2.1. Categories of spaces

The spaces of interest for this paper are (extended) metric spaces¹ and Hausdorff spaces. However, in examples we restrict to Banach spaces, and some constructions extend to arbitrary topological spaces. Figure 1 summarizes the relations among the following categories of spaces.

Figure 1. Categories of spaces.

Recall that, $K:\mathbf{KH}$ is said to be a compactification of $X:\mathbf{Haus}$ , if X can be regarded as a dense sub-space (via a topological embedding) of K. There are several compactifications of $\mathbb{R}$ , for example:

• The one-point (aka Alexandroff) compactification $\mathbb{R}_\infty$ with the carrier $\mathbb{R} \cup {{\left\{{\infty}\right\}}}$ , whose open subsets are either in $\mathsf{O}(\mathbb{R})$ or subsets in $\mathsf{P}(\mathbb{R}_\infty)$ of the form $A \cup {{\left\{{\infty}\right\}}}$ , where $A\in\mathsf{O}(\mathbb{R})$ and its complement $A^c$ in $\mathbb{R}$ is compact.

• The two-point compactification $\overline{\mathbb{R}}$ , which is topologically homeomorphic to the interval [0,1] with the Euclidean topology. The two-point compactification of $\mathbb{R}$ has $\mathbb{R} \cup {{\left\{{ -\infty, +\infty}\right\}}}$ as its carrier, with the order on $\mathbb{R}$ extended so that $\forall x \in \mathbb{R}.\ -\infty < x < +\infty$ . The collection of sets of the form $[-\infty,x)$ , (x,y), $(y,+\infty]$ , with $x,y \in \mathbb{R}$ , forms a base for the two-point compactification.

• The Stone-Čech compactification $\beta\mathbb{R}$ , which is characterized by the universal property that any continuous function from $\mathbb{R}$ to a compact Hausdorff space K can be extended to a continuous function from $\beta \mathbb{R}$ to K in a unique way. For more details see, e.g., Munkres (Reference Munkres2000, Chapter 5).

All three methods of compactification exemplified above have limitations. The two-point compactification may work only for topological spaces induced by a linear order. The one-point compactification works only for locally compact spaces, which do not include infinite-dimensional Banach spaces. The Stone-Čech compactification works for all Tychonoff spaces. The problem is that, even for simple spaces such as $\mathbb{R}$ , there is no concrete description of $\beta\mathbb{R}$ , which makes it impossible to be used in an effective framework as we intend to do.

The following theorems recall some properties of categories and functors in Figure 1.

Theorem 2.3. The categories in the following diagram have finite limits and finite sums, and the functors preserve them:

The categories $\mathbf{CMS}$ , $\mathbf{Met,}$ and $\mathbf{Haus}$ have also small limits and small colimits.

For the existence of sums and infinitary products, it is essential to use extended metric spaces.

Theorem 2.4. The categories in the following diagram have small limits and finite sums, and the functors preserve them:

The categories have also small colimits.

1. (1) $B(S,\delta)\stackrel{\triangle}{=}\{y \mid \exists x \in S.d(x,y)<\delta\} \in \mathsf{O}(\mathbb{S})$ , where $S \in \mathsf{P}(\mathbb{S})$ and $\delta>0$ . The subset $B(S,\delta)$ is open, because it is the union of the open balls $B(x,\delta)\stackrel{\triangle}{=}\{y \mid d(x,y)<\delta\}$ with $x \in S$ .

2. (2) the closure $\overline{S} \in \mathsf{C}(\mathbb{S})$ of $S \in \mathsf{P}(\mathbb{S})$ , i.e., the smallest $C \in \mathsf{C}(\mathbb{S})$ such that $S\subseteq C$ .

3. (3) is the $\delta$ -fattening of $S \in \mathsf{P}(\mathbb{S})$ .

While $S_\delta$ is always closed, the subset $C(S,\delta)\stackrel{\triangle}{=}\{y \mid \exists x \in S.d(x,y)\leq\delta\}$ may fail to be closed, e.g., in the metric space $\mathbb{R}$ we have $B(S,1)=C(S,1)=(-2,2)\subset [-2,2]=S_1$ , when S is the open interval $(-1,1)$ . Moreover, when $C(S,\delta)$ is closed it may fail to coincide with $S_\delta$ , e.g., in a 1-discrete metric space $\mathbb{S}$ (i.e., the metric is either 0 or 1), we have $B(S,1)=S_1=S\subset\mathbb{S}=C(S,1)$ , for any proper non-empty subset S of $\mathbb{S}$ . We summarize some basic properties of $B(S,\delta)$ and $S_\delta$ , see also Moggi et al. (Reference Moggi, Farjudian, Taha, Cherubini, Sabadini and Tini2019).

We now prove two properties of short maps (between metric spaces) and compact metric spaces expressed in terms of $\delta$ -fattening. These properties are relevant to the proof of Theorem 3.10.

Claim (2) follows from (1) and the chain of equivalences $f(A)\subseteq C'\iff A\subseteq f^{-1}(C')\iff \overline{f(\overline{A})}\subseteq C'$ , which holds for any $f:\mathbf{Haus}(\mathbb{S},\mathbb{S}')$ , $A \in \mathsf{P}(\mathbb{S})$ and $C' \in \mathsf{C}(\mathbb{S}')$ . Namely, take $A=B(S,\delta)$ and $C'=f(S)_\delta$ .

For simplicity, we assume that $(x_n \mid n)$ is a Cauchy sequence and x is its limit. Since the Cauchy sequence $(x_n \mid n)$ is eventually in $K_\delta$ for any $\delta>0$ , we conclude that $x\in K$ , and also $x\in O$ . Since O is open, there exists an open ball $B(x,\delta)\subseteq O$ , which contradicts the assumption $\forall n.x_n\not\in O$ .

2.2 Categories of analyses

We define an analysis as a monotonic map between complete lattices. However, we need to consider further properties of analyses, that (with the exception of computability) can be defined as continuity with respect to suitable topologies on (the carrier of) complete lattices. For this reason, we introduce the category $\mathbf{Top}_A$ of topological analyses, which refines the category $\mathbf{Po}_A$ of analyses (see Figure 2).

Figure 2. Poset-enriched categories.

Moreover, when X is finite $\mathbf{Top}(\leq)$ is trivial, i.e., $\tau_\bot=\tau_\top$ .

Therefore, $\tau_\top$ and $\tau_\bot$ are respectively the top and bottom element in $\mathbf{Top}(\leq)$ . Since the topologies on a set X (ordered by reverse inclusion) form a complete lattice, with (non-empty) sups given by intersections, so do the topologies $\tau$ on X such that $\tau_\top\subseteq\tau\subseteq\tau_\bot$ . Moreover, such topologies are $T_0$ , because $\tau_\top$ is.

For every $x \in X,$ we have $\uparrow x=\bigcap\{\not\downarrow y \mid y \in X\land x\not\leq y\}$ . When X is finite, the right-hand side of the equality is a finite intersection of open sets in $\tau_\top$ , thus $\uparrow x\in\tau_\top$ , and therefore, $\tau_\bot\subseteq\tau_\top$ .

When $\leq$ is an object in $\mathbf{Po}_A$ , the topologies in $\mathbf{Top}(\leq)$ are objects in $\mathbf{Top}_A$ .

2.3 Adjunctions and best approximations

A key property of categories of analyses is poset-enrichment, which provides a qualitative criterion for comparing analyses, and allows the definition of adjunctions between two complete lattices.

Given an $\omega^{op}$ -chain $(p_n:D_{n+1} \to D_n \mid n)$ of right adjoints in $\mathbf{Top}_A$ , its limit in $\mathbf{Top}_0$ is the sub-space of $\prod_{n}D_n$ corresponding to the subset $\lvert\thinspace {D} \thinspace\rvert\stackrel{\triangle}{=}\{d \mid \forall n.d_n=p_n(d_{n+1})\}$ , and the maps $\pi_n:D \to D_n$ with $\pi_n(d)=d_n$ form a limit cone. Since $U:\mathbf{Top}_0 \to \mathbf{Po}$ is $\mathbf{Po}$ -enriched and preserves limits, we have that $(Up_n \mid n)$ is an $\omega^{op}$ -chain of right adjoints in $\mathbf{Po}_A$ and $(U\pi_n \mid n)$ is a limit cone in $\mathbf{Po}$ . By the result for $\mathbf{Po}_A$ , we have $UD:\mathbf{Po}_A$ . Hence, $D:\mathbf{Top}_A$ .

A similar result holds if right adjoints are replaced by left adjoints (i.e., $\mathbf{Po}_A$ and $\mathbf{Top}_A$ have limits of $\omega^{op}$ -chains of left adjoints). We recall further properties of adjunctions in $\mathbf{Po}$ -enriched categories (and in $\mathbf{Po}$ ). Each of these properties has a dual, which we do not state explicitly.

In other words, if a left adjoint is monic, then it is split monic and its right adjoint is split epic.

A subset X of a poset Y can be identified with the sub-poset of Y with carrier X and the partial order inherited from Y. Then, the inclusion $f:X \to Y$ is a left adjoint in $\mathbf{Po}$ exactly when every $y \in Y$ has a best X-approximation, and the right adjoint to f maps y to its best X-approximation. When Y is a complete lattice, Proposition 2.18 characterizes the subsets X of Y for which every $y \in Y$ has a best X-approximation.

We are mainly interested in the cases where Y is a complete lattice of analyses $\mathbf{Po}_A(\leq_1,\leq_2)$ and X is a sub-poset of the form $\mathbf{Top}_A(\tau_1,\tau_2)$ , where $\leq_i$ is the specialization order of the topology $\tau_i$ . In these cases, the best X-approximations exist whenever the poset $\mathbf{Top}_A(\tau_1,\tau_2)$ is a complete lattice with sups computed as in $\mathbf{Po}_A(\leq_1,\leq_2)$ .

Clearly, $\tau_S(\leq)\subseteq\tau_\omega(\leq)$ . Given a map $f:\mathbf{Po}_A(\leq_1,\leq_2)$ , there is an order-theoretic characterization of continuity with respect to these two topologies, namely:

• $f:\mathbf{Top}_A(\tau_S(\leq_1),\tau_S(\leq_2)) \iff f$ preserves sups of directed sets, i.e., f is Scott continuous.

• $f:\mathbf{Top}_A(\tau_\omega(\leq_1),\tau_\omega(\leq_2))\iff f$ preserves sups of $\omega$ -chains, i.e., f is $\omega$ -continuous.

These characterizations imply $\mathbf{Top}_A(\tau_S(\leq_1),\tau_S(\leq_2))\subseteq \mathbf{Top}_A(\tau_\omega(\leq_1),\tau_\omega(\leq_2))\subseteq\mathbf{Po}_A(\leq_1,$ $\leq_2)$ and that these subsets are closed with respect to sups computed in $\mathbf{Po}_A(\leq_1,\leq_2)$ . Therefore, every analysis $f:\mathbf{Po}_A(\leq_1,\leq_2)$ has a best Scott-continuous approximation $\Box_S(f)$ , and a best $\omega$ -continuous approximation $\Box_\omega(f)$ , with $\Box_S(f)\leq\Box_\omega(f)\leq f$ .

We introduce two sub-categories of $\mathbf{Po}_A$ , related to the example above.

Proposition 2.22. The $\mathbf{Po}$ -enriched categories in the following diagram have finite products and limits of $\omega^{op}$ -chains of right adjoints and the functors preserve them:

where the full&faithful functor from $\mathbf{CL}$ to $\mathbf{Top}_A$ maps a continuous lattice $\leq$ to the Scott topology $\tau_S(\leq)$ . Moreover, the categories $\omega\mathbf{CL}$ and $\mathbf{CL}$ have exponentials, the functor preserves them, and every $\omega$ -continuous map between $\omega$ -continuous lattices is necessarily Scott continuous.

2.4 From spaces to complete lattices

Given a topological space $\mathbb{S}$ , the set of closed subsets of $\mathbb{S}$ , ordered by reverse inclusion, forms a complete lattice $\mathbb{C}(\mathbb{S})$ , with sups given by intersection. We introduce several topologies on these complete lattices, but first we give the main properties of $\mathbb{C}$ as a functor from $\mathbf{Haus}$ to $\mathbf{Po}_A$ .

Goubault-Larrecq defines $d'(C,C')\stackrel{\triangle}{=}\sup_{x' \in C'}\inf_{x \in C}d(x,x')$ under the assumption that d is a hemi-metric, i.e., d satisfies only the properties $d(x,x)=0$ and $d(x,z)\leq d(x,y)+d(y,z)$ . However, unlike Proposition 1, we do not restrict d’ to the subset of $\mathsf{C}(\mathbb{S})$ of the non-empty compact subsets, thus what is claimed in the proposition may fail for closed subsets.

The topology induced by a hemi-metric d is the smallest topology containing all open balls $B(x,\delta)=\{x' \mid d(x,x')<\delta\}$ . Therefore, the topology induced by d’ on $\mathsf{C}(\mathbb{S})$ coincides with the Robust topology, because the following inclusions hold for every $C \in \mathsf{C}(\mathbb{S})$ and $0<\delta<\delta'$

\begin{align*} B'(C,\delta)\stackrel{\triangle}{=}\{C' \mid d'(C,C')<\delta\} \subseteq\uparrow B(C,\delta)\stackrel{\triangle}{=}\uparrow\{x' \mid \exists x \in C.d(x,x')<\delta\} \subseteq B'(C,\delta'). \end{align*}

In item (3) of Theorem 2.27, when $\mathbb{S}$ is not compact, the inclusions may be strict. One such counter-example is the metric space $\mathbb{R}$ of real numbers.

By the universal property of D, there exists a unique map $\phi:\mathbb{C}(X) \to D$ in $\mathbf{Po}_A$ such that:

namely $\phi(C)=(\pi_n(C) \mid n)$ for every $C \in \mathbb{C}(X)$ . Moreover, $\phi$ preserves infs, since the $\mathbb{C}(\pi_n)$ are right adjoints and preserve infs. Therefore, $\phi$ has a left adjoint $\phi':D \to \mathbb{C}(X)$ , namely $\phi'(d)=\bigcap_n\pi_n^*(d_n)$ . We prove that $\phi'$ is the inverse of $\phi$ , and the limit in $\mathbf{Po}_A$ is actually in $\mathbf{CL}$ , because $\mathbf{CL}$ is replete in $\mathbf{Po}_A$ (i.e., if $A:\mathbf{CL}$ and $f:A\to B$ is an iso in $\mathbf{Po}_A$ , then f is in $\mathbf{CL}$ ). Because of the adjunction $\phi'\dashv\phi$ we have:

1. (1) $\forall n.\forall d \in D.\phi(\phi'(d))_n=\pi_n(\bigcap_n\pi_n^*(d_n))\subseteq d_n$ , and

2. (2) $\forall C \in \mathbb{C}(X).C\subseteq\phi'(\phi(C))=\bigcap_n\pi_n^*(\pi_n(C))$ .

We prove that the inclusions in (1) and (2) are indeed equalities. In the case of (1), given $d \in D$ , we have (by the Axiom of Choice) $\forall n.\forall x \in d_n.\exists y \in d_{n+1}.x=p_n(y)$ , which leads to the construction of a sequence $\hat{y} \in X$ , from which we obtain: $\forall n.\forall x \in d_n.\exists \hat{y} \in X.x=\hat{y}_n\land(\forall i.\hat{y}_i\in d_i)$ . But $\{\hat{y} \in X \mid \forall n.\hat{y}_n\in d_n\}$ is another way of denoting $\bigcap_n\pi_n^*(d_n) \in \mathbb{C}(X)$ . Thus, the first inclusion is an equality.

In the case of (2), given an $x\in\phi'(\phi(C))$ , i.e., $\forall n.x_n\in\pi'_n(C)$ , we prove that $x\in C$ . Since C is closed and a base for the topology on X is given by the subsets $[O]_n=\{y \in X \mid y_n\in O\}$ with $O \in \mathsf{O}(X_n)$ , it suffices to prove that $\forall n.\forall O \in \mathsf{O}(X_n).x\in[O]_n\implies\exists y \in C.y\in[O]_n$ . But, $x_n\in\pi_n(C)$ means that $x_n=y_n$ for some $y\in C$ . Hence, $x\in [O]_n\iff x_n\in O\iff y_n\in O\iff y\in [O]_n$ .

3.1 Idempotents, sections, and retractions

In this section, we establish some general properties of idempotents, sections (aka split monos), and retractions (aka split epis). All these notions are preserved by functors and reflected by full&faithful functors. The properties of section-retraction pairs that we consider are weaker variants of those considered in Domain Theory for embedding-projection pairs, such as the limit-colimit coincidence, see e.g., Abramsky and Jung (Reference Abramsky, Jung, Abramsky, Gabbay and Maibaum1994, Sec 3.3.2). The notion of embedding-projection pair (and insertion-closure pair) is available only in $\mathbf{Po}$ -enriched categories, see e.g., Abramsky and Jung (Reference Abramsky, Jung, Abramsky, Gabbay and Maibaum1994, Sec 3.1). Therefore, in the categories (of spaces) that we consider, one can rely only on the more general notion of section-retraction pair.

We write $\mathbb{I}(X)$ for the poset of idempotents on X and $\mathbb{A}_{sr}$ for the category with sr-pairs as arrows.

When $\mathbb{A}$ is $\mathbf{Po}$ -enriched, there is a partial order $\sqsubseteq$ on the hom-set $\mathbb{A}(X,X)$ of endomorphisms on X, but its restriction to the subset of idempotents on X is unrelated to the partial order $\leq$ . However, the two partial orders coincide when restricted to the idempotents below $\text{id}_X$ , i.e., the idempotents g such that $g\sqsubseteq\text{id}_X$ .

In Category Theory, a definition or result given for a generic category $\mathbb{A}$ can be recast in the dual category $\mathbb{A}^{op}$ . In particular, sections are the dual of retractions and idempotents are self-dual.

The following statements hold:

1. (1) The definition of $g_1\leq g_2$ is self-dual, i.e., $g_1\leq g_2$ in $\mathbb{A}$ $\iff$ $g_1\leq g_2$ in $\mathbb{A}^{op}$ .

2. (2) (e,p) is an sr-pair from X to Y in $\mathbb{A}$ $\iff$ (p,e) is an sr-pair from X to Y in $\mathbb{A}^{op}$ .

The definition of epic (and monic) generalizes from maps (aka arrows) to families of maps.

Definition 3.3. We say that the family of maps $(h_i:X_i\to X \mid i:I)$ into X is jointly epic in $\mathbb{A}$ $\stackrel{\triangle}{\iff}$

\begin{align*} \forall Y:\mathbb{A}.\forall f,f':\mathbb{A}(X,Y),(\forall i.f\circ h_i=f'\circ h_i)\implies f=f'. \end{align*}

Dually, we say that the family of maps $(h_i:X\to X_i \mid i \in I)$ from X is jointly monic in $\mathbb{A}$ $\stackrel{\triangle}{\iff}$

\begin{align*} \forall Y:\mathbb{A}.\forall f,f':\mathbb{A}(Y,X),(\forall i. h_i\circ f= h_i\circ f')\implies f=f'. \end{align*}

The family $(h_i \mid i \in I)$ determines X, provided I is not empty. As expected, a family consisting of one map h (i.e., I is a singleton) is jointly epic exactly when h is epic. The following proposition extends well-known properties of epis (namely, $h\circ g$ epic implies h epic, and compositions of epis are epic) to families of maps.

Moreover, if $(h_i:X_i\to X \mid i \in I)$ is a colimit cone from some diagram D in $\mathbb{A}$ to X, then the family of maps $(h_i \mid i \in I)$ into X is jointly epic.

Item (3).

Consider an idempotent g of X which is an upper-bound of $(g_i \mid i)$ , i.e., $\forall i.g_i\leq g$ . Then, $\forall i.g_i\circ g=g_i$ , which implies $g=\text{id}_X$ , because $(g_i \mid i)$ is jointly monic.

Item (4).

To prove that any arrow (e,p) in $\mathbb{A}_{sr}$ is a mono, it suffices to observe that in $\mathbb{A}$ every section e is monic and every retraction p is epic. If (e’,p’) is the inverse of (e,p) in $\mathbb{A}_{sr}$ , then e’ is the inverse of e in $\mathbb{A}$ , thus $p=p\circ e\circ e'=e'$ , because $p\circ e=\text{id}_X$ .

Item (5).

Uniqueness of (e,p) is immediate, because $(e_2,p_2)$ is a mono in $\mathbb{A}_{sr}$ . For existence, define $e=p_2\circ e_1$ and $p=p_1\circ e_2$ . Then, from the assumption $e_1\circ p_1=g_1\leq g_2=e_2\circ p_2$ we derive:

1. (i) $p\circ e=\text{id}_{X_1}$ (i.e., (e,p) is an sr-pair), because

   $(p_1\circ e_2)\circ(p_2\circ e_1)=$ by definition of $g_2$

   $p_1\circ g_2\circ e_1=$ by $(e_1,p_1)$ sr-pair

   $p_1\circ g_2\circ e_1\circ(p_1\circ e_1)=$ by definition of $g_1$

   $p_1\circ g_2\circ g_1\circ e_1=$ by $g_1\leq g_2$

   $p_1\circ g_1\circ e_1=$ by definition of $g_1$

   $p_1\circ (e_1\circ p_1)\circ e_1=$ by $(e_1,p_1)$ sr-pair

   $\text{id}_{X_1}\circ\text{id}_{X_1}=\text{id}_{X_1}$

2. (ii) $e_2\circ e=e_1$ , because

   $e_2\circ(p_2\circ e_1)=$ by definition of $g_2$

   $g_2\circ e_1=$ by $(e_1,p_1)$ sr-pair

   $g_2\circ e_1\circ(p_1\circ e_1)=$ by definition of $g_1$

   $g_2\circ g_1\circ e_1=$ by $g_1\leq g_2$

   $g_1\circ e_1=$ by definition of $g_1$

   $(e_1\circ p_1)\circ e_1=e_1$ by $(e_1,p_1)$ sr-pair

3. (iii) $p\circ p_2=p_1$ , dual of the previous proof.

Proposition 3.6. ( $\omega$ -colimits of sections). Given an $\omega$ -chain $((e_n,p_n):X_n\to X_{n+1} \mid n \in \omega)$ in $\mathbb{A}_{sr}$ and a colimit cone $(\underline{f}_n:X_n \to \underline{X} \mid n)$ in $\mathbb{A}$ from the $\omega$ -chain $(e_n:X_n \to X_{n+1} \mid n)$ to $\underline{X}$ , there exists a unique cone $((\underline{f}_n,\underline{q}_n):X_n \to \underline{X} \mid n)$ in $\mathbb{A}_{sr}$ from the $\omega$ -chain $((e_n,p_n) \mid n)$ to $\underline{X}$ (see Figure 3 ). If $\underline{g}_n$ is the idempotent $\underline{f}_n\circ\underline{q}_n$ on $\underline{X}$ , then $(\underline{g}_n \mid n)$ is an $\omega$ -chain in $\mathbb{I}(\underline{X})$ and is jointly epic in $\mathbb{A}$ .

Figure 3. Partial diagrammatic recast of Proposition 3.6.

Second, we prove (by induction on $|j-i|$ ) the property:

(1) \begin{equation} \forall i,j.h_{i+1,j}\circ e_i=h_{i,j}=p_j\circ h_{i,j+1}. \end{equation}

:

• case $i=j$ : immediate, since $h_{i+1,i}=p_i$ and $h_{i,i+1}=e_i$ .

• case $i<j$ : $h_{i+1,j}\circ e_i=h_{i,j}$ by definition of $h_{i,j}$ . For the other equality:

  1. $h_{i,j}=$ by definition of $h_{i,j}$

  2. $h_{i+1,j}\circ e_i=$ by IH on $(i+1,j)$

  3. $p_j\circ h_{i+1,j+1}\circ e_i=$ by definition of $h_{i,j+1}$

  4. $p_j\circ h_{i,j+1}$ .

• case $i>j$ : $h_{i,j}=p_j\circ h_{i,j+1}$ by definition of $h_{i,j}$ . For the other equality:

  1. $h_{i+1,j}\circ e_i=$ by definition of $h_{i+1,j}$

  2. $p_j\circ h_{i+1,j+1}\circ e_i=$ by IH on $(i,j+1)$

  3. $p_j\circ h_{i,j+1}=$ by definition of $h_{i,j}$

  4. $h_{i,j}$ .

Property (1) implies that $(h_{i,n} \mid i)$ is a cone in $\mathbb{A}$ from $(e_i \mid i)$ to $X_n$ . Thus, there exists a unique $\underline{q}_n:X_n\to\underline{X}$ such that $\forall i.\underline{q}_n\circ \underline{f}_i=h_{i,n}$ . In particular, $\underline{q}_n\circ \underline{f}_n=h_{n,n}=\text{id}_{X_n}$ , i.e., $(\underline{f}_n,\underline{q}_n):X_n\to \underline{X}$ in $\mathbb{A}_{sr}$ . Since the colimit cone $(\underline{f}_n \mid n)$ is jointly epic (by Proposition 3.4), property (1) implies also that $\underline{q}_n=p_n\circ \underline{q}_{n+1}$ Thus, $((\underline{f}_n,\underline{q}_n) \mid n)$ is a cone in $\mathbb{A}_{sr}$ from $((e_n,p_n) \mid n)$ to $\underline{X}$ .

For uniqueness of $(\underline{q}_n \mid n)$ , we use again that $(\underline{f}_n \mid n)$ is jointly epic and prove (by induction on $|j-i|$ ) that $\forall i,j.q'_j\circ \underline{f}_i=h_{i,j}$ when $(q'_n \mid n)$ is a cone from $\underline{X}$ to $(p_n \mid n)$ such that $\forall n.q'_n\circ \underline{f}_n=\text{id}_{X_n}$ .

• case $i=j$ : immediate, by $q'_i\circ\underline{f}_i=\text{id}_{X_i}=h_{i,i}$ , assumption on $(q'_n \mid n)$ and definition of $h_{i,i}$ .

• case $i<j$ :

  1. $q'_j\circ \underline{f}_i=$ by definition of $(\underline{f}_n \mid n)$

  2. $q'_j\circ \underline{f}_{i+1}\circ e_i=$ by IH on $(i+1,j)$

  3. $h_{i+1,j}\circ e_i=h_{i,j}$ by definition of $h_{i,j}$ .

• case $i>j$ :

  1. $q'_j\circ \underline{f}_i=$ by assumption on $(q'_n \mid n)$

  2. $p_j\circ q'_{j+1}\circ \underline{f}_i=$ by IH on $(i,j+1)$

  3. $p_j\circ h_{i,j+1}=h_{i,j}$ by definition of $h_{i,j}$ .

Consider the idempotents $\underline{g}_n=\underline{f}_n\circ \underline{q}_n$ on $\underline{X}$ . From item (5) of Proposition 3.5, it follows that $\forall n.\underline{g}_n\leq \underline{g}_{n+1}$ . Moreover, by Proposition 3.4, the family $(\underline{g}_n \mid n)$ is jointly epic, because it is the composition of jointly epic families, namely the colimit cone $(\underline{f}_n \mid n)$ and the singleton families consisting of the (split) epic $\underline{q}_n$ .

The following is the dual of Proposition 3.6:

Given an $\omega$ -chain $((e_n,p_n):X_n\to X_{n+1} \mid n \in \omega)$ , in $\mathbb{A}_{sr}$ and a limit cone $(\overline{q}_n: \overline{X}\to X_n \mid n)$ in $\mathbb{A}$ from $\overline{X}$ to the $\omega^{op}$ -chain $ (p_n:X_{n+1} \to X_n \mid n)$ , there exists a unique cone $((\overline{f}_n,\overline{q}_n):X_n \to \overline{X} \mid n)$ in $\mathbb{A}_{sr}$ from the $\omega$ -chain $((e_n,p_n) \mid n)$ to $\overline{X}$ . If $\overline{g}_n$ is the idempotent $\overline{f}_n\circ\overline{q}_n$ on $\overline{X}$ , then $(\overline{g}_n \mid n)$ is an $\omega$ -chain in $\mathbb{I}(\overline{X})$ and is jointly monic in $\mathbb{A}$ .

Theorem 3.9 on the next page implies existence and uniqueness of $\iota:\underline{X} \to \overline{X}$ in $\mathbb{A}$ such that $\forall n.\iota\circ\underline{f}_n=\overline{f}_n$ and $\forall n.\underline{q}_n=\overline{q}_n\circ\iota$ , where $ ((\underline{f}_n,\underline{q}_n):X_n \to \underline{X} \mid n)$ and $((\overline{f}_n,\overline{q}_n):X_n \to \overline{X} \mid n)$ are the cones in $\mathbb{A}_{sr}$ given by Proposition 3.6 and Corollary 3.7. In general, there is no reason for $\iota$ to be monic, epic, or iso, as demonstrated in the examples below.

If we start from an $\omega$ -chain $(g_n \mid n)$ in $\mathbb{I}(X)$ , where X is an object of interest in $\mathbb{A}$ , then we can get, by splitting the idempotents, an $\omega$ -chain in $\mathbb{A}_{sr}$ which is unique up to iso. From this $\omega$ -chain, there are three cones in $\mathbb{A}_{sr}$ , whose vertices are $\underline{X}$ , X and $\overline{X}$ , respectively. The following theorem states that these cones are related by unique maps $\underline{\iota}:\underline{X}\to X$ and $\overline{\iota}:X\to\overline{X}$ in $\mathbb{A}$ . There is also an inverse (which we do not prove), where we start from an $\omega$ -chain $((e_n,p_n) \mid n)$ in $\mathbb{A}_{sr}$ , and consider the unique map $\iota:\underline{X}\to\overline{X}$ in $\mathbb{A}$ , relating the cones in $\mathbb{A}_{sr}$ from the $\omega$ -chain to $\underline{X}$ and $\overline{X}$ . In this case, any factorization $(\underline{\iota},\overline{\iota})$ of $\iota$ through an object X, induces a $\omega$ -chain $(g_n \mid n)$ of idempotents in $\mathbb{I}(X)$ , where $g_n=\underline{\iota}\circ\underline{f}_n\circ\overline{q}_n\circ\overline{\iota}$ .

Item (1).

We rely on the proof of Proposition 3.6, where $\underline{q}_n$ is defined. Since $(\underline{f}_i \mid i)$ is jointly epic, $\forall j.q_j\circ\underline{\iota} = \underline{q}_j$ follows from $\forall i,j.q_j\circ\underline{\iota}\circ\underline{f}_i = \underline{q}_j\circ\underline{f}_i$ , or equivalently, from $\forall i,j.q_j\circ f_i = h_{i,j}$ , which we prove by induction on $|j-i|$ :

• case $i=j$ : immediate, since $q_i\circ f_i=\text{id}_{X_i}=h_{i,i}$ , because $(f_i,q_i)$ is a map in $\mathbb{A}_{sr}$ and by definition of $h_{i,i}$ .

• case $i<j$ :

  1. $q_j\circ f_i=$ because $(f_{i+1},q_{i+1})\circ(e_i,p_i)=(f_i,q_i)$ in $\mathbb{A}_{sr}$

  2. $q_j\circ f_{i+1}\circ e_i=$ by IH on $(i+1,j)$

  3. $h_{i+1,j}\circ e_i=h_{i,j}$ by definition of $h_{i,j}$ .

• case $i>j$ :

  1. $q_j\circ f_i=$ because $(f_{i+1},q_{i+1})\circ(e_i,p_i)=(f_i,q_i)$ in $\mathbb{A}_{sr}$

  2. $p_j\circ q_{j+1}\circ f_i=$ by IH on $(i,j+1)$

  3. $p_j\circ h_{i,j+1}=h_{i,j}$ by definition of $h_{i,j}$ .

Item (2). By duality, since it is the dual of Item (1).

3.2 Extended metric spaces versus compact Hausdorff spaces

The following result requires moving between four categories (and we have added also $\mathbf{Set}$ ) using four functors (where denotes a faithful functor and a full&faithful functor):²

All the categories in the above diagram have:

• finite limits and finite sums (Theorem 2.3);

• enough points, more precisely, the faithful forgetful functor U from $\mathbb{A}$ into $\mathbf{Set}$ is (isomorphic) to the global section functor $\mathbb{A}(1,-)$ , where $\mathbb{A}$ is any of the other four categories ( $\mathbf{KMS}$ , $\mathbf{Met}$ , $\mathbf{KH}$ or $\mathbf{Haus}$ ), and 1 is the terminal object in $\mathbb{A}$ ;

• splittings of idempotents, because each of the five categories has equalizers.

Moreover, all functors in the diagrams preserve finite limits and finite sums; $\mathbf{Haus}$ has all small limits and small colimits (Theorem 2.4), where limits are computed as in $\mathbf{Top}$ , and limits (computed in $\mathbf{Haus}$ ) of diagrams in $\mathbf{KH}$ are in $\mathbf{KH}$ .

In applications, we start from a metric space $\mathbb{S}$ , then we identify an $\omega$ -chain $(g_n \mid n)$ of idempotents on $\mathbb{S}$ in $\mathbf{Met}$ , and by applying Theorem 3.9 in $\mathbf{Haus,}$ we get a map $\overline{\iota}:\mathbb{S}\to\overline{\mathbb{S}}$ in $\mathbf{Haus}$ . The theorems below provide sufficient conditions to ensure that $\overline{\mathbb{S}}$ is compact, $\overline{\iota}$ is monic and epic, and, above all, that the complete lattice $\mathbb{C}(\overline{\mathbb{S}})$ is $\omega$ -continuous and the monotonic map $\mathbb{C}(\overline{\iota})$ is in $\mathbb{A}_{RS}(\mathbb{S},\overline{\mathbb{S}})$ . These properties can be proved for maps $\iota\,:\,\mathbb{S}\to\overline{\mathbb{S}}$ that are not necessarily obtained through Theorem 3.9, and the theorems below capture this greater generality. If we start from an $\omega$ -chain $(g_n \mid n)$ of idempotents on $\mathbb{S}$ in $\mathbf{Met}$ , then Theorem 3.9 provides candidates for $(p_n \mid n)$ and $(q_n \mid n)$ in the following theorem, because $\mathbf{Met}$ has splittings of idempotents.