}}^{\mathfrak{C}}_{+}(d,\sigma(D))$.
Therefore, $\sigma\cup\\{(c,d)\\}\in\mathcal{K}$, concluding by maximality
that $c\in D$. As $c$ is arbitrary, we conclude that
$\sigma\colon\mathfrak{C}\to\mathfrak{C}$ is an endomorphism. By Corollary
5.10, we conclude that $\sigma\in{\mathrm{Aut}}(\mathfrak{C}/A)$. ∎

######  Definition 5.12.

Let $M\models{\mathtt{T}}$ and $\kappa$ a cardinal. We say that $M$ is _local
homomorphism $\kappa$-universal for ${\mathtt{T}}$_ if for every
$N\models{\mathtt{T}}$ with $|N|<\kappa$ there is a homomorphism $f\colon N\to
M$. We say that it is _local homomorphism absolutely universal for
${\mathtt{T}}$_ if it is local homomorphism $\kappa$-universal for every
cardinal $\kappa$.

From now on, for the sake of simplicity, we will abbreviate the terminology by
writing “universal” in place of “local homomorphism universal”.

The following lemma is [Hru22a, Proposition 2.1(6)]. We give the proof for the
sake of completeness.

######  Lemma 5.13.

Assume ${\mathtt{T}}$ has the local joint property and
$\kappa>|{\mathtt{L}}|$. Then, every $\kappa$-ultrahomogeneous model of
${\mathtt{T}}$ is $\kappa$-universal for ${\mathtt{T}}$. In particular, when
${\mathtt{T}}$ satisfies the local joint property, there is at most one
retractor of ${\mathtt{T}}$ up to automorphism and it is absolutely universal
for ${\mathtt{T}}$.

###### Proof.

Let $M$ be $\kappa$-ultrahomogeneous for ${\mathtt{T}}$. Take
$B\models{\mathtt{T}}$ with $|B|<\kappa$. By downwards Löwenheim-Skolem
Theorem, there is $A\preceq M$ with $|A|<\kappa$. By the local joint property,
there are homomorphisms $f\colon A\to N$ and $g\colon B\to N$ to a common
$N\models{\mathtt{T}}$. By downwards Löwenheim-Skolem Theorem again, we can
find $N$ with $|N|<\kappa$. Thus, by ultrahomogeneity, there is $h\colon N\to
M$ such that ${\mathrm{id}}=h\circ f$. In particular, we have a homomorphism
$h\circ g\colon B\to M$. As $B$ is arbitrary, we conclude that $M$ is
$\kappa$-universal for ${\mathtt{T}}$.

Suppose $\mathfrak{C},\mathfrak{C}^{\prime}$ are retractors of ${\mathtt{T}}$.
Then, by homomorphism universality of $\mathfrak{C}$, we have a homomorphism
$f\colon\mathfrak{C}^{\prime}\to\mathfrak{C}$. Now, we have the homomorphism
${\mathrm{id}}\colon\mathfrak{C}^{\prime}\to\mathfrak{C}^{\prime}$. By
ultrahomogeneity of $\mathfrak{C}^{\prime}$, there is a homomorphism
$g\colon\mathfrak{C}\to\mathfrak{C}^{\prime}$ such that ${\mathrm{id}}=g\circ
f$. On the other hand, we have the homomorphism
${\mathrm{id}}\colon\mathfrak{C}\to\mathfrak{C}$. By ultrahomogeneity of
$\mathfrak{C}$, there is
$f^{\prime}\colon\mathfrak{C}^{\prime}\to\mathfrak{C}$ such that
${\mathrm{id}}=f^{\prime}\circ g$. Hence,
$g\colon\mathfrak{C}\to\mathfrak{C}^{\prime}$ is an isomorphism and, _a
posteriori_ , $f=f^{\prime}=g^{-1}$. ∎

Without the local joint property, retractors are not necessarily unique.
However, we can still classify them in a natural way, getting also a bound on
the number of retractors. In [Hru22a], Hrushovski suggested a way of doing so.
In particular, he suggested considering all the possible local positive types
and when they can be realised at the same time. This method seems a bit
complicated in the many sorted case. We provide here a rephrasing of that idea
that seems to work better in the many sorted case.

######  Definition 5.14.

We say that two local models $A$ and $B$ of ${\mathtt{T}}$ are _compatible in
${\mathtt{T}}$_ if there are homomorphisms $f\colon A\to M$ and $g\colon B\to
M$ to a common local model $M\models{\mathtt{T}}$; write
$A\frown_{{\mathtt{T}}}B$. For a local model $A\models{\mathtt{T}}$, we write
${\mathrm{Com}}_{{\mathtt{T}}}(A/{\mathtt{L}})\coloneqq\\{B\models{\mathtt{T}}:A\frown_{{\mathtt{T}}}B\\}$
and
${\mathrm{Com}}^{\mathrm{pc}}_{{\mathtt{T}}}(A/{\mathtt{L}})\coloneqq\\{B\in{\mathrm{Com}}_{{\mathtt{T}}}(A/{\mathtt{L}}):B\models^{\mathrm{pc}}{\mathtt{T}}\\}$.
As usual, we omit ${\mathtt{T}}$ and ${\mathtt{L}}$ if they are clear from the
context.

######  Lemma 5.15.

Let $M\models^{\mathrm{pc}}{\mathtt{T}}$. Then, for any
$A_{1},A_{2}\in{\mathrm{Com}}(M)$, $A_{1}\frown A_{2}$. In particular,
$\frown$ is an equivalence relation in the class
$\mathcal{M}^{\mathrm{pc}}({\mathtt{T}})$ and
${\mathrm{Com}}^{\mathrm{pc}}(M)$ is the equivalence class of $M$ for every
$M\models^{\mathrm{pc}}{\mathtt{T}}$.

###### Proof.

As $A_{1}\frown M$ and $A_{2}\frown M$. There are then homomorphisms
$f_{1}\colon A_{1}\to B_{1}$, $f_{2}\colon M\to B_{1}$, $g_{1}\colon M\to
B_{2}$ and $g_{2}\colon A_{2}\to B_{2}$ with $B_{1},B_{2}\models{\mathtt{T}}$.
As $M\models^{\mathrm{pc}}{\mathtt{T}}$, we have that $f_{2}$ and $g_{1}$ are
positive embeddings so, by Remark 2.3,
$B_{1},B_{2}\models{\mathrm{Th}}_{-}(M/M)$. As $M$ is pointed, by Theorem 3.5,
there are ${\mathtt{L}}(M)$-homomorphisms $h_{1}\colon B_{1}\to N$ and
$h_{2}\colon B_{2}\to N$ to a common local model
$N\models{\mathrm{Th}}_{-}(M/M)$. In particular, $h_{1}\circ f_{1}\colon
A_{1}\to N$ and $h_{2}\circ g_{2}\colon A_{2}\to N$ are homomorphisms with
$N\models{\mathtt{T}}$. Therefore, $A_{1}\frown A_{2}$. ∎

######  Theorem 5.16.

Let $\mathcal{E}$ be an equivalence class of compatibility in
$\mathcal{M}^{\mathrm{pc}}({\mathtt{T}})$. Then:

  1. (1)

$\mathfrak{C}\in\mathcal{E}$ is a retractor of ${\mathtt{T}}$ if and only if
it is universal for $\mathcal{E}={\mathrm{Com}}^{\mathrm{pc}}(\mathfrak{C})$.

  2. (2)

There is at most one retractor in $\mathcal{E}$ up to isomorphism.

  3. (3)

There is a retractor in $\mathcal{E}$ if and only if $\mathcal{E}$ is bounded,
i.e. there is a cardinal $\kappa$ such that every element of $\mathcal{E}$ has
cardinality at most $\kappa$.

###### Proof.

  1. (1)

Pick $A\in\mathcal{E}$. As $A\frown\mathfrak{C}$, there are homomorphisms
$f\colon A\to B$ and $g\colon\mathfrak{C}\to B$ to $B\models{\mathtt{T}}$.
Now, there is a retraction $h\colon B\to\mathfrak{C}$ of $g$. In particular,
$h\circ f\colon A\to\mathfrak{C}$ is a homomorphism. Since $A$ is arbitrary,
we conclude that $\mathfrak{C}$ is universal for $\mathcal{E}$.

Now, suppose $\mathfrak{C}$ is universal for $\mathcal{E}$. By Theorem 5.3,
there is a homomorphism $h\colon\mathfrak{C}\to\mathfrak{C}^{\prime}$ where
$\mathfrak{C}^{\prime}$ is a $\lambda$-ultrahomogeneous local positively
closed model of ${\mathtt{T}}$ for any $\lambda>|\mathfrak{C}|$. As
$\mathfrak{C}^{\prime}\in\mathcal{E}$, we conclude that there is a positive
embedding $g\colon\mathfrak{C}^{\prime}\to\mathfrak{C}$. In particular, we
have that $|\mathfrak{C}^{\prime}|\leq|\mathfrak{C}|<\lambda$. By
$\lambda$-ultrahomogeneity of $\mathfrak{C}^{\prime}$, there is a homomorphism
$f\colon\mathfrak{C}\to\mathfrak{C}^{\prime}$ such that ${\mathrm{id}}=f\circ
g$, in particular, $f$ is surjective. As $\mathfrak{C}$ is local positively
closed for ${\mathtt{T}}$, $f$ is a positive embedding. Hence, $f$ is a
surjective embedding, that is an isomorphism. Hence, $\mathfrak{C}$ is
$\lambda$-ultrahomogeneous for ${\mathtt{T}}$. As $\lambda$ is arbitrary, we
conclude that it is a retractor of ${\mathtt{T}}$.

  2. (2)

Let $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ be compatible retractors of
${\mathtt{T}}$. By (1), there is a homomorphism
$f\colon\mathfrak{C}_{1}\to\mathfrak{C}_{2}$. Now, considering
${\mathrm{id}}\colon\mathfrak{C}_{1}\to\mathfrak{C}_{1}$, by ultrahomogeneity
of $\mathfrak{C}_{1}$, there is a homomorphism
$g\colon\mathfrak{C}_{2}\to\mathfrak{C}_{1}$ such that ${\mathrm{id}}=g\circ
f$. Again, considering
${\mathrm{id}}\colon\mathfrak{C}_{2}\to\mathfrak{C}_{2}$, by ultrahomogeneity
of $\mathfrak{C}_{2}$, there is a homomorphism
$f^{\prime}\colon\mathfrak{C}_{1}\to\mathfrak{C}_{2}$ such that
${\mathrm{id}}=f^{\prime}\circ g$. Hence, $g$ is an isomorphism,
$\mathfrak{C}_{1}\cong\mathfrak{C}_{2}$ and, _a posteriori_ ,
$f=f^{\prime}=g^{-1}$.

  3. (3)

By (1), if $\mathfrak{C}$ is a retractor in $\mathcal{E}$, then $\mathfrak{C}$
is universal for $\mathcal{E}$, so every element of $\mathcal{E}$ has
cardinality at most $|C|$. On the other hand, suppose $\mathcal{E}$ is
bounded. Since it is bounded, we can take a set $\mathcal{E}_{0}$ of
representatives of $\mathcal{E}$ such that every element of $\mathcal{E}$ is
isomorphic to exactly one element of $\mathcal{E}_{0}$. Now, $\mathcal{E}_{0}$
is partially ordered by $\lesssim$, where $A\lesssim B$ if there is an
embedding $f\colon A\to B$. Furthermore, note that $\lesssim$ is a directed
order since any $A_{1},A_{2}\in\mathcal{E}_{0}$ are compatible. Now, take a
chain $\\{A_{i}\\}_{i\in I}$ in $\mathcal{E}_{0}$. By Lemma 2.6,
$A\coloneqq\underrightarrow{\lim}A_{i}$ is a local positively closed model of
${\mathtt{T}}$ and $A_{i}\lesssim A$ for every $i\in I$. Therefore, by Zorn’s
Lemma, there are maximal elements in $\mathcal{E}_{0}$. As $\lesssim$ is
directed, we conclude that, in fact, $\mathcal{E}_{0}$ has a maximum
$\mathfrak{C}$. Thus, $\mathfrak{C}$ is universal for
${\mathrm{Com}}^{\mathrm{pc}}(\mathfrak{C})=\mathcal{E}_{0}$ and, by (1), it
is a retractor. ∎

######  Remark 5.17.

By downwards Löwenheim-Skolem Theorem, using Corollary 2.13, every equivalence
class of compatibility in $\mathcal{M}^{\mathrm{pc}}({\mathtt{T}})$ contains
at least one element $A$ with $|A|\leq|{\mathtt{L}}|$. Therefore, there are at
most $2^{|{\mathtt{L}}|}$ possible equivalence classes. In particular,
${\mathtt{T}}$ has at most $2^{|{\mathtt{L}}|}$ different retractors.

######  Example 5.18.

The bound $2^{|{\mathtt{L}}|}$ on the number of retractors is sharp:

Let $I$ be a set and ${\mathtt{L}}$ the one-sorted local language with unary
predicates $\\{P_{i},Q_{i}\\}_{i\in I}$ and locality relations
$\mathtt{D}=\\{{\mathrm{d}}_{n}\\}_{n\in\mathbb{N}}$ (with the usual ordered
monoid structure induced by the natural numbers). Consider the
${\mathtt{L}}$-structure $M$ with universe $2^{I}$ and interpretations
$M\models{\mathrm{d}}_{n}(\eta,\zeta)\Leftrightarrow|\\{i:\eta(i)\neq\zeta(i)\\}|\leq
n$ for $n\in\mathbb{N}$, $M\models P_{i}(\eta)\Leftrightarrow\eta(i)=1$ and
$M\models Q_{i}(\eta)\Leftrightarrow\eta(i)=0$ for each $i\in I$. Let
${\mathtt{T}}={\mathrm{Th}}_{-}(M)$ be the negative theory of $M$. Since
$M\models{\mathtt{T}}_{\mathrm{loc}}$, ${\mathtt{T}}$ is locally satisfiable
by Theorem 1.9. For $\eta\in M$, write $M_{\eta}\coloneqq\mathtt{D}(\eta)$.

Now, we look at $M$ as no-local structure, i.e. as
${\mathtt{L}}_{\star}$-structure. Note that the positive logic topology in $M$
is coarser than the Tychonoff topology, so $M$ is compact with the positive
logic topology. Thus, by [Seg22, Lemma 2.24], we get that $M$ is homomorphism
universal for $\mathcal{M}_{\star}({\mathtt{T}})$. In particular, by Lemma
5.38, ${\mathtt{T}}$ has retractors.

Let $\eta\in M$ and suppose $M\models\varphi$ with $\varphi$ a positive
formula in ${\mathtt{L}}$. For $I_{0}\subseteq I$ finite, let
${\mathtt{L}}_{I_{0}}$ be the reduct of ${\mathtt{L}}$ consisting of
$\\{{\mathrm{d}}_{k}\\}_{k\in\mathbb{N}}\cup\\{P_{i},Q_{i}:i\in I_{0}\\}$ and
take $I_{0}$ such that $\varphi$ is a formula in ${\mathtt{L}}_{I_{0}}$.
Consider the map $f\colon M\to M_{\eta}$ given by $f(\xi)(i)=\xi(i)$ for $i\in
I_{0}$ and $f(\xi)(i)=\eta(i)$ for $i\notin I_{0}$. Obviously, $P_{i}(\xi)$
implies $P_{i}(f(\xi))$ and $Q_{i}(\xi)$ implies $Q_{i}(f(\xi))$ for any $i\in
I_{0}$. On the other hand, $|\\{i:f(\xi)(i)\neq
f(\zeta)(i)\\}|\leq|\\{i:\xi(i)\neq\zeta(i)\\}|$, so
${\mathrm{d}}_{k}(\xi,\zeta)$ implies ${\mathrm{d}}_{k}(f(\xi),f(\zeta))$ for
any $k$. Thus, $f$ is an ${\mathtt{L}}_{I_{0}}$-homomorphism. By Remark 2.3,
we conclude that $M_{\eta}\models\varphi$. As $\varphi$ is arbitrary, we
conclude that ${\mathrm{Th}}_{-}(M_{\eta})={\mathtt{T}}$ for any $\eta\in M$.
Hence, ${\mathtt{T}}$ is a local theory.

Note that $M_{\eta}\not\frown M_{\zeta}$ for $\eta,\zeta\in M$ with
$\\{i:\zeta(i)\neq\eta(i)\\}$ infinite. In order to reach a contradiction,
suppose that there are homomorphisms $f\colon M_{\zeta}\to N$ and $g\colon
M_{\zeta}\to N$ with $N\models{\mathtt{T}}$. Then, $\bigwedge_{i\in
I}R_{i}(f(\eta),g(\zeta))$ where $R_{i}(x,y)=(P_{i}(x)\wedge
Q_{i}(y))\vee(Q_{i}(x)\wedge P_{i}(y))$ and $I=\\{i:\zeta(i)\neq\eta(i)\\}$.
On the other hand, for any finite subset $J\subseteq I$,
${\mathtt{T}}\models\neg\exists x,y\,\bigwedge_{i\in
J}R_{i}(x,y)\wedge{\mathrm{d}}_{|J|-1}(x,y)$. Thus, we get
$N\not\models{\mathrm{d}}_{n}(f(\eta),g(\zeta))$ for every $n\in\mathbb{N}$,
contradicting that $N$ is a local structure.

The following easy lemma provides a variation of Theorem 5.16(1) which is also
a converse to Corollary 5.10.

######  Lemma 5.19.

Let $\mathfrak{C}\models{\mathtt{T}}$. Suppose that $\mathfrak{C}$ is
homomorphism universal for ${\mathrm{Com}}(\mathfrak{C})$ and every
endomorphism of $\mathfrak{C}$ is an automorphism. Then, $\mathfrak{C}$ is a
retractor of ${\mathtt{T}}$.

###### Proof.

Let $f\colon\mathfrak{C}\to M$ be a homomorphism to $M\models{\mathtt{T}}$. By
homomorphism universality of $\mathfrak{C}$ for
${\mathrm{Com}}(\mathfrak{C})$, there is a homomorphism $g\colon
M\to\mathfrak{C}$. Thus, $h\coloneqq g\circ f$ is an endomorphism, so an
automorphism. Therefore, $h^{-1}\circ g\colon M\to\mathfrak{C}$ is a
retraction of $f$, concluding that $\mathfrak{C}$ is a retractor. ∎

###  5.1. Local positive logic topologies

Recall that, unless otherwise stated, we have fixed a local language
${\mathtt{L}}$ and a locally satisfiable negative local theory ${\mathtt{T}}$.

Let $M$ be a local ${\mathtt{L}}$-structure, $x$ a variable and $A$ a subset.
We say that $D\subseteq M^{x}$ is _local positive $A$-definable_ if there is a
local positive formula $\underline{D}(x)$ with parameters in $A$ defining it,
i.e. $D=\\{a\in M^{x}:M\models\underline{D}(a)\\}$. We say that $D$ is _local
positive $\bigwedge_{A}$-definable_ if there is
$\underline{D}(x)\subseteq{\mathrm{LFor}}^{x}_{+}({\mathtt{L}}(A))$ defining
it, i.e. $D=\\{a\in M^{x}:M\models\underline{D}(a)\\}$. The _local positive
$A$-logic topology_ of $M^{x}$ is the topology on $M^{x}$ given by taking as
closed sets the local positive $\bigwedge_{A}$-definable subsets — we may need
to add the empty set. The local positive $M$-logic topology is simply called
the _local positive logic topology_.

######  Lemma 5.20.

Let $M$ be a local ${\mathtt{L}}$-structure, $x,y$ two disjoint variables and
$A$ a subset. Then, the local positive $A$-logic topology in $M^{x}\times
M^{y}$ is finer than the product topology from the local positive $A$-logic
topologies of $M^{x}$ and $M^{y}$.

###### Proof.

It suffices to show that the Cartesian projections are continuous. By
symmetry, we just show that ${\mathrm{proj}}_{x}\colon M^{x}\times M^{y}\to
M^{x}$ is continuous. Let $V\subseteq M^{x}$ be locally positive
$\bigwedge_{A}$-definable. Then, ${\mathrm{proj}}^{-1}_{x}(V)$ is trivially
locally positive $\bigwedge_{A}$-definable, being defined by the same partial
local positive type $\underline{V}$ as $V$ but with extra dummy variable $y$.
∎

The main aim of this subsection is to prove the following two fundamental
results, one being the converse of the other. The first one was already proved
in [Hru22a, Lemma 2.3]. We give here the proof for the sake of completeness.

######  Theorem 5.21.

Let $\mathfrak{C}$ a retractor of ${\mathtt{T}}$. Then, every ball of
$\mathfrak{C}$ is compact in the positive local logic topology.

###### Proof.

Consider a ball ${\mathrm{d}}(a)$ in $\mathfrak{C}$. Let $\\{V_{i}\\}_{i\in
I}$ be a family of local positive $\bigwedge_{\mathfrak{C}}$-definable subsets
of $\mathfrak{C}$ such that $\bigcap_{i\in
I_{0}}V_{i}\cap{\mathrm{d}}(a)\neq\emptyset$ for any finite $I_{0}\subseteq
I$. Consider $\underline{V}(x)=\bigwedge_{i\in
I}\underline{V}_{i}(x)\wedge{\mathrm{d}}(x,a)$. Note that ${\mathrm{d}}(x,a)$
is a bound of $x$. By hypothesis, every finite subset of $\underline{V}$ is
satisfiable in $\mathfrak{C}$, so $\underline{V}$ is boundedly satisfiable. By
Lemma 4.9, we conclude that $\underline{V}$ is locally satisfiable in
${\mathrm{Th}}_{-}(\mathfrak{C}/\mathfrak{C})$. By saturation of
$\mathfrak{C}$ (Theorem 5.9), we conclude that there is $b\in V=\bigcap
V_{i}\cap{\mathrm{d}}(a)$. As $\\{V_{i}\\}_{i\in I}$ is arbitrary, we conclude
that ${\mathrm{d}}(a)$ is compact. ∎

The next theorem is a stronger version of [Hru22a, Lemma 2.4], closer to the
original version for positive logic.

######  Theorem 5.22.

Let $M\models^{\mathrm{pc}}{\mathtt{T}}$. Suppose that every ball in $M$ is
compact in the local positive logic topology. Then, $M$ is a retractor of
${\mathtt{T}}$.

###### Proof.

Let $f\colon A\to M$ be a positive embedding and $g\colon A\to B$ a
homomorphism with $A,B\models{\mathtt{T}}$. Consider the family $\mathcal{K}$
of partial homomorphisms $h\colon D\subseteq B\to M$ such that $f=h\circ g$.
By Corollary 2.13, as $A\precsim^{+}M\models^{\mathrm{pc}}{\mathtt{T}}$,
$A\models^{\mathrm{pc}}{\mathtt{T}}$. Therefore, $g\colon A\to B$ is a
positive embedding, so $h_{0}=f\circ g^{-1}\colon{\mathrm{Im}}\,g\to M$ is a
partial homomorphism from $B$ to $M$ with $f=h_{0}\circ g$. In particular,
$\mathcal{K}\neq\emptyset$. Obviously, $\mathcal{K}$ is partially ordered by
$\subseteq$ and, for any chain $\Omega\subseteq\mathcal{K}$,
$\bigcup\Omega\in\mathcal{K}$. By Zorn’s Lemma, there is a maximal element
$h\in\mathcal{K}$. Write $D\coloneqq\mathrm{Dom}\ h$ and take an arbitrary
single element $b\in B$. For some $a\in A$ and locality relation
${\mathrm{d}}$, we have $b\in{\mathrm{d}}(g(a))$. Consider
$\Gamma(x,D)={\mathrm{ltp}}^{B}_{+}(b/D)$. Then, for any
$\Delta(x,D)\subseteq\Gamma(x,D)$ finite, we have that $B\models\exists
x{\in}{\mathrm{d}}(g(a))\,\bigwedge\Delta(x,D)$. Using that $h$ is a
homomorphism, by Remark 2.3, we have that $M\models\exists
x{\in}{\mathrm{d}}(a)\,\bigwedge\Delta(x,h(D))$. Therefore,
$\Delta(M,h(D))\cap{\mathrm{d}}(a)\neq\emptyset$ for any
$\Delta\subseteq\Gamma$ finite. By compactness of ${\mathrm{d}}(a)$ in the
local positive logic topology, we conclude that there is $c\in{\mathrm{d}}(a)$
such that $M\models\Gamma(c,h(D))$. Thus, $h\cup\\{(b,c)\\}\in\mathcal{K}$,
concluding by maximality that $b\in D$. As $b$ is arbitrary, we conclude that
$h\colon B\to M$ is a homomorphism with $f=h\circ g$. ∎

Without local positive closedness, $M$ is not necessarily a retractor. Indeed,
the local positive logic topology of a reduct is always coarser than the local
positive topology on the original language, so every reduct of a retractor
still satisfies that balls are compact on the local positive logic topology.
However, we know that the reduct of a local positively closed structure might
be no local positively closed, so the reduct of a retractor is not necessarily
a retractor.

However, it is worth noting that, in the proof of Theorem 5.22, the fact that
$M$ is local positively closed for ${\mathtt{T}}$ is only used to conclude
that $A$ is a local positively closed model of ${\mathtt{T}}$. Therefore,
dropping out this hypothesis, we can get the following useful variation.

######  Theorem 5.23.

Let $M\models{\mathtt{T}}$. Suppose that every ball in $M$ is compact in the
local positive logic topology. Then, $M_{A}$ is universal for
${\mathrm{Th}}_{-}(M/A)$ for any $A\leq M$. Thus, if
$A\models^{\mathrm{pc}}{\mathtt{T}}$, $M$ is universal for
${\mathrm{Com}}^{\mathrm{pc}}(A)$, and so there is a retractor of
${\mathtt{T}}$ at $A$.

###### Proof.

Let $B_{A}\models{\mathrm{Th}}_{-}(M/A)$. Write $f\colon A\to M$ for the
inclusion and $g\colon A\to B$ be the map given by $a\mapsto a^{B_{A}}$.
Consider the family $\mathcal{K}$ of partial homomorphisms $h\colon D\subseteq
B\to M$ such that $f=h\circ g$.

First, note that $\mathcal{K}\neq\emptyset$. If $a_{1}\neq a_{2}$, then $\neg
a_{1}=a_{2}\in{\mathrm{Th}}_{-}(M/A)$, so $a^{B_{A}}_{1}\neq a^{B_{A}}_{2}$.
Hence, $g$ is injective. On the other hand, since
$B_{A}\models{\mathrm{Th}}_{-}(M/A)$, $g$ preserves satisfaction of quantifier
free negative formulas, so $g^{-1}\colon{\mathrm{Im}}\,g\to A$ preserves
satisfaction of quantifier free positive formulas. In other words, $g^{-1}$ is
a homomorphism. Therefore, $h_{0}=f\circ g^{-1}\colon{\mathrm{Im}}\,g\to M$ is
a partial homomorphism from $B$ to $M$ with $f=h_{0}\circ g$, concluding that
$h_{0}\in\mathcal{K}$.

Obviously, $\mathcal{K}$ is partially ordered by $\subseteq$ and, for any
chain $\Omega\subseteq\mathcal{K}$, $\bigcup\Omega\in\mathcal{K}$. By Zorn’s
Lemma, there is a maximal element $h\in\mathcal{K}$. As in the proof of
Theorem 5.22, we conclude that $\mathrm{Dom}\,h=B$, so, in other words,
$h\colon B_{A}\to M_{A}$ is an ${\mathtt{L}}(A)$-homomorphism.

Suppose now that $A\models^{\mathrm{pc}}{\mathtt{T}}$. Then, we have that
$B_{A}\models{\mathrm{Th}}_{-}(M/A)$ if and only if $B\models{\mathtt{T}}$ and
$A\lesssim B$ — where $B$ is the ${\mathtt{L}}$-reduct of $B_{A}$ and $B_{A}$
is the ${\mathtt{L}}(A)$-expansion of $B$ given by the embedding $A\lesssim
B$. Indeed, if $B_{A}\models{\mathrm{Th}}_{-}(M/A)$, obviously
$B\models{\mathtt{T}}$ and $g\colon a\mapsto a^{B}$ defines a homomorphism
from $A$ to $B$. On the other hand, as $A\models^{\mathrm{pc}}{\mathtt{T}}$,
if $B\models{\mathtt{T}}$ and there is a homomorphism $g\colon A\to B$, then
$B_{A}\models{\mathrm{Th}}_{-}(A/A)={\mathrm{Th}}_{-}(M/A)$ via
$a^{B_{A}}=g(a)$.

Now, let $B\models{\mathtt{T}}$ with $A\frown B$. There is
$N\models{\mathtt{T}}$ such that $A\lesssim N$ and $B\lesssim N$. It follows
that $N_{A}\models{\mathrm{Th}}_{-}(M/A)$, so $N\lesssim M$ by universality of
$M$ for ${\mathrm{Th}}_{-}(M/A)$. In particular, $B\lesssim M$. As
$B\in{\mathrm{Com}}(A)$ is arbitrary, we conclude that $M$ is universal for
${\mathrm{Com}}(A)$. In particular, $|N|\leq|M|$ for every
$N\in{\mathrm{Com}}^{\mathrm{pc}}(A)$, so there is a retractor of
${\mathtt{T}}$ at $A$ by Theorem 5.16(3). ∎

######  Definition 5.24.

Let ${\mathrm{d}}$ be a locality relation. An _entourage in ${\mathrm{d}}$_ is
a pair $\psi(x,y)$ and $\varphi(x,y)$ of local positive formulas such that
$\psi(x,y)\leq{\mathrm{d}}(x,y)$, $\varphi(x,y)\perp x=y$ and
$\psi(x,y)\mathrel{\top}\varphi(x,y)$. We say that ${\mathtt{T}}$ has
_entourages_ if it has entourages in all sorts.

######  Remark 5.25.

If $(\psi,\varphi)$ is an entourage, then ${}={}\leq\psi$ by definition of
$\psi\mathrel{\top}\varphi$.

######  Lemma 5.26.

Let ${\mathrm{d}}$ and ${\mathrm{d}}^{\prime}$ be locality relations.

  1. (1)

Let $(\psi,\varphi)$ and $(\psi^{\prime},\varphi^{\prime})$ be entourages in
${\mathrm{d}}$. Then, $(\psi\wedge\psi^{\prime},\varphi\vee\varphi^{\prime})$
is an entourage in ${\mathrm{d}}$.

  2. (2)

Let $(\psi,\varphi)$ be an entourage in ${\mathrm{d}}$. Let $\psi_{0}$ and
$\varphi_{0}$ be local positive formulas with
$\psi\leq\psi_{0}\leq{\mathrm{d}}$ and $\varphi\leq\varphi_{0}$ with
$\varphi_{0}\perp{}={}$. Then, $(\psi_{0},\varphi_{0})$ is an entourage in
${\mathrm{d}}$.

  3. (3)

Suppose ${\mathrm{d}}$ and ${\mathrm{d}}^{\prime}$ are on the same sort. Let
$(\psi,\varphi)$ be an entourage in ${\mathrm{d}}$. If
${\mathrm{d}}\leq{\mathrm{d}}^{\prime}$, then it is an entourage in
${\mathrm{d}}^{\prime}$.

  4. (4)

Let $(\psi(x,y),\varphi(x,y))$ be an entourage in ${\mathrm{d}}(x,y)$ and
$(\psi^{\prime}(x^{\prime},y^{\prime}),\varphi^{\prime}(x^{\prime},y^{\prime}))$
an entourage in ${\mathrm{d}}^{\prime}(x^{\prime},y^{\prime})$. Then,
$(\psi(x,y)\wedge\psi^{\prime}(x^{\prime},y^{\prime}),\varphi(x,y)\vee\varphi^{\prime}(x^{\prime},y^{\prime}))$
is an entourage in ${\mathrm{d}}_{1}\times{\mathrm{d}}_{2}$.

###### Proof.

Elementary by Lemma 2.14. ∎

######  Lemma 5.27.

Let $M\models^{\mathrm{pc}}{\mathtt{T}}$. Let ${\mathrm{d}}$ be a locality
relation and $(\psi,\varphi)$ an entourage in ${\mathrm{d}}$. Then, every
${\mathrm{d}}$-ball is a closed neighbourhood of its centre in the local
positive logic topology of $M$. Furthermore, $\psi(M,a)$ is a closed
neighbourhood of $a$.

###### Proof.

Obviously, $\psi(M,a)$ and $\varphi(M,a)$ are closed subsets of $M^{x}$ in the
local positive logic topology. As $\varphi(x,y)\perp x=y$,
$a\notin\varphi(M,a)$. On the other hand, as $\psi\mathrel{\top}\varphi$ and
$M\models^{\mathrm{pc}}{\mathtt{T}}$, we get that
$\varphi(M,a)\cup\psi(M,a)=M^{x}$ by Lemma 2.14(2). Thus,
$a\in\neg\varphi(M,a)\subseteq\psi(M,a)$, concluding that $\psi(M,a)$ is a
closed neighbourhood of $a$. Finally, as $\psi\leq{\mathrm{d}}$ and
$M\models^{\mathrm{pc}}{\mathtt{T}}$, by Lemma 2.14(2), we get
$\psi(M,a)\subseteq{\mathrm{d}}(a)$, so ${\mathrm{d}}(a)$ is a closed
neighbourhood of $a$ as well. ∎

######  Corollary 5.28.

Let $\mathfrak{C}$ be a retractor of ${\mathtt{T}}$. Suppose ${\mathrm{d}}$
has an entourage. Then, for every $a$, the ${\mathrm{d}}$-ball at $a$ is a
compact neighbourhood of $a$ in the local positive logic topology. In
particular, $\mathfrak{C}$ is weakly locally closed compact in the local
positive logic topology whenever ${\mathtt{T}}$ has entourages.

###### Proof.

By Theorems 5.21 and 5.27. ∎

######  Definition 5.29.

Let ${\mathrm{d}}$ be a locality relation. A _full system of entourages_ is a
family $\\{(\psi_{i},\varphi_{i})\\}_{i\in I}$ of entourages in ${\mathrm{d}}$
such that $\\{\psi_{i}\\}_{i\in I}$ is a full system of approximations of $=$
(see Definition 2.17). We say that ${\mathtt{T}}$ has _full systems of
entourages_ if it has a full system of entourages for every sort.

######  Corollary 5.30.

Let $\mathfrak{C}$ be a retractor of ${\mathtt{T}}$. Let ${\mathrm{d}}(x,y)$
be a locality relation and $\\{(\psi_{i},\varphi_{i})\\}_{i\in I}$ a full
system of entourages in ${\mathrm{d}}$. Then, for every
$a\in\mathfrak{C}^{y}$, the family $\\{\bigwedge_{i\in
I_{0}}\psi_{i}(\mathfrak{C},a):I_{0}\subseteq I\mathrm{\ finite}\\}$ is a
local base of closed compact neighbourhoods of $a$ in the local positive logic
topology. In particular, $\mathfrak{C}$ is locally closed compact in the local
positive logic topology whenever ${\mathtt{T}}$ has full systems of
entourages.

###### Proof.

By Lemma 5.26(1), $(\bigwedge_{i\in I_{0}}\psi_{i},\bigvee_{i\in
I_{0}}\varphi_{i})$ is an entourage of ${\mathrm{d}}$ for any $I_{0}\subseteq
I$ finite. Therefore, by Corollary 5.28, $\\{\bigwedge_{i\in
I_{0}}\psi_{i}(\mathfrak{C},a):I_{0}\subseteq I\mathrm{\ finite}\\}$ is family
of compact neighbourhoods of $a$ in the local positive logic topology. Let
$\phi(x)$ be a local positive formula with $\mathfrak{C}\not\models\phi(a)$.
Since $\\{\psi_{i}\\}_{i\in I}$ is a full system of approximations,
$\bigcap_{i\in I}\psi_{i}(\mathfrak{C},a)=\\{a\\}$. Thus, $\bigcap_{i\in
I}\psi_{i}(\mathfrak{C},a)\cap\phi(\mathfrak{C})=\emptyset$. By compactness of
${\mathrm{d}}(a)$ (Theorem 5.21), we conclude that there is $I_{0}\subseteq I$
finite with $\bigcap_{i\in
I_{0}}\psi_{i}(\mathfrak{C},a)\cap\phi(\mathfrak{C})=\emptyset$, so
$\bigwedge_{i\in
I_{0}}\psi_{i}(\mathfrak{C},a)\subseteq\neg\phi(\mathfrak{C})$. ∎

###  5.2. Local positive compactness

Recall that, unless otherwise stated, we have fixed a local language
${\mathtt{L}}$ and a locally satisfiable negative local theory ${\mathtt{T}}$.

We say that ${\mathtt{T}}$ is _local positive compact_ 333Since Hrushovski
preferably works with (local) primitive positive formulas, this property is
called “(local) primitive positive compactness”, abbreviated $\mathrm{(ppC)}$,
in [Hru22a]. In his paper, Hrushovski attributes the origin of this property
to Walter Taylor. if for any local positive formula $\varphi(x,y,z)$ with
$\exists z\,\varphi(x,y,z)\perp x=y$, any bound $\mathrm{B}(x,z)$ and any
locality relation symbol ${\mathrm{d}}(x,y)$, where $x$ and $y$ are different
single variables on the same sort and $z$ is an arbitrary finite variable,
there is a number $k\in\mathbb{N}_{>0}$ such that

${\mathtt{T}}\models\neg\exists x_{1},\ldots,x_{k},z\,\bigwedge_{i\neq
j}\mathrm{B}(x_{i},z)\wedge{\mathrm{d}}(x_{i},x_{j})\wedge\varphi(x_{i},x_{j},z).$

In [Hru22a, Proposition 2.1(1)], Hrushovski proved that every local positively
compact theory has retractors. Furthermore, in [Hru22a], Hrushovski briefly
argues (assuming the local joint property) that local positive compactness,
existence of retractors and boundedness of local positively closed models are
equivalent properties — these equivalences are true under weak completeness.
Here, in this subsection, we provide the full proof of these equivalences and
show how they vary under different completeness conditions.

We start reproving that local positive compactness implies the existence of
retractors. Although this was already proved in [Hru22a, Proposition 2.1(1)],
we provide the full proof for the sake of completeness.

######  Theorem 5.31.

Suppose ${\mathtt{T}}$ is local positively compact. Then, every
$|{\mathtt{L}}|^{+}$-ultrahomogeneous local positively closed model of
${\mathtt{T}}$ is a retractor. In particular, ${\mathtt{T}}$ has retractors.

###### Proof.

Let $\mathfrak{C}$ be an $|{\mathtt{L}}|^{+}$-ultrahomogeneous local
positively closed model of ${\mathtt{T}}$. By Theorem 5.3, find a homomorphism
$f\colon\mathfrak{C}\to\mathfrak{C}^{\prime}$ to a $\kappa$-ultrahomogeneous
local positively closed model $\mathfrak{C}^{\prime}$ of ${\mathtt{T}}$ with
$\kappa\geq|{\mathtt{L}}|^{+}$. As $\mathfrak{C}$ is local positively closed,
we know that $f$ is a positive embedding. To simplify the notation, we
identify $\mathfrak{C}$ with its image, so
$\mathfrak{C}\preceq^{+}\mathfrak{C}^{\prime}$. We claim that
$\mathfrak{C}=\mathfrak{C}^{\prime}$ — in other words, $f$ is an isomorphism.
As $\kappa$ is arbitrary, this concludes that $\mathfrak{C}$ is a retractor of
${\mathtt{T}}$.

In order to reach a contradiction, suppose falsity of our claim. There is then
some (single) element $b$ in $\mathfrak{C}^{\prime}$ outside $\mathfrak{C}$.
Pick $A_{0}$ pointed subset of $\mathfrak{C}$ such that
$|A_{0}|\leq|{\mathtt{L}}|$. We define by recursion a sequence
$(a_{i})_{i\in\mathbb{N}}$ in $\mathfrak{C}$ such that
${\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(a_{i}/A_{i})={\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(b/A_{i})$
for every $i\in\mathbb{N}$, where $A_{i}=A_{0}\cup\\{a_{0},\ldots,a_{i-1}\\}$.
Indeed, suppose $a_{0},\ldots,a_{i-1}$ are already defined and denote
$\mathfrak{C}_{i}\coloneqq\mathfrak{C}_{A_{i}}$ and
$\mathfrak{C}^{\prime}_{i}\coloneqq\mathfrak{C}^{\prime}_{A_{i}}$. Consider
the local positive type
$p(x)={\mathrm{ltp}}^{\mathfrak{C}^{\prime}_{i}}_{+}(b/A_{i})$ of
$\mathfrak{C}^{\prime}$ over $A_{i}$. As
$\mathfrak{C}\preceq^{+}\mathfrak{C}^{\prime}$,
${\mathrm{Th}}_{-}(\mathfrak{C}^{\prime}_{i})={\mathrm{Th}}_{-}(\mathfrak{C}_{i})$.
Then, $p(x)$ is a local positive type of $\mathfrak{C}$ over $A_{i}$. By
saturation of $\mathfrak{C}$ (Lemma 5.7), we conclude that there is $a_{i}$ in
$\mathfrak{C}$ realising $p$. Hence,
${\mathrm{ltp}}^{\mathfrak{C}}_{+}(a_{i}/A_{i})={\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(b/A_{i})$.
As $\mathfrak{C}\preceq^{+}\mathfrak{C}^{\prime}$, we have
${\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(a_{i}/A_{i})={\mathrm{ltp}}^{\mathfrak{C}}_{+}(a_{i}/A_{i})={\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(b/A_{i})$.

Let $\mathrm{B}(x,c)$ be any bound realised by $b$ over $A_{0}$. Also, as
$\mathfrak{C}^{\prime}$ is a local structure, pick a locality relation with
$\mathfrak{C}^{\prime}\models{\mathrm{d}}(a_{0},b)$. It follows that
$\mathfrak{C}^{\prime}\models\mathrm{B}(a_{i},c)$ and
$\mathfrak{C}^{\prime}\models{\mathrm{d}}(a_{i},a_{j})$ for any
$i,j\in\mathbb{N}$. By Lemma 4.31, and using saturation of $\mathfrak{C}$, we
can find a local positive $A_{0}$-indiscernible sequence
$(b_{i})_{i\in\mathbb{N}}$ in $\mathfrak{C}$ such that
${\mathrm{ltp}}^{\mathfrak{C}}_{+}(b_{i}/A_{0},b_{0},\ldots,b_{i-1})={\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(b/A_{0},b_{0},\ldots,b_{i-1})$
for any $i\in\mathbb{N}$. Again, as
$\mathfrak{C}\preceq^{+}\mathfrak{C}^{\prime}$, we conclude that
$(b_{i})_{i\in\mathbb{N}}$ is local positive $A_{0}$-indiscernible in
$\mathfrak{C}^{\prime}$ and
${\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(b_{i}/A_{0},b_{0},\ldots,b_{i-1})={\mathrm{ltp}}^{\mathfrak{C}^{\prime}}_{+}(b/A_{0},b_{0},\ldots,b_{i-1})$
for any $i\in\mathbb{N}$.

By Lemma 2.12, as $\mathfrak{C}^{\prime}$ is local positively closed and
$b_{0}\neq b$, there is a positive formula
$\varphi_{0}(x,y)\in{\mathrm{LFor}}^{x,y}_{+}({\mathtt{L}})$ such that
$\varphi_{0}(x,y)\perp x=y$ and
$\mathfrak{C}^{\prime}\models\varphi_{0}(b,b_{0})$. Say $\varphi_{0}=\exists
w\,\phi_{0}(x,y,w)$ and take $c$ in $A_{0}$ on the sort of $w$. There is a
locality relation ${\mathrm{d}}_{1}$ such that
$\mathfrak{C}^{\prime}\models\varphi_{1}(b,b_{0},c)$ with
$\varphi_{1}(x,y,z)=\exists w{\in}{\mathrm{d}}_{1}(z)\,\phi_{0}(x,y,w)$. Take
$\varphi(x,y,z)=\varphi_{1}(x,y,z)\vee\varphi_{1}(y,x,z)$ and note that
$\varphi(x,y,z)$ is a local positive formula with $\exists
z\,\varphi(x,y,z)\perp x=y$ and
$\mathfrak{C}^{\prime}\models\varphi(b,b_{0},c)$. Hence,
$\mathfrak{C}^{\prime}\models\varphi(b_{i},b_{0},c)$ for any
$i\in\mathbb{N}_{>0}$. As $(b_{i})_{i\in\mathbb{N}}$ is local positive
$A_{0}$-indiscernible, we conclude that
$\mathfrak{C}^{\prime}\models\varphi(b_{i},b_{j},c)$ for any $i\neq j$. Hence,
$\mathfrak{C}^{\prime}\models\bigwedge_{i\neq
j}\mathrm{B}(b_{i},c)\wedge{\mathrm{d}}(b_{i},b_{j})\wedge\varphi(b_{i},b_{j},c)$.
Thus, we have $\exists x_{1},\ldots,x_{k},z\,\bigwedge_{i\neq
j}\mathrm{B}(x_{i},z)\wedge{\mathrm{d}}(x_{i},x_{j})\wedge\varphi(x_{i},x_{j},z)\in{\mathtt{T}}_{+}$
for every $k\in\mathbb{N}$ as witnessed by $\mathfrak{C}^{\prime}$. In
particular, ${\mathtt{T}}$ is not local positively compact, contradicting our
initial assumption. ∎

We now list all the conditions equivalent to existence of retractors. We start
with a version that does not require irreducibility conditions:

######  Theorem 5.32.

Let $M\models^{\mathrm{pc}}{\mathtt{T}}$. Then, the following are equivalent:

  1. (1)

There is a homomorphism $h\colon M\to N$ with $N\models{\mathtt{T}}$ whose
balls are compact in the local positive topology.

  2. (2)

${\mathtt{T}}$ has a retractor at $M$.

  3. (3)

${\mathrm{Com}}^{\mathrm{pc}}(M)$ is bounded.

  4. (4)

${\mathrm{Th}}_{-}(M/M)$ is local positively compact (in ${\mathtt{L}}(M)$).

  5. (5)

Every $|{\mathtt{L}}(M)|^{+}$-ultrahomogeneous local positively closed model
$N$ of ${\mathtt{T}}$ with $M\precsim^{+}N$ is a retractor of ${\mathtt{T}}$.

###### Proof.

  1. (1)$\Rightarrow$(2)

By Theorem 5.23.

  2. (2)$\Leftrightarrow$(3)

By Theorem 5.16(3).

  3. (2)$\Rightarrow$(4)

Let $x$ be a single variable and $z$ a finite variable. Take a bound
$\mathrm{B}(x,z)$ of $x$ in ${\mathrm{Th}}_{-}(M/M)$, a local positive formula
$\varphi(x,y,z)\in{\mathrm{LFor}}_{+}({\mathtt{L}}(M))$ with
${\mathrm{Th}}_{-}(M/M)\models\exists z\,\varphi(x,y,z)\perp x=y$ and a
locality relation ${\mathrm{d}}(x,y)$. In order to reach a contradiction,
suppose that

$M\models\exists x_{1},\ldots,x_{k},z\,\bigwedge_{i\neq
j}\mathrm{B}(x_{i},z)\wedge{\mathrm{d}}(x_{i},x_{j})\wedge\varphi(x_{i},x_{j},z)$

for every $k\in\mathbb{N}$. Consider the partial local positive type

$\Gamma(\bar{x},z)=\bigwedge_{i\neq
j}\mathrm{B}(x_{i},z)\wedge{\mathrm{d}}(x_{i},x_{j})\wedge\varphi(x_{i},x_{j},z)$

with $\bar{x}=\\{x_{i}\\}_{i\in\mathbb{N}}$ countable set of single variables.
By supposition, $\Gamma(\bar{x},z)$ is finitely locally satisfiable in
${\mathrm{Th}}_{-}(M/M)$. As $\Gamma(\bar{x},z)$ implies a bound of
$\bar{x},z$, by Lemma 4.9, we conclude that $\Gamma(\bar{x},z)$ is locally
satisfiable in ${\mathrm{Th}}_{-}(M/M)$. Let $\mathfrak{C}$ be a retractor of
${\mathtt{T}}$ for $M$. As $M$ is local positively closed,
${\mathrm{Th}}_{-}(\mathfrak{C}/M)={\mathrm{Th}}_{-}(M/M)$. By Lemma 5.2,
$\mathfrak{C}_{M}$ is a retractor of ${\mathrm{Th}}_{-}(M/M)$. By Theorem
5.9(iii), $\mathfrak{C}$ satisfies $\Gamma(\bar{x},z)$. Therefore, there are
$(a_{i})_{i\in\mathbb{N}}$ and $c$ in $\mathfrak{C}$ with
$a_{i}\in{\mathrm{d}}(a_{0})$ for each $i\in\mathbb{N}$ and
$\mathfrak{C}\models\varphi(a_{i},a_{j},c)$ for each $i\neq j$ — in
particular, $a_{i}\neq a_{j}$ for $i\neq j$. Applying Zorn’s Lemma, find
$(a_{i})_{i\in\lambda}$ maximal such that $a_{i}\in{\mathrm{d}}(a_{0})$ for
every $i\in\lambda$ and $\mathfrak{C}\models\varphi(a_{i},a_{j},c)$ for each
$i\neq j$ — in particular, $a_{i}\neq a_{j}$ for $i\neq j$, so
$\lambda\leq|C|$. Now, consider
$U_{i}=\neg\varphi(\mathfrak{C},a_{i},c)\cup\neg\varphi(a_{i},\mathfrak{C},c)$.
By maximality of $(a_{i})_{i\in\lambda}$, it follows that
${\mathrm{d}}(a_{0})\subseteq\bigcup_{i\in\lambda}U_{i}$. By Theorem 5.21,
${\mathrm{d}}(a_{0})$ is compact in the local positive topology. As
$\\{U_{i}\\}_{i\in\lambda}$ is an open covering, there is
$I_{0}\subseteq\lambda$ finite such that
${\mathrm{d}}(a_{0})\subseteq\bigcup_{i\in I_{0}}U_{i}$. As $I_{0}$ is finite,
there is $n\in\mathbb{N}$ with $n\notin I_{0}$. Then, by choice of
$(a_{i})_{i\in\lambda}$, we have that $a_{n}\in{\mathrm{d}}(a_{0})$ and
$a_{n}\notin\bigcup_{i\in I_{0}}U_{i}$, getting a contradiction.

  4. (4)$\Rightarrow$(5)

By Theorem 5.31.

  5. (5)$\Rightarrow$(1)

By Theorem 5.3, there is a retractor at $M$. Hence, by Theorem 5.21, we
conclude.∎

######  Corollary 5.33.

We have $|\mathfrak{C}|\leq 2^{|{\mathtt{L}}|}$ for every retractor
$\mathfrak{C}$ of ${\mathtt{T}}$.

###### Proof.

By Löwenheim-Skolem, take $M\preceq\mathfrak{C}$ with $|M|\leq|{\mathtt{L}}|$.
Using Theorems 5.3 and 5.4, find an $|{\mathtt{L}}|^{+}$-ultrahomogeneous
$\mathfrak{C}_{0}$ model of ${\mathtt{T}}$ with $|\mathfrak{C}_{0}|\leq
2^{|{\mathtt{L}}|}$ and $M\leq\mathfrak{C}_{0}$. Since $M\preceq\mathfrak{C}$
and $\mathfrak{C}\models^{\mathrm{pc}}{\mathtt{T}}$, we have
$M\models^{\mathrm{pc}}{\mathtt{T}}$ by Corollary 2.13. Thus,
$M\preceq^{+}\mathfrak{C}_{0}$. As $\mathfrak{C}$ is a retractor of
${\mathtt{T}}$ at $M$, by Theorem 5.32, ${\mathrm{Th}}_{-}(M)$ is local
positively compact. As $M\preceq^{+}\mathfrak{C}_{0}$,
${\mathrm{Th}}_{-}(\mathfrak{C}_{0})={\mathrm{Th}}_{-}(M)$. By Lemma 5.2,
$\mathfrak{C}_{0}$ is $|{\mathtt{L}}|^{+}$-ultrahomogeneous for
${\mathrm{Th}}_{-}(M)$. By Theorem 5.31, we conclude that $\mathfrak{C}_{0}$
is a retractor of ${\mathrm{Th}}_{-}(M)$. By Lemma 5.2, as
$\mathfrak{C}_{0}\models^{\mathrm{pc}}{\mathtt{T}}$, we conclude that it is a
retractor of ${\mathtt{T}}$ too. Note that $\mathfrak{C}\frown
M\frown\mathfrak{C}_{0}$ and $M\models^{\mathrm{pc}}{\mathtt{T}}$. By Lemma
5.15, $\mathfrak{C}\frown\mathfrak{C}_{0}$. By Theorem 5.16(2), we conclude
that $\mathfrak{C}\cong\mathfrak{C}_{0}$. In particular, $|\mathfrak{C}|\leq
2^{|{\mathtt{L}}|}$. ∎

######  Corollary 5.34.

The following are equivalent:

  1. (1)

${\mathtt{T}}$ has retractors.

  2. (2)

There is some cardinal $\kappa$ such that $|M|<\kappa$ for every
$M\models^{\mathrm{pc}}{\mathtt{T}}$.

  3. (3)

For every $M\models^{\mathrm{pc}}{\mathtt{T}}$, $|M|\leq 2^{|{\mathtt{L}}|}$.

We now state the equivalences under irreducibility assumptions.

######  Corollary 5.35.

Assume ${\mathtt{T}}$ is weakly complete. Then, ${\mathtt{T}}$ has retractors
if and only if it is local positively compact.

###### Proof.

By Theorem 5.31, if ${\mathtt{T}}$ is local positively compact, then
${\mathtt{T}}$ has retractors. On the other hand, suppose that ${\mathtt{T}}$
is weakly complete and has retractors. Let $\mathfrak{C}$ be a retractor at
$M$ with $M\models^{\mathrm{pc}}{\mathtt{T}}_{\pm}$. By Remark 2.3, we get
${\mathrm{Th}}_{-}(\mathfrak{C})={\mathtt{T}}$. By Theorem 5.32,
${\mathrm{Th}}_{-}(\mathfrak{C}/\mathfrak{C})$ is local positively compact so,
in particular, ${\mathtt{T}}={\mathrm{Th}}_{-}(\mathfrak{C})$ is local
positively compact. ∎

######  Corollary 5.36.

Assume ${\mathtt{T}}$ is complete. Then, ${\mathtt{T}}$ has retractors if and
only if it has at least one retractor.

###### Proof.

Suppose ${\mathtt{T}}$ is complete and has at least one retractor
$\mathfrak{C}$. By Theorem 5.32, we conclude that
${\mathrm{Th}}_{-}(\mathfrak{C}/\mathfrak{C})$ is local positively compact. In
particular, ${\mathrm{Th}}_{-}(\mathfrak{C})$ is local positively compact. By
completeness, ${\mathtt{T}}={\mathrm{Th}}_{-}(\mathfrak{C})$ is local
positively compact. Then, by Theorem 5.31, ${\mathtt{T}}$ has retractors. ∎

######  Example 5.37.

Corollary 5.36 may not hold if ${\mathtt{T}}$ is merely weakly complete.
Indeed, take the one-sorted local language ${\mathtt{L}}$ with locality
relations $\mathtt{D}=\\{{\mathrm{d}}_{n}\\}_{n\in\mathbb{N}}$ (with the usual
ordered monoid structure induced by the natural numbers), unary predicates
$\\{P_{n},Q_{n}\\}_{n\in\mathbb{N}}$ and a binary relation $\prec$.

Consider the local structure in $\mathbb{R}$ given by $P_{n}(x)\Leftrightarrow
x\geq n$, $Q_{n}(x)\Leftrightarrow x\leq-n$, $x\prec y\Leftrightarrow 0<x<y$
and ${\mathrm{d}}_{n}(x,y)\Leftrightarrow|x-y|\leq n$. Let
${\mathtt{T}}={\mathrm{Th}}_{-}(\mathbb{R})$. Then, ${\mathtt{T}}$ is not
local positive compact. Indeed, we can take $\varphi(x,y)\coloneqq x\prec
y\vee y\prec x$, that satisfies $\varphi(x,y)\perp x=y$, and the locality
relation ${\mathrm{d}}_{1}$. In that case, we find that
$S=\\{\nicefrac{{1}}{{i}}\\}_{i\in\mathbb{N}_{+}}$ satisfies that, for any two
distinct $i,j$, we have $\varphi(i,j)\wedge{\mathrm{d}}_{1}(i,j)$. Thus, by
Corollary 5.35, ${\mathtt{T}}$ does not have retractors (that is, there is
some models of ${\mathtt{T}}$ incompatible with all retractors).

On the other hand, let $\widetilde{\mathbb{R}}$ be an $\omega$-saturated
elementary extension of $\mathbb{R}$. Then, since $\bigwedge_{n}Q_{n}(x)$ is
finitely satisfiable, there is some $a\in\widetilde{\mathbb{R}}$ realising it.
Consider the substructure $A$ with universe $\\{a\\}$. Since
$\widetilde{\mathbb{R}}\cong\mathbb{R}$, we have
$\widetilde{\mathbb{R}}\mathrel{\mathord{\models}_{\star}}{\mathtt{T}}$, so
$A\models{\mathtt{T}}$. Furthermore, $A$ is a retractor as, if $g\colon A\to
M$ is a homomorphism with $M\models{\mathtt{T}}$, then the constant function
$f\colon M\to A$ mapping everything to $a$ is homomorphism too (so a
retractor). Indeed, if there is a homomorphism $g\colon A\to M$, we must have
that $P_{n}(M)=\prec(M)=\emptyset$, since $M\models Q_{m+1}(h(a))$ while
$\neg\exists x,y\,Q_{m+1}(x)\wedge
P_{n}(y)\wedge{\mathrm{d}}_{m+1}(x,y)\in{\mathtt{T}}$ and $\neg\exists
x,y,z\,Q_{m+1}(x)\wedge y\prec z\wedge{\mathrm{d}}_{m}(x,y)\in{\mathtt{T}}$
for all $m\in\mathbb{N}$. On the other hand, for any $n\in\mathbb{N}$, we have
$A\models Q_{n}(a)$ and $A\models{\mathrm{d}}_{n}(a,a)$, thus $f$ is a
homomorphism.

###  5.3. Retractors and non-local homomorphism universal models

As covered in [Hru22b, Subsection 2.5], negative theories have a (non-local)
homomorphism universal positively closed model if and only if there is a bound
on the cardinality of their (non-local) positively closed models. Since every
negative local theory is, in particular, a negative theory, it is natural to
ask whether there is a connection between the existence of local retractors
and the existence of non-local homomorphism universal positively closed
models. One direction is straightforward:

######  Lemma 5.38.

Assume ${\mathtt{T}}$ is a negative local theory that has a non-local
homomorphism universal model $\mathfrak{U}$. Then, ${\mathtt{T}}$ has local
retractors, and every local retractor embeds into $\mathfrak{U}$.

###### Proof.

Let $M\models^{\mathrm{pc}}{\mathtt{T}}$. Then, since
$M\mathrel{\mathord{\models}_{\star}}{\mathtt{T}}$, there is some homomorphism
$h\colon M\to\mathfrak{U}$. Since $h(M)\models{\mathtt{T}}$, we conclude that
$h$ is a positive embedding into $h(M)$ and, in particular, injective. We get
that $|M|\leq|\mathfrak{U}|$ for all $M\models^{\mathrm{pc}}{\mathtt{T}}$.
Thus, by Theorem 5.16(3), ${\mathtt{T}}$ has retractors. ∎

Unfortunately, this is the only connection that holds — a theory with local
retractors does not need to have a non-local homomorphism universal model.

######  Example 5.39.

Consider the one-sorted local language ${\mathtt{L}}=\\{0,1,+,\cdot\\}$ with
locality relations $\mathtt{D}=\\{{\mathrm{d}}_{n}\\}_{n\in\mathbb{N}}$ (with
the usual ordered monoid structure induced by the natural numbers), where
$0,1$ are constants and $+,\cdot$ are ternary relations.

Consider the structure $\mathbb{R}$ with the usual field interpretation
($+,\cdot$ are the graphs of addition and multiplication respectively) and
metric interpretation of the locality relations. By Theorem 5.23, $\mathbb{R}$
is local homomorphism universal for
${\mathtt{T}}_{\mathbb{R}}={\mathrm{Th}}_{-}(\mathbb{R})$. Since every field
endomorphism of $\mathbb{R}$ is the identity, it is in particular an
automorphism thus $\mathbb{R}$ is a retractor by Lemma 5.19. Since
$\mathbb{R}$ is local homomorphism universal, it is the unique retractor (up
to isomorphism).

However, as a negative theory, ${\mathtt{T}}_{\mathbb{R}}$ is unbounded.
Indeed, we have the positive formula $x<y$, which is expressed by $\exists
z,w\,z\cdot w=1\wedge y=x+z^{2}$. Now, $x<y$ denials equality in
${\mathtt{T}}_{\mathbb{R}}$. Since $\mathbb{R}$ has infinite ordered
sequences, by compactness, there is a positively closed model of
${\mathtt{T}}_{\mathbb{R}}$ of any cardinality.

More generally, in fields we can use the formula $\exists z,w\,z\cdot
w=1\wedge y=x+z$, which positively defines inequality in every field. Hence,
by compactness, for any infinite field $K$, the negative theory
${\mathtt{T}}_{-}(K)$ must have positively closed models of arbitrarily large
cardinality. Thus, for instance, we can consider $\mathbb{C}$ or
$\mathbb{Q}_{p}$. Indeed:

The $p$-adic numbers $\mathbb{Q}_{p}$ as ${\mathtt{L}}$-structure where the
locality predicates are interpreted according to the $p$-adic metric is again
local homomorphism universal in their own theory
${\mathtt{T}}_{p}={\mathrm{Th}}_{-}(\mathbb{Q}_{p})$ by Theorem 5.23. Since
every field endomorphism of $\mathbb{Q}_{p}$ is also the identity,
$\mathbb{Q}_{p}$ is the unique retractor of ${\mathtt{T}}_{p}$ by Lemma 5.19.
However, it has arbitrarily large (non-local) positively closed models as
inequality is positively definable.

Similarly, we can consider $\mathbb{C}$ as ${\mathtt{L}}$-structure and
observe that, again, ${\mathtt{T}}_{\mathbb{C}}={\mathrm{Th}}_{-}(\mathbb{C})$
has arbitrarily large (non-local) positively closed models, and $\mathbb{C}$
is local homomorphism universal for ${\mathtt{T}}_{\mathbb{C}}$ by Theorem
5.23. If $h\colon\mathbb{C}\to\mathbb{C}$ is an ${\mathtt{L}}$-endomorphism,
it fixes $\mathbb{Q}$ (since it is in particular a field endomorphism). For
any $a,b\in\mathbb{C}$, we have that
$\mathbb{C}\models{\mathrm{d}}_{m}(na,nb)$ if and only if
$\|a-b\|\leq\nicefrac{{m}}{{n}}$. Consequently, $h$ is continuous. Since
$\mathbb{R}$ is the closure of $\mathbb{Q}$, $h$ setwise fixes $\mathbb{R}$,
so $h_{\mid\mathbb{R}}$ is an endomorphism of $\mathbb{R}$, so it pointwise
fixes $\mathbb{R}$. Thus, it is either the identity or complex conjugation,
and in any case an automorphism. We thus get here as well that $\mathbb{C}$ is
the unique retractor for ${\mathtt{T}}_{\mathbb{C}}$ by Lemma 5.19.

If there is a non-local homomorphism universal positively closed model
$\mathfrak{U}$, one might expect that the local retractors are precisely the
maximal local substructures of $\mathfrak{U}$ (i.e. the local components
containing the constants). This is the case for instance in Example 3.7(3).
However, the following example shows that this is not true in general:

######  Example 5.40.

Consider the one-sorted local language ${\mathtt{L}}$ with locality predicates
$\mathtt{D}=\\{{\mathrm{d}}_{n}\\}_{n\in\mathbb{N}}$ (with the usual ordered
monoid structure induced by the natural numbers) and binary predicates
$\\{S\\}\cup\\{E_{n},U_{n}\\}_{n\in\mathbb{N}}$. Consider the
${\mathtt{L}}$-structure $M$ with universe $2^{\omega}$ and interpretations
$M\models{\mathrm{d}}_{n}(\eta,\xi)\Leftrightarrow|\\{i:\eta(i)\neq\xi(i)\\}|\leq
n$, $M\models E_{n}(\eta,\xi)\Leftrightarrow\eta(n)=\xi(n)$, $M\models
U_{n}(\eta,\xi)\Leftrightarrow\eta(n)\neq\xi(n)$ and $M\models
S(\eta,\xi)\Leftrightarrow\eta,\xi\mathrm{\ have\ no\ common\ zeros}$. Let
${\mathtt{T}}={\mathrm{Th}}_{-}(M)$ be the negative theory of $M$. Since
$M\models{\mathtt{T}}_{\mathrm{loc}}$, ${\mathtt{T}}$ is locally satisfiable
by Theorem 1.9. For $\eta\in M$, write $M_{\eta}\coloneqq\mathtt{D}(\eta)$.
Let $1\in M$ be the constant sequence $1$.

Now, we look at $M$ as non-local structure, i.e. as
${\mathtt{L}}_{\ast}$-structure. Note that the positive logic topology in $M$
is coarser than the Tychonoff topology, so $M$ is compact with the positive
logic topology. Thus, by [Seg22, Lemma 2.24], we get that $M$ is homomorphism
universal for $\mathcal{M}_{\ast}({\mathtt{T}})$. Furthermore, note that, in
$M$, the unique element $\eta$ satisfying $M\models S(\eta,\eta)$ is precisely
$\eta=1$. Thus, $\eta(n)=1\Leftrightarrow M\models\exists x\ S(x,x)\wedge
E_{n}(x,\eta)$ and $\eta(n)=0\Leftrightarrow M\models\exists x\ S(x,x)\wedge
U_{n}(x,\eta)$. Consequently, the only endomorphism of $M$ is the identity. By
Lemma 5.19 applied to ${\mathtt{L}}_{\star}$-structures, $M$ is a non-local
homomorphism universal positively closed model of ${\mathtt{T}}$.

Suppose $M\models\varphi$ with $\varphi$ a positive formula in ${\mathtt{L}}$.
For $n\in\mathbb{N}$, let ${\mathtt{L}}_{n}$ be the reduct of ${\mathtt{L}}$
consisting of
$\\{S\\}\cup\\{{\mathrm{d}}_{k}\\}_{k\in\mathbb{N}}\cup\\{E_{k},U_{k}\\}_{k\leq
n}$ and take $n$ such that $\varphi$ is a formula in ${\mathtt{L}}_{n}$.
Consider the map $f\colon M\to M_{1}$ given by $f(\xi)(k)=\xi(k)$ for $k\leq
n$ and $f(\xi)(k)=1$ for $k>n$. Obviously, $E_{k}(\xi,\zeta)$ implies
$E_{k}(f(\xi),f(\zeta))$ and $U_{k}(\xi,\zeta)$ implies
$U_{k}(f(\xi),f(\zeta))$ for any $k\leq n$. Also,
$\\{i:f(\xi)(i)=f(\zeta)(i)=0\\}\subseteq\\{i:\xi(i)=\zeta(i)=0\\}$, so
$S(\xi,\zeta)$ implies $S(f(\xi),f(\zeta))$. On the other hand,
$|\\{m:f(\xi)(m)\neq f(\zeta)(m)\\}|\leq|\\{m:\xi(m)\neq\zeta(m)\\}|$, so
${\mathrm{d}}_{k}(\xi,\zeta)$ implies ${\mathrm{d}}_{k}(f(\xi),f(\zeta))$ for
any $k$. Thus, $f$ is an ${\mathtt{L}}_{n}$-homomorphism. By Remark 2.3, we
conclude that $M_{1}\models\varphi$. As $\varphi$ is arbitrary, we conclude
that ${\mathrm{Th}}_{-}(M_{1})={\mathtt{T}}$. Hence, ${\mathtt{T}}$ is a
weakly complete local theory.

Pick $\eta\in M\setminus M_{1}$. We claim that $M_{\eta}$ is not a retracter
of ${\mathtt{T}}$. In particular, we claim that $M_{\eta}$ is not local
positively closed. Consider the map $f\colon M_{\eta}\to M_{1}$ given by
$f(\xi)(i)=1$ if $x(i)=\eta(i)$ and $f(\xi)(i)=0$ if $\xi(i)\neq\eta(i)$. We
claim that $f$ is a homomorphism from $M_{\eta}$ to $M_{1}$ which is not a
positive embedding. Obviously,

$\displaystyle\\{i:\zeta(i)=\xi(i)\\}$
$\displaystyle=\\{i:\zeta(i)=\xi(i)=\eta(i)\\}\cup\\{i:\zeta(i)=\xi(i)\neq\eta(i)\\}$
$\displaystyle=\\{i:f(\zeta)(i)=f(\xi)(i)=1\\}\cup\\{i:f(\zeta)(i)=f(\xi)(i)=0\\}$
$\displaystyle=\\{i:f(\zeta)(i)=f(\xi)(i)\\},$

so $f$ preserves $E_{n},U_{n},{\mathrm{d}}_{n}$ for all $n$. As $\eta\notin
M_{1}$, for any $\zeta,\xi\in M_{\eta}$, we have $\neq S(\zeta,\xi)$. Hence,
$f$ is a homomorphism. However, $M_{1}\models S(1,1)$ and $M_{\eta}\not\models
S(\eta,\eta)$, so $f$ is not a positive embedding.

On the other hand, in Example 5.40, there is one local component which is
local positively closed, so a local retractor. In fact, it turns out that
Example 5.40 has the local joint property, so this is the only local
retractor.

######  Question 5.41.

Suppose ${\mathtt{T}}$ has a non-local homomorphism universal positively
closed model $\mathfrak{U}$. Is there always at least one local component of
$\mathfrak{U}$ which is a local retractor of ${\mathtt{T}}$? For instance,
consider the pointed case: Is the local component at the constants of
$\mathfrak{U}$ a local retractor?

######  Example 5.42.

Without assuming ${\mathtt{T}}$ is a local theory (i.e. closed under local
implications), Question 5.41 is obviously false.

Consider the single sorted language ${\mathtt{L}}=\\{S,R\\}$ where $S,R$ are
binary relations, and with $\mathtt{D}=\\{=,{\mathrm{d}}_{1}\\}$ (with trivial
ordered monoid structure). Consider $N$ with universe $\\{a,b,c,d\\}$ and
interpretations $R=\\{(a,d),(b,c),(d,a),(c,b)\\}$,
$S=\\{(a,b),(b,a),(c,d),(d,c)\\}$, and
${\mathrm{d}}_{1}=\Delta_{N}\cup\\{(a,c),(c,a),(b,d),(d,b)\\}$, where
$\Delta_{N}$ is the diagonal in $N$.

Then, $N$ has two maximal local substructures — $\\{a,c\\}$ and $\\{b,d\\}$ —
but neither is a local positively closed model of ${\mathrm{Th}}_{-}(N)$. In
fact, the unique (local) retractor for $N$ is a singleton with both $R$ and
$S$ empty.

###  5.4. Automorphisms of retractors

Recall that, unless otherwise stated, we have fixed a local language
${\mathtt{L}}$ and a locally satisfiable negative local theory ${\mathtt{T}}$.
From now on, unless otherwise stated, we also fix a retractor $\mathfrak{C}$
of ${\mathtt{T}}$.

The _pointwise topology_ on ${\mathrm{Aut}}(\mathfrak{C})$ is the initial
topology from the local positive topology given by the evaluation maps
$\mathrm{ev}_{a}\colon{\mathrm{Aut}}(\mathfrak{C})\to\mathfrak{C}^{x}$ for
$a\in\mathfrak{C}^{x}$ with $x$ finite. In other words, a base of open sets is
given by the sets

$\\{\sigma\in{\mathrm{Aut}}(\mathfrak{C}):\mathfrak{C}\models\varphi(\sigma(a))\\}$

where $\varphi(x)\in{\mathrm{LFor}}_{-}({\mathtt{L}}(\mathfrak{C}))$ and
$a\in\mathfrak{C}^{x}$.

######  Remark 5.43.

Let $x$ be a variable and $a\in\mathfrak{C}^{x}$. Then,
$\mathrm{ev}_{a}\colon{\mathrm{Aut}}(\mathfrak{C})\to\mathfrak{C}^{x}$ is
continuous. For $x$ finite this is precisely the definition of the pointwise
topology. For $x$ infinite it follows from the fact that the local positive
logic topology of $\mathfrak{C}^{x}$ is the initial topology given by the
projections
${\mathrm{proj}}_{x^{\prime}}\colon\mathfrak{C}^{x}\to\mathfrak{C}^{x^{\prime}}$
for $x^{\prime}\subseteq x$ finite.

Let ${\mathrm{d}}=({\mathrm{d}}_{s})_{s\in\mathtt{S}}$ be a choice of locality
relations and $a=(a_{s})_{s\in\mathtt{S}}$ a choice of points for each sort.
The _${\mathrm{d}}$ -ball at $a$ in ${\mathrm{Aut}}(\mathfrak{C})$_ is the
subset
${\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)\coloneqq\\{f:f(a_{s})\in{\mathrm{d}}_{s}(a_{s})\mathrm{\
for\ each\ }s\in\mathtt{S}\\}$.

We now prove the main result of this section. In the one-sort case, Theorem
5.44(4) was proved in [Hru22a, Lemma 2.6] by Hrushovski. His proof works
without modifications in the many sorts case as well. In any case, we recall
it here for the sake of completeness. The trivial statements Theorem 5.44(3,5)
were already noted in [Hru22a] too. The easy fact Theorem 5.44(1) is original.
Finally, Theorem 5.44(2) corresponds to [Hru22b, Remark 3.27] in the non-local
case, but was not discussed in [Hru22a].

######  Theorem 5.44.

The following hold:

  1. (1)

${\mathrm{Aut}}(\mathfrak{C})$ is a closed subspace of the set of all function
on $\mathfrak{C}$ with the _pointwise topology_ , i.e. the initial topology
given by the evaluation maps on tuples.

  2. (2)

Let $B$ be a ball in $\mathfrak{C}^{x}$ and $F$ a closed subset of $B$ and $U$
an open subset. Then, $\\{\sigma:\sigma(F)\subseteq U\\}$ is open. In
particular, if there is an entourage on $\mathfrak{C}^{x}$, then
$\\{\sigma:\sigma(F)\subseteq U\\}$ is open for every closed compact subset
$F$ and open set $U$.

  3. (3)

${\mathrm{Aut}}(\mathfrak{C})$ is $\mathrm{T}_{1}$, i.e. Fréchet.

  4. (4)

Every ball of ${\mathrm{Aut}}(\mathfrak{C})$ is compact.

  5. (5)

The inversion map $g\mapsto g^{-1}$ is continuous and left and right
translations are continuous, i.e. ${\mathrm{Aut}}(\mathfrak{C})$ is a semi-
topological group.

###### Proof.

  1. Note that the set of endomorphisms of $\mathfrak{C}$ is given by

$\mathrm{End}(\mathfrak{C})=\bigcap_{a}\mathrm{ev}_{a}^{-1}({\mathrm{qftp}}_{+}(a)(\mathfrak{C})),$

so it is a closed subset in the pointwise topology. Since
${\mathrm{Aut}}(\mathfrak{C})=\mathrm{End}(\mathfrak{C})$ by Corollary 5.10,
we conclude.

  2. (2)

Let $\underline{F}$ be the partial local positive type defining $F$ and
$\underline{U}$ the set of local negative formulas such that
$\neg\underline{U}$ defines $\mathfrak{C}^{x}\setminus U$. Assume
$\underline{F}$ is closed under finite conjunctions and $\underline{U}$ is
closed under finite disjuctions. By compactness of $B$ (Theorem 5.21),
$\\{\sigma:\sigma(F)\subseteq
U\\}=\bigcup_{\varphi\in\underline{F},\psi\in\underline{U}}\\{\sigma:\sigma(\varphi(\mathfrak{C}))\subseteq\psi(\mathfrak{C})\\}$.
Hence, it suffices to show that
$\\{\sigma:\sigma(\varphi(\mathfrak{C}))\subseteq\psi(\mathfrak{C})\\}$ is
open for $\psi$ local negative and $\varphi$ local positive with
$\varphi(\mathfrak{C})\subseteq B$.

Say $B={\mathrm{d}}(b)$ and $c$ are the parameters of $\varphi$ and $\psi$.
Then, $\sigma(\varphi(\mathfrak{C},c))\subseteq\psi(\mathfrak{C},c)$ if and
only if $\mathfrak{C}\models\varphi(a,c)$ implies
$\mathfrak{C}\models\psi(\sigma(a),c)$. Now, since
$\sigma\in{\mathrm{Aut}}(\mathfrak{C})$, it follows that
$\mathfrak{C}\models\psi(\sigma(a),c)$ if and only if
$\mathfrak{C}\models\psi(a,\sigma^{-1}(c))$. Therefore,
$\sigma(\varphi(\mathfrak{C},c))\subseteq\psi(\mathfrak{C},c)$ if and only if
$\mathfrak{C}\models\forall x\,\varphi(x,c)\rightarrow\psi(x,\sigma^{-1}(c))$.
Now, since $\varphi(\mathfrak{C},c)\subseteq B$, we have that $\forall
x\,\varphi(x,c)\rightarrow\psi(x,\sigma^{-1}(c))$ is equivalent to
$\phi(b,c,\sigma^{-1}(c))\coloneqq\neg\exists
x{\in}{\mathrm{d}}(b)\,\varphi(x,c)\wedge\neg\psi(x,\sigma^{-1}(c))$. Hence,

$\displaystyle\\{\sigma:\sigma(\varphi(\mathfrak{C},c))\subseteq\psi(\mathfrak{C},c)\\}=$
$\displaystyle\\{\sigma:\mathfrak{C}\models\phi(b,c,\sigma^{-1}(c))\\}$
$\displaystyle=$
$\displaystyle\\{\sigma:\mathfrak{C}\models\phi(\sigma(b),\sigma(c),c)\\}=\mathrm{ev}^{-1}_{b,c}(\phi(\mathfrak{C},\mathfrak{C},c)).$

If there is an entourage on $\mathfrak{C}^{x}$, then every compact subset of
$\mathfrak{C}^{x}$ is contained in some ball. Indeed, if $F$ is compact and
${\mathrm{d}}$ is an entourage on $\mathfrak{C}^{x}$, as
$F\subseteq\bigcup_{a\in F}{\mathrm{d}}(a)$, we find a finite subset
$a_{0},\ldots,a_{n-1}\in F$ such that
$F\subseteq\bigcup_{i<n}{\mathrm{d}}(a_{i})$ by compactness of $F$ and Lemma
5.27. By locality axiom A5, there are ${\mathrm{d}}_{i}$ for $0<i<n$ such that
$a_{i}\in{\mathrm{d}}_{i}(a_{0})$. Hence, by A2 and A3, we get that
$F\subseteq{\mathrm{d}}^{\prime}(a_{0})$ for
${\mathrm{d}}^{\prime}\coloneqq{\mathrm{d}}\ast{\mathrm{d}}_{1}\ast\cdots\ast{\mathrm{d}}_{n-1}$.

  3. (3)

Obvious as $\mathfrak{C}$ is $\mathrm{T}_{1}$. Indeed, we have
$\\{f\\}=\bigcap_{a}\mathrm{ev}^{-1}_{a}(\\{f(a)\\})$ for any
$f\in{\mathrm{Aut}}(\mathfrak{C})$.

  4. (4)

Consider a ball ${\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)$ in
${\mathrm{Aut}}(\mathfrak{C})$. It suffices to show that
${\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)$ is closed under ultralimits.

Pick a sequence $(g_{i})_{i\in I}$ in
${\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)$ and $\mathfrak{u}$ an
ultrafilter in $I$. Let $\mathfrak{C}^{\mathfrak{u}}$ be the corresponding
(non-local) ultrapower. Consider the diagonal inclusion
${\iota}\colon\mathfrak{C}\to\mathfrak{C}^{\mathfrak{u}}$ and $\mathtt{D}(a)$
the local substructure of $\mathfrak{C}^{\mathfrak{u}}$ at
$a\coloneqq{\iota}(a)$. Note that ${\iota}\colon\mathfrak{C}\to\mathtt{D}(a)$.
By Łoś’s Theorem,
$\mathfrak{C}^{\mathfrak{u}}\mathrel{\mathord{\models}_{\star}}{\mathtt{T}}$.
By Lemma 1.8, $\mathtt{D}(a)\models{\mathtt{T}}$. Consider the map
$g_{\mathfrak{u}}\colon x\mapsto[g_{i}(x)]_{\mathfrak{u}}$. By Łoś’s Theorem,
$g_{\mathfrak{u}}$ is a homomorphism. As
$\mathfrak{C}\models{\mathrm{d}}(g_{i}(a),a)$ for all $i\in I$ (and
$\mathfrak{C}$ is local),
$g_{\mathfrak{u}}\colon\mathfrak{C}\to\mathtt{D}(a)$. On the other hand, since
$\mathfrak{C}$ is a retractor, there is $r\colon\mathtt{D}(a)\to\mathfrak{C}$
such that ${\mathrm{id}}=r\circ{\iota}$. Set $g\coloneqq r\circ
g_{\mathfrak{u}}\colon\mathfrak{C}\to\mathfrak{C}$.

By Corollary 5.10, $g\in{\mathrm{Aut}}(\mathfrak{C},a)$. Since $(g_{i})_{i\in
I}$ is in ${\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)$, we get that
$\mathfrak{C}\models{\mathrm{d}}_{s}(g_{i}(a_{s}),a_{s})$ for all $i\in I$ and
all sort $s$. Thus, by Łoś’s Theorem,
$\mathfrak{C}^{\mathfrak{u}}\models{\mathrm{d}}_{s}(g_{\mathfrak{u}}(a_{s}),a_{s})$
for all $s$, concluding that
$\mathfrak{C}\models{\mathrm{d}}_{s}(g(a_{s}),a_{s})$ by Remark 2.3.
Therefore, $g\in{\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)$.

Let $\varphi(x)\in{\mathrm{LFor}}^{x}_{-}({\mathtt{L}}(\mathfrak{C}))$ and
$b\in\mathfrak{C}^{x}$ such that $\mathfrak{C}\models\varphi(g(b))$. Then,
$\mathtt{D}(a)\models\varphi(g_{\mathfrak{u}}(b))$ by Remark 2.3. By Lemma
1.8, as $\varphi$ is local, we conclude that
$\mathfrak{C}^{\mathfrak{u}}\models\varphi(g_{\mathfrak{u}}(b))$. Hence, by
Łoś’s Theorem, $\\{i:\mathfrak{C}\models\varphi(g_{i}(b))\\}\in\mathfrak{u}$.
Therefore, as $\varphi$ and $b$ are arbitrary, we conclude that $g$ is the
$\mathfrak{u}$-limit of $(g_{i})_{i\in I}$.

  5. (5)

$\mathrm{ev}_{a}^{-1}(\varphi(\mathfrak{C},b))^{-1}=\\{\sigma^{-1}:\mathfrak{C}\models\varphi(\sigma(a),b)\\}=\\{\sigma^{-1}:\mathfrak{C}\models\varphi(a,\sigma^{-1}(b))\\}=\mathrm{ev}_{b}^{-1}(a,\mathfrak{C})$.
Thus, the inverse map is a homeomorphism. On the other hand, pick
$\alpha\in{\mathrm{Aut}}(\mathfrak{C})$. Then,
$\mathrm{ev}_{a}^{-1}(\varphi(\mathfrak{C}))\circ\alpha=\\{\sigma\circ\alpha:\mathfrak{C}\models\varphi(\sigma(a))\\}=\\{\sigma\circ\alpha:\mathfrak{C}\models\varphi(\sigma\circ\alpha(\alpha^{-1}(a)))\\}=\mathrm{ev}_{\alpha^{-1}(a)}(\varphi(\mathfrak{C}))$.
Thus, right translations are homeomorphism. As the inverse map and right
translations are continuous, left translations are continuous too. ∎

######  Remark 5.45.

By Theorem 5.44(1), it seems natural to try to apply Tychonoff’s Theorem to
prove Theorem 5.44(4). Unfortunately, the pointwise topology in the space of
function of $\mathfrak{C}$ to itself is not the usual Tychonoff topology, as
we are considering valuations under arbitrary finite tuples and not only under
single elements. Still, it is obvious from the definition that the space of
functions of $\mathfrak{C}$ to itself with the pointwise topology can be
canonically identified with the topological subspace

$\left\\{f\in\prod\mathfrak{C}^{a}:\mathrm{There\ is\
}g\colon\mathfrak{C}\to\mathfrak{C}\mathrm{\ with\ }f_{a}=g(a)\mathrm{\ for\
all\ }a\right\\}$

of $\prod\mathfrak{C}^{a}$ with the Tychonoff topology, where
$\mathfrak{C}^{a}$ is denoting a copy of $\mathfrak{C}^{s}$ for each
$a\in\mathfrak{C}^{s}$ of finite sort $s$. Unfortunately, it seems that the
space of functions from $\mathfrak{C}$ to itself is not a closed subset of
$\prod\mathfrak{C}^{a}$, as in general $\prod\mathfrak{C}^{a}$ is not
Hausdorff.

For the rest of the section, we adapt to the local case several results from
[Hru22b] which where not discussed at all in [Hru22a]. We start with [Hru22b,
Lemma 3.23].

######  Lemma 5.46.

Any evaluation $\mathrm{ev}_{a}$ on a pointed tuple $a$ is a continuous closed
map.

###### Proof.

Continuity was already noted in Remark 5.43. Consider a non-empty basic closed
subset $D$ of ${\mathrm{Aut}}(\mathfrak{C})$, so
$D=\mathrm{ev}_{b}^{-1}\varphi(\mathfrak{C})$ where $\varphi(y)$ is a local
positive formula with parameters in $\mathfrak{C}$ and $b\in\mathfrak{C}^{y}$.
We now show that $\mathrm{ev}_{a}(D)$ is closed. By Corollary 5.11,
$a^{\prime}\in\mathrm{ev}_{a}(D)$ if and only if there is
$b^{\prime}\in\mathfrak{C}^{y}$ with $\mathfrak{C}\models\varphi(b^{\prime})$
such that ${\mathrm{ltp}}_{+}(a^{\prime},b^{\prime})={\mathrm{ltp}}_{+}(a,b)$.
Write $p(x,y)\coloneqq{\mathrm{ltp}}_{+}(a,b)$. In other words, we have

$\mathrm{ev}_{a}(D)=\\{a^{\prime}:\mathrm{There\ is\ }y\mathrm{\ such\ that\
}\varphi(y,c)\wedge p(a^{\prime},y)\\}.$

Consider $\Sigma(x)=\\{\exists y\,\varphi(y)\wedge\psi(x,y):\psi\in
p\\}\subseteq{\mathrm{LFor}}^{x}_{+}({\mathtt{L}}(\mathfrak{C}))$. We conclude
that $\Sigma(\mathfrak{C})=\mathrm{ev}_{a}(D)$. Indeed, if
$a^{\prime}\in\mathrm{ev}_{a}(D)$, then there is $b^{\prime}$ such that
$\mathfrak{C}\models\varphi(b^{\prime})\wedge p(a^{\prime},b^{\prime})$, so,
in particular, $a^{\prime}$ satisfies $\Sigma$. On the other hand, if
$a^{\prime}$ satisfies $\Sigma$, then $\varphi(y)\wedge p(a^{\prime},y)$ is
finitely locally satisfiable. As $a^{\prime}$ is pointed, $p(a^{\prime},y)$
implies some bound of $y$ with parameters in $a^{\prime}$. Thus,
$\varphi(y,c)\wedge p(a^{\prime},y)$ is boundedly satisfiable in
${\mathrm{Th}}_{-}(\mathfrak{C}/\mathfrak{C})$, so it is locally satisfiable
by Lemma 4.9. Hence, by saturation of $\mathfrak{C}$ (Theorem 5.9), we get
that $\varphi(y)\wedge p(a^{\prime},y)$ is realised by some $b^{\prime}$ in
$\mathfrak{C}$, concluding that $a^{\prime}\in\mathrm{ev}_{a}(D)$. In
particular, $\mathrm{ev}_{a}(D)$ is closed. As $D$ is arbitrary, we conclude
that the image of every basic closed subset of ${\mathrm{Aut}}(\mathfrak{C})$
is closed.

Let $F$ be an arbitrary closed subset of ${\mathrm{Aut}}(\mathfrak{C})$. Then,
$F=\bigcap_{i\in I}D_{i}$ with $D_{i}$ basic closed sets. Suppose $a^{\prime}$
is in the closure of $\mathrm{ev}_{a}(F)$. Then, $a^{\prime}$ is in the
closure of $\mathrm{ev}_{a}(\bigcap_{i\in I_{0}}D_{i})$ for all
$I_{0}\subseteq I$. Since the basic closed subsets are closed under finite
intersections, we get that $\mathrm{ev}_{a}(\bigcap_{i\in I_{0}}D_{i})$ is
closed for $I_{0}\subseteq I$ finite. Hence,
$\mathrm{ev}^{-1}_{a}(a^{\prime})\cap\bigcap_{i\in I_{0}}D_{i}\neq\emptyset$
for any $I_{0}\subseteq I$ finite. Now, pick
${\mathrm{d}}=({\mathrm{d}}_{s})_{s\in\mathtt{S}}$ such that
${\mathrm{d}}(a,a^{\prime})$. Then,
$\mathrm{ev}^{-1}_{a}(a^{\prime})\subseteq{\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)$.
By Theorem 5.44(4), we get that $\mathrm{ev}^{-1}_{a}(a^{\prime})$ is compact.
Therefore, $\mathrm{ev}^{-1}_{a}(a^{\prime})\cap F\neq\emptyset$, concluding
that $a^{\prime}\in\mathrm{ev}_{a}(F)$. As $a^{\prime}$ is arbitrary, we get
that $\mathrm{ev}_{a}(F)$ is closed. ∎

######  Lemma 5.47.

Suppose ${\mathtt{T}}$ has full systems of entourages. Then,
${\mathrm{Aut}}(\mathfrak{C})$ is a Hausdorff topological group and evaluation
maps on finite sorts are jointly continuous.

###### Proof.

Let us show first that composition in ${\mathrm{Aut}}(\mathfrak{C})$ is
continuous. By Theorem 5.44(2,3), we have the topological base

$\\{B(F,U):U\mathrm{\ open\ and\ }F\mathrm{\ closed\ compact\ in\ the\ same\
sort\ of\ }\mathfrak{C}\\},$

where $B(F,U)\coloneqq\\{\sigma:\sigma(F)\subseteq U\\}$. Thus, it suffices to
show that $\\{(f,g):f\circ g\in B(F,U)\\}$ is open for every closed compact
$F$ and open $U$.

Pick $\alpha\circ\beta\in B(F,U)$. Then, $\beta(F)\subseteq\alpha^{-1}(U)$.
Since $\mathfrak{C}$ is locally closed compact by Corollary 5.30, there are
$V$ open and $K$ closed compact such that $\beta(F)\subseteq V\subseteq
K\subseteq\alpha^{-1}(U)$. Hence, $(\alpha,\beta)\in B(K,U)\times
B(F,V)\subseteq\\{(f,g):f\circ g\in B(F,U)\\}$. Indeed, if $f\in B(K,U)$ and
$g\in B(F,V)$, then $g(F)\subseteq V\subseteq K\subseteq f^{-1}(U)$,
concluding that $f\circ g\in B(F,U)$. Since $\alpha,\beta$ and $B(F,U)$ are
arbitrary, we conclude that composition is continuous. Since we already know
that the inverse map is continuous by Theorem 5.44(5), we conclude that
${\mathrm{Aut}}(\mathfrak{C})$ is a topological group. Since
${\mathrm{Aut}}(\mathfrak{C})$ is Fréchet by Theorem 5.44(3), it is also
Hausdorff by [Hus66, §21 Theorem 4].

Let us now show that evaluations are jointly continuous. Pick a finite
variable $x$, $a\in\mathfrak{C}^{x}$, $\sigma\in{\mathrm{Aut}}(\mathfrak{C})$
and $U$ open subset of $\mathfrak{C}^{x}$ such that $\sigma(a)\in U$. Then,
$a\in\sigma^{-1}(U)$. By local closed compactness of $\mathfrak{C}^{x}$, find
an open set $V$ and a closed compact set $F$ such that $a\in V\subseteq
F\subseteq\sigma^{-1}(U)$. Then, $(\sigma,a)\in B(F,U)\times
V\subseteq\\{(f,b):f(b)\in U\\}$. Indeed, if $f\in B(F,U)$ and $b\in V$, then
$b\in V\subseteq F\subseteq f^{-1}(U)$, concluding that $f(b)\in U$. Since
$\sigma$, $a$ and $U$ are arbitrary, we conclude that evaluations are jointly
continuous. ∎

######  Lemma 5.48.

Suppose that ${\mathtt{L}}$ only has finitely many single sorts and set an
enumeration $s$ of all of them. Let ${\mathrm{d}}$ be a locality relation of
sort $s$.

  1. (1)

Suppose $(\psi,\varphi)$ is an entourage in ${\mathrm{d}}$. Then,
$\mathrm{ev}_{a}^{-1}\psi(\mathfrak{C},a)=\\{\sigma:\mathfrak{C}\models\psi(\sigma(a),a)\\}$
is a compact neighbourhood of the identity for any $a\in\mathfrak{C}^{s}$. In
particular, ${\mathrm{Aut}}(\mathfrak{C})$ is weakly locally compact if
$\mathfrak{C}$ has entourages.

  2. (2)

Suppose $\\{(\psi_{i},\varphi_{i})\\}_{i\in I}$ is a full system of entourages
in ${\mathrm{d}}$. Then,

$\left\\{\bigcap_{a\in A,\,i\in
I_{0}}\mathrm{ev}_{a}^{-1}\psi_{i}(\mathfrak{C},a):I_{0}\subseteq I,\
A\subseteq\mathfrak{C}^{s}\mathrm{\ both\ finite}\right\\}$

is a base of compact neighbourhoods of the identity. In particular,
${\mathrm{Aut}}(\mathfrak{C})$ is locally compact if $\mathfrak{C}$ has full
systems of entourages.

###### Proof.

  1. (1)

By Corollary 5.28, $\psi(\mathfrak{C},a)\subseteq{\mathrm{d}}(a)$ is a closed
neighbourhood of $a={\mathrm{id}}(a)$. Since $\mathrm{ev}_{a}$ is continuous,
we conclude that $\mathrm{ev}_{a}^{-1}\psi(\mathfrak{C},a)$ is a closed
neighbourhood of ${\mathrm{id}}$. By Theorem 5.44(4),
${\mathrm{d}}({\mathrm{Aut}}(\mathfrak{C}),a)=\mathrm{ev}^{-1}_{a}{\mathrm{d}}(a)$
is compact, concluding that $\mathrm{ev}_{a}^{-1}\psi(\mathfrak{C},a)$ is a
closed compact neighbourhood of ${\mathrm{id}}$. Since translations are
continuous, we get that ${\mathrm{Aut}}(\mathfrak{C})$ is weakly locally
closed compact.

  2. (2)

By compactness and the fact that translations are continuous, it suffices to
show that $\\{{\mathrm{id}}\\}=\bigcap_{a\in\mathfrak{C}^{s},\,i\in
I}\mathrm{ev}_{a}^{-1}\psi_{i}(\mathfrak{C},a)$. This is obvious since
$\bigcap_{i\in
I}\mathrm{ev}_{a}^{-1}\psi_{i}(\mathfrak{C},a)=\\{\sigma:\sigma(a)=a\\}$ when
$\\{\psi_{i}\\}$ a full system of approximations of $=$.∎

As Lemmas 5.47 and 5.48 show, the existence of full systems of entourages is a
very strong hypothesis and, even in the non-local context, there is a plenty
of examples that do not satisfy it. In its absence, one considers Hausdorff
quotients of ${\mathrm{Aut}}(\mathfrak{C})$. We conclude this subsection
adapting the study of these Hausdorff quotients from [Hru22b].

Let $x$ be a variable and $X$ a local positive $\bigwedge_{0}$-definable
subset of $\mathfrak{C}^{x}$. Consider

$\mathfrak{g}_{X}\coloneqq\\{\sigma:\sigma(U)\cap U\neq\emptyset\mathrm{\ for\
every\ nonempty\ open\ subset\ }U\subseteq X\\},$

In other words, according to [Hru22b, Appendix C], $\mathfrak{g}_{X}$ is the
infinitesimal subgroup of the evaluation action of
${\mathrm{Aut}}(\mathfrak{C})$ on $X$. In the case $X=\mathfrak{C}^{s}$, write
$\mathfrak{g}_{s}\coloneqq\mathfrak{g}_{\mathfrak{C}^{s}}$. Finally, let
$\mathfrak{g}\coloneqq\bigcap\mathfrak{g}_{s}$ and
$\mathfrak{g}_{!}\coloneqq\bigcap_{X}\mathfrak{g}_{X}$. So, according to
[Hru22b, Appendix C], $\mathfrak{g}$ is the infinitesimal subgroup of the
evaluation action on $\bigsqcup\mathfrak{C}^{s}$ and $\mathfrak{g}_{!}$ is the
infinitesimal subgroup of the evaluation action on $\bigsqcup X$.

By [Hru22b, Lemma C.1], we know that $\mathfrak{g}$, $\mathfrak{g}_{X}$ and
$\mathfrak{g}_{!}$ are closed normal subgroups of
${\mathrm{Aut}}(\mathfrak{C})$. Let
$\mathscr{G}_{X}\coloneqq\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{X}$}}$,
$\mathscr{G}_{s}\coloneqq\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{s}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{s}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{s}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{s}$}}$,
$\mathscr{G}\coloneqq\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}$}}$ and
$\mathscr{G}_{!}\coloneqq\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{Aut}}(\mathfrak{C})$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$. Then, by [Hru22b,
Lemma C.1], [AT08, Theorem 1.5.3] and Theorem 5.44(5), $\mathscr{G}_{X}$,
$\mathscr{G}_{s}$, $\mathscr{G}$ and $\mathscr{G}_{!}$ are Hausdorff
semitopological groups. Furthermore, by [AT08, Theorem 1.5.3], the quotient
homomorphisms from ${\mathrm{Aut}}(\mathfrak{C})$ are continuous and open.

######  Lemma 5.49.

The following hold:

  1. (1)

Let $X$ and $Y$ be local positive $\bigwedge_{0}$-definable with $Y\subseteq
X$. Then, $\mathfrak{g}_{Y}\subseteq\mathfrak{g}_{X}$. In particular, the
quotient homomorphism $\pi\colon\mathscr{G}_{Y}\to\mathscr{G}_{X}$ is a
surjective continuous open map.

  2. (2)

Let $X\subseteq\mathfrak{C}^{x}$ and $Y\subseteq\mathfrak{C}^{y}$ be local
positive $\bigwedge_{0}$-definable with $x\subseteq y$. Suppose
${\mathrm{proj}}(Y)\subseteq X$, where ${\mathrm{proj}}$ is the restriction of
the projection of $\mathfrak{C}^{y}$ to $\mathfrak{C}^{x}$. Then,
$\mathfrak{g}_{Y}\subseteq\mathfrak{g}_{X}$. In particular, the quotient
homomorphism $\pi\colon\mathscr{G}_{Y}\to\mathscr{G}_{X}$ is a surjective
continuous open map.

  3. (3)

$\mathfrak{g}_{!}\subseteq\mathfrak{g}$. In particular, the quotient
homomorphism $\pi\colon\mathscr{G}_{!}\to\mathscr{G}$ is a surjective
continuous open map.

###### Proof.

  1. (1)

By definition, the inclusion ${\iota}\colon Y\to X$. is continuous. Also, note
that the evaluation action of ${\mathrm{Aut}}(\mathfrak{C})$ commutes with
${\iota}$, i.e. ${\iota}(\sigma(a))=\sigma({\iota}(a))$. Suppose
$\sigma\in\mathfrak{g}_{Y}$. Pick $U\subseteq X$ open. As
$\sigma\in\mathfrak{g}_{X}$, there is
$a\in\sigma({\iota}^{-1}(U))\cap{\iota}^{-1}(U)$. Hence, ${\iota}(a)\in U$ and
$\sigma^{-1}({\iota}(a))={\iota}(\sigma^{-1}(a))\in U$, concluding
${\iota}(a)\in\sigma(U)\cap U$. As $U$ and $\sigma$ are arbitrary,
$\mathfrak{g}_{Y}\subseteq\mathfrak{g}_{X}$. By [AT08, Theorem 1.5.3], the
quotient homomorphism $\pi\colon\mathscr{G}_{Y}\to\mathscr{G}_{X}$ is
continuous and open.

  2. (2)

By Lemma 5.20, ${\mathrm{proj}}$ is continuous. Also, note that the evaluation
action of ${\mathrm{Aut}}(\mathfrak{C})$ commutes with ${\mathrm{proj}}$, i.e.
${\mathrm{proj}}(\sigma(a))=\sigma({\mathrm{proj}}(a))$. Suppose
$\sigma\in\mathfrak{g}_{Y}$. Pick $U\subseteq X$ open. As
$\sigma\in\mathfrak{g}_{Y}$, there is
$a\in\sigma({\mathrm{proj}}^{-1}(U))\cap{\mathrm{proj}}^{-1}(U)$. Hence,
${\mathrm{proj}}(a)\in U$ and
$\sigma^{-1}({\mathrm{proj}}(a))={\mathrm{proj}}(\sigma^{-1}(a))\in U$,
concluding ${\mathrm{proj}}(a)\in\sigma(U)\cap U$. As $U$ and $\sigma$ are
arbitrary, $\mathfrak{g}_{Y}\subseteq\mathfrak{g}_{X}$. By [AT08, Theorem
1.5.3], the quotient homomorphism $\pi\colon\mathscr{G}_{Y}\to\mathscr{G}_{X}$
is continuous and open.

  3. (3)

By (1), $\mathfrak{g}_{!}\subseteq\mathfrak{g}$. By [AT08, Theorem 1.5.3], the
quotient homomorphism $\pi\colon\mathscr{G}_{!}\to\mathscr{G}$ is continuous
and open. ∎

######  Lemma 5.50.

Suppose that ${\mathtt{L}}$ only has finitely many single sorts and
${\mathtt{T}}$ has entourages. Then, $\mathscr{G}_{X}$ is a locally compact
Hausdorff topological group for any local positive $\bigwedge_{0}$-definable
set $X$. Similarly, $\mathscr{G}$ and $\mathscr{G}_{!}$ are locally compact
Hausdorff topological groups too.

###### Proof.

By Lemma 5.48(1), ${\mathrm{Aut}}(\mathfrak{C})$ is weakly compact. Then,
since $\pi\colon{\mathrm{Aut}}(\mathfrak{C})\to\mathscr{G}_{X}$ is continuous
and open, we get that $\mathscr{G}_{X}$ is weakly locally compact. Hence, by
Hausdorffness, it is locally compact. As it is a semitopological group, by
Ellis’ Theorem [Ell57, Theorem 2], we conclude that it is a locally compact
Hausdorff topological group. Similar for $\mathscr{G}_{!}$ and $\mathscr{G}$.
∎

Let $X$ be a local positive $\bigwedge_{0}$-definable subset. Let
$\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{X}$}}$ denote the quotient
space of $\mathfrak{g}_{X}$-orbits of $X$. In [Hru22b], it was proved that
this quotient space is Hausdorff and, if $X$ is defined by a local positive
type, the $\mathfrak{g}_{X}$-conjugacy relation in $X$ is local positive
$\bigwedge_{0}$-definable — see [Hru22b, Propositions 3.24]. We adapt here
these results.

######  Lemma 5.51.

Let $X$ be a local positive $\bigwedge_{0}$-definable subset of
$\mathfrak{C}^{x}$ on a pointed variable $x$. Write $\underline{X}(x)$ for a
partial local positive type without parameters defining it. Then:

  1. (1)

$\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{X}$}}=\mathchoice{\raise
2.15277pt\hbox{$\displaystyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ is Hausdorff.

  2. (2)

Suppose $\underline{X}$ is a local positive type. Then, the
$\mathfrak{g}_{X}$-conjugacy equivalence relation on $X$ is local positive
$\bigwedge_{0}$-definable.

###### Proof.

  1. (1)

By [Hru22b, Lemma C.1], $\mathscr{G}_{!}$ and $\mathscr{G}_{X}$ are Hausdorff.
By Lemma 5.46, ${\mathrm{Aut}}(\mathfrak{C})$ acts by closed maps. By [Hru22b,
Remark C.3], it follows that $\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}=\mathchoice{\raise
2.15277pt\hbox{$\displaystyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{X}$}}$. By [Hru22b, Lemma
C.2(1)], $\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{X}$}}$ is Hausdorff.

  2. (2)

By (1), $\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\textstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{X}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle X$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{X}$}}$ is Hausdorff, so the
diagonal is closed. By Lemma 5.20, we conclude that the
$\mathfrak{g}_{X}$-conjugacy equivalence relation $E$ is closed in $X\times
X$. Let $\Sigma(x,y,c)$ be a partial local positive type with parameters $c$
defining $\mathfrak{g}_{X}$-conjugacy in $X$. Without loss of generality, take
$c$ pointed. Then, $\Sigma(a,b,c)$ if and only if $\Sigma(a,b,c^{\prime})$ for
any $c^{\prime}$ such that
${\mathrm{ltp}}_{+}(c)={\mathrm{ltp}}_{+}(c^{\prime})$. One direction is
obvious. On the other hand, by Corollary 5.11, since
${\mathrm{ltp}}_{+}(c)={\mathrm{ltp}}(c^{\prime})$, there is
$\sigma\in{\mathrm{Aut}}(\mathfrak{C})$ such that $\sigma(c^{\prime})=c$.
Then, as $\Sigma(a,b,c^{\prime})$, we get $\Sigma(\sigma(a),\sigma(b),c)$, so
there is $g\in\mathfrak{g}_{X}$ with $g\sigma(a)=\sigma(b)$, so
$\sigma^{-1}g\sigma(a)=b$. By normality of $\mathfrak{g}_{X}$, we get
$\sigma^{-1}g\sigma\in\mathfrak{g}_{X}$, so $\Sigma(a,b,c)$ by definition of
$E$.

Let $p(z)\coloneqq{\mathrm{ltp}}_{+}(c)$. Picking any $a\in X$, by A5, there
is some bound $\mathrm{B}(x,z)$ such that
$\mathfrak{C}\models\mathrm{B}(a,c)$. Then, by Corollary 5.11, for any
$a^{\prime}\in X$, there is $c^{\prime}\in p(\mathfrak{C})$ with
$\mathfrak{C}\models\mathrm{B}(a^{\prime},c^{\prime})$. Hence, we conclude
that $\mathfrak{g}_{X}$-conjugacy is defined by

$\displaystyle\underline{E}(x,y)\coloneqq$ $\displaystyle\exists
z\,\Sigma(x,y,z)\wedge p(z)\wedge\mathrm{B}(x,z)$ $\displaystyle\coloneqq$
$\displaystyle\left\\{\exists z\,\varphi(x,y,z)\wedge\psi(z)\wedge\bigwedge
B_{0}(x,z):\varphi\in\Sigma,\ \psi\in p,\ B_{0}\subseteq B\mathrm{\
finite}\right\\}.$

Indeed, if $a$ and $b$ are $\mathfrak{g}_{X}$-conjugates, then
$\mathfrak{C}\models\Sigma(a,b,c^{\prime})$ for any $c^{\prime}\in
p(\mathfrak{C})$. Now, there is $c^{\prime}\in p(\mathfrak{C})$ such that
$\mathfrak{C}\models\mathrm{B}(a,c^{\prime})$, so we get that
$\mathfrak{C}\models\underline{E}(a,b)$ witnessed by $z=c^{\prime}$. On the
other hand, suppose $a$ and $b$ satisfy $\underline{E}$. Then, the partial
local positive type $\Sigma(a,b,z)\wedge p(z)\wedge\mathrm{B}(a,z)$ is
finitely locally satisfiable. Now, $\mathrm{B}(a,z)$ is a bound, so we get by
Lemma 4.9, that there is $c^{\prime}$ realising it. In other words,
${\mathrm{ltp}}_{+}(c^{\prime})={\mathrm{ltp}}_{+}(c)$ with
$\Sigma(a,b,c^{\prime})$, so $a$ and $b$ are $\mathfrak{g}_{X}$-conjugates.

Therefore, we conclude that the $\mathfrak{g}_{X}$-conjugacy equivalence
relation is local positive $\bigwedge_{0}$-definable. ∎

We finish with [Hru22b, Proposition 3.25 and Corollary 3.26].

######  Lemma 5.52.

Let $p(x)$ be a local positive type without parameters of $\mathfrak{C}$ on a
pointed variable $x$. Slightly abusing of the notation, write $p\coloneqq
p(\mathfrak{C})$. Then, $\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ has a dense subset of
size $|{\mathtt{L}}|^{|x|}$. In particular,
$\left|{\mathrm{Aut}}\left(\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}\right)\right|\leq
2^{|{\mathtt{L}}|^{|x|}}$.

###### Proof.

Pick $c\in P$ arbitrary. By A5, $p=p\cap\mathtt{D}(c)$. As there are at most
$|{\mathtt{L}}|^{|x|}$ balls centred at $c$, it suffices to show that
$\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ has a dense subset of
size $|{\mathtt{L}}|$ for any ball ${\mathrm{d}}(c)$, where
${\mathrm{d}}(p,c)\coloneqq p\cap{\mathrm{d}}(c)$.

Let $E$ be the equivalence relation of $\mathfrak{g}_{!}$-conjugacy in
${\mathrm{d}}(p,c)$. By Lemma 5.51(2), $E$ is local positive
$\bigwedge_{c}$-definable. Thus, by Lemma 2.12, as
$\mathfrak{C}\models^{\mathrm{pc}}{\mathtt{T}}$, if $(a,b)\notin E$, there is
a local positive $c$-definable subset $D$ such that $(a,b)\in D$ and $D\cap
E=\emptyset$. Since there are at most $|{\mathtt{L}}|$ local positive formulas
over $c$, we have a family $\\{D_{i}\\}_{i<|{\mathtt{L}}|}$ of local positive
$c$-definable subsets disjoint to $E$ such that, for any $(a,b)\notin E$,
there is $i<|{\mathtt{L}}|$ with $(a,b)\in D_{i}$.

For each $i<|{\mathtt{L}}|$, consider

$\widetilde{D}_{i}\coloneqq\\{(a^{\prime},b^{\prime}):\mathrm{There\ are\
}(a,b)\in D_{i}\mathrm{\ with\ }(a,a^{\prime}),(b,b^{\prime})\in E\\}.$

Let $\varphi_{i}$ be a local positive formula over $c$ defining $D_{i}$ and
$\Psi$ be a partial local positive $0$-type closed under finite conjunctions
defining $E$. Then, $\widetilde{D}_{i}$ is defined by the partial local
positive $0$-type

$\Phi_{i}(x,y)\coloneqq\\{\exists
x^{\prime},y^{\prime}\,\varphi_{i}(x^{\prime},y^{\prime})\wedge\psi(x,x^{\prime})\wedge\psi(y,y^{\prime})\wedge{\mathrm{d}}(x^{\prime},c)\wedge{\mathrm{d}}(y^{\prime},c):\psi\in\Psi\\}.$

Indeed, obviously $\widetilde{D}_{i}\subseteq\Phi_{i}(\mathfrak{C})$. On the
other hand, if $\mathfrak{C}\models\Phi_{i}(a,b)$, then
$\\{\varphi_{i}(x^{\prime},y^{\prime})\wedge\psi(a,x^{\prime})\wedge\psi(b,y^{\prime})\wedge{\mathrm{d}}(x^{\prime},c)\wedge{\mathrm{d}}(y^{\prime},c):\psi\in\Psi\\}$
is finitely satisfiable. As it implies a bound, we conclude that it is
boundedly satisfiable, so there are $(a^{\prime},b^{\prime})\in D_{i}$ with
$(a,a^{\prime}),(b,b^{\prime})\in E$ by saturation of $\mathfrak{C}$. In other
words, $(a,b)\in\widetilde{D}_{i}$.

Since $E$ is an equivalence relation and $D_{i}\cap E=\emptyset$, we conclude
that $\widetilde{D}_{i}\cap E=\emptyset$. Also, note that
$\widetilde{D}_{i}=\bar{\pi}^{-1}\bar{\pi}(D_{i})$ where $\bar{\pi}=(\pi,\pi)$
is the quotient map $\pi\colon{\mathrm{d}}(p,c)\to\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$. Hence,
$\bar{\pi}^{-1}\bar{\pi}({\mathrm{d}}(p,c)\setminus\widetilde{D}_{i})={\mathrm{d}}(p,c)\setminus\widetilde{D}_{i}$,
so $\pi({\mathrm{d}}(p,c)\setminus\widetilde{D}_{i})$ is open in the quotient
topology. Now, $\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ is Hausdorff, and the
quotient topology in $\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}\times\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ is compact and finer
than the product topology by Lemma 5.20. Therefore, the quotient and product
topologies agree, concluding that the diagonal in $\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ is the intersection of
$|{\mathtt{L}}|$ open sets in the product topology. Thus, by compactness,
$\mathchoice{\raise
2.15277pt\hbox{$\displaystyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle{\mathrm{d}}(p,c)$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ has a dense subset of
size $|{\mathtt{L}}|$.

As $\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ is Hausdorff, any
automorphism fixing a dense subset is the identity. Hence, as
$\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}$ has a dense subset of
size $|{\mathtt{L}}|^{|x|}$, we conclude that
$\left|{\mathrm{Aut}}\left(\mathchoice{\raise 2.15277pt\hbox{$\displaystyle
p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\textstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{!}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{!}$}}\right)\right|\leq
2^{|{\mathtt{L}}|^{|x|}}$. ∎

######  Corollary 5.53.

Let $p(x)$ be a local positive type without parameters of $\mathfrak{C}$ on a
pointed variable $x$. Then, $|\mathscr{G}_{p}|\leq 2^{|{\mathtt{L}}|^{|x|}}$.
Consequently, $|\mathscr{G}_{!}|\leq 2^{|{\mathtt{L}}|^{\lambda}}$ where
$\lambda$ is the number of sorts of ${\mathtt{L}}$.

###### Proof.

Slightly abusing of the notation, write $p\coloneqq p(\mathfrak{C})$. Consider
the group homomorphism
$f\colon{\mathrm{Aut}}(\mathfrak{C})\to{\mathrm{Aut}}\left(\mathchoice{\raise
2.15277pt\hbox{$\displaystyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\displaystyle\mathfrak{g}_{p}$}}{\raise
2.15277pt\hbox{$\textstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\textstyle\mathfrak{g}_{p}$}}{\raise
2.15277pt\hbox{$\scriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptstyle\mathfrak{g}_{p}$}}{\raise
2.15277pt\hbox{$\scriptscriptstyle p$}\kern-1.00006pt/\kern-1.00006pt\lower
1.07639pt\hbox{$\scriptscriptstyle\mathfrak{g}_{p}$}}\right)$ given by
$f(\sigma)([a])=[\sigma(a)]$, where
$[\text{\raisebox{0.25pt}{\scalebox{0.6}{$\bullet$}}}]$ denotes the
$\mathfrak{g}_{p}$-conjugacy class. By Lemma 5.51(2),
$\mathfrak{g}_{p}$-conjugacy in $p$ is local positive
$\bigwedge_{0}$-definable, so $[\sigma(a)]=[\sigma(b)]$ when $[a]=[b]$. In