2.1 The setup

Throughout the paper, we find ourselves in the following situation: We have a category of ‘small’ objects ${\mathfrak {S}}$ and a fixed ‘large’ object U, both living in an ambient category ${\mathfrak {L}}$ . The main theorems relate properties of ${\mathfrak {S}}$ , like the (weak) Ramsey property, with properties of U, like extreme amenability of its automorphism group. Note that (weak) Fraïssé theory follows the same pattern: Properties of ${\mathfrak {S}}$ , like the (weak) amalgamation property or existence of a (weak) Fraïssé sequence, are related to properties of the ‘generic’ object U, like (weak) homogeneity and (weak) injectivity.

A connection between ${\mathfrak {S}}$ and U in ${\mathfrak {L}}$ is established by fixing a coned sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ : a triple $(\vec {u}, \vec {u}^{\,\infty }, U)$ , where $\vec {u}$ is a sequence in ${\mathfrak {S}}$ and $(\vec {u}^{\,\infty }, U)$ is a cone for $\vec {u}$ in ${\mathfrak {L}}$ . The object U can be thought of as a ‘quasi-limit’ of $\vec {u}$ in ${\mathfrak {L}}$ . Sometimes, it even is an actual colimit, but it is not necessary.

In this subsection, we impose a minimalistic set of conditions on the coned sequence $(\vec {u}, \vec {u}^{\,\infty }, U)$ for our theory to work. A concise list is given in the following definition. A rather long remark (that can be safely skipped) giving some insight follows.

Remark 2.2. Let us give some insight to the conditions defining a matching sequence. Given a category ${\mathfrak {S}}$ , let us define the induced category of sequences $\sigma _0{\mathfrak {S}}$ . The objects are all sequences $\vec {u}$ in ${\mathfrak {S}}$ . The morphisms are transformations between sequences modulo a certain equivalence. A transformation $\vec {\varphi }\colon \vec {u} \to \vec {v}$ is a sequence of ${\mathfrak {S}}$ -arrows $\{\varphi _n\colon u_n \to v_{\varphi (n)}\}_{n \in \omega }$ , where $\{\varphi (n)\}_{n \in \omega } \subseteq \omega $ is an increasing cofinal sequence such that $v^{\varphi (m)}_{\varphi (n)} \circ \varphi _n = \varphi _m \circ u^m_n$ for every $n \leqslant m \in \omega $ , as in Figure 2, that is, it is a natural transformation from $\vec {u}$ to a subsequence of $\vec {v}$ . The composition is obvious: $(\vec {\psi } \circ \vec {\varphi })_n = \psi _{\varphi (n)} \circ \varphi _n$ for every $n \in \omega $ . We say that two transformations $\vec {\varphi }, \vec {\psi }\colon \vec {u} \to \vec {v}$ are equivalent (and we write $\vec {\varphi } \approx \vec {\psi }$ ) if for every $n \in \omega $ there is $m \geqslant \varphi (n), \psi (n)$ such that $v^m_{\varphi (n)} \circ \varphi _n = v^m_{\psi (n)} \circ \psi _n$ . It is easy to see that this is a well-defined congruence of a category, and so defining morphisms of $\sigma _0{\mathfrak {S}}$ as transformations modulo this equivalence is correct.

Figure 2 A morphism between sequences and a matching morphism between associated cones.

Note that we may identify every ${\mathfrak {S}}$ -object x with the constant sequence $\vec {\operatorname {i\!d}}_{x}$ and every ${\mathfrak {S}}$ -arrow ${f\colon x \to y}$ with the constant transformation $\vec {\operatorname {i\!d}}_{f}\colon \vec {\operatorname {i\!d}}_{x} \to \vec {\operatorname {i\!d}}_{y}$ . This way we identify ${\mathfrak {S}}$ with a subcategory of $\sigma _0{\mathfrak {S}}$ . Moreover, this subcategory is full since every $\sigma _0{\mathfrak {S}}$ -arrow $\vec {\varphi }\colon x \to \vec {u}$ from a constant identity sequence is uniquely determined (as a transformation up to the equivalence) by the ${\mathfrak {S}}$ -arrow $\varphi _0\colon x \to u_{\varphi (0)}$ . Also note that every constant sequence $\vec {\operatorname {i\!d}}_{x}$ admits the canonical limit cone $(\vec {\operatorname {i\!d}}_{x}^\infty , x)$ in ${\mathfrak {L}}$ and that $(\vec {\operatorname {i\!d}}_{x}, \vec {\operatorname {i\!d}}_{x}^\infty , x)$ is a matching sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ .

For two coned sequences $(\vec {u}, \vec {u}^{\,\infty }, U)$ and $(\vec {v}, \vec {v}^{\,\infty }, V)$ in $({\mathfrak {S}}, {\mathfrak {L}})$ we consider the matching relation $\vartriangleright $ between $\sigma _0{\mathfrak {S}}$ -arrows $\vec {u} \to \vec {v}$ and ${\mathfrak {L}}$ -arrows $U \to V$ : We put

$$ \begin{align*}\vec{\varphi} \vartriangleright \varphi_\infty \quad\text{ if }\quad v^\infty_{\varphi(n)} \circ \varphi_n = \varphi_\infty \circ u^\infty_n \text{ for every}\ n \in \omega;\end{align*} $$

see Figure 2. The matching relation is functorial: We have ${\operatorname {i\!d}_{\vec {u}}} \vartriangleright {\operatorname {i\!d}_{U}}$ , and if $\vec {\varphi } \vartriangleright \varphi _\infty $ and $\vec {\psi } \vartriangleright \psi _\infty $ , then $\vec {\psi } \circ \vec {\varphi } \vartriangleright \psi _\infty \circ \varphi _\infty $ .

Now, let $(\vec {u}, \vec {u}^{\,\infty }, U)$ be a coned sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ . For every $x \in \operatorname {Obj}({\mathfrak {S}})$ , the matching relation for $(\vec {\operatorname {i\!d}}_{x}, \vec {\operatorname {i\!d}}_{x}^\infty , x)$ and $(\vec {u}, \vec {u}^{\,\infty }, U)$ is a function $\sigma _0{\mathfrak {S}}(x, \vec {u}) \to {\mathfrak {L}}(x, U)$ : Every $\sigma _0{\mathfrak {S}}$ -arrow $\varphi \colon x \to \vec {u}$ is determined by an ${\mathfrak {S}}$ -arrow $f\colon x \to u_n$ for some n, and the unique ${\mathfrak {L}}$ -map $\varphi _\infty $ such that $\vec {\varphi } \vartriangleright \varphi _\infty $ is $u^\infty _n \circ f$ . The condition (F1) says that this matching function is surjective, and the condition (F2) says that the matching function is one-to-one. Together, (F1) and (F2) hold if and only if for every $x \in \operatorname {Obj}({\mathfrak {S}})$ the matching relation is a bijection between $\sigma _0{\mathfrak {S}}(x, \vec {u})$ and ${\mathfrak {L}}(x, U)$ .

Moreover, under (F1), for every ${\mathfrak {L}}$ -arrow $\varphi _\infty \colon U \to U$ there is a transformation $\vec {\varphi }\colon \vec {u} \to \vec {u}$ such that $\vec {\varphi } \vartriangleright \varphi _\infty $ : For every $n \in \omega $ , there is an ${\mathfrak {S}}$ -arrow $\varphi _n\colon u_n \to u_{\varphi (n)}$ with $u^\infty _{\varphi (n)} \circ \varphi _n = \varphi _\infty \circ u^\infty _n$ and we can make sure that the sequence $\{\varphi (n)\}_{n \in \omega }$ is increasing and cofinal. Under (F2), for every ${\mathfrak {L}}$ -arrow $\varphi _\infty \colon U \to U$ there is at most one transformation $\vec {\varphi }\colon \vec {u} \to \vec {u}$ up to the equivalence such that $\vec {\varphi } \vartriangleright \varphi _\infty $ : For a different such transformation $\vec {\psi }$ and $n \in \omega $ , we have $u^\infty _{\varphi (n)} \circ \varphi _n = \varphi _{\infty } \circ u^\infty _n = u^\infty _{\psi (n)} \circ \psi _n$ , and so by (F2) there is m such that $u^m_{\varphi (n)} \circ \varphi _n = u^m_{\psi (n)} \circ \psi _n$ . Together, the matching relation for $(\vec {u}, \vec {u}^{\,\infty }, U)$ is a function ${\mathfrak {L}}(U, U) \to \sigma _0{\mathfrak {S}}(\vec {u}, \vec {u})$ . By the functoriality, the function is a monoid homomorphism, and it restricts to a group homomorphism $F\colon \operatorname {Aut}_{{\mathfrak {L}}}(U) \to \operatorname {Aut}_{\sigma _0{\mathfrak {S}}}(\vec {u})$ . Observe that $\sigma _0{\mathfrak {S}}$ -automorphisms $\vec {u} \to \vec {u}$ correspond to the zig-zag sequences used in condition (BF), and so (BF) states that the mapping F is surjective (or even without (F1) and (F2) that for every automorphism $\vec {\varphi }\colon \vec {u} \to \vec {u}$ there is a matching automorphism $\varphi _\infty \colon U \to U$ ). Similarly, (H) states that the mapping F is one-to-one (here we cannot omit (F2)). Together, under (F1) and (F2), the conditions (BF) and (H) hold if and only if the matching relation is a group isomorphism between $\operatorname {Aut}_{\sigma _0{\mathfrak {S}}}(\vec {u})$ and $\operatorname {Aut}_{{\mathfrak {L}}}(U)$ .

In the following lemma, we observe that the notion of a matching sequence is robust under isomorphism and that the large object U determines the sequence of small objects $\vec {u}$ uniquely.

Lemma 2.3. Let $(\vec {u}, \vec {u}^{\,\infty }, U)$ be a matching sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ .

1. (i) If $f\colon U \to V$ is an ${\mathfrak {L}}$ -isomorphism, then $(\vec {u}, f \circ \vec {u}^{\,\infty }, V)$ is a matching sequence.

2. (ii) If $\vec {\varphi }\colon \vec {v} \to \vec {u}$ is an $\sigma _0{\mathfrak {S}}$ -isomorphism, then $(\vec {v}, \vec {u}^{\,\infty } \circ \vec \varphi , U)$ is a matching sequence.

3. (iii) For every other matching sequence $(\vec {v}, \vec {v}^{\,\infty }, V)$ and every ${\mathfrak {L}}$ -isomorphism $\varphi _\infty \colon U \to V$ , there is a matching $\sigma _0{\mathfrak {S}}$ -isomorphism $\vec {\varphi }\colon \vec {u} \to \vec {v}$ .

Proof. The proof is straightforward, though technical. For example, to obtain (BF) in (ii), for an $\sigma _0{\mathfrak {S}}$ -automorphism $\psi \colon \vec {v} \to \vec {v}$ , we consider the automorphism $(\vec {\varphi } \circ \vec {\psi } \circ \vec {\varphi }^{\,-1})\colon \vec {u} \to \vec {u}$ and its matching automorphism $h\colon U \to U$ . For every $n \in \omega $ , we have

$$ \begin{align*}u^\infty_{\varphi(\psi(\varphi^{-1}(n)))} \circ (\vec\varphi \circ \vec\psi \circ \vec{\varphi}^{\,-1})_n = h \circ u^\infty_n.\end{align*} $$

Also, $\vec {\varphi }^{\,-1} \circ \vec {\varphi }$ is equivalent to ${\operatorname {i\!d}_{\vec {u}}}$ , and so we have

$$ \begin{align*} (\vec{u}^{\,\infty} \circ \vec\varphi)_{\psi(n)} \circ \psi_n &= (\vec{u}^{\,\infty} \circ \vec\varphi \circ \vec\psi \circ \vec{\varphi}^{\,-1} \circ \vec\varphi)_n \\ &= u^\infty_{\varphi(\psi(\varphi^{-1}(\varphi(n))))} \circ (\vec\varphi \circ \vec\psi \circ \vec{\varphi}^{\,-1})_{\varphi(n)} \circ \varphi_n = h \circ u^\infty_{\varphi(n)} \circ \varphi_n = h \circ (\vec{u}^{\,\infty} \circ \vec\varphi)_n, \end{align*} $$

which we wanted.

Claim (iii) in the case of automorphisms was already discussed in Remark 2.2, and the proof for isomorphisms is analogous.

It is not always the case that a sequence $\vec {u}$ can be completed to a matching sequence at most one way (up to an isomorphism). However, it may have at most one colimit $(\vec {u}^{\,\infty }, U)$ in ${\mathfrak {L}}$ . Moreover, if $(\vec {u}^{\,\infty }, U)$ is a colimit of $\vec {u}$ in ${\mathfrak {L}}$ , then $(\vec {u}, \vec {u}^{\,\infty }, U)$ satisfies (BF) and (H). In fact, for every transformation $\vec {\varphi }\colon \vec {u} \to \vec {v}$ and every coned sequence $(\vec {v}, \vec {v}^{\,\infty }, V)$ there is a unique matching ${\mathfrak {L}}$ -arrow $\varphi _\infty \colon U \to V$ since such matching arrow is the same thing as the colimit factorizing arrow for the cone $(\vec {v}^{\,\infty } \circ \vec {\varphi }, V)$ for $\vec {u}$ .

Let us continue discussing how to obtain the conditions defining a matching sequence. Observe that if ${\mathfrak {L}}$ consists of monomorphisms, we obtain (F2) for free: If $u^\infty _n \circ f = u^\infty _n \circ f'$ , then $f = f'$ since $u^\infty _n$ is a monomorphism. Let us also recall the notion of finitely presentable object [Reference Adámek and Rosický1, Definition 1.1] in a category ${\mathfrak {L}}$ : It is an object x such that for every directed colimit $\vec {u} = (u^j_i, u^\infty _i, U)_{i \leqslant j \in I}$ every ${\mathfrak {L}}$ -arrow $f\colon x \to U$ essentially uniquely factorizes through $\vec {u}$ , that is, there is an arrow $\tilde {f}\colon x \to u_i$ for some i such that $u^\infty _i \circ \tilde {f} = f$ , and for every other such arrow $g\colon x \to u_j$ , there is $k \geqslant i, j$ such that $u^k_i \circ \tilde {f} = u^k_j \circ g$ . In other words, a finitely presented object satisfies analogues of (F1) and (F2) for every directed system with a colimit. Hence, if ${\mathfrak {S}} \subseteq {\mathfrak {L}}$ is a full subcategory whose objects are finitely presented in ${\mathfrak {L}}$ , and $(\vec {u}, \vec {u}^{\,\infty }, U)$ is a sequence in ${\mathfrak {S}}$ with a colimit in ${\mathfrak {L}}$ , then it satisfies (F1) and (F2).

We summarize this discussion in the following lemma.

A classical example of the described situation are categories of finitely generated L-structures in a first-order language L, with embeddings or one-to-one homomorphisms as arrows between objects. Let ${\mathfrak {L}}$ be the category of all L-structures and homomorphisms. Recall that a sequence $\vec {u}$ in ${\mathfrak {L}}$ consisting of embeddings can be without loss of generality viewed as an increasing $\subseteq $ -chain of substructures. Then its colimit in ${\mathfrak {L}}$ is the union of the chain. In particular, the colimit maps $u^\infty _n$ can be taken to be inclusion maps. So if we consider the wide subcategory ${\mathfrak {L}}' \subseteq {\mathfrak {L}}$ of all embeddings between L-structures, then ${\mathfrak {L}}$ -colimits of ${\mathfrak {L}}'$ -sequences (or of any directed systems in ${\mathfrak {L}}'$ ) are also ${\mathfrak {L}}'$ -colimits. Moreover, finitely generated L-structures are finitely presentable in ${\mathfrak {L}}'$ : If an L-structure A is a directed union of its substructures $(A_i \subseteq A)_{i \in I}$ and $B \subseteq A$ is a substructure generated by a finite set $F \subseteq A$ , then every generator $x \in F$ is contained in some $A_{i_x}$ , and so all of them are contained in some $A_i$ . Hence, $B \subseteq A_i$ . Together, we obtain the following.

Construction 2.6 (The topology of $\operatorname {Aut}(U)$ .

Let $(\vec {u}, \vec {u}^{\,\infty }, U)$ be a matching sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ . The automorphism group $G := \operatorname {Aut}(U)$ has a natural topology defined by the following neighborhood base of the identity:

$$\begin{align*}V_n = \Bigl\{g \in G\colon g \circ u_n^\infty = u_n^\infty\Bigr\}. \end{align*}$$

The topology is Hausdorff, thanks to condition (H). Note that each $V_n$ is a subgroup of G, therefore we obtain a non-Archimedean topological group. Note also that G is completely metrizable and that it may not be separable unless all the hom-sets ${\mathfrak {S}}(u_n,u_m)$ are countable (see the next lemma).

To show that the open subgroups $V_n$ indeed form a base of the identity of the group topology, we need to show that for every $n \in \omega $ and $g \in G$ there is $m \in \omega $ such that $g^{-1} \circ V_m \circ g \subseteq V_n$ , that is, for every $h \in G$ , if $h \circ u_m^\infty = u_m^\infty $ , then $h \circ g \circ u_n^\infty = g \circ u_n^\infty $ . By (F1), $g \circ u_n^\infty = u^\infty _m \circ f$ for some $m \in \omega $ and an ${\mathfrak {S}}$ -arrow $f\colon u_n \to u_m$ . Hence,

$$\begin{align*}h \circ (g \circ u_n^\infty) = (h \circ u^\infty_m) \circ f = u^\infty_m \circ f = g \circ u^\infty_n. \end{align*}$$

Analogously, we can show that the topology on G does not depend on the choice of a matching sequence for U: If $(\vec {v}, \vec {v}^{\,\infty }, U)$ is another matching sequence, then the subgroups $V^{\prime }_n = \{g \in G: g \circ v^\infty _n = v^\infty _n\}$ induce the same topology. For every $n \in \omega $ , we have $v^\infty _n = u^\infty _m \circ f$ for some $m \in \omega $ and an ${\mathfrak {S}}$ -arrow $f\colon v_n \to u_m$ . Hence, $V_m \subseteq V^{\prime }_n$ .

Note that in the case when ${\mathfrak {S}}$ is a category of finite first-order structures and all embeddings, and $(\vec {u}^{\,\infty }, U)$ is a colimit, then U is essentially the union of the chain of finite structures $u_n$ , and the condition $g \circ u^\infty _n = u^\infty _n$ says that the automorphism g fixes the elements of $u_n$ . Therefore, the induced topology is the topology of pointwise convergence inherited from $U^U$ where U has the discrete topology.

Lemma 2.7. Let $(\vec {u}, \vec {u}^{\,\infty }, U)$ be a matching sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ .

1. (i) The topology on $\operatorname {Aut}(U)$ is completely metrizable.

2. (ii) If all the hom-sets ${\mathfrak {S}}(u_n, u_m)$ are countable, then $\operatorname {Aut}(U)$ is also separable and so a Polish group.

Proof. The group $G := \operatorname {Aut}(U)$ is metrizable by Birkhoff-–Kakutani theorem [Reference Arhangel’skii and Tkachenko2, Theorem 3.3.12] since we have a countable neighborhood base of the identity. We show that the two-sided uniformity of G is induced by a complete metric. Let $\{h_n\}_{n \in \omega }$ be a Cauchy sequence with respect to the two-sided uniformity, that is, for every $m \in \omega $ there is $\varphi (m) \in \omega $ such that $h_{n'} \in (h_n \circ V_m) \cap (V_m \circ h_n)$ for every $n, n' \geqslant \varphi (m)$ , and so $h_n \circ u^\infty _m = h_{n'} \circ u^\infty _m$ and $h_n^{-1} \circ u^\infty _m = h_{n'}^{-1} \circ u^\infty _m$ .

Put $k_0 := 0$ . By (F1), there is an ${\mathfrak {S}}$ -arrow $f_0\colon u_{k_0} \to u_{\ell _0}$ for some $\ell _0> k_0$ such that $u^\infty _{\ell _0} \circ f_0 = h_{\varphi (k_0)} \circ u^\infty _{k_0}$ . Then there is an ${\mathfrak {S}}$ -arrow $g^{\prime }_0\colon u_{\ell _0} \to u_{k^{\prime }_1}$ for some $k^{\prime }_1> \ell _0$ such that $u^\infty _{k^{\prime }_1} \circ g^{\prime }_0 = h_{\varphi (\ell _0)}^{-1} \circ u^\infty _{\ell _0}$ . Together, we have

$$\begin{align*}u^\infty_{k^{\prime}_1} \circ g^{\prime}_0 \circ f_0 = h_{\varphi(\ell_0)}^{-1} \circ u^\infty_{\ell_0} \circ f_0 = h_{\varphi(\ell_0)}^{-1} \circ h_{\varphi(k_0)} \circ u^\infty_{k_0} = u^\infty_{k_0} \end{align*}$$

since $h_{\varphi (k_0)} \circ u^\infty _{k_0} = h_{\varphi (\ell _0)} \circ u^\infty _{k_0}$ since $\varphi (\ell _0) \geqslant \varphi (k_0)$ . By (F2), there is $k_1 \geqslant k^{\prime }_1$ such that $u^{k_1}_{k^{\prime }_1} \circ g^{\prime }_0 \circ f_0 = u^{k_1}_{k_0}$ . We put $g_0 := u^{k_1}_{k^{\prime }_1} \circ g^{\prime }_0$ so that $g_0 \circ f_0 = u^{k_1}_{k_0}$ . We continue this way and build ${\mathfrak {S}}$ -sequences $\{f_n\}_{n \in \omega }$ , $\{g_n\}_{n \in \omega }$ as in (BF). Hence, there is an automorphism $h_\infty \in G$ such that for every $m \in \omega $ we have

$$ \begin{align*} h_\infty \circ u^\infty_{k_m} &= u^\infty_{\ell_m} \circ f_m = h_{\varphi(k_m)} \circ u^\infty_{k_m} = h_n \circ u^\infty_{k_m} && \text{for every}\ n \geqslant \varphi(k_m), \\ h_\infty^{-1} \circ u^\infty_{\ell_m} &= u^\infty_{k_{m + 1}} \circ g_m = h_{\varphi(\ell_m)}^{-1} \circ u^\infty_{\ell_m} = h_n^{-1} \circ u^\infty_{\ell_m} && \text{for every}\ n \geqslant \varphi(\ell_m). \end{align*} $$

Since the sequences $\{k_m\}_{m \in \omega }$ , $\{\ell _m\}_{m \in \omega }$ are strictly increasing, it follows that $\lim _{n \to \infty } h_n = h_\infty $ .

If the hom-sets ${\mathfrak {S}}(u_n, u_m)$ for $n \leqslant m$ are countable, then G is separable since for every n, $\{g \circ V_n: g \in G\}$ is a countable cover by basic open sets. This is because we have the one-to-one map $g \circ V_n \mapsto g \circ u^\infty _n$ , and every arrow $g \circ u^\infty _n$ is of the form $u^\infty _m \circ f$ for some $f \in {\mathfrak {S}}(u_n, u_m)$ .

Construction 2.8 (The topological group $G(\vec {u}, {\mathfrak {S}})$ ).

We observe that given matching sequences $(\vec {u}, \vec {u}^{\,\infty }, U)$ in $({\mathfrak {S}}, {\mathfrak {L}})$ and $(\vec {v}, \vec {v}^{\,\infty }, V)$ in $({\mathfrak {S}}, {\mathfrak {L}}')$ , every isomorphism of sequences $\vec {\varphi }\colon \vec {u} \to \vec {v}$ canonically induces an isomorphism of the topological groups $\operatorname {Aut}_{{\mathfrak {L}}}(U)$ and $\operatorname {Aut}_{{\mathfrak {L}}'}(V)$ . Moreover, any sequence $\vec {u}$ in any category ${\mathfrak {S}}$ admits a canonical matching sequence $(\vec {u}, \{\vec {\imath }_n\}_{n \in \omega }, \vec {u})$ in $({\mathfrak {S}}, \sigma _0{\mathfrak {S}})$ (see Remark 2.2). Therefore, up to a canonical isomorphism, every ${\mathfrak {S}}$ -sequence $\vec {u}$ determines a topological group $G(\vec {u}, {\mathfrak {S}})$ that is the automorphism group of every associated matching sequence $(\vec {u}, \vec {u}^{\,\infty }, U)$ in every $({\mathfrak {S}}, {\mathfrak {L}})$ . Moreover, isomorphic sequences determine isomorphic topological groups.

Proof. Recall that $\vec {u}$ is both a sequence in ${\mathfrak {S}} \subseteq \sigma _0{\mathfrak {S}}$ and an object in $\sigma _0{\mathfrak {S}}$ . For every n, we let $\vec {\imath }_n\colon u_n \to \vec {u}$ be the $\sigma _0{\mathfrak {S}}$ -arrow corresponding to ${\operatorname {i\!d}_{u_n}}\colon u_n \to u_n$ , that is, $(\vec {\imath }_n)_k = u^{\max (k, n)}_n$ for $k \in \omega $ . It is easy to see that $\vec {u}$ (viewed as a $\sigma _0 {\mathfrak {S}}$ -object) is the colimit of itself (viewed as an ${\mathfrak {S}}$ -sequence). Also, for every sequences $\vec {u}, \vec {v}$ the corresponding matching relation is the identity on $\sigma _0{\mathfrak {S}}(\vec {u}, \vec {v})$ . Hence, by Remark 2.2, $(\vec {u}, \{\vec {\imath }_n\}_{n \in \omega }, \vec {u})$ is a colimiting matching sequence in $({\mathfrak {S}}, \sigma _0{\mathfrak {S}})$ .

For every matching sequence $(\vec {u}, \vec {u}^{\,\infty }, U)$ in $({\mathfrak {S}}, {\mathfrak {L}})$ the matching relation between $\operatorname {Aut}(\vec {u})$ and $\operatorname {Aut}(U)$ is an isomorphism of groups by Remark 2.2. Recall that we write $\vec {\varphi } \approx \vec {\psi }$ for every two transformations between sequences that are equivalent, that is, representing the same $\sigma _0{\mathfrak {S}}$ -arrow. Since for every matching pair of automorphisms $\vec {\varphi }\colon \vec {u} \to \vec {u}$ and $\varphi _\infty \colon U \to U$ we have

$$\begin{align*}\varphi_\infty \circ u^\infty_n = u^\infty_n \,\Longleftrightarrow\, u^\infty_{\varphi(n)} \circ \varphi_n = u^\infty_n \,\Longleftrightarrow\, \exists m\; u^m_{\varphi(n)} \circ \varphi_n = u^m_n \,\Longleftrightarrow\, \vec{\varphi} \circ \vec{\imath}_n \approx \vec{\imath}_n \end{align*}$$

for every $n \in \omega $ , it follows that matching relation is also a homeomorphism between the automorphism groups. To see $\vec {\varphi } \circ \vec {\imath }_n \approx \vec {\imath }_n$ , let $k \in \omega $ and $k' := \max (k, n)$ , and note that for every $m \geqslant \varphi (k')$ we have $u^m_{\varphi (k')} \circ (\vec {\varphi } \circ \vec {\imath }_n)_k = u^m_{\varphi (k')} \circ (\varphi _{k'} \circ u^{k'}_n) = u^m_{\varphi (n)} \circ \varphi _n$ , while $u^m_{k'} \circ (\vec {\imath }_n)_{k'} = u^m_{k'} \circ u^{k'}_n = u^m_n$ .

Finally, the isomorphism $\vec {\varphi }\colon \vec {u} \to \vec {v}$ of ${\mathfrak {S}}$ -sequences induces an isomorphism $\vec {\psi } \mapsto \vec {\varphi } \circ \vec {\psi } \circ \vec {\varphi }^{\,-1}$ between $\operatorname {Aut}(\vec {u})$ and $\operatorname {Aut}(\vec {v})$ . This isomorphism is also a homeomorphism. Let $\{\vec {\imath }_n\}_{n \in \omega }$ and $\{\vec {\jmath }_n\}_{n \in \omega }$ be the cones of the canonical matching sequences for $\vec {u}$ and $\vec {v}$ . For every $n \in \omega $ , we put $m := (\vec {\varphi }^{\,-1})(n)$ , and we have $\vec {\jmath }_{\varphi (m)} \circ \varphi _m \circ (\vec {\varphi }^{\,-1})_n \approx \vec {\jmath }_n$ . Hence, if $\vec {\psi }$ fixes $\vec {\imath }_m$ , then $\vec {\varphi } \circ \vec {\psi } \circ \vec {\varphi }^{\,-1}$ fixes $\vec {\varphi } \circ \vec {\imath }_m \approx \vec {\jmath }_{\varphi (m)} \circ \varphi _m$ , and so it fixes also $\vec {\jmath }_{\varphi (m)} \circ \varphi _m \circ (\vec {\varphi }^{\,-1})_n \approx \vec {\jmath }_n$ .

2.2 Weak Fraïssé theory

In this section, we summarize the definitions and theorems of the weak Fraïssé theory [Reference Kubiś12] that will be used later in the paper. We recall the notion of a weak Fraïssé category which is a generalization of a Fraïssé category [Reference Kubiś13] (which is itself an abstraction of the classical notion of a Fraïssé class [Reference Hodges6, §7] of first-order structures). A weak Fraïssé category ${\mathfrak {S}}$ can be characterized by existence of a weak Fraïssé sequence $\vec {u}$ . First, we need to recall the concepts of (weak) amalgamation and (weak) domination.

The weak amalgamation property was introduced by Ivanov [Reference Ivanov7] and later independently by Kechris and Rosendal [Reference Kechris and Rosendal14] in connection with existence of generic automorphisms of homogeneous first-order structures. Since then, ideas of weak Fraïssé theory have been developed and used in many works; see, for example, [Reference Kruckman11], [Reference Di Liberti4], [Reference Krawczyk and Kubiś9], [Reference Kubiś12], [Reference Krawczyk, Kruckman, Kubiś and Panagiotopoulos10], [Reference Panagiotopoulos and Tent23], [Reference Malicki17].

Definition 2.10 (Amalgamable arrows).

An arrow $e\colon z \to {z'}$ is called amalgamable if for every arrows $f\colon z' \to x$ , $g\colon z' \to y$ there exist arrows $f'\colon x \to w$ , $g'\colon y \to w$ satisfying

$$ \begin{align*}f' \circ f \circ e = g' \circ g \circ e.\end{align*} $$

An object z is amalgamable if ${\operatorname {i\!d}_{z}}$ is an amalgamable arrow. A category ${\mathfrak {S}}$ has the amalgamation property if all identity arrows are amalgamable. A category with a weak Fraïssé sequence has the weak amalgamation property; namely, for every object a there exists an amalgamable arrow with domain a.

Remark 2.12. Note that a number $m \geqslant n$ works in (W1) if and only if the arrow $u^m_n$ is amalgamable. To prove that $u^m_n$ is amalgamable one can consider $f_0\colon u_m \to y_0$ , $f_1\colon u_m \to y_1$ , obtaining $g_0\colon y_0 \to u_\ell $ , $g_1\colon y_1 \to u_\ell $ satisfying

$$ \begin{align*}g_0 \circ f_0 \circ u_n^m = u_n^\ell = g_1 \circ f_1 \circ u_n^m.\end{align*} $$

The other implication is essentially [Reference Kubiś12, Lemma 3.10].

The next definition is analogous to the definition of weak Fraïssé sequence but applies to subcategories. In fact, both concepts could be unified by a notion of a weakly dominating functor.

The following lemma, which we shall use later, was essentially proved in the proof of [Reference Kubiś12, Proposition 2.7].

In the classical case of categories of structures and embeddings, being directed is often called the joint embedding property. Note that a category is identified with its collection of morphisms, so a category is countable if it has countably many morphisms (as opposed to just having countably many objects). However, for a category of finite structures, having countably many isomorphism types is sufficient for being dominated by a countable subcategory.

This is [Reference Kubiś12, Theorem 3.7] together with the fact (implicitly used in the proof) that every countable weakly dominating subcategory of a weak Fraïssé category can be extended by countably many arrows to become directed and to have the weak amalgamation property.

Next, we recall the key properties of a weak Fraïssé limit, generalizing the extension property / injectivity and homogeneity from Fraïssé theory. Again, see [Reference Kubiś12] for details.

The theorem is essentially proved in [Reference Kubiś12]: see Theorem 4.2, Corollary 4.5, Theorem 4.6 and Theorem 5.1. However, we use weaker assumptions here — we neither assume that $(\vec {u}^{\,\infty }, U)$ is a colimit of $\vec {u}$ nor that ${\mathfrak {S}}$ -sequences have colimits in ${\mathfrak {L}}$ nor that ${\mathfrak {L}}$ -arrows are monic. The only assumptions that are used in the original proof are covered by the notion of a matching sequence. Given all the necessary definitions and conditions, the proof is quite direct.

3.1 The weak Ramsey property

For an arrow $\alpha \colon a \to a'$ and an object b in a category ${\mathfrak {C}}$ , we shall write ${\mathfrak {C}}(\alpha , b)$ as a shortcut for ${\mathfrak {C}}(a', b) \circ \alpha $ . Below, we introduce the main definition.

The following lemma generalizes [Reference Nešetřil21, Theorem 4.2 (i)].

Lemma 3.2. Let ${\mathfrak {S}}$ be a directed category, and let $\alpha \colon a \to a'$ be an ${\mathfrak {S}}$ -arrow. If $\alpha $ is a Ramsey arrow, then $\alpha $ is amalgamable. Hence, a directed category with the weak Ramsey property has the weak amalgamation property.

Proof. Suppose $\alpha \colon a \to a'$ is a Ramsey arrow, and let $k = 2$ . Fix $f_0, f_1 \in {\mathfrak {S}}$ with $\operatorname {dom}(f_0) = a' = \operatorname {dom}(f_1)$ . Using directedness, choose $b \in \operatorname {Obj}({\mathfrak {S}})$ and $g_0, g_1 \in {\mathfrak {S}}$ such that $g_i \circ f_i \in {\mathfrak {S}}(a',b)$ for $i=0,1$ . Of course, $g_0,g_1$ are independent of $f_0,f_1$ and there is no reason for the equality $g_0 \circ f_0 = g_1 \circ f_1$ . Let $F = \{g_0 \circ f_0 \circ \alpha ,g_1 \circ f_1 \circ \alpha \}$ .

Now, find $v \in \operatorname {Obj}({\mathfrak {S}})$ from the weak Ramsey property applied to F. Define $\varphi \colon {\mathfrak {S}}(\alpha ,v) \to 2$ by setting $\varphi (g) = 1$ if and only if $g = g' \circ f_1 \circ \alpha $ for some $g' \in {\mathfrak {S}}$ . By (wR), there exists $e\colon b \to v$ such that $\varphi $ is constant on $e \circ F$ . Note that $\varphi (e \circ (g_1 \circ f_1) \circ \alpha ) = 1$ by associativity. Thus, also $\varphi (e \circ (g_0 \circ f_0) \circ \alpha ) = 1$ , which means that there exists h such that

$$ \begin{align*}e \circ g_0 \circ f_0 \circ \alpha = h \circ f_1 \circ \alpha.\end{align*} $$

We are done, because $e\circ g_0$ and h witness the weak amalgamation.

Recall that if one of arrows $\alpha , \beta $ is amalgamable, then so is $\beta \circ \alpha $ ; see [Reference Kubiś12, Lemma 2.5]. The same composition behavior is true also for Ramsey arrows.

Theorem 3.10. Let $(\vec {u}, \vec {u}^{\,\infty }, U)$ be a matching sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ for some categories ${\mathfrak {S}} \subseteq {\mathfrak {L}}$ , and let $\alpha \colon a \to a'$ be an ${\mathfrak {S}}$ -arrow. If $\vec {u}$ is a weak Fraïssé sequence, then the following conditions are equivalent:

1. (i) $\alpha $ is a Ramsey arrow, that is, it satisfies (wR),

2. (ii) $\alpha $ satisfies (wA),

3. (iii) $\alpha $ satisfies (wB).

It follows that ${\mathfrak {S}}$ has the (weak) Ramsey property if and only if U has the (weak) finitary big Ramsey property in $({\mathfrak {S}}, {\mathfrak {L}})$ .

Proof. By (F1), for every $f \in {\mathfrak {L}}(\alpha , U)$ there exists $n_f \in \omega $ and $f' \in {\mathfrak {S}}(\alpha , u_n)$ such that $u^\infty _n \circ f' = f$ . For every $n \geqslant n_f$ , let us put $\Delta ^n(f) := u^n_{n_f} \circ f'$ , so we have $f = u^\infty _n \circ \Delta ^n(f)$ and $\Delta ^{n'}(f) = u^{n'}_n \circ \Delta ^n(f)$ for every $n' \geqslant n \geqslant n_f$ . Note that by (F2) we have also $\Delta ^n(u^\infty _m \circ f) = u^n_m \circ f$ for every compatible ${\mathfrak {S}}$ -arrow f and every sufficiently large n.

The implication (wA) $\implies $ (wB) is trivial.

Next, we prove (wB) $\implies $ (wR) by contradiction. Suppose $\alpha $ fails (wR) and it is witnessed by $k \in \omega $ , $b \in \operatorname {Obj}({\mathfrak {S}})$ and a finite $F \subseteq {\mathfrak {S}}(\alpha , b)$ . Specifically, for every $n \in \omega $ there is $\varphi _n\colon {\mathfrak {S}}(\alpha , u_n) \to k$ such that $\varphi _n[e \circ F]$ is not a singleton whenever $e \in {\mathfrak {S}}(b, u_n)$ . Fix a nonprincipal ultrafilter p on $\omega $ , and define $\varphi \colon {\mathfrak {L}}(\alpha , U) \to k$ by setting

$$ \begin{align*}\varphi(f) = i \,\Longleftrightarrow\, \{{n\in\omega}\colon \varphi_n(\Delta^n(f)) = i\} \in p.\end{align*} $$

Note that $\Delta ^n(f)$ is defined for all but finitely many numbers n. Since $\vec {u}$ is a weak Fraïssé sequence, there is an ${\mathfrak {S}}$ -map $e_0\colon b \to u_{n_0}$ for some $n_0 \in \omega $ . We consider $F' := u^\infty _{n_0} \circ e_0 \circ F \subseteq {\mathfrak {L}}(\alpha , U)$ . Since $\alpha $ satisfies (wB), there is $j \in k$ and $e' \in {\mathfrak {L}}(U, U)$ such that $\varphi [e' \circ F'] = \{j\}$ . By (F1), we find $m \in \omega $ and $e \in {\mathfrak {S}}(b, u_m)$ such that $e' \circ u^\infty _{n_0} \circ e_0 = u_m^\infty \circ e$ , as in the diagram below.

Given $f \in F$ , we have $\Delta ^n(u_m^\infty \circ e \circ f) = u_m^n \circ e \circ f$ for every sufficiently large n, and hence

$$ \begin{align*} j = \varphi(e' \circ u_{n_0}^\infty \circ e_0 \circ f) &= \varphi(u_m^\infty \circ e \circ f) = \lim_{n \to p} \varphi_n(\Delta^n(u_m^\infty \circ e \circ f)) = \lim_{n \to p} \varphi_n(u_m^n \circ e \circ f). \end{align*} $$

The last limit along p means that the set $A_f := \{n \geqslant m\colon \varphi _n(u_m^n \circ e \circ f) = j\} \in p$ . Since F is finite, we may find $\ell> m$ such that $\varphi _\ell (u_m^\ell \circ e \circ f) = j$ for every $f \in F$ . It is enough to choose $\ell \in \bigcap _{f \in F} A_f$ . Together, $\varphi _\ell $ restricted to $(u_m^\ell \circ e) \circ F$ is constant, which is a contradiction.

To prove (wR) $\implies $ (wA), let $\alpha \colon a \to a'$ be a Ramsey ${\mathfrak {S}}$ -arrow. Fix $k \in \omega $ , fix finite $F \subseteq {\mathfrak {L}}(\alpha , U)$ and fix $\varphi \colon {\mathfrak {L}}(\alpha , U) \to k$ . Our goal is to find $e \in \operatorname {Aut}(U)$ such that $\varphi $ is constant on $e \circ F$ .

Since F is finite, there is $m \in \omega $ such that $f = u^\infty _m \circ \Delta ^m(f)$ for every $f \in F$ . Since $\vec {u}$ is a weak Fraïssé sequence, there is $m' \geqslant m$ such that $\beta := u_m^{m'}$ works in (W1), or equivalently is amalgamable. Let $b := u_{m'}$ and $F' := \{\beta \circ \Delta ^m(f)\colon f \in F\} \subseteq {\mathfrak {S}}(\alpha , b)$ . Using (wR) with $F'$ and b, we obtain $v \in \operatorname {Obj}({\mathfrak {S}})$ such that for every $\psi \colon {\mathfrak {S}}(\alpha , v) \to k$ there is $e'\colon b \to v$ with $\psi $ constant on the set $e' \circ F'$ . Note that there exists at least one $\psi $ as above (unless $k = 0$ , in which case (wA) is trivially true). Consequently, there exists an ${\mathfrak {S}}$ -arrow $\gamma \colon b \to v$ . Recalling that $\beta = u^{m'}_m$ is amalgamable, we find an ${\mathfrak {L}}$ -arrow $\delta \colon v \to U$ such that

$$ \begin{align*}u_m^\infty = \delta \circ \gamma \circ \beta.\end{align*} $$

The following diagram should clarify the situation.

Define $\tilde {\varphi }\colon {\mathfrak {S}}(\alpha ,v) \to k$ by

$$ \begin{align*}\tilde{\varphi}(\xi) = \varphi(\delta \circ \xi) \qquad \text{ for every } \xi \in {\mathfrak{S}}(\alpha,v).\end{align*} $$

The weak Ramsey property gives $e'\colon b \to v$ such that

$$ \begin{align*}\tilde{\varphi}(e' \circ \beta \circ \Delta^m(f)) = j\end{align*} $$

for every $f \in F$ , where $j \in k$ is fixed. Now, we use the weak homogeneity of U (Theorem 2.21), knowing that $\beta $ is amalgamable. Namely, there exists $e \in \operatorname {Aut}(U)$ such that

$$ \begin{align*}e \circ \delta \circ \gamma \circ \beta = \delta \circ e' \circ \beta.\end{align*} $$

Finally, given $f \in F$ , we have

$$ \begin{align*} \varphi(e \circ f) &= \varphi(e \circ u_m^\infty \circ \Delta^m(f)) = \varphi(e \circ \delta \circ \gamma \circ \beta \circ \Delta^m(f)) \\ &= \varphi(\delta \circ e' \circ \beta \circ \Delta^m(f)) = \tilde{\varphi}(e' \circ \beta \circ \Delta^m(f)) = j. \end{align*} $$

This completes the proof.

3.2 Extreme amenability

Recall that an action $G \curvearrowright X$ of a group G on a set X is a group homomorphism $\eta \colon G \to \operatorname {Aut}(X)$ , where $\operatorname {Aut}(X)$ is the the group of all bijections of X. We shall write $g x$ or $g \cdot x$ instead of $\eta (g)(x)$ . This way an action can be equivalently viewed as a map $G \times X \to X$ such that $1 x = x$ and $g (h x) = (g h) x$ for $g, h \in G$ and $x \in X$ . Given $G \curvearrowright X$ , the orbit of $x_0 \in X$ is $G x_0 := \{g x_0\colon g \in G\}$ . The action is transitive if $G x_0 = X$ for some $x_0$ (equivalently: for every $x_0$ ).

A morphism $\pi \colon \eta \to \eta '$ of actions $\eta \colon G \curvearrowright X$ and $\eta '\colon G \curvearrowright Y$ is a mapping $\pi \colon X \to Y$ such that $\pi (gx) = g\pi (x)$ for every $g \in G$ and $x \in X$ , or equivalently $\pi \circ \eta (g) = \eta '(g) \circ \pi $ for every $g \in G$ .

An action $G \curvearrowright X$ of a topological group G on a topological space X is continuous if it is continuous when viewed as a mapping $G\times X \to X$ with respect to the product topology on $G \times X$ . Recall that a topological group G is called extremely amenable if every continuous action $G \curvearrowright X$ on a compact space X has a fixed point, that is, there is a point $x_0 \in X$ such that $g x_0 = x_0$ for every $g \in G$ .

The proof can be found essentially in [Reference Kechris, Pestov and Todorcevic8, Prop. 4.2], where it is assumed that G is a closed subgroup of $S_\infty $ ; however, the proof uses exclusively the existence of a neighborhood base ${\mathcal {V}}$ as above (see also the remarks after [Reference Kechris, Pestov and Todorcevic8, Prop. 4.2]). Recall that a topological group G embeds into $S_\infty $ as a closed subgroup if and only if it is a non-Archimedean Polish group, that is, if it has a countable neighborhood base ${\mathcal {V}}$ of the unit consisting of open subgroups and is separable.

Remark 3.13. Let $(\vec {u}, \vec {u}^{\,\infty }, U)$ be a matching sequence in $({\mathfrak {S}}, {\mathfrak {L}})$ . Recall that a basic neighborhood of the identity ${\operatorname {i\!d}_{U}} \in G := \operatorname {Aut}(U)$ is of the form

$$\begin{align*}V_m = \{g \in G\colon g \circ u_m^\infty = u_m^\infty\}, \end{align*}$$

where $m \in \omega $ . This is obviously a subgroup of G. In fact, it is the stabilizer of $u^\infty _m$ of the action $G \curvearrowright {\mathfrak {L}}(u_m, U)$ . Hence, the map $\pi \colon G /{V}_m \to G \circ u^\infty _m$ defined by $h \circ V_m \mapsto h \circ u^\infty _m$ is an isomorphism of the action $G \curvearrowright G /{V}_m$ on left cosets and the action $G \curvearrowright G \circ u^\infty _m$ of the automorphism group on the orbit $G \circ u^\infty _m \subseteq {\mathfrak {L}}(u_m, U)$ . Moreover, if $m' \geqslant m$ is such that $u^{m'}_m$ is amalgamable, then by the weak homogeneity the orbit $G \circ u^\infty _m$ is the whole ${\mathfrak {L}}(u^{m'}_m, U)$ .

Recalling that the topological group $\operatorname {Aut}(U)$ does not depend on the choice of ${\mathfrak {L}}$ and of weak Fraïssé matching sequence $(\vec {u}, \vec {u}^{\,\infty }, U)$ in $({\mathfrak {S}}, {\mathfrak {L}})$ (see Construction 2.19), we obtain the following.

In the classical case when ${\mathfrak {S}}$ is a family of finite first-order structures with all embeddings, our category is locally finite, which allows us to simplify the weak Ramsey property. In this case, when expanded, we obtain the following corollary. Also, recall that the topology on $\operatorname {Aut}(U)$ does not depend on the choice of a weak Fraïssé sequence.

3.3 Amalgamation extension and arrow extension

We briefly discuss the phenomenon of weak versions of certain notions like the amalgamation property and the Ramsey property from theoretical perspective. In both situations, the core property is localized at individual objects of a category and then generalized from objects to arrows. Here, we note that amalgamable/Ramsey arrows may be viewed as arrows factorizing through amalgamable/Ramsey objects in a certain extension category.

First, observe that for a full cofinal subcategory ${\mathfrak {C}} \subseteq {\mathfrak {C}}'$ we have that a ${\mathfrak {C}}$ -arrow is amalgamable in ${\mathfrak {C}}$ if and only if it is amalgamable in ${\mathfrak {C}}'$ . In this section, we shall work with full subcategories, and they will be sometimes identified with their classes of objects, for example, ${\mathfrak {C}}' \setminus {\mathfrak {C}}$ denotes the full subcategory of ${\mathfrak {C}}'$ consisting of objects in $\operatorname {Obj}({\mathfrak {C}}') \setminus \operatorname {Obj}({\mathfrak {C}}')$ .

Let ${\mathfrak {C}}$ be a category. By $\operatorname {Am}({\mathfrak {C}})$ , we denote the full subcategory of ${\mathfrak {C}}$ consisting of all amalgamable objects. Suppose that both $\operatorname {Am}({\mathfrak {C}})$ and ${\mathfrak {C}} \setminus \operatorname {Am}({\mathfrak {C}})$ are cofinal. Then $\operatorname {Am}({\mathfrak {C}})$ has the amalgamation property, ${\mathfrak {C}}$ has the cofinal amalgamation property [Reference Kubiś12] but not the amalgamation property and ${\mathfrak {C}} \setminus \operatorname {Am}({\mathfrak {C}})$ has the weak amalgamation property but not the cofinal amalgamation property. In fact, ${\mathfrak {C}} \setminus \operatorname {Am}({\mathfrak {C}})$ has no amalgamable objects. So sometimes we may obtain a category with the weak amalgamation property without any amalgamable objects simply by removing the amalgamable objects. Sometimes, it even happens that every amalgamable arrow in ${\mathfrak {C}} \setminus \operatorname {Am}({\mathfrak {C}})$ factorizes through an object in $\operatorname {Am}({\mathfrak {C}})$ . Let us capture this situation by a definition.

As seen in the example, amalgamation extensions may arise naturally. On the other hand, every category admits at least the following ‘artificial’ amalgamation extension.

Proposition 3.21. A ${\mathfrak {C}}$ -arrow $\alpha \colon a \to a'$ is amalgamable if and only if it is an amalgamable object in ${\mathfrak {C}}^\uparrow $ . Hence, ${\mathfrak {C}} \cup \operatorname {Am}({\mathfrak {C}}^\uparrow )$ is an amalgamation extension of ${\mathfrak {C}}$ .

Proof. For the first part, it is enough to translate amalgamation spans (i.e., diagrams of the form $b \leftarrow a \to c$ ) and their solutions between ${\mathfrak {C}}$ and ${\mathfrak {C}}^\uparrow $ , as shown in Figure 3 for the first implication. If $\alpha $ is an amalgamable arrow in ${\mathfrak {C}}$ , $\beta \colon b \to b'$ and $\gamma \colon c \to c'$ are ${\mathfrak {C}}^\uparrow $ objects, and $f\colon \alpha \to \beta $ and $g\colon \alpha \to \gamma $ are ${\mathfrak {C}}^\uparrow $ -arrows, then there are ${\mathfrak {C}}$ -arrows $\tilde {f}\colon a' \to b$ and $\tilde {g}\colon a' \to c$ such that $f = \tilde {f} \circ \alpha $ and $g = \tilde {g} \circ \alpha $ (or one of $f, g$ is an identity arrow, in which case the amalgamation is trivial), and there are ${\mathfrak {C}}$ -arrows $f'$ and $g'$ to a ${\mathfrak {C}}$ -object d such that $f' \circ \beta \circ \tilde {f} \circ \alpha = g' \circ \gamma \circ \tilde {g} \circ \alpha $ .

Figure 3 A span in ${\mathfrak {C}}^\uparrow $ and the corresponding span in ${\mathfrak {C}}$ .

If $\alpha $ is an amalgamable object in ${\mathfrak {C}}^\uparrow $ and $f\colon a' \to b$ and $g\colon a' \to c$ are ${\mathfrak {C}}$ -arrows, then $f \circ \alpha \colon \alpha \to b$ and $g \circ \alpha \colon \alpha \to c$ , and so there is a ${\mathfrak {C}}^\uparrow $ -object $(d, \delta )$ and ${\mathfrak {C}}^\uparrow $ -arrows $f'\colon b \to \delta $ and $g'\colon c \to \delta $ such that $f' \circ f \circ \alpha = g' \circ g \circ \alpha $ . Since $f'$ and $g'$ may be also viewed as ${\mathfrak {C}}$ -arrows $b \to d$ and $c \to d$ , we are done.

It follows that ${\mathfrak {C}} \cup \operatorname {Am}({\mathfrak {C}}^\uparrow )$ is and amalgamation extension of ${\mathfrak {C}}$ . ${\mathfrak {C}}$ is full and cofinal in ${\mathfrak {C}} \cup \operatorname {Am}({\mathfrak {C}}^\uparrow )$ since it is so in ${\mathfrak {C}}^\uparrow $ . Every object in $\operatorname {Am}({\mathfrak {C}}^\uparrow )$ is amalgamable in ${\mathfrak {C}} \cup \operatorname {Am}({\mathfrak {C}}^\uparrow )$ since ${\mathfrak {C}} \cup \operatorname {Am}({\mathfrak {C}}^\uparrow )$ is full and cofinal in ${\mathfrak {C}}^\uparrow $ . Finally, every amalgamable arrow $\alpha \colon a \to a'$ in ${\mathfrak {C}}$ factorizes as $g \circ f$ , where $f\colon a \to \alpha $ corresponds to ${\operatorname {i\!d}_{a}}$ and $g\colon \alpha \to a'$ corresponds to $\alpha $ .

Together, we obtain the following.

4.1 Monoids as categories

Recall that a monoid is a triple $(M, \cdot , 1)$ , where $\cdot $ is an associative operation on a set M, and $1 \in M$ is the unit: We have $x \cdot 1 = 1 \cdot x = x$ for every $x \in M$ . A monoid M can be viewed as a category with a single object: The elements of M become the endomorphisms, the multiplication $\cdot $ becomes the composition, and the unit $1$ becomes the identity on the single unnamed object.

Then, an element $\alpha \in M$ corresponds to a Ramsey arrow if and only if for every $k \in \omega $ , for every finite $F \subseteq M\alpha $ , and for every $\varphi \colon M\alpha \to k$ there exists $e \in M$ such that $\varphi \mathop {\upharpoonright }{eF}$ is constant. The monoid M has the weak Ramsey property if and only if there exists a Ramsey arrow, and M has the Ramsey property if and only if the unit (and so every element) is a Ramsey arrow.