We continue the study of projective limits of trees initiated by Charatonik and Roe and we construct many generalized dendrites as the topological realization of a projective limit of families of finite trees with (weakly) coherent epimorphisms. Moreover we use the categorical approach to limits developed by to construct all generalized dendrites as topological realizations of limits of suitable categories of finite structures. As an application we recover a homogeneity result for countable dense sets of endpoints in generalized dendrites.

§ PROJECTIVE LIMITS AND TOPOLOGICAL GRAPHS

Projective limits where introduced by Irwin and Solecki in <cit.> and generalised to restricted classes of epimorphisms by Solecki and Panagiotopolous in <cit.>. They have since been used to study various continua, for example the Lelek fan <cit.>, The Poulsen simplex <cit.>, the Menger curve <cit.> and a new class of compact metric spaces named fences <cit.>. More recently Charatonik and Roe studied projective limits of trees in <cit.> by generalising some classes of maps from continuum theory to finite graphs. In particular they constructed the dendrite $W_3$ (denoted by $D_3$ in their paper) as a projective limit of the family of finite trees with monotone maps. Building on their work we construct many generalized dendrites $W_P$ for $P\subseteq \{3,4,5,\ldots,\infty\}$ as the topological realization of projective limits of appropriately chosen families of finite trees and epimorphisms between them. For convenience we begin with a quick review of projective limits; a more detailed exposition can be found in the aforementioned papers.

Let $L$ be a first order language consisting of a set $\{R_i\}_{i\in I}$ of relation symbols, with $R_i$ of arity $n_i\in \Bbb N$, and a set $\{f_j\}_{j\in J}$ of function symbols, with $f_j$ of arity $m_j$. A topological $L$-structure is a zero-dimensional, compact, second countable topological space $A$  together with closed subsets $R_i^A\subseteq A^{n_i}$ for every $i\in I$ and continuous functions $f_j^A\colon A^{m_j}\to A$ for every $j\in J$.

Given two topological $L$-structures $A$ and $B$ we say that $g\colon B\to A$ is an epimorphism if $g$ is a continuous surjection satisfying

* For all $j\in J$, $f_j^A(g(b_1),\ldots,g(b_{m_j}))=g(f_j^B(b_1,\ldots,b_{m_j}))$

* For all $i\in I$, we have $R_i^A(a_1,\ldots,a_{n_i})$ if and only if there are $b_1\in g^{-1}(a_1),\ldots,b_{n_i}\in g^{-1}(a_{n_i})$ such that $R_i^B(b_1,\ldots,b_{n_i})$ holds.

We can now give the definition of a projective family and its projective limit. We follow the construction from <cit.>. The main difference compared to the original construction from <cit.> is allowing a restricted class of epimorphisms between the structures under consideration.

Let $\mathcal F$ be a class of finite topological $L$-structures with a fixed class of epimorphisms between them. We say that $\mathcal F$ is a projective family if it satisfies the following conditions:

* Up to isomorphism there are countably many $L$-structures in $\mathcal F$.

* The epimorphisms in $\mathcal F$ are closed under composition and the identity epimorphisms are in $\mathcal F$.

* For all $A,B\in\mathcal F$ there are $C\in\F$ and epimorphisms $f\colon C\to A$ and $g\colon C\to B$ in $\mathcal F$. In analogy with the classical limits this property is known as the joint projection property.

* For all $A,B,C\in\mathcal F$ and epimorphisms $f\colon B\to A$ and $g\colon C\to A$ in $\mathcal F$, there exist $D\in\F$ and epimorphisms $h\colon D\to B$ and $k\colon D\to C$ in $\mathcal F$ such that $f\circ h=g\circ k$. This property is known as the projective amalgamation property.

Following <cit.> we introduce $\mathcal F^\omega$ as the class of structures that can be approximated in $\mathcal F$, indeed the projective limit will be an element of $\mathcal F^\omega$ satisfying appropriate universality and homogeneity properties.

Let $\mathcal F$ be a class of finite topological $L$-structures. We define $\mathcal F^\omega$ to be the class of all topological $L$-structures in $\F$ that are realized as the inverse limits of an $\omega$-indexed inverse system of topological $L$-structures with bonding epimorphisms in $\mathcal F$. More explicitely $F\in\mathcal F^\omega$ iff there is an inverse system of topological $L$-structures $\langle F_i,f^n_m\colon F_n\to F_m,i,n,m<\omega\rangle$ with $F_i,f^n_m\in\mathcal F$, such that $F=\varprojlim F_i$ (note in particular that $\mathcal F\subseteq\mathcal F^\omega$ by picking constant sequences).

Note that, given an inverse system $\langle F_i,f^n_m\colon F_n\to F_m,i,n,m<\omega\rangle$ in $\mathcal F$ as above, $F=\varprojlim F_i$ exists. Its underlying space is the inverse limit of the $F_i$ in the category of topological spaces, while for $f^n\colon F\to F_n$, the continuous maps obtained by construction of the limit of topological spaces, we have
$$R_i^F(a_1,\ldots,a_{n_i})\iff \forall n R_i^{F_n}(f^n(a_1),\ldots,f^n(a_{n_i}))$$

$$f_j^F(a_1,\ldots,a_{m_j})=a\iff\forall n f_j^{F_n}(f^n(a_1),\ldots,f^n(a_{m_j}))=f^n(a).$$

If $F=\varprojlim F_i\in\mathcal F^\omega$ and $E\in \mathcal F$, an epimorphism $f\colon F\to E$ is in $\mathcal F^\omega$ if and only if there exist $i\in\Bbb N$ and an epimorphism $f'\colon F_i\to E$ with $f=f'\circ f^i$. If $G$ is another object in $F^\omega$ an epimorphism $g\colon G\to F$ is in $\mathcal F^\omega$ iff $f^i\circ g$ is in $\mathcal F^\omega$ for all $i$.

Let $\mathcal F$ be a projective family of topological $L$-structures. Then there exists a unique topological $L$-structure $\mathbb F\in\mathcal F^\omega$ satisfying the following properties:

* For every $A\in\mathcal F$ there is an epimorphism $f\colon\mathbb F\to A$ in $\mathcal F^\omega$.

* For all $A,B\in\mathcal F^\omega$ and all epimorphisms $f\colon \mathbb F\to A$ and $g\colon B\to A$ in $\mathcal F^\omega$, there exist an epimorphism $h\colon \mathbb F\to B$ in $\mathcal F^\omega$ such that $f=g\circ h$.

Moreover, as pointed out in Theorem 1.3 of <cit.>, we also have that $\mathbb F$ satisfies

* For every $\varepsilon>0$ there is $A\in\mathcal F$ and an epimorphism $f\colon \mathbb F\to A$ in $\mathcal F^\omega$ such that $\mathrm{diam}(f^{-1}(a))\leq\varepsilon$ for every $a\in A$.

We give a sketch of the argument, the details can be found in <cit.> and <cit.>. The idea is to obtain $\Bbb F$ as the inverse limit of a sequence $\langle F_i \mid i\in\Bbb N\rangle$ with maps $f^n_m\colon F_n\to F_m$ of elements and epimorphisms of $\F$. We will call this sequence a sequence for $\F$. Since there are only countably many finite structures in $\F$ up to isomorphism, we can build two countable lists $\langle A_i\mid i\in\Bbb N\rangle$ and $\langle e_n\colon B_n\to C_n\rangle$ so that

* Every structure in $\F$ is isomorphic to $A_i$ for some $i$.

* Every epimorphism type $e\colon B\to C$ in $\F$ appears infinitely many times in $\langle e_n\mid n\in\Bbb N\rangle$.

Now set $F_0=A_0$. Inductively assume that $F_n$ together with the epimorphisms $f^n_m\colon F_n\to F_m$ for all $m<n$ have been defined. By the joint projection property we can find $H\in\F$ with epimorphisms $g\colon H\to A_{n+1}$ and $f\colon H\to F_n$. Since $H$ is finite there are only finitely many epimorphism types $s_i\colon H\to C_{n+1}$, $i=1,\ldots,k$. Using the projective amalgamation property, we can find $H_1'$ and epimorphisms $h_1\colon H_1'\to H$, $c_1\colon H_1'\to B_{n+1}$ such that $s_1\circ h_1= e_{n+1}\circ c_1$. Iterating this construction, for all $n\leq k$, we can find $H_n'$ with epimorphisms $h_n\colon H_n'\to H_{n-1}'$ and $c_n\colon H_n'\to B_{n+1}$ such that $s_n\circ h_1\circ\ldots\circ h_n=e_{n+1}\circ c_n$. Setting now $H'=H_k'\in\F$, $h=h_1\circ\ldots h_k\colon H'\to H$, $d_i=c_i\circ h_{i+1}\circ\ldots\circ h_k\colon H'\to B_{n+1}$, $i=1,\ldots,k$, we have that the following diagram commutes, that is $s_i\circ h=e_{n+1}\circ d_i$ for all $i$.

F_n         H' [d, "h"] [r, "d_1"] [rr, "d_2"', bend left] [rrrr, "d_k", bend left]                        B_n+1 [d, "e_n+1"]     B_n+1 [d, "e_n+1"]     ⋯    B_n+1 [d, "e_n+1"]

A_n+1     H [r, "s_1"] [rr, "s_2", bend right] [rrrr, "s_k", bend right] [l, "g"] [lu, "f"']     C_n+1                          C_n+1                          ⋯    C_n+1

Once this is done set $F_{n+1}=H'$, and set $f^{n+1}_m=f^n_m\circ f\circ h$ for all $m<n$. It is easy to verify that $\Bbb F=\varprojlim F_i$ satisfies the first two properties of a projective limit of $\mathcal F$. The third one follows from the definition of the product metric on $\prod F_i$.

From the proof of Theorem <ref> we extract the following important notion.

Let $\F$ be a projective family of topological $L$-structures. A sequence $\langle F_n\mid n\in\Bbb N\rangle$ with epimorphisms $f^m_n\colon F_m\to F_n$ is called a sequence for $\F$ if the following conditions hold:

* for every $A\in\F$ there exists $n\in\Bbb N$ such that there is an epimorphism $f\colon F_n\to A$ in $\F$,

* whenever $f\colon A\to F_m$ is an epimorphism in $\F$, there exists $n\in\Bbb N$ such that there is an epimorphism $g\colon F_n\to A$ in $\F$ with $f\circ g=f^n_m$.

§ TOPOLOGICAL GRAPHS AND MONOTONE EPIMORPHISMS

In this section we reproduce for convenience definition and results about topological graphs that will be used in Section 3 and Section 4. They are mostly taken from <cit.>.

A graph $G$ is a pair $(V(G),E(G))$ consisting of a set of vertices $V(G)$ and a set of edges $E(G)\subseteq V(G)^2$ so that

* For all $a,b\in V(G)$, $\langle a,b\rangle\in E(G)\implies\langle b,a\rangle\in E(G)$.

* For all $a\in V(G)$, $\langle a,a\rangle\in E(G)$.

We will use interchangeably $a\in V(A)$ and $a\in A$ and we call the edges $\langle a,b\rangle\in E(A)$ with $a\neq b$ nontrivial.
A topological graph is a graph $G$ equipped with a compact, Hausdorff, zero-dimensional, second countable topology on $V(G)$ so that $E(G)$ is a closed subspace of $V(G)^2$. Every topological graph is a topological $L$-structure for the language $L=\{R\}$ consisting of a single binary relation in an obvious way.  We will keep using the notation $\langle a,b\rangle\in E(G)$ rather than $aRb$ even though we will often think about topological graphs as topological $L$-structures. An epimorphism $f\colon A\to B$ of topological graphs is an epimorphism in the sense of Definition <ref>, so in particular it maps edges to edges.

If $A$ is a topological graph and $B\subseteq V(A)$, $A\setminus B$ denotes the topological graph with $V(A\setminus B)=V(A)\setminus V(B)$ and for all $a_1,a_2\in V(A\setminus B)$,
$$\langle a_1,a_2\rangle\in E(A\setminus B)\iff \langle a_1,a_2\rangle\in E(A).$$

A topological graph $G$ is said to have a transitive set of edges if
$$\langle a,b\rangle,\langle b,c\rangle\in E(G)\implies\langle a,c\rangle\in E(G).$$
It is called a prespace if the edge relation $aRb\iff \langle a,b\rangle\in E(G)$ is an equivalence relation. In that case we call the quotient space $|G|=G/R$ the topological realization of $G$.

Let $\mathcal C\subseteq [0,1]$ be the standard middle-thirds Cantor set. Consider the topological graph $G$ with $V(G)=\mathcal C$ and for distinct $x,y\in \mathcal C$ we have $\langle x,y\rangle\in E(G)$ if and only if $x,y$ are the endpoints of one of the intervals removed from $[0,1]$ in the construction of $\mathcal C$ (see the picture below for the first few stages of the construction, with the edges represented by solid black lines).

(4/3,0) – (8/3,0) (4,0) – (8,0) (28/3,0) – (32/3,0);
(12/27,0) – (24/27,0) (84/27,0) – (96/27,0)
(228/27,0) – (240/27,0) (300/27,0) – (312/27,0);
in  1,2,3,6,7,8
[shift=(12/9*,0),color=black] (0pt,3pt) – (0pt,-3pt);
[shift=(12/9*,0),color=black] (0pt,0pt) – (0pt,-3pt) node[below] $\frac{\x}{9}$;

[shift=(0,0),color=black] (0pt,3pt) – (0pt,-3pt) node[below] $0$;
[shift=(12,0),color=black] (0pt,3pt) – (0pt,-3pt) node[below] $1$;

in 1,2,7,8,19,20,25,26
[shift=(12/27*,0),color=black] (0pt,3pt) – (0pt,-3pt);
[shift=(12/27*,0),color=black] (0pt,0pt) – (0pt,-3pt) node[below] $\frac{\x}{27}$;

Then $G$ is a prespace, in particular $|G|\cong [0,1]$.

A topological graph $G$ is called disconnected if it is possible to partition $V(G)$ into two nonempty disjoint closed sets $A,B$ such that whenever $a\in A$, $b\in B$, we have $\langle a,b\rangle\not\in E(G).$
A topological graph is called connected if it is not disconnected. A topological graph $A$ is called an arc if it is connected, but for all $a\in V(A)$, except at most two vertices, called the endpoints, $A\setminus\{a\}$ is disconnected.

The Cantor graph of Example <ref> is a connected and hereditarily unicoherent topological graph. In fact it is an arc. Note that while there is a unique topological arc, namely the interval $[0,1]$, there are many nonisomorphic topological graphs which are arcs. Indeed, there are finite, countable and uncountable topological graphs which are arcs.

The notion of monotone epimorphism from continuum theory, where a continuous map $f\colon X\to Y$ between continua is called monotone if $f^{-1}(y)$ is connected for every $y\in Y$, was generalised to topological graphs by Panagiotopoulos and Solecki in <cit.> (where those maps were called connected epimorphism instead) and studied in detail by Charatonik and Roe in <cit.>. The epimorphism we will consider in the following sections will always be monotone so we will use various results from <cit.>, whose statements we repeat here for convenience, but refer to <cit.> for their proofs.

An epimorphism $f\colon G\to H$ between topological graphs is called monotone if $f^{-1}(h)$ is connected in $G$ for every $h\in H$.

By Lemma 1.1 of <cit.> we can equivalently require that $f^{-1}(C)$ is connected in $G$ whenever $C\subseteq H$ is connected in $H$. From this equivalent definition it is clear that if $f\colon G\to H$ is monotone and $H$ is connected then $G$ is too. The converse of this statement actually holds for arbitrary epimorphisms:

Let $f\colon G\to H$ be an epimorphism between topological graphs. If $G$ is connected then $H$ is too.

Let $\mathcal G$ be a projective family of graphs such that for every $G\in\mathcal G$ and $a,b,c\in V(G)$ pairwise distinct with $\langle a,b\rangle,\langle b,c\rangle\in E(G)$, there is a graph $H\in\mathcal G$ and an epimorphism $f^H_G\colon H\to G$ so that whenever $p,q,r\in V(H)$ are such that $f^H_G(p)=a,f^H_G(q)=b$ and $f^H_G(r)=c$, we have $\langle p,q\rangle\not\in E(H)$ or $\langle q,r\rangle \not\in E(H)$. Then, if $\G$ denotes the projective limit of $\mathcal G$, for every $a\in\G$ there is at most one $b\in\G\setminus\{a\}$ with $\langle a,b\rangle\in E(\G)$. In particular, the edge relation of $\G$ is an equivalence relation and $\G$ is a prespace.

Suppose for a contradiction that $a,b,c\in\G$ are pairwise distinct and such that $\langle a,b\rangle\in E(\G),\langle b,c\rangle\in E(G)$. Find $G\in\mathcal G$ and an epimorphism $f_G\colon\G\to G$ such that $f_G(a),f_G(b)$ and $f_G(c)$ are pairwise distinct. Since $f_G$ maps edges to edges we have
$$\langle f_G(a),f_G(b)\rangle\in E(G),\langle f_G(b),f_G(c)\rangle\in E(G).$$

Now let $H\in\mathcal G$ and $f^H_G\colon H\to G$ be as in the assumptions of the theorem. Since $\G$ is the projective limit of $\mathcal G$ we can find $f_H\colon\G\to H$ with $f^H_G\circ f_H=f_G$. But for this to be the case we must have either $\langle f_H(a),f_H(b)\rangle\not\in E(H)$ or $\langle f_H(b),f_H(c)\rangle\not\in E(H)$, contradicting the fact that $f_H$ maps edges to edges.

Let $\T$ be a projective family of finite graphs. We say that $\T$ allows splitting edges if for all $G\in\T$ and all distinct $a,b\in V(G)$ with $\langle a,b\rangle\in E(G)$ the graph $H$ defined by $V(H)=V(G)\sqcup\{\ast\}$ and
$$E(H)=(E(G)\setminus\{\langle a,b\rangle\})\cup\{\langle a,\ast\rangle,\langle\ast, b\rangle\}$$
and the two morphisms $H\to G$ that map $\ast$ to either $a$ or $b$ and are the identity otherwise, are in $\T$.

Allowing to split edges is a basic closure property that will be satisfied by all the projective families considered in what follows. Despite its simplicity it has important consequences, such as the following lemma, and it will be used for some arguments in the next sections.

Suppose that $\T$ is a projective family of finite graphs that allows splitting edges. If $\G$ is the projective limit of $\T$, then $\G$ is a prespace.

We will show that $\T$ satisfies the hypothesis of Theorem <ref>, from which the conclusion follows immediately. Fix $G\in\T$ and pairwise distinct $a,b,c\in V(G)$ with $\langle a,b\rangle,\langle b,c\rangle\in E(G)$ and let $H$ be the graph obtained from $G$ by splitting the edge $\langle a,b\rangle$ twice. More formally we have $V(H)=V(G)\sqcup\{a',b'\}$ and
$$E(H)=(E(G)\setminus\{\langle a,b\rangle\})\cup\{\langle a,a'\rangle,\langle a',b'\rangle,\langle b',b\rangle\},$$
with the epimorphism $f^H_G$ such that $f^H_G(a')=a$, $f^H_G(b')=b$ and $f^H_G$ is the identity otherwise. We have that $H$ and $f^H_G$ are in $\T$ since $\T$ allows splitting edges, but if $p,q,r\in V(H)$ are such that $f^H_G(p)=a$, $f^H_G(q)=b$ and $f^H_G(r)=c$, then either $\langle p,q\rangle\not\in E(H)$, or $\langle q,r\rangle\not\in E(H)$.

A topological graph $A$ is called arcwise connected if for all $a,b\in V(A)$, there is a subgraph of $A$ containing $a$ and $b$ which is an arc. A topological graph is called hereditarily unicoherent if whenever $A,B$ are closed connected subsets of $V(G)$, $A\cap B$ is connected. Note that a hereditarily unicoherent and arcwise connected graph is uniquely arcwise connected.

A topological graph $G$ is called a dendroid if it is hereditarily unicoherent and arcwise connected. A dendroid $G$ is called a dendrite if it is locally connected.

Let $G$ be a topological graph. A vertex $x\in G$ is called an endpoint of $G$ if whenever $H$ is a topological graph which is an arc and $f\colon H\to G$ is an embedding, $x$ is the image of an endpoint of $H$.

If $f\colon G\to H$ is a monotone epimorphism between topological graphs and $G$ is an arc, then $H$ is an arc and the images of endpoints of $G$ are endpoints of $H$.

Our goal is to construct a large class of dendrites as projective limits of finite graphs. The following result from <cit.> will be crucial.
Let $\mathcal T$ be a projective family of trees with monotone epimorphisms, let $\G$ be its projective limit, then $\G$ is a dendrite. Moreover if $\G$ has a transitive set of edges, then $|\G |$ is a dendrite.

Note that the hypotheses of Theorem <ref> are satisfied by all projective families of trees that allow splitting edges. In particular they are satisfied by all projective families considered in the following sections. The following result from <cit.> will also be important.

Let $\T$ be a projective family of graphs with monotone epimorphisms. Let $\langle  T_i\mid i<\omega\rangle$ with maps $f^{i+1}_i\colon T_{i+1}\to T_i$ be a sequence for $\T$ and let $\mathbb T$ be its projective limit. Then for $G\in\T$ all epimorphisms $\mathbb T\to G$ in $\T^\omega$ are monotone.

We conclude this section by recalling some facts concerning dendrites from general topology. A compact metrizable space is called a dendrite if it is hereditarily unicoherent and uniquely arcwise connected. Equivalently a dendrite is a connected, metrizable, compact, locally connected topological space in which any two points are joined by a unique arc. For a proof of this equivalence and more characterizations of dendrites see <cit.>. In a dendrite $X$ there are three types of points $x\in X$, based on how many connected components $X\setminus\{x\}$ has:

* If $X\setminus \{x\}$ is connected, then $x$ is called an endpoint of $X$.

* If $X\setminus \{x\}$ has two connected components, then $x$ is called a regular point of $X$.

* If $X\setminus\{x\}$ has more than two connected components, then $x$ is called a ramification point of $X$.

In all of those cases the (potentially infinite) number of connected components of $X\setminus\{x\}$ is called the order or degree of $x$, denoted by $\ord(x)$, and it coincides with the Menger-Urysohn order of $x$ in $X$, that is with the maximum number of arcs in $X$ meeting only in $x$ <cit.>. For $P\subseteq \{3,4,5,\ldots,\omega\}$ the generalized dendrite $W_P$ is a dendrite satisfying the following two properties:

* If $x\in W_P$ is a ramification point, $\ord(x)\in P$.

* If $p\in P$, the set of ramification points in $W_P$ of order $p$ is arcwise dense. This means that for all $x,y\in W_P$, there is a ramification point of order $p$ in the unique arc joining $x$ and $y$.

For a fixed $P$ those two properties characterize a unique dendrite up to homeomorphism <cit.>, so we can actually talk about the generalized dendrite $W_P$.

§ WEAKLY COHERENT EPIMORPHISMS AND GENERALIZED DENDRITES.

In this section, we introduce a new class of maps between trees, which we call (weakly) coherent and study the relationship between weakly coherent points in a prespace and ramification points of its topological realization. Once this is done we will show how to construct many generalized dendrites as the topological realization of a projective limit.

Let $A,B$ be finite trees and $f\colon B\to A$ a monotone epimorphism. Fix $a\in A$  with $\ord(a)=n\geq 3$ and enumerate as $A_0,\ldots,A_{n-1}$ the connected components of $A\setminus\{a\}$. We say that $a$ is a point of weak coherence for $f$ if there exists $b\in f^{-1}(a)$ such that $m=\ord(b)\geq n$ and an injection $p\colon n\to m$ such that if $B_0,\ldots,B_{m-1}$ are the connected components of $B\setminus\{b\}$, then $f^{-1}(A_i)\subseteq B_{p(i)}$ for all $0\leq i\leq n-1$. In this case we call $b$ the witness for the weak coherence of $f$ at $a$. We say that $f$ is weakly coherent if every $a\in A$ with $\ord(a)\geq 3$ is a point of weak coherence for $f$.

This notion can be strengthened by requiring $\ord(b)=\ord(a)$ and $p$ to be a bijection. In that case we say that $a$ is a point of coherence of $f$ and that $b$ is a witness for the coherence of $f$ at $a$. We say that $f$ is coherent if every $a\in A$ with $\ord(a)\geq 3$ is a point of coherence for $f$.

Let $A,B$ be trees, $f\colon B\to A$ and $a\in A$ a ramification point. There exists at most one $b\in B$ witnessing the weak coherence of $f$ at $a$. Consider now a third tree $C$ with an epimorphism $g\colon C\to B$. If $f\circ g$ is weakly coherent at $a$ with witness $c$, then $f$ is weakly coherent at $a$. Moreover if $b$ witnesses the weak coherence of $f$ at $a$, $c$ witnesses the weak coherence of $g$ at $b$.

(-1,2) – (0,1) (1,2) – (0,1) (0,1) – (0,0)
(-1,-1) – (0,0) (1,-1) – (0,0);

[fill,circle,scale=0.5,label=below:$a_1$] at (0,0) ;
[fill,circle,scale=0.5,label=$b$] at (-1,2) ;
[fill,circle,scale=0.5,label=$c$] at (1,2) ;
[fill,circle,scale=0.5,label=$a_2$] at (0,1) ;
[fill,circle,scale=0.5,label=below:$d$] at (-1,-1) ;
[fill,circle,scale=0.5,label=below:$e$] at (1,-1) ;

(4,0.5) – (3,1.5) (4,0.5) – (5,1.5) (4,0.5) – (3,-0.5)
(4,0.5) – (5,-0.5);

[->] (1,0.5) – (3,0.5) node[midway,above] f;
[fill,circle,scale=0.5,label=below:$a$] at (4,0.5) ;
[fill,circle,scale=0.5,label=$b$] at (3,1.5) ;
[fill,circle,scale=0.5,label=$c$] at (5,1.5) ;
[fill,circle,scale=0.5,label=below:$d$] at (3,-0.5) ;
[fill,circle,scale=0.5,label=below:$e$] at (5,-0.5) ;

The map $f$ defined by $f(a_1)=f(a_2)=a$ and mapping every other node to the node with the same name is a monotone epimorphism which is not weakly coherent at $a$.

(-1,1) – (0,0) (1,1) – (0,0)
(-1,-1) – (0,0) (1,-1) – (0,0);

[fill,circle,scale=0.5,label=below:$a_1$] at (0,0) ;
[fill,circle,scale=0.5,label=$b$] at (-1,1) ;
[fill,circle,scale=0.5,label=$c$] at (1,1) ;
[fill,circle,scale=0.5,label=below:$d$] at (-1,-1) ;
[fill,circle,scale=0.5,label=below:$a_2$] at (1,-1) ;

(4,0) – (3,1) (4,0) – (5,1) (4,0) – (4,-1);

[->] (1,0) – (3,0) node[midway,above] f;
[fill,circle,scale=0.5,label=$a$] at (4,0) ;
[fill,circle,scale=0.5,label=$b$] at (3,1) ;
[fill,circle,scale=0.5,label=$c$] at (5,1) ;
[fill,circle,scale=0.5,label=below:$d$] at (4,-1) ;

The map $f$ defined by $f(a_1)=f(a_2)=a$ and mapping every other node to the node with the same name is a weakly coherent epimorphism which is not coherent at $a$.

Let $A,B$ be finite trees in which every vertex has degree at most $3$. Then any monotone epimorphism $f\colon A\to B$ is coherent. Indeed fix $b\in B$ of order $3$ and let $b_1$,$b_2$ and $b_3$ be its distinct neighbours. Let $b'_1\in f^{-1}(b_1),b'_2\in f^{-1}(b_2)$ and let $P=(p_0=b_1',p_1,\ldots,p_n=b'_2)$ be the unique path from $b'_1$ to $b'_2$. By Lemma <ref> $f(P)$ is connected and since it contains $b_1,b_2$ and $B$ is uniquely arcwise connected it must also contain $b$. Hence $P\cap f^{-1}(b)\neq\varnothing$. Let $b'_3\in f^{-1}(b_3)$ and let $Q$ be the shortest path from $b'_3$ to a point in $P$, let $p_i=P\cap Q$ and note that, by another application of Lemma <ref> similar to the one above, $p_i\in f^{-1}(b)$. It is now easy to check that $p_i$ witnesses the coherence of $f$ at $b$.

Note that if $A,B,C$ are finite trees and $f\colon A\to B,\,g\colon B\to C$ are (weakly) coherent epimorphisms, then $g\circ f\colon A\to C$ is also (weakly) coherent. Indeed if $c\in C$ with $\ord(c)\geq 3$ then there is some $b\in g^{-1}(c)$ witnessing the (weak) coherence of $g$ at $c$ and some $a\in f^{-1}(b)$ witnessing the (weak) coherence of $f$ at $b$. It is easy to check that $a$ also witnesses the (weak) coherence of $g\circ f$ at $c$.

Note also that if $A$ is a finite tree with at least two vertices and $B$ with $V(B)=V(A)\sqcup\{\ast\}$ obtained from $A$ by splitting an edge as in Definition <ref>, then the two maps $B\to A$ mapping $\ast$ to either endpoint of the split edge are coherent.

Given an inverse system $\langle A_i,i<\omega\rangle$ of graphs and monotone epimorphisms $f^i_j\colon A_i\to A_j$ we say that $a=(a_i)_i\in\varprojlim A_i\subseteq\prod A_i$ is a point of weak coherence or a weakly coherent point if there is a $k\in\omega$ such that for all $l>k$, $a_l$ is a point of weak coherence of $f^{l+1}_l$ and $a_{l+1}$ witnesses this. We say that $a$ is a point of coherence or a coherent point if there is a $k\in\omega$ such that for all $l>k$, $a_l$ is a point of coherence of $f^{l+1}_l$, and $a_{l+1}$ witnesses this.

Let $\T$ be a projective family of finite trees with monotone epimorphisms that allows splitting edges, let $\G$ be its projective limit and let $\pi\colon\G\to |\G|$ be its topological realization, which is a dendrite by Theorem <ref>. Moreover let $p\in |\G|$ be a ramification point. Then there is a unique $p'\in\G$ with $\pi(p')=p$.

Suppose for a contradiction that $|\pi^{-1}(p)|>1$ and, using that equivalence classes in $\G$ have at most two elements by Theorem <ref>, write $\pi^{-1}(p)=\{p_1,p_2\}$. Let $\{A_i'\mid i\leq n\}$ be the connected components of $|\Bbb G|\setminus\{p\}$, where $\ord(p)=n\in\Bbb N\cup\{\infty\}$. Let $A_i=A_i'\cup\{p\}$ for $i=1,\ldots,n$  and note that $\{\pi^{-1}(A_i)\mid i\leq n\}$ are connected sets in $\G$ intersecting only in $\{p_1,p_2\}$. Find a finite tree $G$ and a monotone epimorphism $f_G\colon \G\to G$ such that $f_G(\pi^{-1}(A_i))$ is nontrivial (meaning not contained in $\{f_G(p_1),f_G(p_2)\}$) for $i\in\{1,2,3\}$ and such that $f_G(p_1)\neq f_G(p_2)$. We claim that
$$f_G(\pi^{-1}(A_i))\cap f_G(\pi^{-1}(A_j))\subseteq\{f_G(p_1),f_G(p_2)\},$$
for all distinct $i,j\in\{1,2,3\}$. Suppose for a contradiction that
$$r\in f_G(\pi^{-1}(A_1))\cap f_G(\pi^{-1}(A_2))\setminus\{f_G(p_1),f_G(p_2)\}.$$
Then $f_G^{-1}(r)$ and $\{p_1,p_2\}$ are disjoint, nonempty, closed, connected subsets of $\Bbb G$. However $f_G^{-1}(r)$ meets both $\pi^{-1}(A_1)\setminus\{p_1,p_2\}$ and $\pi^{-1}(A_2)\setminus\{p_1,p_2\}$, which is a contradiction since $f_G^{-1}(r)$ is connected, while
$(\pi^{-1}(A_1)\cup\pi^{-1}(A_2))\setminus\{p_1,p_2\}$ is not.

The idea now is that we can always move to a bigger graph $H$ obtained from $G$ by splitting an edge in order to force $f_H(\pi^{-1}(A_1))\cap f_H(\pi^{-1}(A_3))\not\subseteq\{f_H(p_1),f_H(p_2)\}$. Indeed there must be at least two of $f_G(\pi^{-1}(A_1)),\,f_G(\pi^{-1}(A_2)),\,f_G(\pi^{-1}(A_3))$ that are connected to the same vertex $f_G(p_1)$ or $f_G(p_2)$, suppose without loss of generality that $f_G(\pi^{-1}(A_1))$ and $f_G(\pi^{-1}(A_3))$ are both connected to $f_G(p_2)$. Now construct a graph $H$ by taking $H=G\sqcup\{r\}$, where $r$ is a vertex not in $G$ and
$$E(H)=(E(G)\setminus\{\langle f_G(p_1),f_G(p_2)\rangle\})\cup\langle f_G(p_1),r\rangle,\langle r,f_G(p_2)\rangle,$$
together with the epimorphism $f^H_G\colon H\to G$ defined by $f^H_G(r)=f_G(p_2)$ and the identity otherwise. Call $a=(f^H_G)^{-1}(f_G(p_1))$ and $b$ the vertex in $(f^H_G)^{-1}(f_G(p_2))$ distinct from $r$. Note that $H,f^H_G\in\T$ since $\T$ allows splitting edges. By definition of projective limit there must be an epimorphism $f_H\colon\Bbb G\to H$ such that $f^H_G\circ f_H=f_G$. The last identity combined with the fact that $f_H$ maps edges to edges forces $f_H(p_1)=a$ and $f_H(p_2)=r$. Now we can use the same argument as above to show that
$$f_H(\pi^{-1}(A_1))\cap f_H(\pi^{-1}(A_3))\subseteq\{f_H(p_1),f_H(p_2)\}.$$
This is a contradiction, since by construction those two sets also meet in $b$.

Let $\T$ be a projective family of trees with monotone epimorphisms that allows splitting edges, let $\langle G_i\mid i<\omega\rangle$ with epimorphisms $f^m_n\colon G_m\to G_n$ be a sequence for $\T$, let $\G=\varprojlim G_i$ be the projective limit of $\T$ and let $\pi\colon \G\to |\G|$ be its topological realization, which is a dendrite by Theorem <ref>. Then $\pi$ maps points of weak coherence of $\G\subseteq\prod G_i$ to ramification points of $|\G|$.

Suppose that $p=(p_i)$ is a point of weak coherence of $\G$, we want to show that $\pi(p)$ is a ramification point of $|\G|$. Let $f_k\colon\G\to G_k$ be the projection epimorphisms obtained by construction of the inverse limit, let $A^1_k,\ldots,A^{n_k}_k$ be the connected components of $G_k\setminus\{p_k\}$, enumerated so that $(f^{k+1}_k)^{-1}(A^i_k)\subseteq A^i_{k+1}$ for all $i$. Let $k(i)$ be the least $k$ such that $G_k\setminus\{p_k\}$ has at least $i$ connected components, if such an $i$ exists, and $k(i)=\infty$ otherwise. Note that $p=\bigcap f_k^{-1}(p_k)$ and that
$$\{A^i=\bigcup_{k\geq k(i)} f_k^{-1}(A^i_k)\mid 1\leq i< \infty\}$$
are the connected components of $\G\setminus \{p\}$ (with $A^i$ being empty if $k(i)=\infty$. We will show that the equivalence class of $p$ in $\G$ is a singleton, from which we can conclude that $\pi(A^i)$ are the connected components of $|\G|\setminus\{\pi(p)\}$, so that $\pi(p)$ is a ramification point of $|\G|$ of order equal to the least $n$ such that $k(n+1)=\infty$ or to $\infty$ if there is no such $n$. Suppose for a contradiction that there exist $q=(q_i)\in\G$ with $\langle p,q\rangle\in E(\G)$, so in particular $\langle p_i,q_i \rangle \in E(G_i)$ for all $i$. Let $k$ be big enough so that $p_k\neq q_k$ and let $A^j_k$ be the component of $G_k\setminus \{p_k\}$ containing $q_k$. Note that by weak coherence for all $m\geq k$, $q_m\in A^j_m$. Consider the tree $H$ obtained from $G_k$ by splitting the $\langle p_k,q_k\rangle$ edge into two edges $\langle p_k,p_k'\rangle$ and $\langle p_k',q_k\rangle$, with the epimorphism $f^H_k\colon H\to G_k$ such that $f^H_k(p'_k)=p_k$ and the identity otherwise. Note that $f^H_k$ is coherent at $p_k\in G_k$, as witnessed by $p_k\in H$.
By definition of sequence we can find $l$ big enough and an epimorphism $f^l_H\colon G_l\to H$ such that $f^l_k=f^H_k\circ f^l_H$, which implies that $f^l_H(q_l)=q_k\in H$. On one hand we now must have $f^l_H(p_l)=p_k\in H$ by weak coherence, since $p_l$ witnesses the weak coherence of $f^l_k$ at $p_k\in G_k$ and $p_k\in H$ witnesses the weak coherence of $f^H_k$ at $p_k\in G_k$, on the other hand we must have $f^l_H(p_l)=p_k'$, since $f^l_H$ maps edges to edges, a contradiction.

Note in particular that if $x=(x_i)$ is a point of weak coherence in $\prod G_i$ then either $\ord(x_i)$ stabilizes after some index $j\in\Bbb N$ to some finite value $n$ or it grows unboundedly. In the former case $\pi(x)$ also has order $n$ as a ramification point of $|\Bbb G|$, while in the latter $\pi(x)$ is a ramification point of $|\Bbb G|$ of infinite order.

The converse to Theorem <ref> is false in general, but it is true when monotone epimorphisms are replaced with weakly coherent ones, as shown in the following theorem.

Let $\T$ be a projective family of trees with weakly coherent epimorphisms, let $\langle G_i\mid i<\omega\rangle$ with epimorphisms $f^m_n\colon G_m\to G_n$ be a sequence for $\T$ and let $\pi\colon\G\to|\G|$ be its topological realization, which is a dendrite by theorem <ref>. Then $\pi^{-1}$ maps ramification points of $|\G|$ to points of weak coherence of $\G\subseteq\prod G_i$.

As above let $f_k\colon\G\to G_k$ be the projection epimorphisms obtained by construction of the limit and let $x'\in|\G|$ be a ramification point. By Lemma <ref> there is a unique $x=(x_i)\in\pi^{-1}(x')$ and we want to show that it is a point of weak coherence in $\prod G_i$. Let $\{A^i\mid i<\ord(x)\}$ with $\ord(x)\in\Bbb N\cup\{\infty\}$ be the connected components of $\G\setminus\{x\}$ and let $k$ be big enough so that $f_k(A^1),f_k(A^2),f_k(A^3)$ are not singletons in $G_k$. Let $A^1_k,A^2_k$ and $A^3_k$ be connected components of $G_k\setminus \{x_k\}$, numbered so that $f_k^{-1}(A^i_k)\subseteq A^i$ for $i=1,2,3$, which is always possible since $f_k^{-1}(A^i_k)$ is connected by Lemma <ref> and doesn't meet $f_k^{-1}(x_k)$ by construction. By assumption $f^{k+1}_k\colon G_{k+1}\to G_k$ is weakly coherent at $x_k$, as witnessed by some vertex $a\in G_{k+1}$. Suppose by contradiction that $x_{k+1}\neq a$. Note that if $A^i_{k+1}$ is a connected component of $G_{k+1}\setminus\{ x_{k+1}\}$, then there is a unique $j$ such that $f_{k+1}^{-1}(A^i_{k+1})\subseteq A^j$, once again because $f_{k+1}^{-1}(A^i_{k+1})$ is connected and doesn't meet $f^{-1}_{k+1}(x_{k+1})$.
Since $x_{k+1}\neq a$ and $G_{k+1}$ is a tree, there is a connected component $A^n_{k+1}$ of $G\setminus \{x_{k+1}\}$, that contains all connected components of $G\setminus \{a\}$ except at most one, so it also contains $(f^{k+1}_k)^{-1}(A^i_k)$ for at least two distinct $i$. Assume without loss of generality that

\begin{align*}
(f^{k+1}_k)^{-1}(A^1_k)&\subseteq A^n_{k+1} \\
(f^{k+1}_k)^{-1}(A^2_k)&\subseteq A^n_{k+1},
\end{align*}

and let $j$ be the unique index such that $f_{k+1}^{-1}(A^n_{k+1})\subseteq A^j$. But now we have both $f^{k+1}_k\circ f_{k+1}=f_k$ and

\begin{align*}
(f_{k+1}^{-1}\circ(f^{k+1}_k)^{-1})(A^1_k)&\subseteq A^j\\
(f_{k+1}^{-1}\circ(f^{k+1}_k)^{-1})(A^2_k)&\subseteq A^j\\
f_k^{-1}(A^1_k)&\subseteq A^1\\
f_k^{-1}(A^2_k)&\subseteq A^2,
\end{align*}

which cannot all hold at the same time regardless of the value of $j$, a contradiction.

Let $\T$ be the family of all trees with monotone epimorphisms. As shown in <cit.> this is a projective family, whose projective limit is the dendrite $D_3$. In light of the previous two theorems if $\langle T_i\mid i<\omega\rangle$ with maps $f^m_n\colon T_m\to T_n$ is a sequence for $\T$, then any weakly coherent point $x=(x_i)\in\prod T_i$ must satisfy $\ord(x_i)=3$ for big enough $i$. Indeed this is case, suppose for a contradiction that $\ord(x_i)\geq 4$ for some $i$, we will find $k>i$ such that the map $f^k_i\colon T_k\to T_i$ is not weakly coherent at $x_i$, contradicting our assumptions. Construct a graph $G$ from $T_i$ by replacing the vertex $x_i$ with two new vertices $x^1_i,x^2_i$ both of order at least $3$ as in Example <ref>. Consider the function $f_i\colon G\to T_i$ defined by $f_i(x^1_i)=f_i(x^2_i)=x_i$ and the identity otherwise, which is monotone but not weakly coherent at $x_i$. Since the $T_i$ are a sequence for $\T$ we can find $k>i$ and a monotone epimorphism $f^k\colon T_k\to G$ such that $f_i\circ f^k=f^k_i$, which implies that $f^k_i$ is not weakly coherent at $x_i$.

We now introduce some families of finite trees. The remainder of this section will be dedicated to showing that those are projective families and that the topological realizations of their projective limits are generalized dendrites.

Let $P\subseteq\{3,4,5,\ldots,\infty\}$. We consider two cases. If $\infty\in P$ we consider the family $\mathcal F_P$ whose elements are finite trees with no vertices of order $2$. Given $A,B\in\mathcal F_P$, an epimorphism of graphs $f\colon B\to A$ is in $\mathcal F_P$ if:

* $f$ is monotone;

* If $a\in A$ is such that $\ord(a)\in P$, then $f$ is coherent at $a$.

* If $a\in A$ is such that $\ord(a)\not\in P$ and $\ord(a)\geq 3$ then $f$ is weakly coherent at $a$, and if $b\in f^{-1}(a)$ is the witness for the weak coherence of $f$ at $a$, then $\ord(b)\not\in P$.

On the other hand, if $\infty\not\in P$, we consider the family $\mathcal G_P$ whose elements are finite trees all of whose vertices are either endpoints or have order in $P$, with coherent maps.

The families $\F_P$ and $\mathcal G_P$ don't allow splitting edges as defined earlier, since the trees in those families don't have any vertices of degree two. However if $T$ is a tree in $\F_P$ or $\mathcal G_P$ and $a,b\in T$ with $\langle a,b\rangle\in E(T)$ it is still possible to split the edge $\langle a,b\rangle$ by removing it and then adding a new point $x$ connected to $a$ and $b$. We have to also add enough new neighbours $x_i$ to $x$ until $\ord(x)\in P$. This is enough for all the arguments in the previous sections that used the edge splitting property.

We will now verify that $\mathcal F_P$ and $\mathcal G_P$ are projective families and later we will identify the topological realizations of their projective limits. It seems natural to guess that if $\mathbb F_P$ and $\G_P$ are the projective limits of $\F_P$ and $\mathcal G_P$, then $|\mathbb F_P|\cong W_P$ and $|\G_P|\cong W_P$, but unfortunately this is not always the case. It is true for all families of the form $\mathcal G_P$, but for families of the form $\F_P$ it only holds when $\{3,4,5,\ldots,\omega\}\setminus P$ is infinite, as we will see later. Clearly both $\mathcal F_P$ and $\mathcal G_P$ contain countably many structures up to isomorphism, contain the identity morphisms and their morphisms are closed under composition by Remark <ref>. We begin by verifying the joint projection property for $\F_P$ and $\mathcal G_P$, with a construction that works for both families.

Let $A,B\in\F_P$ (respectively $A,B\in\mathcal G_P$). Then there exists $C\in\F_P$ (respectively $C\in\mathcal G_P$) with epimorphisms $f_1\colon C\to A,\,f_2\colon C\to B$ in $\F_P$ (respectively in $\mathcal G_P$).

Let $a\in A,\,b\in B$ be two endpoints. Consider the tree $C'$ obtained by taking the disjoint union of $A$ and $B$ and identifying $a$ with $b$. Let $C$ be the tree obtained from $C'$ by adding new vertices $x_i$ connected only to $x$ until $\ord(x)\in P$, where $x$ is the vertex $a$ and $b$ have been identified in. Define $f\colon C\to A$ to be the identity on $V(A)$, while any other vertex is mapped to $x$. Analogously define $G\colon C\to B$ to be the identity on $V(B)$, while any other vertex is mapped to $x$. It is immediate to check that $C$ witnesses the joint projection property for $A,B$.

It remains to verify that the projective amalgamation property is satisfied. We do so for families of the form $\mathcal F_P$ and note that the procedure described for those to produce an amalgam $D$ from a diagram $C\to A\leftarrow B$ of structures in $\mathcal F_P$ can also be applied to diagrams of the same shape in $\mathcal G_P$, and the amalgam $D$ produced in that case is itself an element of $\mathcal G_P$.

The family $\F_P$ satisfies the projective amalgamation property, so it is a projective family. 	Explicitely for all $A,B,C\in\F_P$ with epimorphisms $f\colon B\to A$ and $g\colon C\to A$ in $\F_P$, there exist $D\in\F_P$ with epimorphisms $h_1\colon D\to B$ and $h_2\colon D\to C$ in $\F_P$ such that $f\circ h_1=g\circ h_2$.

We proceed by induction on $|E(B)|+|E(C)|$. If there are no nontrivial edges in $B$ and $C$ then $A$ must also be a singleton and we can take $D$ to be a singleton as well.  For the inductive step suppose now that $B$ and $C$ are not both singletons and that the lemma is proved for all diagrams in $\F_P$ of the form $F\to A\leftarrow G$ with $|E(F)|+|E(G)|<|E(B)|+|E(C)|$. If $A=B=C$ and $f=g=\mathrm{Id}_A$ we can simply take $D=A$ and $h_1=h_2=\mathrm{Id}_A$, so we can assume that at least one of $f$ and $g$ is nontrivial. Suppose without loss of generality that $f$ is nontrivial. There are two possibilities now, $a$ could be a ramification point or and endpoint of $A$, and we deal with those two cases separately.

Case 1: $\mathbf{a}$ is a ramification point. Find $a\in A$ such that $|f^{-1}(a)|>1$. Let $b\in B$ be the witness for the (weak) coherence of $f$ at $a$ and let $B_1$ be a component of $B\setminus\{b\}$ with $B_1\cap f^{-1}(a)\neq\varnothing$. Let $B_2=B_1\cap f^{-1}(a)$. We need to distinguish two cases, based on whether $B_2=B_1$ or not.

Case 1a: $\mathbf{B_2\neq B_1}$ We start with the harder case, that is $B_2\neq B_1$ and explain how to deal with the easier case later. Let $b_1$ be the only element of $B_2$ with $\langle b,b_1\rangle\in E(B)$ and let $b_2$ be the only element of $B_2$ for which there exist a $b_3\in B_1\setminus B_2$ with $\langle b_2,b_3\rangle\in E(B)$ (uniqueness of $b_1,b_2,b_3$ follows from $B$ being a tree). Consider now the graph $B'$ obtained from $B$ by collapsing $B_2$ to $b$, formally $V(B')=V(B)\setminus V(B_2)$ and
$$E(B')=(E(B)\setminus\left( E(B_2)\cup\{\langle b,b_1\rangle,\langle b_2,b_3\rangle\})\right)\cup\{\langle b,b_3\rangle\}.$$
There is a natural map $\pi\colon B\to B'$ given by $\pi(x)=b$ if $x\in B_2$ and the identity otherwise. Note that $B'\in\F_P$ since both $b$ and $b_3$ have the same order as $\pi(b)$ and $\pi(b_3)$, and $\pi$ is a coherent epimorphism, so in particular it is in $\F_P$. Let $f'\colon B'\to A$ be the unique map with $f'\circ \pi=f$ and note that $f'$ is still  in $\F_P$ by construction. We reach the following picture, where the important vertices and edges are drawn, while the dashed sections represent parts of the trees whose precise structure is not important.

at (0,1) $A$;
[dotted] (0,0) circle (0.65cm);
[dotted] (2,0) circle (0.65cm);
[dotted] (1,1) circle (0.65cm);
(0,0) – (2,0);
(1,0) – (1,1);
[fill,circle,scale=0.5,label=below:$a$] at (1,0) ;
[fill,circle,scale=0.5,label=below:$a_1$] at (0,0) ;
[fill,circle,scale=0.5] at (1,1) ;
[fill,circle,scale=0.5] at (2,0) ;

[->] (4.2,0) – (2.8,0) node[midway,above] $f$;

at (9.85,1) $B$;
at (7,1) $B_2$;
[dotted] (5,0) circle (0.65cm);
(5,0) – (6.5,0);
(7.35,0) – (8.85,0);
[rounded corners,dotted] (5.85, -0.65) rectangle (8, 0.65) ;
[fill,circle,scale=0.5,label=below:$b_3$] at (5,0) ;
[fill,circle,scale=0.5,label=below:$b_2$] at (6.5,0) ;
[fill,circle,scale=0.5,label=below:$b_1$] at (7.35,0) ;
[fill,circle,scale=0.5,label=below:$b$] at (8.85,0) ;
(8.85,0) – (9.85,0);
(8.85,0) – (8.85,1);
[dotted] (8.85,1) circle (0.65cm);
[dotted] (9.85,0) circle (0.65cm);
[fill,circle,scale=0.5] at (8.85,1) ;
[fill,circle,scale=0.5] at (9.85,0) ;

[->] (1,-2.2) – (1,-0.65) node[midway,left] $g$;

at (0,-3) $C$;
[dotted] (0,-4) circle (0.65cm);
[dotted] (2,-4) circle (0.65cm);
[dotted] (1,-3) circle (0.65cm);
[dotted] (1,-5) circle (0.65cm);
(0,-4) – (2,-4);
(1,-5) – (1,-3);
[fill,circle,scale=0.55,label=[xshift=0.1cm, yshift=-0.15cm]$c$] at (1,-4) ;
[fill,circle,scale=0.5,label=below:$c_1$] at (0,-4) ;
[fill,circle,scale=0.5] at (1,-3) ;
[fill,circle,scale=0.5] at (2,-4) ;
[fill,circle,scale=0.5] at (1,-5) ;

at (9,-3) $B'$;
[dotted] (7,-4) circle (0.65cm);
(7,-4) – (9,-4);
[fill,circle,scale=0.5,label=[yshift=-0.8cm]$\pi(b_3)$] at (7,-4) ;
[fill,circle,scale=0.5,label=[yshift=-0.8cm]$\pi(b)$] at (8,-4) ;
(8,-4) – (8,-3);

[dotted] (8,-3) circle (0.65cm);
[dotted] (9,-4) circle (0.65cm);
[fill,circle,scale=0.5] at (8,-3) ;
[fill,circle,scale=0.5] at (9,-4) ;

[->] (8,-0.75) – (8, -2.2) node[midway,right] $\pi$;

[->] (7,-3) – (2.6,-0.6) node[midway,above] $f'$;

Since $|E(B')|<|E(B)|$ we can amalgamate $f'$ and $g$ over $A$ by inductive hypothesis, so we can find $D'\in\F_P$ and epimorphisms $h'_1\colon D'\to B'$ and $h'_2\colon D'\to C$ in $\F_P$ such that $f'\circ h_1'=g\circ h_2'$. Now we want to construct $D$ together with $h_1\colon D\to B$ and $h_2\colon D\to C$ from $D'$, $h'_1$ and $h'_2$, the intuitive idea is that we just need to paste back $B_2$ in $D'$ in the right spot, in particular $B_2$ should be inserted on the edge of $D'$ that corresponds to the edge $\langle \pi(b),\pi(b_3)\rangle$ in $B'$ (analogously to how $B$ is obtained from $B'$ by inserting $B_2$ on the $\langle \pi(b),\pi(b_3)\rangle$ edge). Let $a_1$ be the unique vertex of $A$ with $\langle a,a_1\rangle\in E(A)$ and $f^{-1}(a_1)\subseteq B_1$ and let $A_1$ be the connected component of $A\setminus \{a\}$ containing $a_1$. Let $c\in C$ witness the (weak) coherence of $g$ at $a$ and let $C_1$ be the connected component of $C\setminus \{c\}$ containing $g^{-1}(a_1)$. Let $c_1\in C_1$ be the unique vertex of $C$ with $\langle c,c_1\rangle\in E(C)$. Let $d\in D'$ witness the (weak) coherence of $h'_2$ at $c$ and let $D_1$ be the connected component of $D'\setminus \{d\}$ containing $(h'_2)^{-1}(c_1)$. Note that, since $f'\circ h'_1=g\circ h'_2$ and witnesses for (weak) coherence are unique, $d$ also witnesses the (weak) coherence of $h'_1$ at $b$ and $(h'_1)^{-1}(\pi(B_1))\subseteq D_1$ by Remark <ref>. Now let $d_1\in \{d\}\cup(D_1\cap (h'_1)^{-1}(\pi(b)))$ and $d_2\in D_1\cap (h'_1)^{-1}(\pi(b_3))$ be the unique such vertices with $\langle d_1,d_2\rangle\in E(D')$. We can now construct $D$ by inserting $B_2$ between $d_1$ and $d_2$, formally $V(D)=V(D')\sqcup V(B_2)$ and
$$E(D)=(E(D')\setminus\{\langle d_1,d_2\rangle\})\cup E(B_2)\cup \{\langle b_1,d_1\rangle,\langle b_2,d_2\rangle\},$$

note that $B_2\in\F_P$ and we didn't change the degrees of $d_1$ and $d_2$, so $D\in\F_P$ as well. Now consider the projection $\pi'\colon D\to D'$ collapsing $B_2$ to $d_1$, so $\pi'(x)=d_1$ if $x\in B_2$ and the identity otherwise, which is also an epimorphism in $\F_P$. We are now in a situation that looks like this:

at (0,1) $D'$;
at (2.8,1) $(h_1')^{-1}(\pi(b))$;
at (4.75,-1) $(h_1')^{-1}(\pi(b_3))$;
[dotted] (0,0) circle (0.65cm);
[dotted] (1,1) circle (0.65cm);
(0,0) – (2,0);
(1,0) – (1,1);
[fill,circle,scale=0.5,label=below:$d$] at (1,0) ;
[rounded corners,dotted] (1.55, -0.65) rectangle (3.7, 0.65) ;
[fill,circle,scale=0.5] at (2,0) ;
[fill,circle,scale=0.5,label=below:$d_1$] at (3.05,0) ;
[fill,circle,scale=0.5] at (0,0) ;
[fill,circle,scale=0.5] at (1,1) ;
(3.05,0) – (4.55,0);
[fill,circle,scale=0.5,label=below:$d_2$] at (4.55,0) ;
[dotted] (4.55,0) circle (0.65cm);

at (0,-3) $D$;
[dotted] (0,-4) circle (0.65cm);
[dotted] (1,-3) circle (0.65cm);
(0,-4) – (2,-4);
(1,-4) – (1,-3);
[fill,circle,scale=0.5,label=below:$d$] at (1,-4) ;
[rounded corners,dotted] (1.55, -4.65) rectangle (3.7, -3.35) ;
[rounded corners,dotted] (3.9, -4.65) rectangle (6.05, -3.35) ;
[fill,circle,scale=0.5] at (2,-4) ;
[fill,circle,scale=0.5,label=below:$d_1$] at (3.05,-4) ;
[fill,circle,scale=0.5] at (0,-4) ;
[fill,circle,scale=0.5] at (1,-3) ;
(3.05,-4) – (4.55,-4);
[fill,circle,scale=0.5,label=below:$b_1$] at (4.55,-4) ;
[fill,circle,scale=0.5,label=below:$b_2$] at (5.6,-4) ;
[dotted] (7.1,-4) circle (0.65cm);
(5.6,-4) – (7.1,-4);
[fill,circle,scale=0.5,label=below:$d_2$] at (7.1,-4) ;

[->] (1,-2.2) – (1,-0.65) node[midway,left] $\pi'$;

Define $h_2=h_2'\circ \pi'$ and $h_1\colon D\to B$ by

x & \text{if }x\in B_2 \\
h'_1(x) & \text{otherwise},
\end{cases}$$

where we identify points of $B\setminus B_2$ with their image in $B'$ through $\pi$. Note that $h_1$ is an epimorphism in $\F_P$, indeed it is (weakly) coherent at points of $B_2$ because they are witnesses for their own (weak) coherence and it is (weakly) coherent elsewhere because $h'_1$ is. Moreover we have $h_1'\circ\pi'=\pi\circ h_1$ so that the following diagram commutes:

D \arrow[r, "h_1"] \arrow[d, "\pi'"']& B \arrow[d, "\pi"'] &   \\
D' \arrow[r, "h_1'"] \arrow[rd, "h_2'"', bend right] & B' \arrow[r, "f'"]                             & A \\
& C \arrow[ru, "g"', bend right]                 &
\end{tikzcd}$$

It only remains to check that $(f\circ h_1)(x)=(g\circ h_2)(x)$ for all $x\in D$, for which there are two cases:

* If $x\in B_2$ then $h_1(x)=x$ and $f(h_1(x))=a$, while $h_2(x)=h'_2(\pi'(x))= h_2'(d_1)$, but $g\circ h_2'=f'\circ h_1'$ and $f'(h_1'(d_1))=a$ so $g(h_2(x))=a$.

* Otherwise $h_1(x)$ and $h_2(x)$ agree with $h'_1(x)$ and $h'_2(x)$ respectively, so we immediately obtain $f(h_1(x))=g(h_2(x))$ by construction.

This concludes the argument when $B_2\neq B_1$.

Case 1b: $\mathbf{B_2=B_1}$. Let $b\in B$ be the witness for the (weak) coherence of $f$ at $a$ and let $c\in C$ be the witness for the (weak) coherence of $g$ at $a$. Suppose now that for every component $\hat B$ of $B\setminus\{b\}$ that meets $f^{-1}(a)$, we have $\hat B\subseteq f^{-1}(a)$. Similarly we can assume that for every component $\hat C$ of $C\setminus\{c\}$ that meets $g^{-1}(a)$, we have $\hat C\subseteq g^{-1}(a)$, since otherwise we would be in the previous case modulo renaming $B$ and $C$. Let $a_1^A,\ldots,a_n^A$ enumerate the connected components of $A\setminus\{a\}$. By the weak coherence of $f$ we can find $a_1^B,\ldots,a_n^B$ components of $B\setminus\{b\}$ such that $f^{-1}(a_i^A)\subseteq a_i^B$ for all $i$. We define $a_1^C,\ldots,a_n^C$ analogously. Now let $B_1=f^{-1}(a)$ and $C_1=g^{-1}(a)$.

The idea now is to amalgamate the diagram
\begin{equation}a_i^B\cup\{b\}\to \{a\}\cup\{a_i^A\}\leftarrow a_i^C\cup\{c\}\label{diagram}\end{equation}
for every $i$, then to obtain $D_1$ from $B_1$ and $C_1$ by the joint projection property, and to glue everything back together at the end.
By inductive hypothesis we can amalgamate diagrams of the form (<ref>) (note that the trees involved are still members of $\F_P$ and so are the maps) obtaining an amalgam $a_i^D$ with maps $h_1^i\colon a_i^D\to a_i^B\cup\{b\}$ and $h_2^i\colon a_i^D\to a_i^C\cup\{c\}$. Note that, since $b$ and $c$ are the only preimages of $a$ in $a_i^B\cup\{b\}$ and $a_i^C\cup\{c\}$ respectively, there must be an endpoint $x_i\in a^D_i$ such that $h_1^i(x_i)=b$ and $h_2^i(x_i)=c$ for every $i$.

We now need to deal with $B_1$ and $C_1$. Enumerate as $\widehat{c_1},\ldots,\widehat{c_r}$ the components of $C_1\setminus\{c\}$ and as $\widehat{b_1},\ldots,\widehat{b_s}$ those of $B_1\setminus\{b\}$. Assume without loss of generality $s\leq r$. For $i\leq r$, let $c_i$ be the only vertex of $\widehat{c_i}$ such that $\langle c_i,c\rangle\in E(C)$ and let $b_i$, for $i\leq s$, be defined analogously. Consider now the tree $D_1$ built from
$$\left(\bigsqcup_{i\leq r}\widehat{c_i}\right)\sqcup\left(\bigsqcup_{i\leq s}\widehat{b_i}\right)$$
by adding $r+1$ new vertices $z,z_1,\ldots,z_r$ and edges $\langle z_i,c_i\rangle$ for all $i\leq r$, $\langle z_i,b_i\rangle$ for all $i\leq s$ and $\langle z,z_i\rangle$ for all $i\leq r$

Finally build a tree $D$ by identifying $z\in D_1$ and $x_i\in a_i^D$ for every $i$, and calling $d$ the image of all those points in $D$, add neighbours to $ z_i, b_i$ and $c_i$ until they have order in $P$. Define a map $h_1\colon D\to B$ by
h_1^i(y) & \text{if } y\in a_i^D \\
y & \text{if } y\in\widehat{b_i}  \\
b & \text{otherwise}
\end{cases}.$$

Similarly define a map $h_2\colon D\to C$ by

h_2^i(y) & \text{if } y\in a_i^D \\
y & \text{if } y\in \widehat{c_i} \\
c & \text{otherwise}
\end{cases}.$$

It is easy to check that $D$ together with the maps $h_1,h_2$ is an amalgam of $B$ and $C$ over $A$.

This concludes the argument for a ramification point $a\in A$.

Case 2: $\mathbf{a}$ is an endpoint. Suppose now that $a\in A$ is an endpoint. The construction is essentially an easier version of the one we just carried out for the case in which $B_1\neq B_2$. Since $B\setminus f^{-1}(a)$ is connected we can let $b\in f^{-1}(a)$ be the unique vertex for which there is a $b'\in B\setminus f^{-1}(a)$ with $\langle b,b'\rangle\in E(B)$ and let $B_2$ be $f^{-1}(a)$. As in the previous case consider $\pi\colon B\to B'$ collapsing $B_2$ to $b$ and let $f'\colon B'\to A$ be the unique map with $f'\circ\pi=f$. Moreover let $D'$ be an amalgam of $f'$ and $g$ over $A$ with maps $h_1'\colon D'\to B'$ and $h_2'\colon D'\to C$ such that $g\circ h_2'=f'\circ h_1'$. Once again we build $D$ from $D'$ by glueing $B_2$ to $D'$ in the right spot.  Let $\alpha\in(h_1')^{-1}(b)$ be an endpoint and take the disjoint union of $B_2$ and $D'$, identifying $\alpha\in D'$ and $b\in B_2$. Let $d$ be the image of $\alpha$ and $b$ through this identification.

As in the previous case, we obtain a map $\pi'\colon D\to D'$ by mapping the whole of $B_2$ to $d$, and a map $h_1\colon D\to B$ defined by

x & \text{if }x\in B_2 \\
h'_1(x) & \text{otherwise}.
\end{cases}$$

Now by considering $h_2=h_2'\circ\pi'$ we see that $D$ with the maps $h_2\colon D\to C$ and $h_1\colon D\to B$ defined above gives an amalgam of $f$ and $g$ over $A$, as desired.

The above proof of amalgamation is probably best explained through some pictures of a simple explicit case. To avoid excessive cluttering in the drawings some points are drawn with order $2$ even though trees in $\F_P$ don't have any. Consider the trees $A,B,C$ depicted below and weakly coherent epimorphisms (in fact coherent for simplicity in the drawing) $f\colon B\to A$ and $g\colon C\to A$ that send all vertices that are not circles to $a$ and are the identity otherwise. The weak coherence of $f$ at $a$ is witnessed by $b\in B$ and $b_1,b_2\in B$ as constructed in the proof are also named, while $B_2\subseteq B$ is represented by star shaped vertices. The weak coherence of $g$ at $a$ is witnessed by $c\in C$. The point $c_1\in C$ as constructed in the proof above is named.

(-1,0) – (1,0) (0,-1) – (0,1);

[fill,circle,scale=0.5,label=above right:$a$] at (0,0) ;
[fill,circle,scale=0.5,label=[yshift=-0.3cm] above right:$a_1$] at (0,-1) ;
[fill,circle,scale=0.5] at (-1,0) ;
[fill,circle,scale=0.5] at (0,1) ;
[fill,circle,scale=0.5] at (1,0) ;
at (-1,1) $A$;

[->] (3.5,0) – (1.5,0) node[midway,above] $f$;

(5,0) – (6,0) (6,1) – (6,-1);
(5,0) – (5,-2) (5,-1) – (4,-1);
(4,0) – (5,0) (5,0) – (5,1) (6,0) – (7,0) (5,-2) – (5,-3);
[fill,circle,scale=0.5] at (4,0) ;
[fill,diamond,scale=0.5,label=above right:$b$] at (5,0) ;
[fill,star, star points=5,scale=0.5,label=[yshift=-0.3cm] above right:$b_1$] at (5,-1) ;
[fill,star, star points=5,scale=0.5,label=[yshift=-0.3cm] above right:$b_2$] at (5,-2) ;
[fill,diamond,scale=0.5] at (6,0) ;
[fill,diamond,scale=0.5] at (6,-1) ;
[fill,diamond,scale=0.5] at (6,1) ;
[fill,star, star points=5,scale=0.5] at (4,-1) ;
[fill,circle,scale=0.5] at (7,0) ;
[fill,circle,scale=0.5,label=below:$f^{-1}(a_1)=b_3$] at (5,-3) ;
[fill,circle,scale=0.5] at (5,1) ;
at (4,1) $B$;

[->] (0,-3.5) – (0,-1.5) node[midway,right] $g$;

(0,-5) – (0,-8) – (1,-8) (0,-6) – (1,-6);
(0,-6) – (-2,-6) (-2,-7) – (-2,-5);
(-3,-6) – (-2,-6) (0,-4) – (0,-5) (0,-8) – (0,-9) (1,-6) – (2,-6);

[fill,circle,scale=0.5] at (0,-4) ;
at (-1,-5) $C$;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (0,-5) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5,label=above right:$c$] at (0,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5,label=[yshift=-0.3cm] above right:$c_1$] at (0,-7) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (0,-8) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (1,-8) ;
[fill,circle,scale=0.5,label=below:$g^{-1}(a_1)$] at (0,-9) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (1,-6) ;
[fill,circle,scale=0.5] at (2,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-1,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-5) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-7) ;
[fill,circle,scale=0.5] at (-3,-6) ;

Now following the argument of the proof above we construct $B'$ from $B$ and $\pi\colon B\to B'$ by collapsing $B_2$, thus obtaining the following graph:

(1,0) – (2,0) (2,1) – (2,-1);
(0,0) – (1,0) (1,1) – (1,-1) (2,0) – (3,0);
at (0,1) $B'$;
[fill,circle,scale=0.5] at (0,0) ;
[fill,circle,scale=0.5] at (1,1) ;
[fill,circle,scale=0.5,label=below:$(f')^{-1}(a_1)=b_3$] at (1,-1) ;
[fill,circle,scale=0.5] at (3,0) ;
[fill,diamond,scale=0.5,label=above right:$b$] at (1,0) ;
[fill,diamond,scale=0.5] at (2,0) ;
[fill,diamond,scale=0.5] at (2,1) ;
[fill,diamond,scale=0.5] at (2,-1) ;

We also obtain a weakly coherent epimorphism $f'\colon B'\to A$ defined as the unique function so that $f\circ \pi=f'$. In this case the map $f'$ is mapping the diamond shaped vertices of $B'$ to $a$ and is the identity otherwise. Since $|V(B')|<|V(B)|$ we can amalgamate $f'$ and $g$ over $A$. One possible amalgam is the graph $D'$ depicted below, with square and diamond shaped vertices for convenience.

(0,-5) – (0,-8) – (1,-8) (1,-6) – (2,-6);
(0,-6) – (-2,-6) (-2,-7) – (-2,-5);
(-3,-6) – (-2,-6) (0,-4) – (0,-5) (0,-8) – (0,-9) (2,-6) – (3,-6);
(0,-6) – (1,-6) (1,-5) – (1,-7);

[fill,circle,scale=0.5] at (0,-4) ;
at (-1,-5) $D'$;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (0,-5) ;
[fill,diamond,scale=0.5,label=above right:$d$] at (0,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (0,-7) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5,label=[yshift=-0.3cm] above left:$d_1$] at (0,-8) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (1,-8) ;
[fill,circle,scale=0.5,label=[yshift=-0.3cm] above left:$d_2$] at (0,-9) ;
[fill,diamond,scale=0.5] at (1,-6) ;
[fill,diamond,scale=0.5] at (1,-5) ;
[fill,diamond,scale=0.5] at (1,-7) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (2,-6) ;
[fill,circle,scale=0.5] at (3,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-1,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-5) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-7) ;
[fill,circle,scale=0.5] at (-3,-6) ;

with the map $h_2'\colon D'\to C$ being given by collapsing the diamond shaped vertices of $D'$ to $d$, and the map $h_1'\colon D'\to B$ being given by collapsing every square shaped vertex to the closest diamond shaped one. The only step that remains is, as in the proof above, inserting $B_2$ back between $d$ and $d_1$. This yields the following graph $D$, which is an amalgam of $f$ and $g$ over $A$, as desired.

(0,-5) – (0,-6) (1,-6) – (2,-6) (0,-8) – (0,-10);
(0,-6) – (-2,-6) (-2,-7) – (-2,-5);
(-3,-6) – (-2,-6) (0,-4) – (0,-5) (2,-6) – (3,-6) (0,-10) – (0,-11);
(0,-6) – (1,-6) (1,-5) – (1,-7);
(0,-6) – (0,-8);
(0,-8) – (1,-8);
(0,-9) – (-1,-9);

[fill,circle,scale=0.5] at (0,-4) ;
at (-1,-5) $D$;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (0,-5) ;
[fill,diamond,scale=0.5,label=above right :$d$] at (0,-6) ;
[fill,star, star points=5,scale=0.5] at (0,-9) ;
[fill,star, star points=5,scale=0.5] at (0,-10) ;
[fill,star, star points=5,scale=0.5] at (-1,-9) ;
[fill,circle,scale=0.5,label=[yshift=-0.3cm] above left:$d_2$] at (0,-11) ;
[fill,diamond,scale=0.5] at (1,-6) ;
[fill,diamond,scale=0.5] at (1,-5) ;
[fill,diamond,scale=0.5] at (1,-7) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (2,-6) ;
[fill,circle,scale=0.5] at (3,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-1,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-5) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (-2,-7) ;
[fill,circle,scale=0.5] at (-3,-6) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (0,-7) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5] at (1,-8) ;
[fill,regular polygon, regular polygon sides=4,scale=0.5,label=[yshift=-0.3cm] above left:$d_1$] at (0,-8) ;

As mentioned before Lemma <ref>, the construction in its proof also shows the following:

The family $\mathcal G_P$ satisfies the projective amalgamation property, so it is a projective family.

Now that we know that $\F_P$ and $\mathcal G_P$ are projective families let $\mathbb F_P$ and $\G_P$ denote their projective limits. We know from Lemma <ref> and Theorem <ref> that $|\mathbb F_P|$ and $|\G_P|$ are dendrites, and in the rest of this section we will identify which dendrites they are exactly. Let's start with the easiest case, namely $\infty\not\in P$, so that we're looking at the family $\mathcal G_P$ with projective limit $|\G_P|$.

Fix $P\subseteq\{3,4,5,\ldots\}$. Then $|\G_P|\cong W_P$.

For the remainder of this proof we fix a sequence $G_i$ for $\mathcal G_P$, with epimorphisms $f^i_j\colon G_i\to G_j$. We consider $\G_P$ as a subspace of $\prod G_i$, and call $f_i\colon\G_P\to G_i$ the canonical projections. We know that $|\G_P|$ is a dendrite, so we need to check that all ramification points of $|\G_P|$ have order in $P$, and that for all $p\in P$ the set
$$\{x\in|\G_P|\mid \ord(x)=p\}$$
is arcwise dense in $|\G_P|$. The first part follows quickly from previous results: if $x\in|\G_P|$ is a ramification point, then $\pi^{-1}(x)=(x_i)$ is a coherent point in $\G_P$ by Theorem <ref>, where $\pi\colon\G_P\to|\G_P|$ is the topological realization. Note that $\ord(x_i)\in P$ for large enough $i$, so by Remark <ref> we have that, for $k$ big enough, $\ord(x)=\ord(x_k)\in P$. It remains to show that for every $p\in P$ the set of ramification points in $|\G_P|$ is arcwise dense in $|\G_P|$. Fix $p\in P$ and $X\subseteq|\G|$ an arc. Let $a',b'\in X$ be two distinct points and fix $a\in\pi^{-1}(a'),b\in\pi^{-1}(b')$. Let $i$ be big enough to have $f_i(a)\neq f_i(b)$ and let $\langle w,y\rangle$ be any edge on the unique path in $G_i$ joining $f_i(a)$ and $f_i(b)$. Consider the graph $H$ obtained from $G_i$ by splitting the edge $\langle w,y\rangle$ into two edges $\langle w,x^i_p\rangle$,$\langle x^i_p,y\rangle$ and by adding $p-2$ new vertices $z_1,\ldots,z_{p-2}$ with edges $\langle x^i_p,z_j\rangle$ for all $1\leq j\leq p$. Note that there is an epimorphism $f^H_i\colon H\to G_i$ in $\mathcal G_P$ with $f^H_i(v)=w$ for all $v\in V(H)\setminus V(G_i)$ and the identity otherwise, and note that in $H$ there is a point of order $p$ on the path connecting $f_i(a)$ to $f_i(b)$. We can now find $k$ big enough so that there is an epimorphism $f^k_H\colon G_k\to H$ in $\mathcal G_P$ with $f^H_i\circ f^k_H=f^k_i$ and note that in $G_k$ there is a point of order $p$ on the path connecting $f_k(a)$ and $f_k(b)$, namely the point $x^k_p\in (f^k_H)^{-1}(x^i_p)$ witnessing the coherence of $f^k_H$ at $x^i_p$. This is because $f_k(a)$, $f_k(b)$ are in different components of $G_k\setminus\{x_p^k\}$ by coherence, and $G_k$ is uniquely arcwise connected. Now for all $m>k$ let $x^m_p\in G_m$ be a point witnessing the coherence of $f^m_{m-1}\colon G_m\to G_{m-1}$ at $x^{m-1}_p$ and note that for all $m>k$, $x^m_p$ is a point of order $p$ on the unique path in $G_m$ joining $f_m(a)$ and $f_m(b)$. Since $x=(x^m_p)_{m\in\Bbb N}\in\G_P$ is a coherent sequence it determines a ramification point of order $p$ in $|\G_P|$ by Theorem <ref> and Remark <ref>. By construction $a$ and $b$ belong to different connected components of $\G_P\setminus\{x\}$, and since $\G_P$ is hereditarily unicoherent this implies that any connected set in $\G_P$ containing both $a$ and $b$ must contain also $x$. In particular $x\in\pi^{-1}(X)$ and so $\pi(x)\in X$.

We can now analyze what happens in the case $\infty\in P$, so that we are looking at the families $\mathcal F_P$ with projective limit $\mathbb F_P$ instead. The situation is more subtle.

Fix $P\subseteq\{3,4,5,\ldots,\infty\}$. If $\{3,4,5,\ldots,\infty\}\setminus P$ is infinite and $\infty\in P$, then $|\mathbb F_P|\cong W_P$.

The proof is very similar to that of Theorem <ref>. For the remainder of this proof we fix a sequence $F_i$ for $\mathcal F_P$, with epimorphisms $f^i_j\colon F_i\to F_j$. We consider $\mathbb F_P$ as a subspace of $\prod F_i$, and call $f_i\colon\mathbb F_P\to F_i$ the canonical projections. We already know that $|\mathbb F_P|$ is a dendrite so we need to check that all ramification points of $|\mathbb F_P|$ have order in $P$, and that for all $p\in P$ the set
$$\{x\in|\mathbb F_P|\mid \ord(x)=p\}$$
is arcwise dense in $|\mathbb F_P|$. The first part is very similar to the argument in the proof of Theorem <ref>: if $x\in|\mathbb F_P|$ is a ramification point, then $\pi^{-1}(x)=(x_i)$ is a weakly coherent point of $\mathbb F_P$ by Theorem <ref>, where $\pi\colon\mathbb F_P\to|\mathbb F_P|$ is the topological realization. Now the way the family $\F_P$ is constructed allows two options: either $\ord(x_i)$ becomes constant for big enough $i$, so that $(x_i)$ is actually a coherent point of $\mathbb F_P$ and we are in the same case as in the proof of Theorem <ref>, or $\ord(x_i)\to\infty$ as $i\to\infty$, which by Remark <ref> implies that $x$ is a ramification point of $|\Bbb F_P|$ of infinite order. In either case $\ord(x)\in P$.

It remains to show that for every $p\in P$ the set of ramification points in $|\mathbb F_P|$ is arcwise dense in $|\mathbb F_P|$. If $p\neq\infty$ we can repeat verbatim the argument given in the proof of Theorem <ref>, so we need to show that ramification points of infinite order are arcwise dense in $|\mathbb F_P|$. Fix $X\subseteq|\mathbb F_P|$ an arc, distinct $a',b'\in X$, $a\in\pi^{-1}(a'),b\in\pi^{-1}(b')$ and $i$ big enough to have $f_i(a)\neq f_i(b)$. Fix also a strictly increasing sequence $(n_j)_{j\in\Bbb N}$ of natural numbers, such that $n_j\not\in P$ for all $j$ and $n_j\to\infty$ as $j\to\infty$. Let $\langle w,y\rangle$ be any edge on the unique path in $F_i$ joining $f_i(a)$ and $f_i(b)$. Consider the graph $H$ obtained from $F_i$ by splitting the edge $\langle w,y\rangle$ into two edges $\langle w,x_1\rangle$,$\langle x_1,y\rangle$ and by adding $n_1-2$ new vertices $z_1,\ldots,z_{n_1-2}$ with edges $\langle x_1,z_j\rangle$ for all $1\leq j\leq n_1-2$. Note that there is an epimorphism $f^H_i\colon H\to F_i$ in $\F_P$ with $f^H_i(v)=w$ for all $v\in V(H)\setminus V(F_i)$ and the identity otherwise, and note that in $H$ there is a point of order exactly $n_1$ on the path connecting $f_i(a)$ to $f_i(b)$. We can now find $k$ big enough so that there is an epimorphism $f^k_H\colon F_k\to H$ in $\F_P$ with $f^H_i\circ f^k_H=f^k_i$ and note that in $F_k$ there is a point of order at least $n_1$ on the path connecting $f_k(a)$ and $f_k(b)$, namely the point $x_2\in (f^k_H)^{-1}(x_1)$ witnessing the weak coherence of $f^k_H$ at $x_1$. Now we can iterate this construction, by considering the tree $H_2$ obtained from $F_k$ by adding at least $n_s-n_1-2$ neighbours to $x_2$, with the map $f\colon H_2\to F_k$ sending all of those new vertices to $x_2$, where $s$ is big enough to have $\ord(x_2)<n_s-n_1-2$. Then we can find some $l>k$ and an epimorphism $f^l_H\colon F_l\to F_H$ such that $f^{H_2}_k\circ f^l_{H_2}=f^l_k$ and as above we can find a point $x_3\in F_l$ of order at least $n_s$ on the path between $f_l(a)$ and $f_l(b)$. Repeating this construction we build a weakly coherent sequence $(x_i)$ such that $\ord(x_i)\to\infty$ and $x_i$ lies on the unique path in $F_i$ between $f_i(a)$ and $f_i(b)$. The same argument used at the end of the proof of Theorem <ref> allows us to conclude that the ramification point of infinite order $\pi(x_i)\in|\mathbb F_P|$ belongs to the arc $X$.

In Theorem <ref> we had to assume that $\{3,4,5,\ldots,\infty\}\setminus P$ is infinite, which was used to find a sequence $n_j$ of natural numbers with $\lim n_j\to\infty$ and $n_j\not\in P$, in order to construct weakly coherent points of $\mathbb F_P$ corresponding to ramification points of $|\mathbb F_P|$ of infinite order. When $P$ is cofinite in $\{3,4,5,\ldots,\infty\}$ the family $\mathcal F_P$ is still a projective family, as we showed above, but the problem is that the order of the points in weakly coherent sequences must stabilize instead of growing unboundedly. Indeed the techniques used in the proof of Theorem <ref> easily establish that $|\mathbb F_P|=W_{P'}$ where $P'=P\setminus\{\infty\}$ if $P=\{3,4,5,\ldots,\infty\}$ and $P'=(P\setminus\{\infty\})\cup \{a\}$ otherwise, where $a$ is the smallest integer such that for all $n>a$, $n\in P$.

§ CATEGORIES AND PROJECTION-EMBEDDING PAIRS

In this section, we construct all generalized dendrites as projective limits by moving to the more general setting of categories developed by in <cit.>. Those are categories in which the amalgamation and joint embedding properties hold (when expressed in terms of the appropriate diagrams). As an application we recover a countable dense homogeneity result for $\End(W_P)$. We quickly recall some definitions from <cit.>, but stress that the approach followed there is far more general than what we need. In particular, since we are only working with countable collections of finite objects (with finitely many morphisms between any two of them) we can ignore all the issues regarding existence of sequences and uniqueness of limits that arise when looking at the uncountable case. Of particular interest to us is section 6 of <cit.> in which projection-embedding pairs are introduced as a tool to produce objects that are universal both in an injective and a projective sense.

The following definitions and results are taken from <cit.>, where they are presented in a more general way, which works in the uncountable case as well.

As a warning let us point out that while 's approach is phrased in terms of injective limits we decided to phrase everything in terms of projective limits. There is no real difference between the two approaches, since it is enough to look at the opposite category $\K^{\mathrm{op}}$ to translate between the two.

Given a category $\K$ and two objects $a,b\in\K$, let $\K(a,b)$ denote the morphisms $a\to b$ in $\K$.

Let $\K$ be a category. We say that $\K$ has the amalgamation property if for all $a,b,c\in\K$ and all morphisms $f\in\K(b,a),g\in\K(c,a)$, there exist $d\in\K$ and morphisms $f'\in\K(d,b),g'\in\K(d,c)$ with $f\circ f'=g\circ g'$. We say that $\K$ is directed or that $\K$ has the joint projection property if for all $a,b\in\K$ there exists $c\in\K$ such that $\K(c,a)\neq\varnothing\neq\K(c,b)$.

We now wish to define a category of projective sequences in $\K$, which we denote by $\sigma\K$, whose morphisms will be defined in a way that guarantees the following property. Suppose that $\bar{x},\bar{y}\in\sigma\K$ and $\K$ is embedded in a category in which those two sequences have a limit. Given an arrow in $\sigma\K(\bar{x},\bar{y})$ we want an induced arrow $\varprojlim \bar{x}\to\varprojlim\bar{y}$. Given $\bar{x}\in\sigma\K$ we denote $x(n)$ by $x_n$ and the morphism $x(n)\to x(m)$ for $n\geq m$ by $x^n_m$. By definition a sequence in $\K$ is a functor $\omega\to\K$, where we think about the former as a poset category with the reverse order, so a transformation $\bar{x}\to\bar{y}$ is by definition a natural transformation $F\colon\bar{x}\to\bar{y}\circ\varphi$, where $\varphi\colon\omega\to\omega$ is order preserving. To define
an arrow $\bar{x}\to\bar{y}$ in $\sigma\K$ we need to identify some of those natural transformations.

Let $F\colon\bar{x}\to\bar{y}\circ\varphi$ and $G\colon\bar{x}\to\bar{y}\circ\psi$ be two natural transformations. We say that $F$ and $G$ are equivalent if

* For every $n$ there exist $m\geq n$ such that $\varphi(m)\geq\psi(n)$ and
$$y_{\psi(n)}^{\varphi(m)}\circ F(m)=G(n)\circ x^m_n;$$

* For every $n$ there exist $m\geq n$ such that $\psi(m)\geq\varphi(n)$ and
$$y^{\psi(m)}_{\varphi(n)}\circ G(m)=F(n)\circ x^m_n.$$

An arrow $\bar{x}\to\bar{y}$ in $\sigma\K$ is an equivalence class of natural transformations $\bar{x}\to\bar{y}$.

Note that the category $\sigma\K$ in 's approach is the analogue of the class of structures $\mathcal F^\omega$ in the approach by Panagiotopolous and Solecki. We can now define a sequence for a category $\K$, which is the analogue of the sequence in the classical or projective limit construction. defined in an analogous manner a category of sequences of length $\kappa$ for any cardinal $\kappa$. Similarly the following definition is a special case of the definition of a $\kappa$-sequence from <cit.>.

Let $\K$ be a category. A sequence in $\K$ is a projective sequence $\bar{u}\colon \omega\to\K$ satisfying:

* For every $x\in\K$ there exist $n<\omega$ with $\K(u_n,x)\neq\varnothing$.

* For every $n<\omega$ and every arrow $f\in\K(y,u_n)$, with $y\in\K$, there exist $m\geq n$ and $g\in\K(u_m,y)$ such that $f\circ g=u^m_n$.

Note that if $\K$ has the amalgamation property then $\bar{u}$ also satisfies

* For all arrows $f\in\K(a,b)$, $g\in\K(u_n,b)$, there exist $m\geq n$ and $h\in\K(u_m,a)$ with $g\circ u^m_n=f\circ h$.

The following is a special case of Corollary 3.8 of <cit.> and shows existence of sequences:

Let $\K$ be a category with the amalgamation and joint embedding property with countably many objects and such that between any two objects there are finitely many arrows. Then $\K$ has a sequence.

The following is Theorem 3.15 of <cit.> and shows the uniqueness (up to isomorphism) of sequences, hence also the uniqueness of limits when we embed into a category in which the sequence has a limit:

Assume $\overline{u}$ and $\overline{v}$ are sequences in a given category $\K$. Assume further that $k,l<\omega$ and $f\in\K(u_k,v_l)$. Then there exists an isomorphism $F\colon\overline{u}\to\overline{v}$ in $\sigma\K$ such that the diagram

u [d,"u_k"'][r, "F"]     v [d,"v_l"']

u_k [r,"f"']     v_l

commutes. In particular $\overline{u}\approx\overline{v}$.

We now have all the tools needed to talk about embedding-projection categories and construct the generalized dendrites not obtained in previous sections.

Given $P\subseteq\{3,4,\ldots,\infty\}$ let $\L_P'$ denote the language with a binary relation $R$ and unary relations $U_p$ for every $p\in P$. Let $\mathcal L_R=\{R\}\subseteq\mathcal L_P'$ and let $\mathcal L_P=\{U_p\mid p\in P\}\subseteq\mathcal L_P'$. Let $\dF$ denote the following category:

* $A\in\dF$ iff $A$ is a finite tree (with $R$ as the edge relation), $A$ contains at least one ramification point and every vertex of $A$ is either an endpoint or a ramification point. Moreover for every ramification point $a\in A$ there is exactly one $p\in P$ such that $U_p(a)$ holds, and in that case $\ord(a)\leq p$. When $a$ is a ramification point we will refer to the unique $p\in P$ for which $U_p(a)$ holds as the label of $a$. If $a$ instead is an endpoint of $A$, then $U_p(a)$ doesn't hold for any $p\in P$.

* Given $A,B\in\dF$ an arrow $f\in\dF(B,A)$ is a pair $(p(f),e(f))$ where $p(f)\colon B\to A$ is a weakly coherent epimorphism from $B$ to $A$ viewed as $\mathcal L_R$-structures with the discrete topology. Moreover if for $a\in A$, $U_p(a)$ holds and $b\in B$ witnesses the weak coherence of $p(f)$ at $a$, then $U_p(b)$ holds. In particular, we are not requiring that if $U_p(b)$ holds, then $U_p(p(f)(b))$ also holds. On the other hand $e(f)\colon \End(A)\to \End(B)$ is a map associating to every $a\in \End(A)$ a $b\in\End(B)$ with $p(f)\circ e(f)=\Id_{\End(A)}$, where $\End(X)$ is the set of endpoints of $X$. In other words $p(f)$ is a weakly coherent epimorphism between the $\mathcal L_R$-reducts, while $e(f)$ is a partial right inverse defined on the endpoints.

Given two arrows $f\colon B\to A$ and $g\colon C\to B$ in $\dF$ we define their composition in the obvious way: $f\circ g$ is the pair $(p(f)\circ p(g),e(g)\circ e(f))$.
It is clear that composition is associative and that the pair $(\mathrm{Id}_A,\mathrm{Id}_A)$ is the identity morphism for $A$ in $\dF$, so $\dF$ is a category.

The argument used to prove Lemma <ref> shows that $\dF$ is directed, so in order to check that it is a category we only need to check that it has the amalgamation property.

The category $\dF$ has amalgamation. In other words whenever we have arrows $f\in\dF(B,A)$, $g\in\dF(C,A)$, there exists $D\in\dF$ with arrows $h_1\in\dF(D,B)$, $h_2\in\dF(D,C)$ such that $f\circ h_1=g\circ h_2$.

The proof is very similar to the proof of amalgamation given in the previous section for $\mathcal F_P$, so we don't write all the details. Instead we point out the extra steps that need to be taken to adapt the proof of <ref> to the current setting. Let $A,B,C,f,g$ be as in the statement of the lemma. We proceed by induction on $|E(B)|+|E(C)|$. The smallest structures in $\dF$ are trees with a single ramification point labelled with any $p\in P$ that has $3$ neighbours. Clearly if $B$ and $C$ are of that form then $A$ must be the same tree on $4$ vertices as well, and all the ramification points in $A,B,C$ must have the same label, so we can take $D=A=B=C$ with identity maps to be the amalgam.
Our induction hypothesis now becomes that not only diagrams of the form $B\to A\leftarrow C$ with smaller $|E(B)|+|E(C)|$ can be amalgamated in $\dF$, but that the amalgamation can be carried on in a way that preserves labels and is compatible with the embedding part of the arrows. With this assumption we can use the same construction as in the proof of Lemma <ref>. Indeed as in that lemma suppose without loss of generality that $p(f)^{-1}(a)$ is nontrivial for some $a\in A$. Suppose first that $a$ is a ramification point and we are in the $B_2\neq B_1$ case. We can produce $B'$ with $\pi\in\dF(B,B')$, $f'\in\dF(B',A)$ and $D'$ with $h_1'\in\dF(D',B')$ and $h_2'\in\dF(D',C)$ as in the proof of Lemma <ref>. Note that in this case $p(\pi)$ is the identity map between $\End(B)$ and $\End(B')$, so that $e(f')$ can be defined to agree with $e(f)$. We then build $D$ from $D'$ by glueing back $B_2$ along an edge. Points in $D$ are already labeled correctly, with labels coming from either $B_2$ or $D'$, and since glueing $B_2$ into $D'$ doesn't collapse any endpoints in $D'$ we can take $e(h_2)=e(h_2')$, while $e(h_1)=e(h_1')$ on $\End(B')$ and is the identity on $\End(B_2)\setminus\{b_1,b_2\}$.

The $B_2=B_1$ is easier to deal with: we follow the same construction as in the proof of Lemma <ref> and we never run into issues with the embedding part of the morphisms, since endpoints are never collapsed during the construction.

If instead $a$ is an endpoint there are still no issues mimicking the construction from Lemma <ref> as far as labels are concerned, but a little more care is needed for the embedding part of the arrows, so we will write out some of the details. Let $b\in p(f)^{-1}(a)$ be the unique vertex for which there is a $b'\in B\setminus p(f)^{-1}(a)$ with $\langle b',b\rangle\in E(B)$ and let $B_2$ be $p(f)^{-1}(a)$. As in the previous case we construct $B'$ from $B$ by collapsing $B_2$ to $b$ through the map $p(\pi)\colon B\to B'$ but now we also need to define the embedding part $e(\pi)$ in order to get a morphism in $\dF$. Let $e(\pi)$ be the identity on $\End(B')\setminus\{b\}$ and $e(\pi)(p(\pi)(b))=e(f)(a)$. Similarly we obtain a coherent epimorphism $p(f')\colon B'\to A$ as the only such morphism satisfying $p(f')\circ p(\pi)=p(f)$. There is only one choice for $e(f')$, namely

$$e(f')(x)=\begin{cases}e(f)(x) & \text{ if $x\in\End(A)\setminus\{a\}$} \\ p(\pi)(b) & \text{ if $x=a$}\end{cases}.$$

Once again we obtain, by inductive hypothesis, an amalgam $D'$ with $h_1'\in\dF(D',B')$ and $h_2'\in\dF(D',C)$ such that $f'\circ h_1'=g\circ h_2'$. We now have to glue back $B_2$ to $D'$ in order to build an amalgam $D$ with maps $h_1\in\dF(D,B)$ and $h_2\in\dF(D,C)$. We already know how to build $D$ as a graph and how to define $p(h_1),p(h_2)$, since the construction is  the same as the one in the proof of Lemma <ref>. As in the previous section we obtain $D$ from $D'$ and $B_2$ by identifying $b\in B_2$ with an endpoint of $D'$. Here we take $\alpha\in\End(D')$ to be $e(h_1')(b)$ (which is also equal to $e(h_2')(e(g)(a))$). Points of $D$ are already correctly labeled, with labels coming from either $B_2$ or $D$, and we define $p(h_1),p(h_2)$ exactly as above. We only need to define $e(h_1),e(h_2)$. There is nothing to be done for $e(h_2)$ which we simply define to be equal to $e(h_2')$ on all endpoints except $e(g)(a)\in B_2$, where it now takes value $e(f)(a)$. To define $e(h_1)$ there are a few cases to consider, as follows:

e(h_1')(x) & \text{if } x\in\End(B)\setminus\End(B_2) \\
x & \text{if } x\in\End(B_2)
\end{cases}.$$

It is now easy to check that the resulting square diagram is commutative, as desired.

Since $\dF$ has countably many objects and between any two objects of $\dF$ there are finitely many arrows, Lemma <ref> and Theorem <ref> immediately imply the following:

The category $\dF$ has a sequence $\overline{u}$ and any two sequences are isomorphic as elements of $\sdF$. In particular whenever $\dF$ is embedded in a category in which sequences have colimits, the colimits arising from two sequences are isomorphic.

As mentioned in the introduction to this section we want to embed $\sdF$ in a category in which those sequences actually have a limit. We do so by considering $\sdF$ as a subcategory of a category $\mathcal G$ of labeled topological graphs defined as follows: The objects of $\mathcal G$ are pairs $G=(B,E)$ where $B$ is a topological graph which is a dendrite, equipped with unary predicates $U_n$ such that if $a$ is a ramification point of $G$, there is a unique $n$ for which $U_n(a)$ holds and $n\leq\ord(a)$, and $E$ is a countable subset of $\Bbb U$. The morphisms of $\mathcal G$ from $(B,E)$ to $(A,F)$ are pairs $f=(p(f),e(f))$ such that $e(f)\colon F\to E$ is an injection, while $p(f)\colon B\to A$ is an epimorphism of topological graphs such that $p(f)\circ e(f)=\mathrm{Id}_F$.

To a sequence $\overline{u}\in\sdF$ we associate the pair $(\Bbb U,E)$, where $\Bbb U$ is the topological space obtained as the projective limit of the projection part of $\overline{u}$, while $E$ is the direct limit of the embedding part of $\overline{u}$. Moreover every ramification point of $\Bbb U$ is labeled with its order as label. Given another $\overline{v}\in\sdF$, a morphism $\psi\colon \overline{u}\to\overline{v}$ is by definition a natural transformation $\psi$ from $\overline{u}$ to $\overline{v}\circ\varphi$, where $\varphi\colon\omega\to\omega$ is order-preserving. It is easy to check that if $(\Bbb U,E)$ and $(\Bbb V,F)$ are the objects associated to $\overline{u}$ and $\overline{v}$ as described above, then the projection part of $\psi$ converges to a continuous surjection $p(\psi)\colon\Bbb U\to\Bbb V$, while the embedding part converges to an injection $e(\psi)\colon F\to E$ such that $p(\psi)\circ e(\psi)=\mathrm{Id}_F$. Note that $p(\psi)$ has the property that if $v\in\Bbb V$ is a ramification point with label $l$, then there is a ramification point $u\in p(\psi)^{-1}(v)$ with the same label.
This describes the desired functor $\sdF\to\mathcal G$.

To a sequence $\overline{u}\in\sdF$ we associated a pair $(\Bbb U,E)$, where $\Bbb U$ is a topological graph. We will show that whenever $\overline{u}$ is a sequence for $\dF$, then $|\Bbb U|$ is the generalized dendrite $W_P$, where $\pi\colon\Bbb U\to|\Bbb U|$ is the topological realization.

Note that the embedding part of $\overline{u}$ plays no role in the following theorem. This observation will be made precise and become relevant later.

Let $\overline{u}=\langle F_i\mid i<\omega\rangle$ with maps $f^{i+1}_i\in\dF(F_{i+1},F_i)$ be a sequence for $\dF$ and let $(\mathbb F_P,E)$ be its limit. Then $|\Bbb F_P|=W_P$.

We know that $|\Bbb F_P|$ is a dendrite by Theorem <ref> since $p(f^{i+1}_i)$ is monotone and (weakly) coherent, moreover we still have the correspondence between (weakly) coherent points of $\Bbb F_P$ and ramification points of $|\Bbb F_P|$ established in Theorem <ref> and Theorem <ref>. The fact that each ramification point has order in $P$ follows from an argument similar to the one done in the proof of Theorem <ref>: if $x=(x_n)\in\Bbb F_P$ is labelled with $U_p$, then $x_n$ must be labelled with $U_p$ from some index on. If $p<\omega$, then $\ord(x_n)$ must also be $p$ from some (possibly bigger) index on, which implies that $\ord(x)=p$, and so $\ord(\pi(x))=p$. If $x=(x_n)\in\Bbb F_P$ is labelled with $U_\omega$ instead, then $\ord(x_n)\to\infty$ as $n\to\infty$ as in the proof of Theorem <ref>, hence $\ord(\pi(x))=\omega$. The fact that the ramification points of each order in $P$ are arcwise dense in $|\Bbb F_p|$ follows from exactly the same argument as the one at the end of the proof of Theorem <ref>.

We now want to prove the following homogeneity result for the endpoints of $W_P$, first stated in <cit.> for $W_3$:

Let $P\subseteq\{3,4,\ldots,\omega\}$ and let $Q_1,Q_2$ be countable dense subsets of $\End(W_P)$. Then there is a homeomorphism $h\colon W_P\to W_P$ with $h(Q_1)=Q_2$.

We begin with a more precise description of the endpoints of a projective limit of trees, in particular we want to obtain that if $(\Bbb U,E)$ is the projective limit of $\dF$, then $\pi(E)$ is a dense set of endpoints in $|\Bbb U|$. We first show that endpoints are edge related only to themselves.

Let $\mathcal F$ be a projective family of trees with (weakly) coherent epimorphisms that allows splitting edges and let $\Bbb F$ be its projective limit. If $e\in\End(\Bbb F)$, then there is no $f\in\Bbb F\setminus\{e\}$ with $\langle e,f\rangle\in E(\Bbb F)$.

Suppose for a contradiction that there is $e\neq f\in\Bbb F$ with $\langle e,f\rangle\in E(\Bbb F)$. Let $F_n$ with maps $f^m_n\colon F_m\to F_n$ be a sequence for $\mathcal F$ and let $g_n\colon \Bbb F\to F_n$ be the canonical projections.
Let $e_n,f_n$ be $g_n(e)$ and $g_n(f)$ respectively. First we claim that for all but finitely many $n$ either $e_n\in\End(F_n)$ or $f_n\in\End(F_n)$. Indeed suppose for a contradiction that this is not the case and let $m$ be big enough so that $e_m\neq f_m$ and neither of them is in $\End(F_m)$. Let $a_m,b_m$ be two points in $F_m$ such that $e_m, f_m$ are on the unique path joining $a_m$ and $b_m$ (for example $a_m=e_m$ and $b_m=f_m$ will work). Without loss of generality we can assume that $a_m,b_m$ are ramification points and let, for $k>m$, $a_k,b_k$ denote the vertices of $F_k$ witnessing the (weak) coherence of $f^k_m$ at $a_m$ and $b_m$ respectively. By monotonicity of $f^k_m$ and since $\langle e_k,f_k\rangle\in E(F_k)$, we have that for all $k>m$, $f_k$ and $e_k$ are on the unique path joining $a_k$ and $b_k$. As in the proof of arcwise density in Theorem <ref> we obtain two points $a,b\in\Bbb F$ such that $e$ and $f$ are contained in the unique arc joining $a$ and $b$, contradicting the fact that $e$ and $f$ are endpoints of $\Bbb F$.

We can thus assume without loss of generality that $e_n\in\End(F_n)$ for all $n\geq m$. Now let $H$ be the tree obtained by splitting the $\langle e_m,f_m\rangle$ edge, so that $V(H)=V(F_m)\sqcup\{x\}$ and
$$E(H)=(E(F_m)\setminus\{\langle e_m,f_m\rangle\})\cup\{\langle e_m,x\rangle,\langle x,f_m\rangle\}$$

and consider the epimorphism $\varphi\colon H\to F_m$ defined by $\varphi(x)=e_m$ and $\varphi$ is the identity on $H$ otherwise (when applying this result to the $\mathcal F=\dF$ case we will also need to add neighbours to $x$ until it has the correct order). By assumption we can find $n>m$ and an epimorphism $\psi\colon F_n\to H$ such that $\varphi\circ\psi=f^n_m$. Since $\psi$ respects the edge relation we must have $\psi(e_n)=x,\psi(f_n)=f_m$. But now, since $e_n\in\End(F_n)$ and $\psi$ is monotone, it is impossible for $e_m$ to be in the image of $\psi$, contradicting that the latter is an epimorphism.

Let $\F$ be a projective family of trees with (weakly) coherent epimorphisms that allows splitting edges. Let $\langle F_i\mid i<\omega\rangle$ be a sequence for $\F$ and let $\Bbb F\subseteq\prod F_i$ be its projective limit. Let $e=(e_n)_{n\in\Bbb N}\in\Bbb F$. If there is $N\in\Bbb N$ such that for every $m\geq N$, $e_m$ is an endpoint of $F_m$, then $e\in\End(\Bbb F)$.

Assume that $(e_n)_{n\in\Bbb N}$ is such that $e_m$ is an endpoint of $F_m$ for all $m\geq N$. Suppose by contradiction that $e$ is not an endpoint of $\Bbb F$ and let $A\subseteq\Bbb F$ be an arc with $e\in A$, but $e\neq a_1,a_2$, where $a_1,a_2$ are the endpoints of $A$. Let $k$ be big enough to that $e_k$ is an endpoint of $F_k$, and $f_k(a_1),\,f_k(a_2),\,f_k(e)$ are pairwise distinct. By Theorem <ref> $f_k$ is monotone, so by Lemma <ref> $f_k(a_1),\,f_k(a_2)$ and $e_k$ all belong to $\End(f_k(A))$, a contradiction.

Let $\dF$ be as above, let $\overline{u}=\langle F_i\mid i<\omega\rangle$ be a sequence for $\dF$ with maps $f^m_n\in\dF(F_m,F_n)$ and let $(\Bbb F,E)$ be its projective limit. Then $E\subseteq\End(\Bbb F)$ and $E$ is dense in $\Bbb F$.

It follows from Lemma <ref>, that $E\subseteq\End(\Bbb F)$. Let now $U\subseteq\Bbb F$ be a nonempty open set. We want to show that $E\cap U\neq\varnothing$. Let $n\in\omega$ be sufficiently big so that there is $x\in F_n$ such that $f_n^{-1}(x)\subseteq U$. Consider any finite tree $T\in\dF$ with an epimorphism $f\in\dF(T,F_n)$ such that $p(f)^{-1}(x)$ contains an endpoint of $T$. By definition of a sequence we can find $m\geq n$ and $g\in\dF(F_m,T)$ such that $f\circ g=f^m_n$. Since the preimage of an endpoint through a (weakly) coherent map always contains an endpoint, this implies that there is an endpoint $y\in\End(F_m)$ with $f_m^{-1}(y)\subseteq U$. Now the point of the limit determined by the sequence $e(f^k_m)(y)$ for $k\geq m$ is an endpoint by Lemma <ref> and is contained in both $E$ and $U$ by construction.

As a consequence of the two previous lemma we obtain the following corollary:

If $(\Bbb U,E)$ is the projective limit of $\dF$, then $\pi(E)$ is a dense set of endpoints in $|\Bbb U|\simeq W_P$, where $\pi\colon\Bbb U\to|\Bbb U|$ is the topological realization.

The last piece we need before being able to prove the desired countable dense homogeneity result for the endpoints of $W_P$ is Theorem <ref>. Intuitively this theorem will say that the embedding parts of the morphisms is not important in determining whether a sequence is for $\dF$. This is not surprising, since in Theorem <ref> the embedding part of the morphisms played no role. We first need to introduce new definitions and make a few simple remarks.

Let $X$ be a topological dendrite. Given any three distinct $x,y,z\in X$, their center $C(x,y,z)$ is the unique point in $[x,y]\cap[y,z]\cap[x,z]$. A set $F\subseteq X$ is called center-closed if $C(f_1,f_2,f_3)\in F$ whenever $f_1,f_2,f_3\in F$ are distinct. Given a finite set $A\subseteq X$, its center closure is the smallest center-closed $B\subseteq X$ with $A\subseteq B$. Given a finite center-closed $F\subseteq X$ let, for $a\in F$, $\hat{a}_F$ denote the set of connected components of $X\setminus\{a\}$ that contain no point of $F$. Given distinct $a,b\in F$ let $C_a(b)$ be the connected component of $X\setminus\{a\}$ containing $b$, $C_b(a)$ the connected component of $X\setminus\{b\}$ containing $a$, and $C_{a,b}=C_b(a)\cap C_a(b)$. The partition associated to $F$ is then
$$\Omega_F=\bigcup_{a\in F}\hat{a}_F\cup\{C_{a,b}\mid a\neq b\in F \text{ with } [a,b]\cap F=\{a,b\}\},$$

which is exactly the set of connected components of $X\setminus F$.

An immersion of a finite graph $A$ in a topological graph $B$ is a map $f\colon A\to B$ (which will usually not be a graph homomorphism) such that for all distinct $a,b,c\in A$,
$$b\in[a,c]\iff f(b)\in[f(a),f(c)].$$
In other words there exist a graph $A'$ obtained from $A$ by replacing every edge of $A$ with an arc (as in Definition <ref>) and a graph embedding $f'\colon A'\to B$ with $f=\restr{f'}{A}$.

Note that the image of an immersion $f\colon A\to B$ is center-closed, so it determines a partition $\Omega_f=\Omega_{f(A)}$ of $B\setminus f(A)$ as above. There are two particular instances of immersions that will be relevant. They are described in the following examples.

An arrow $f\in\dF(B,A)$ determines an immersion $i(f)\colon A\to B$ by setting

e(f)(a) & a\in\End(A) \\
w(a) & \text{otherwise},
\end{cases}$$

where $w(a)\in B$ is the witness for the (weak) coherence of $p(f)$ at $a$.

Let $\langle F_n\mid n<\omega\rangle$ with maps $f^m_n\in\dF(F_m,F_n)$ be a sequence in $\dF$ and let $\Bbb F$ with maps $f^\infty_n\in\dF(\Bbb F,F_n)$ be its limit. We have immersions $h_n\colon F_n\to \Bbb F$ given by
$$x\mapsto (p(f^n_0)(x),p(f^n_1)(x),\ldots,p(f^n_{n-1})(x),x,w^n_{n+1}(x),w^n_{n+2}(x),\ldots),$$
where $w^i_j(x)$ denotes the witness for the (weak) coherence of $f^j_i\colon F_j\to F_i$ at $x$ for $x$ a ramification point. For endpoints we follow the embeddings instead of the witnesses, that is
$$x\mapsto (p(f^n_0)(x),p(f^n_1)(x),\ldots,p(f^n_{n-1})(x),x,e(f^{n+1}_n)(x),e(f^{n+2}_n)(x),\ldots),$$
when $x$ is an endpoint.

Let $i\colon A\to B$ be an immersion of finite trees. If $f\colon B\to A$ is a monotone map such that

* for every $a\in A$, $f(i(a))=a$,

* for every $a\in A$ and every $b\in\hat{a}$, $f(b)=a$,

* for every $a\neq a'\in A$ and every $b\in C_{a,a'}$, $f(b)\in\{a,a'\}$.

Then $f$ is weakly coherent.

We say that a topological graph $G$ is a $W_P$-prespace if $G$ is a dendrite and for every $p\in P$ the set of ramification points of order $p$ is arcwise dense in $G$. In particular if $G$ is a $W_P$-prespace, then $|G|=W_P$.

We can now characterize sequences in $\dF$.

A sequence $\langle F_n\mid n<\omega\rangle$ with maps $f^m_n\in\dF(F_m,F_n)$ is if and only if the inverse limit of $p(f^m_n)$ as a topological graph is a $W_P$-prespace.

The if part of the statement was already proved in Theorem <ref>. Suppose that $\langle F_n\mid n<\omega\rangle$ with epimorphisms $f^m_n\in\dF(F_m,F_n)$ is a sequence such that

$$\Bbb F=\varprojlim\left(\cdots\to F_{n+1}\xrightarrow{p\left(f^{n+1}_{n}\right)} F_n\xrightarrow{p\left(f^{n}_{n-1}\right)} \cdots\xrightarrow{p\left(f^1_0\right)} F_0\right)$$

is a $W_P$-prespace. We want to show that $\langle F_n\mid <\omega\rangle$ is a sequence for $\dF$. In other words given $A\in\dF$ and an epimorphism $g\in\dF(A,F_n)$, we want to find $m\geq n$ and an epimorphism $h\in\dF(F_m,A)$ such that $g\circ h=f^m_n$. As in Example <ref> we obtain an immersion $h_n\colon F_n\to\Bbb F$. Moreover we also have an immersion $j\colon F_n\to A$ as in Example <ref>. Using the fact that $\Bbb F$ is a $W_P$-prespace we can find an immersion $k\colon A\to\Bbb F$ such that $k\circ j=h_n$. Now we can find $m\geq n$ such that, calling $f^\infty_m\in\dF(\Bbb F,F_m)$ the canonical projection, all vertices in $k(A)$ have distinct images through $f^\infty_m$. Note that $i=p(f^\infty_m)\circ k$ is an immersion of $A$ in $F_m$. Thanks to Remark <ref> we can now define a coherent map from $F_m$ to $A$, we only need to be careful and make sure that it makes the appropriate diagram commute. Explicitely we define $h\in\dF(F_m,A)$ as follows. Let $x\in F_m$. If $x=i(a)$ for some $a\in A$, then $p(h)(x)=a$. If $x\in \widehat{a}_{i(A)}$ (as defined in Definition <ref>) for some $a\in A$, then $p(h)(x)=a$. If $x\in C_{a,a'}$ for some $a,a'\in A$, then we check whether $p(g)(a)=f^m_n(x)$ or $p(g)(a')=f^m_n(x)$ and define $p(h)(x)$ accordingly. For the embedding part of $h$ consider $a\in\End(A)$. If $a\in e(g)(\End(F_n))$ set $e(h)(a)=e(f^m_n)(b)$, where $b\in\End(F_n)$ is such that $e(g)(b)=a$. If instead $a$ is not in the image of $e(g)$, then $e(h)(a)$ can be any endpoint in $p(h)^{-1}(a)$.

We conclude by proving Theorem <ref>.

We only need to prove that for every countable dense $Q\subseteq\End(W_P)$, there is a sequence $\overline{u}=\langle F_i\mid i<\omega\rangle$ with morphisms $f^m_n\in\dF(F_m,F_n)$ such that if $(\Bbb U,D)$ is its limit, then $\pi(D)=Q$, where $\pi\colon \Bbb U\to |\Bbb U|\cong W_P$ is the topological realization. Once this is done the theorem follows immediately from the uniqueness of the limit. Enumerate $Q$ as $\{q_1,q_2,q_3,\ldots\}$. Let $A_0=\{q_1,q_2,q_3,c\}$ where $c=C(q_1,q_2,q_3)$ and, for all $0<n<\omega$, let $A_n$ be the center closure of $\{q_1,\ldots,q_{k_n}\}$, where $(k_n)_{n<\omega}$ is an increasing sequence chosen so that $|A_{n+1}\cap(a,a')|\geq 2$, whenever $a,a'\in A_n$ are such that $[a,a']\cap A_n=\{a,a'\}$. We now build a finite graph $F_n$ with a vertex $v_a$ for every $a\in A_n$, and an edge $\langle v_a,v_{a'}\rangle$ iff $[a,a']\cap A_n=\{a,a'\}$. We label every $v_c\in F_{n+1}$ that comes from a ramification point $c\in A_{n+1}$ with the order of $c$ in $W_P$. It only remains to define morphisms $f^{n+1}_n\in\dF(F_{n+1},F_n)$. Since we have $A_n\subseteq A_{n+1}$, we also get an immersion $i\colon F_n\to F_{n+1}$ that maps $v_a\in F_n$ to $v_a\in F_{n+1}$, for every $a\in A_n$. The embedding part of $f^{n+1}_n$ is exactly $\restr{i}{\End(F_n)}$. For the projection part we rely on Remark <ref>, by letting $p(f^{n+1}_n)$ be any monotone, hence weakly coherent, map compatible with $i$ as in the above remark, with some care needed in ensuring that $p(f^{n+1}_n)^{-1}(v_a)$ contains at least two points for every $v_a\in F_n$, which can be easily guaranteed since we ensured that $|A_{n+1}\cap[a,a']|>1$ for all $a\neq a'\in A_n$. This last condition shows that the limit of the sequence is a prespace, thanks to Theorem <ref>, and since one can easily check that we obtain a $W_P$-prespace, the sequence is by Theorem <ref>. It is also clear by construction that if $D$ is the limit of the embedding part, then $\pi(D)=Q$, as desired.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleTransactions of the American Mathematical Society
titleFences, their endpoints, and projective Fraïssé theory

journaltitleFundamenta Mathematicae
titleLelek fan from a projective Fraïssé limit

journaltitleTopology and its Applications
titleOn self-homeomorphic spaces

titleProjective Limits of Trees

American Mathematical Society

journaltitleTransactions of the American Mathematical Society
titleProjective Fraïssé Limits and the Pseudo-Arc

journaltitleRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
titleThe Lelek fan and the Poulsen simplex as Fraïssé limits

journaltitleAnnals of Pure and Applied Logic
titleFraïssé sequences: category-theoretic approach to universal homogeneous structures

Monografie Matematyczne, Tom 21. Państwowe Wydawnictwo Naukowe, Warsaw

titleTopologie. Vol. II. Troisième édition, corrigèe et complétée de deux appendice

CRC Press

seriesChapman & Hall/CRC Pure and Applied Mathematics
titleContinuum Theory: An Introduction

journaltitleJournal of Topology and Analysis
titleA combinatorial model for the Menger curve