Equivalents of the finitary non-deterministic inductive definitions

TL;DR

This paper explores the logical equivalences of various fragments of non-deterministic inductive definitions within a weak constructive set theory, revealing their interrelations and implications for constructive topology.

Contribution

It establishes equivalences among different NID fragments and their variants, clarifying their roles in constructive topology within CZF.

Findings

Elementary NID is equivalent to a variant based on biclosed subsets.

Finitary NID is equivalent to its binary fragment.

Proving certain constructive topology statements requires extensions of CZF with NID.

Abstract

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID.

Equations

$$a\subseteq \alpha ⟹b\between \alpha ,$$

$$\forall \alpha \in \mathcal{C}\forall x\in \alpha \exists \beta \in G\left(x\in \beta \subseteq \alpha \right).$$

$$\forall \alpha \in \mathcal{C}\forall \sigma \in \mathrm{Fin}\left(\alpha \right)\exists \beta \in G\left(\sigma \subseteq \beta \subseteq \alpha \right).$$

$$\mathcal{M}(Z)=\left\{\alpha \in \mathrm{Pow}(S)\mid \forall (\sigma ,\Gamma )\in Z\left(\sigma \subseteq \alpha \to \exists U\in \Gamma \left(U\subseteq \alpha \right)\right)\right\}$$

$$\bigwedge \sigma ⊢\underset{U_{n-1}\in {\Gamma }_{n-1}}{\bigvee }\underset{{\Gamma }_{n-2}\in U_{n-1}}{\bigwedge }\cdots \underset{U_0\in {\Gamma }_0}{\bigvee }\bigwedge U_0$$

$$a\between \alpha ⟺b\between \alpha .$$

$$R^{\prime }\stackrel{\mathrm{def}}{=}\left\{(\{x\},b)\mid (a,b)\in R,x\in a\}\cup \{(\{y\},a)\mid (a,b)\in R,y\in b\right\}.$$

$$R^{\prime }\stackrel{\mathrm{def}}{=}\left\{(a\cup b,b)\mid (a,b)\in R\right\}.$$

$${\Delta }_X\stackrel{\mathrm{def}}{=}\left\{(x,x)\mid x\in X\right\},$$

$$\mathcal{E}\stackrel{\mathrm{def}}{=}\left\{U\in \mathrm{Pow}(X)\mid r_{1}U=r_{2}U\right\},$$

$$R\stackrel{\mathrm{def}}{=}\left\{(r_1^{-}y,r_2^{-}y)\mid y\in Y\right\},$$

$$U\in \mathcal{E}⟺\forall y\in Y\left(y\in r_{1}U\leftrightarrow y\in r_{2}U\right)⟺\forall y\in Y\left(r_1^{-}y\between U\leftrightarrow r_2^{-}y\between U\right),$$

$$Urx\stackrel{\mathrm{def}}{⟺}x\in U.$$

$$z\overline{s}U\stackrel{\mathrm{def}}{⟺}U\subseteq sz,$$

$$x{r}_1(a,b)$$

$$x{r}_2(a,b)$$

$$G\stackrel{\mathrm{def}}{=}\left\{re\mid e\in E\right\}.$$

$$\ast r_{\alpha }x\stackrel{\mathrm{def}}{⟺}x\in \alpha ,$$

$$S_1$$

$${⊩}_2\circ r_1={⊩}_2\circ r_2.$$

$$u_1$$

$$S_1$$

$$T$$

$$\bigwedge A⊢\underset{i\in I}{\bigvee }\bigwedge B_i$$

$$A\subseteq m⟹\exists i\in I\left(B_i\subseteq m\right)$$

$$R\stackrel{\mathrm{def}}{=}$$

$$\cup \left\{(\left\{A\right\},\left\{B\right\})\mid B\subseteq A\right\}$$

$$\cup \left\{(\left\{A,B\right\},\left\{A\cup B\right\})\mid A,B\in \mathrm{Fin}(S)\right\}$$

$$\displaystyle\cup\Bigl{\{}\left(\left\{A\right\},\left\{B_{i}\mid i\in I\right\}\right)\mid\operatorname*{{\textstyle{\bigwedge}}}A\vdash\bigvee_{i\in I}\operatorname*{{\textstyle{\bigwedge}}}B_{i}\in T\Bigr{\}}.$$

$$\Phi (\alpha )$$

$$\Phi (\Psi (m))=\cup \Psi (m)=\cup \mathrm{Fin}(m)=m$$

$$\bigwedge a⊢\underset{y\in b}{\bigvee }y.$$

Full text

Keywords: Constructive set theory; Non-deterministic inductive definition; Set-generated class; Basic pair; Formal topology

MSC2010: 03E70; 03F50; 54A05; 06D22

1 Introduction

Many objects in mathematics are defined as subsets of some given set, e.g., an open set of a topological space; a prime ideal of a commutative ring; a sub-algebra of a certain algebraic structure. The totality of these objects, however, does not necessarily form a set in predicative constructive foundations such as Martin-Löf’s type theory [11] or Aczel’s constructive set theory $\mathbf{CZF}$ [3]. In particular, the lack of power-sets in these foundations makes some of the standard constructions in general topology substantially difficult to carry out, which requires a certain amount of ingenuity [1, 10, 12, 13].

The crucial element of these predicative results consists in constructing a subset of the totality of a certain type of objects, called a generating subset, in such a way that every object of that type can be expressed as the union of the elements of the generating subset. The problems of constructing such generating subsets in constructive topology motivated van den Berg [17] and Aczel, Ishihara, Nemoto, and Sangu [2] to independently introduce principles of $\mathbf{CZF}$ which allow us to show that a wide range of collections of mathematical objects are set-generated.

The focus of this paper is on the principle introduced by van den Berg [17], called non-deterministic inductive definitions ($\mathrm{NID}$), or more specifically, its elementary and finitary fragments. The $\mathrm{NID}$ principle asserts that the class of models of an infinitary propositional theory consisting of formulas (or rules) of the form $\bigwedge U\rightarrow\bigvee V$ has a generating subset, where $U$ and $V$ are subsets of the set of propositional variables. The elementary and finitary $\mathrm{NID}$ principles are obtained by restricting the propositional theory to those rules whose premise is singleton and finitely enumerable, respectively. In fact, van den Berg [17] showed that the finitary $\mathrm{NID}$ is equivalent to the principle introduced by Aczel et al. [2], called the set generation axiom ($\mathrm{SGA}$).

The purpose of this paper is to extend the scope of reverse mathematics, classical [16] or constructive [7, 18], in a set-theoretic foundation. Here, we develop the reverse mathematics of the $\mathrm{NID}$ principle initiated in [9] much further by showing that several statements in constructive topology which were initially proved using $\mathrm{NID}$ or $\mathrm{SGA}$ are in fact equivalent to the elementary $\mathrm{NID}$ or the finitary $\mathrm{NID}$. Specifically, we show that the elementary $\mathrm{NID}$ is equivalent to (1) completeness and cocompleteness of the category of basic pairs by Sambin [15] and (2) the existence of weak equalisers in the category of sets and relations. Moreover, the finitary $\mathrm{NID}$ is equivalent to (1) completeness and cocompleteness of the category of concrete spaces and (2) set-generation of the class of formal points of an inductively generated formal topology. We also show that the finitary $\mathrm{NID}$ is equivalent to its binary fragment and that the elementary $\mathrm{NID}$ is equivalent to a symmetric variant of $\mathrm{NID}$, called $\mathrm{NID}_{\mathrm{bi}}$, formulated with respect to the notion of biclosed subset.

Our result suggests that proving these statements in constructive topology requires genuine extensions of $\mathbf{CZF}$ with the elementary $\mathrm{NID}$ or the finitary $\mathrm{NID}$, which are thought to be independent of $\mathbf{CZF}$ (cf. van den Berg [17, Section 8]). It remains to settle the exact relation between $\mathbf{CZF}$, and its extensions with the elementary $\mathrm{NID}$ or the finitary $\mathrm{NID}$, which would also settle the relative strength of the equivalents of these principles.

Organisation

Section 2 introduces the base set theory for our work; Section 3 recalls the $\mathrm{NID}$ principle and its relation to $\mathrm{SGA}$; Section 4 gives equivalents of the elementary $\mathrm{NID}$; and Section 5 gives equivalents of the finitary $\mathrm{NID}$.

2 Elementary constructive set theory

We work in a weak subsystem of $\mathbf{CZF}$, called the elementary constructive set theory $\mathbf{ECST}$ [4], where none of the known fragments of the $\mathrm{NID}$ principle seems to be derivable.

The language of $\mathbf{ECST}$ contains variables for sets and binary predicates $=$ and $\in$. The axioms and rules of $\mathbf{ECST}$ are the axioms and rules of intuitionistic predicate logic with equality, and the following set-theoretic axioms:

Extensionality:

$\forall a\forall b\left(\forall x\left(x\in a\leftrightarrow x\in b\right)\rightarrow a=b\right).$

Pairing:

$\forall a\forall b\exists y\forall u\left(u\in y\leftrightarrow u=a\vee u=b\right).$

Union:

$\forall a\exists y\forall x\left(x\in y\leftrightarrow\exists u\in a\left(x\in u\right)\right).$

Restricted Separation:

$\forall a\exists b\forall x\left(x\in b\leftrightarrow x\in a\wedge\varphi(x)\right)$

where $\varphi(x)$ is restricted. Here, a formula is said to be restricted if all quantifiers in the formula occur in the forms $\forall x\in a$ or $\exists x\in a$.

Replacement:

$\forall a\left(\forall x\in a\exists!y\,\varphi(x,y)\rightarrow\exists b\forall y\left(y\in b\leftrightarrow\exists x\in a\,\varphi(x,y)\right)\right)$

where $\varphi(x,y)$ is any formula.

Strong Infinity:

\begin{multlined}\exists a\left[0\in a\wedge\forall x\left(x\in a\rightarrow x+1\in a\right)\right.\\ \qquad\qquad\left.\wedge\forall y\left(0\in y\wedge\forall x\left(x\in y\rightarrow x+1\in y\right)\rightarrow a\subseteq y\right)\right]\end{multlined}\exists a\left[0\in a\wedge\forall x\left(x\in a\rightarrow x+1\in a\right)\right.\\ \qquad\qquad\left.\wedge\forall y\left(0\in y\wedge\forall x\left(x\in y\rightarrow x+1\in y\right)\rightarrow a\subseteq y\right)\right]

where $x+1$ denotes $x\cup\left\{x\right\}$ and [math] is the empty set $\emptyset$. The set $a$ asserted to exist will be denoted by $\omega$.

This completes the description of $\mathbf{ECST}$.

The constructive Zermelo–Fraenkel set theory $\mathbf{CZF}$ [3] is obtained from $\mathbf{ECST}$ by substituting Strong Collection for Replacement and adding Subset Collection and $\in$-Induction. For the details of these axioms, the reader is referred to Aczel and Rathjen [3, 4].

In $\mathbf{ECST}$, Subset Collection implies

Fullness:

$\forall a\forall b\exists c\left[c\subseteq\mathrm{mv}(a,b)\wedge\forall s\in\mathrm{mv}(a,b)\exists r\in c\left(r\subseteq s\right)\right],$

where $\mathrm{mv}(a,b)$ is the class of total relations from $a$ to $b$. In practice, Fullness is very important as it implies Exponentiation, which asserts that the class $B^{A}$ of functions between sets $A$ and $B$ is a set. In particular, Fullness implies the following weak form of Exponentiation:

Finite Powers Axiom ($\mathrm{FPA}$):

For any set $S$, the class $S^{n}$ of functions from $\left\{0,\dots,n-1\right\}$ to $S$ is a set for all $n\in\omega$.

The extension of $\mathbf{ECST}$ with $\mathrm{FPA}$ is denoted by $\mathbf{ECST}+\mathrm{FPA}$.

A notable consequence of $\mathrm{FPA}$ is that the class $\mathrm{Fin}(S)$ of finitely enumerable subsets of a set $S$ is a set. Here, a set $A$ is finitely enumerable if there is a surjection $f\colon\left\{0,\dots,n-1\right\}\to A$ for some $n\in\omega$. Note that we can decide whether a finitary enumerable set is empty or inhabited by inspecting the domain of $f$.

3 $\mathrm{NID}$ principles

The subjects of our investigation are classes closed under some sets of rules on a set. $\mathrm{NID}$ principles say that such a class has a generating subset $G$ so that every member of the class arises as the union of elements of $G$.

Definition 1**.**

A rule on a set $S$ is a pair $(a,b)$ of subsets of $S$. A rule $(a,b)$ is said to be nullary if $a$ is empty, elementary if $a$ is singleton, and finitary if $a$ is finitely enumerable. A subset $\alpha\subseteq S$ is said to be closed under a rule $(a,b)$ if

[TABLE]

where $b\between\alpha$ means that $b\cap\alpha$ is inhabited. If $R$ is a set of rules on $S$, then a subset $\alpha\subseteq S$ is said to be $R$-closed if it is closed under every rule in $R$.

Definition 2**.**

Let $S$ be a set and $\mathrm{Pow}(S)$ be the class of subsets of $S$. A subclass $\mathcal{C}$ of $\mathrm{Pow}(S)$ is said to be set-generated if there exists a subset $G\subseteq\mathcal{C}$, called a generating subset, such that

[TABLE]

The principle $\mathrm{NID}$ reads:

$\mathrm{NID}$:

For each set $S$ and a set $R$ of rules on $S$, the class of $R$-closed subsets of $S$ is set-generated.

The nullary, elementary, and finitary $\mathrm{NID}$ are the principles obtained from $\mathrm{NID}$ by restricting $R$ to nullary, elementary, and finitary rules, respectively. These principles are denoted by $\mathrm{\mathrm{NID}}_{0}$, $\mathrm{\mathrm{NID}}_{1}$, and $\mathrm{\mathrm{NID}}_{<\omega}$.

Remark 3*.*

$\mathrm{\mathrm{NID}}_{<\omega}$ clearly implies $\mathrm{\mathrm{NID}}_{1}$. Moreover, Ishihara and Nemoto [9] showed that $\mathrm{\mathrm{NID}}_{1}$ implies $\mathrm{\mathrm{NID}}_{0}$ and that $\mathrm{\mathrm{NID}}_{0}$ is equivalent to Fullness over $\mathbf{ECST}$. In particular, $\mathrm{\mathrm{NID}}_{0}$ implies $\mathrm{FPA}$.

We recall the connection between $\mathrm{\mathrm{NID}}_{<\omega}$ and the set generation axiom ($\mathrm{SGA}$) introduced by Aczel et al. [2].

Definition 4**.**

For any set $S$, a subclass $\mathcal{C}$ of $\mathrm{Pow}(S)$ is said to be strongly set-generated if there exists a subset $G\subseteq\mathcal{C}$ such that

[TABLE]

The principle $\mathrm{SGA}$ reads:

$\mathrm{SGA}$:

For each set $S$ and each subset $Z\subseteq\mathrm{Fin}(S)\times\mathrm{Pow}(\mathrm{Pow}(S))$, the class

[TABLE]

of models of $Z$ is strongly set-generated.

A subset $Z\subseteq\mathrm{Fin}(S)\times\mathrm{Pow}(\mathrm{Pow}(S))$ is called a generalised geometric theory over $S$ (of rank $1$), and its element is written $\operatorname*{{\textstyle{\bigwedge}}}\sigma\vdash\bigvee_{U\in\Gamma}\bigwedge U$ instead of $(\sigma,\Gamma)$.

The principle $\mathrm{SGA}$ looks stronger than $\mathrm{\mathrm{NID}}_{<\omega}$, but they are equivalent.

Proposition 5** (van den Berg [17, Theorem 7.3]).**

$\mathrm{SGA}$* and $\mathrm{\mathrm{NID}}_{<\omega}$ are equivalent over $\mathbf{ECST}$. ∎*

Remark 6*.*

In $\mathrm{SGA}$, we can take $Z$ to be a set of elements of the form

[TABLE]

where $\sigma\in\mathrm{Fin}(S),U_{0}\in\mathrm{Pow}(S),U_{1}\in\mathrm{Pow}(\mathrm{Pow}(\mathrm{Pow}(S))),\dots$, and the right-hand-side is a finite nesting of $\bigvee\bigwedge$ pairs. The resulting principle is still equivalent to $\mathrm{\mathrm{NID}}_{<\omega}$ over $\mathbf{ECST}$. See van den Berg [17, Theorem 4.2].

4 Elementary $\mathrm{NID}$

In this section, we give some statements equivalent to $\mathrm{\mathrm{NID}}_{1}$.

4.1 $\mathrm{NID}_{\mathrm{bi}}$ principle

We introduce a symmetric variant of $\mathrm{NID}$, which seems to be quite natural and useful in practice (cf. Section 4.2 and Section 4.3 below).

Definition 7**.**

Let $(a,b)$ be a rule on a set $S$. A subset $\alpha\subseteq S$ is said to be biclosed under $(a,b)$ if

[TABLE]

If $R$ is a set of rules on $S$, then a subset $\alpha\subseteq S$ is said to be $R$-biclosed if $\alpha$ is biclosed under every rule in $R$. Then, the principle $\mathrm{NID}_{\mathrm{bi}}$ reads:

$\mathrm{NID}_{\mathrm{bi}}$:

For each set $S$ and a set $R$ of rules on $S$, the class of $R$-biclosed subsets of S is set-generated.

Proposition 8**.**

$\mathrm{\mathrm{NID}}_{1}$* is equivalent to $\mathrm{NID}_{\mathrm{bi}}$.*

Proof.

Assume $\mathrm{\mathrm{NID}}_{1}$, and let $R$ be a set of rules on a set $S$. Define a set $R^{\prime}$ of elementary rules on $S$ by

[TABLE]

Then, a subset $\alpha\subseteq S$ is $R$-biclosed if and only if it is $R^{\prime}$-closed. This proves one direction.

Conversely, assume $\mathrm{NID}_{\mathrm{bi}}$, and let $R$ be a set of elementary rules on a set $S$. Define another set $R^{\prime}$ of rules on $S$ by

[TABLE]

Then, a subset $\alpha\subseteq S$ is $R$-closed if and only if it is $R^{\prime}$-biclosed. ∎

4.2 Weak equalisers in the category of sets and relations

We show that $\mathrm{\mathrm{NID}}_{1}$ is equivalent to the existence of weak equalisers in the category of sets and relations.

Let $\mathbf{Rel}$ be the category of sets and relations between them: the identity on a set $X$ is a diagonal relation

[TABLE]

and the composition of morphisms is the relational composition.

Definition 9**.**

Let $f,g\colon A\to B$ be a parallel pair of morphisms in a category $\mathbb{C}$. A weak equaliser of $f$ and $g$ is an object $E$ together with a morphism $e\colon E\to A$ such that

$f\circ e=g\circ e$, 2. 2.

for any morphism $h\colon C\to A$ such that $f\circ h=g\circ h$, there exists a morphism $\overline{h}\colon C\to E$ such that $e\circ\overline{h}=h$.

Proposition 10**.**

The following are equivalent over $\mathbf{ECST}$:

$\mathrm{NID}_{\mathrm{bi}}$; 2. 2.

$\mathbf{Rel}$* has weak equalisers.*

Proof.

(1 $\rightarrow$ 2) Assume $\mathrm{NID}_{\mathrm{bi}}$, and let $r_{1},r_{2}\subseteq X\times Y$ be a parallel pair of relations. Consider a class

[TABLE]

where $r_{i}U\overset{\mathrm{def}}{\,=\,}\left\{y\in Y\mid\exists x\in U\left(x\mathrel{r_{i}}y\right)\right\}\;(i=0,1)$. Define a set of rules on $X$ by

[TABLE]

where $r_{i}^{-}y\overset{\mathrm{def}}{\,=\,}\left\{x\in X\mid x\mathrel{r_{i}}y\right\}\;(i=0,1)$. Since

[TABLE]

$\mathcal{E}$ is the class of $R$-biclosed subsets. Thus $\mathcal{E}$ has a generating subset $G$ by $\mathrm{NID}_{\mathrm{bi}}$. Define a relation $r\subseteq G\times X$ by

[TABLE]

Clearly, we have $r_{1}\circ r=r_{2}\circ r$. We show that $r$ is a weak equaliser of $r_{1}$ and $r_{2}$. Let $s\subseteq Z\times X$ be a relation such that $r_{1}\circ s=r_{2}\circ s$. Define a relation $\overline{s}\subseteq Z\times G$ by

[TABLE]

where $sz\overset{\mathrm{def}}{\,=\,}\left\{x\in X\mid z\mathrel{s}x\right\}$. Obviously, we have $r\circ\overline{s}\subseteq s$. Conversely, suppose that $z\mathrel{s}x$. Since $sz$ is an $R$-biclosed subset of $X$, there exists a $U\in G$ such that $x\in U\subseteq sz$. Thus $z\mathrel{(r\circ\overline{s})}x$, and so $s\subseteq r\circ\overline{s}$. Hence $s=r\circ\overline{s}$.

(2 $\rightarrow$ 1) Assume that $\mathbf{Rel}$ has weak equalisers, and let $R$ be a set of rules on a set $S$. Then, $R$ corresponds to two relations $r_{1},r_{2}\subseteq S\times R$ given by

[TABLE]

Let $r\subseteq E\times S$ be a weak equaliser of $r_{1}$ and $r_{2}$ in $\mathbf{Rel}$, and put

[TABLE]

Since $r_{1}\circ r=r_{2}\circ r$, elements of $G$ are $R$-biclosed subsets of $S$. We show that $G$ generates the class of $R$-biclosed subsets. Let $\alpha\subseteq S$ be an arbitrary $R$-biclosed subset, and let $z\in\alpha$. Define a relation $r_{\alpha}\subseteq\left\{*\right\}\times S$ by

[TABLE]

where $\left\{*\right\}$ is a fixed one-element set. Since $r_{\alpha}*=\alpha$ and $\alpha$ is $R$-biclosed, we have $r_{1}\circ r_{\alpha}=r_{2}\circ r_{\alpha}$. Thus, $r_{\alpha}$ factors through $r$ via some relation $\overline{r_{\alpha}}\subseteq\left\{*\right\}\times E$. Since $*\mathrel{r_{\alpha}}z$, there exists an $e\in E$ such that $*\mathrel{\overline{r_{\alpha}}}e$ and $e\mathrel{r}z$. Then, for any $y\in re$, we have $*\mathrel{(r\circ\overline{r_{\alpha}})}y$, i.e., $*\mathrel{r_{\alpha}}y$, and so $y\in\alpha$. Hence $z\in re\subseteq\alpha$. ∎

4.3 Equalisers in the category of basic pairs

We show that $\mathrm{\mathrm{NID}}_{1}$ is equivalent to the existence of equalisers in the category of basic pairs described in the forthcoming book by Sambin [15]. The result in this subsection refines the result by Ishihara and Kawai [8, Proposition 3.8], where they showed that the category of basic pairs has coequalisers using $\mathrm{SGA}$ (cf. Remark 13).

Definition 11**.**

A basic pair is a triple $(X,\Vdash,S)$ where $X$ and $S$ are sets, and $\Vdash$ is a relation from $X$ to $S$. A relation pair between basic pairs $\mathcal{X}_{1}=(X_{1},\Vdash_{1},S_{1})$ and $\mathcal{X}_{2}=(X_{2},\Vdash_{2},S_{2})$ is a pair $(r,s)$ of relations $r\subseteq X_{1}\times X_{2}$ and $s\subseteq S_{1}\times S_{2}$ such that ${\Vdash_{2}}\circ r=s\circ{\Vdash_{1}},$ i.e., the following diagram commutes in $\mathbf{Rel}$:

[TABLE]

Two relation pairs $(r_{1},s_{1})$ and $(r_{2},s_{2})$ between basic pairs $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are said to be equivalent (or equal) if

[TABLE]

In this case, we write ${(r_{1},s_{1})}\sim{(r_{2},s_{2})}$.

Basic pairs and relation pairs with equality defined by (4.1) form a category $\mathbf{BP}$: the identity on a basic pair $\mathcal{X}=(X,\Vdash,S)$ is $(\Delta_{X},\Delta_{S})$ and the composition of two relation pairs $(r,s)\colon\mathcal{X}_{1}\to\mathcal{X}_{2}$ and $(u,v)\colon\mathcal{X}_{2}\to\mathcal{X}_{3}$ is $(u\circ r,v\circ s)$, where each component is a relational composition. It is easy to check that compositions respect equality of relation pairs; see [8, Proposition 2.3] for more details.

Proposition 12**.**

The following are equivalent over $\mathbf{ECST}$:

$\mathbf{Rel}$* has weak equalisers;* 2. 2.

$\mathbf{BP}$* has equalisers.*

Proof.

(1 $\rightarrow$ 2) This follows from a general result on the Freyd completion of categories [6, Section 2.5 (b)]. We recall the proof for the particular case of $\mathbf{Rel}$. Assume that $\mathbf{Rel}$ has weak equalisers. Let $(r_{1},s_{1}),(r_{2},s_{2})\colon\mathcal{X}_{1}\to\mathcal{X}_{2}$ be relation pairs between basic pairs $\mathcal{X}_{1}=(X_{1},\Vdash_{1},S_{1})$ and $\mathcal{X}_{2}=(X_{2},\Vdash_{2},S_{2})$. Put

[TABLE]

Let $e\colon E\to X_{1}$ be a weak equaliser of $u_{1}$ and $u_{2}$ in $\mathbf{Rel}$. Consider a basic pair $\mathcal{E}=(E,\Vdash_{1}\mathrel{\circ}{e},S_{1})$. Then $(e,\Delta_{S_{1}})$ is a relation pair from $\mathcal{E}$ to $\mathcal{X}_{1}$, and we have $(r_{1},s_{1})\circ(e,\Delta_{S_{1}})\sim(r_{2},s_{2})\circ(e,\Delta_{S_{1}})$; see the diagram below.

[TABLE]

We show that $(e,\Delta_{S_{1}})\colon\mathcal{E}\to\mathcal{X}_{1}$ is an equaliser of $(r_{1},s_{1})$ and $(r_{2},s_{2})$. Let $\mathcal{Z}=(Z,\Vdash,T)$ be a basic pair, and let $(u,v)\colon\mathcal{Z}\to\mathcal{X}_{1}$ be a relation pair such that $(r_{1},s_{1})\circ(u,v)\sim(r_{2},s_{2})\circ(u,v)$. Then, $u_{1}\circ u=u_{2}\circ u$ in $\mathbf{Rel}$, so there exists a relation $\overline{u}\subseteq Z\times E$ such that $e\circ\overline{u}=u$. Then, $(\overline{u},v)$ is a relation pair from $\mathcal{Z}$ to $\mathcal{E}$. It is also straightforward to check that $(e,\Delta_{S_{1}})\circ(\overline{u},v)\sim(u,v)$ and that $(\overline{u},v)$ is a unique relation pair from $\mathcal{Z}$ to $\mathcal{E}$ with this property (cf. the diagram below).

[TABLE]

(2 $\rightarrow$ 1) In the following, we write $S_{\Delta}$ for the basic pair $(S,\Delta_{S},S)$ on a set $S$. Assume that $\mathbf{BP}$ has equalisers, and let $r_{1},r_{2}\subseteq X\times Y$ be a parallel pair of relations. Then, $(r_{1},r_{1})$ and $(r_{2},r_{2})$ are relation pairs from $X_{\Delta}$ to $Y_{\Delta}$. Let $(r,s)\colon\mathcal{E}\to X_{\Delta}$ be an equaliser of $(r_{1},r_{1})$ and $(r_{2},r_{2})$ in $\mathbf{BP}$, and write $\mathcal{E}=(E,\Vdash,S)$. We show that $r$ is a weak equaliser of $r_{1}$ and $r_{2}$ in $\mathbf{Rel}$. First, since $(r_{1},r_{1})\circ(r,s)\sim(r_{2},r_{2})\circ(r,s)$, we have $r_{1}\circ r=r_{2}\circ r$. Next, let $u\subseteq Z\times X$ be a relation such that $r_{1}\circ u=r_{2}\circ u$. Then, $(u,u)$ is a relation pair from $Z_{\Delta}$ to $X_{\Delta}$, and we have $(r_{1},r_{1})\circ(u,u)\sim(r_{2},r_{2})\circ(u,u)$. Thus, there exists a unique relation pair $(v,w)\colon Z_{\Delta}\to\mathcal{E}$ such that $(r,s)\circ(v,w)\sim(u,u)$. Then, $r\circ v=u$. ∎

Remark 13*.*

The categories $\mathbf{Rel}$ and $\mathbf{BP}$ are self dual, i.e., $\mathbf{Rel}$ is equivalent to its opposite $\mathbf{Rel}^{\mathrm{op}}$ (and similarly for $\mathbf{BP}$). Thus, $\mathbf{Rel}$ has weak equalisers if and only if it has weak coequalisers, and $\mathbf{BP}$ has equalisers if and only if it has coequalisers. Moreover, since $\mathbf{Rel}$ has small products and hence small coproducts as well, $\mathbf{BP}$ has small products and coproducts [6, Section 2.5 (a)] (see also [8, Proposition 3.2]). Hence, the following are equivalent over $\mathbf{ECST}$:

$\mathbf{BP}$ has (co)equalisers; 2. 2.

$\mathbf{BP}$ is (co)complete.

We summarise the equivalents of $\mathrm{\mathrm{NID}}_{1}$.

Theorem 14**.**

The following are equivalent over $\mathbf{ECST}$:

$\mathrm{\mathrm{NID}}_{1}$; 2. 2.

$\mathrm{NID}_{\mathrm{bi}}$; 3. 3.

$\mathbf{Rel}$* has weak (co)equalisers;* 4. 4.

$\mathbf{BP}$* has (co)equalisers;* 5. 5.

$\mathbf{BP}$* is complete and cocomplete.*

5 Finitary $\mathrm{NID}$

In this section, we give some statements equivalent to $\mathrm{\mathrm{NID}}_{<\omega}$.

5.1 Models of geometric theories

The connection between $\mathrm{NID}$ principles and set-generation of the class of models of a game theory was studied by van den Berg [17, Section 4]. In particular, he showed that $\mathrm{\mathrm{NID}}_{<\omega}$ is equivalent to the statement that the class of models of any finitary game theory is set-generated [17, Corollary 4.4]. Since geometric theories form a subclass of finitary game theories, the result in this subsection follows from his result. However, we provide a small refinement, which serves as a stepping-stone for Section 5.2.

Definition 15**.**

A (propositional) geometric theory over a set $S$ is a set of axioms of the form

[TABLE]

where $I$ is a set and $A,B_{i}$ are finitely enumerable subsets of $S$. If $T$ is a geometric theory over $S$, a model of $T$ is a subset $m\subseteq S$ such that

[TABLE]

for each axiom $\operatorname*{{\textstyle{\bigwedge}}}A\vdash\bigvee_{i\in I}\operatorname*{{\textstyle{\bigwedge}}}B_{i}$ in $T$. The class of models of $T$ is denoted by $\mathcal{M}(T)$.

Definition 16**.**

Let $\mathrm{\mathrm{NID}}_{\leq 2}$ be the principle obtained from $\mathrm{NID}$ by restricting the set $R$ to those rules $(a,b)$ where $a$ is a surjective image of $\{0,\dots,n-1\}$ for some $n\leq 2$.

Proposition 17**.**

The following are equivalent over $\mathbf{ECST}$:

$\mathrm{\mathrm{NID}}_{\leq 2}$; 2. 2.

$\mathrm{\mathrm{NID}}_{<\omega}$; 3. 3.

The class of models of any geometric theory is set-generated.

Proof.

Clearly 2 implies 1. The equivalence of 2 and 3 is a corollary of van den Berg [17, Corollary 4.4]. We give a proof for the sake of completeness.

(1 $\rightarrow$ 3) Assume $\mathrm{\mathrm{NID}}_{\leq 2}$. Let $T$ be a geometric theory over a set $S$. Define a set $R$ of rules on $\mathrm{Fin}(S)$ by

[TABLE]

Note that $\mathrm{Fin}(S)$ is a set since $\mathrm{\mathrm{NID}}_{\leq 2}$ implies $\mathrm{\mathrm{NID}}_{0}$ (cf. Remark 3). Note also that $R$ consists of nullary and binary rules.

Let $\mathcal{C}$ be the class of $R$-closed subsets. Define functions $\Phi\colon\mathcal{C}\to\mathcal{M}(T)$ and $\Psi\colon\mathcal{M}(T)\to\mathcal{C}$ by

[TABLE]

It is straightforward to show that these functions are well-defined. We show that they are inverses of each other. First, we have

[TABLE]

for each $m\in\mathcal{M}(T)$. Next, for each $\alpha\in\mathcal{C}$, we have $\alpha\subseteq\mathrm{Fin}(\mathop{\cup}\alpha)=\Psi(\Phi(\alpha))$. Conversely, let $A\in\mathrm{Fin}(\mathop{\cup}\alpha)$, and write $A=\left\{x_{0},\dots,x_{n-1}\right\}$. For each $i<n$, there is a $B_{i}\in\alpha$ such that $x_{i}\in B_{i}$, so $\{x_{i}\}\in\alpha$ by (5.2). Then $A=\left\{x_{0},\dots,x_{n-1}\right\}\in\alpha$ by (5.3) and induction. Note that if $A=\emptyset$, then $A\in\alpha$ by (5.1). Thus, $\Psi(\Phi(\alpha))=\alpha$ for each $\alpha\in\mathcal{C}$. By $\mathrm{\mathrm{NID}}_{\leq 2}$, the class $\mathcal{C}$ has a generating subset $G$. Then, $\left\{\Phi(\alpha)\mid\alpha\in G\right\}$ is a generating subset of $\mathcal{M}(T)$.

(3 $\rightarrow$ 2) Assume that the class of models of any geometric theory is set-generated, and let $R$ be a set of finitary rules on a set $S$. We can identify each rule $(a,b)$ in $R$ with the following geometric axiom over $S$:

[TABLE]

Let $T_{R}$ be the geometric theory over $S$ with the axioms of the above form for each rule in $R$. Then, a model of $T_{R}$ is just an $R$-closed subset of $S$. Hence, the class of $R$-closed subsets is set-generated. ∎

5.2 $n$-ary $\mathrm{NID}$

By setting a uniform bound on the size of the premise of finitary rules, we have countably many fragments of $\mathrm{\mathrm{NID}}_{<\omega}$.

Definition 18**.**

Let $n\in\omega$. A rule $(a,b)$ on a set $S$ is said to be $n$-ary if there exists a surjection $f\colon\left\{0,\dots,n-1\right\}\to a$. The $n$-ary $\mathrm{NID}$, denoted by $\mathrm{\mathrm{NID}}_{n}$, is the principle obtained from $\mathrm{NID}$ by restricting the set $R$ to $n$-ary rules.

Lemma 19**.**

$\mathrm{\mathrm{NID}}_{\leq 2}$* and $\mathrm{\mathrm{NID}}_{2}$ are equivalent over $\mathbf{ECST}$.*

Proof.

It suffices to show that $\mathrm{\mathrm{NID}}_{2}$ implies $\mathrm{\mathrm{NID}}_{\leq 2}$. Assume $\mathrm{\mathrm{NID}}_{2}$, and let $R$ be a set of rules on $S$ consisting of nullary and binary rules. Choose any set $*_{S}$ not in $S$, and define a set $R^{\prime}$ of binary rules on $S\cup\left\{*_{S}\right\}$ by

[TABLE]

By $\mathrm{\mathrm{NID}}_{2}$, the class of $R^{\prime}$-closed subsets of $S\cup\left\{*_{S}\right\}$ has a generating subset $G$. Put

[TABLE]

We show that $H$ generates the class of $R$-closed subsets of $S$. First, let $\alpha\in G$ such that $*_{S}\in\alpha$. Consider any $(a,b)\in R$ such that $a\subseteq\alpha\cap S$. If $a=\emptyset$, then $b\between(\alpha\cap S)$ because $*_{S}\in\alpha$. If $a$ is inhabited, then obviously $b\between(\alpha\cap S)$. Thus, the elements of $H$ are $R$-closed. Let $\beta$ be any $R$-closed subset of $S$, and let $x\in\beta$. Then, $\beta\cup\left\{*_{S}\right\}$ is an $R^{\prime}$-closed subset of $S\cup\left\{*_{S}\right\}$. Thus, there exists an $\alpha\in G$ such that $x\in\alpha\subseteq\beta\cup\left\{*_{S}\right\}$. Then, $x\in\alpha\cap S\subseteq\beta$. ∎

Lemma 19, together with Proposition 17, yields the following.

Proposition 20**.**

The following are equivalent over $\mathbf{ECST}$:

$\mathrm{\mathrm{NID}}_{<\omega}$; 2. 2.

$\mathrm{\mathrm{NID}}_{n}\;(n\geq 2)$.

5.3 Formal points of formal topologies

The initial motivation of the $\mathrm{NID}$ principle comes from the problem of constructing generating subsets in constructive point-free topology, where some of its results require various extensions of $\mathbf{CZF}$. Van den Berg [17, Section 5] and Aczel et al. [2, Section 7.2] illustrated the power of $\mathrm{NID}$ by providing a uniform solution to these problems using $\mathrm{NID}$ and $\mathrm{SGA}$, respectively. With some adjustments to the setting of $\mathbf{ECST}$, we turn some of their results into equivalents of $\mathrm{\mathrm{NID}}_{<\omega}$.

We adopt the following definition of formal topology [14], which is the notion of point-free topology in constructive and predicative foundations.

Definition 21** **(Coquand et al.  [5, Definition

2.1]).

A formal topology is a triple $\mathcal{S}=(S,\leq,\vartriangleleft)$ where $(S,\leq)$ is a preordered set and $\vartriangleleft$ is a relation from $S$ to $\mathrm{Pow}(S)$ such that $\left\{a\in S\mid a\vartriangleleft U\right\}$ is a set for each $U\subseteq S$ and

$a\in U\implies a\vartriangleleft U$, 2. 2.

$a\vartriangleleft U\mathrel{\&}U\vartriangleleft V\implies a\vartriangleleft V$, 3. 3.

$a\vartriangleleft U\mathrel{\&}a\vartriangleleft V\implies a\vartriangleleft U\downarrow V$, 4. 4.

$a\leq b\implies a\vartriangleleft\left\{b\right\}$,

where

[TABLE]

A subset $\alpha\subseteq S$ is a formal point of $\mathcal{S}$ if

1. (P1)

$\alpha$ is inhabited, 2. (P2)

$a,b\in\alpha\implies\alpha\between(a\downarrow b)$, 3. (P3)

$a\in\alpha\mathrel{\&}a\vartriangleleft U\implies\alpha\between U$,

where $a\downarrow b\overset{\mathrm{def}}{\,=\,}\left\{a\right\}\downarrow\left\{b\right\}$. The class of formal points of $\mathcal{S}$ is denoted by $\mathcal{P}\mathit{t}(\mathcal{S})$.

Our main interest is in inductively generated topologies, which allow us to reason about formal topologies using selected sets of axioms.

Definition 22**.**

An axiom-set on a set $S$ is a pair $(I,C)$, where $(I(a))_{a\in S}$ is a family of sets indexed by $S$, and $C$ is a family $(C(a,i))_{a\in S,i\in I(a)}$ of subsets of $S$ indexed by $\sum_{a\in S}I(a)$. A formal topology $(S,\leq,\vartriangleleft)$ is inductively generated by $(I,C)$ if $\vartriangleleft$ is the smallest among the relations $\vartriangleleft^{\prime}$ such that

$a\leq b\vartriangleleft^{\prime}U\implies a\vartriangleleft^{\prime}U$, 2. 2.

$a\vartriangleleft^{\prime}C(a,i)$ for each $i\in I(a)$,

and which makes $(S,\leq,\vartriangleleft^{\prime})$ a formal topology.

A formal point of an inductively generated formal topology can be characterised by an axiom-set, where condition 3 is replaced by

$a\leq b\mathrel{\&}a\in\alpha\implies b\in\alpha$, 2. 2.

$a\in\alpha\implies\alpha\between C(a,i)$ for each $i\in I(a)$.

Remark 23*.*

The construction of an inductively generated formal topology requires $\mathbf{CZF}$ extended with the Regular Extension Axiom [3], which is much stronger than $\mathbf{ECST}$. However, in Proposition 24 and Proposition 28, all we need is a preorder equipped with an axiom-set. Hence, in this paper, we identify inductively generated formal topologies with preorders equipped with axiom-sets, and adopt the notions of formal point and formal topology map formulated in terms of axiom-sets.

Proposition 24**.**

The following are equivalent over $\mathbf{ECST}+\mathrm{FPA}$:

$\mathrm{\mathrm{NID}}_{<\omega}$; 2. 2.

The class of formal points of any inductively generated formal topology is set-generated.

Proof.

(1 $\rightarrow$ 2) Assume $\mathrm{\mathrm{NID}}_{<\omega}$. Let $\mathcal{S}=(S,\leq,\vartriangleleft)$ be a formal topology inductively generated by an axiom-set $(I,C)$ on $S$. Then, formal points of $\mathcal{S}$ are closed subsets of the following set of finitary rules on $S$:

[TABLE]

Thus $\mathcal{P}\mathit{t}(\mathcal{S})$ is set-generated.

(2 $\rightarrow$ 1) Assume that the class of formal points of any inductively generated formal topology is set-generated. Let $R$ be a set of finitary rules on a set $S$. Using $\mathrm{FPA}$, define an axiom-set $(I,C)$ on $\mathrm{Fin}(S)$ by

[TABLE]

Let $\mathcal{S}=(\mathrm{Fin}(S),\supseteq,\vartriangleleft)$ be a formal topology inductively generated by $(I,C)$ using the reverse inclusion order on $\mathrm{Fin}(S)$, and let $\mathcal{C}$ be the class of $R$-closed subsets of $S$. As in the proof of (1 $\rightarrow$ 3) in Proposition 17, one can show that the mappings

[TABLE]

are well-defined and that they are inverses of each other. By the assumption, $\mathcal{P}\mathit{t}(\mathcal{S})$ has a generating subset $G$. Then, $H=\left\{\mathop{\cup}\alpha\mid\alpha\in G\right\}$ is a generating subset of the class of $R$-closed subsets of $S$.

Note that $\mathrm{FPA}$ is needed only in the direction (2 $\rightarrow$ 1). ∎

A formal point is an instance of morphisms between formal topologies.

Definition 25**.**

Let $\mathcal{S}=(S,\leq,\vartriangleleft)$ and $\mathcal{S}^{\prime}=(S^{\prime},\leq^{\prime},\vartriangleleft^{\prime})$ be formal topologies. A relation $r\subseteq S\times S^{\prime}$ is a formal topology map from $\mathcal{S}$ to $\mathcal{S}^{\prime}$ if

1. (FTM1)

$S\vartriangleleft r^{-}S^{\prime}$, 2. (FTM2)

$r^{-}a\downarrow r^{-}b\vartriangleleft r^{-}(a\downarrow^{\prime}b)$, 3. (FTM3)

$a\vartriangleleft^{\prime}U\implies r^{-}a\vartriangleleft r^{-}U$.

Two formal topology maps $r,s\colon\mathcal{S}\to\mathcal{S}^{\prime}$ are equal if $r^{-}a\vartriangleleft s^{-}a$ and $s^{-}a\vartriangleleft r^{-}a$ for all $a\in S^{\prime}$.

If $r\colon\mathcal{S}\to\mathcal{S}^{\prime}$ is a formal topology map and $\mathcal{S}^{\prime}$ is inductively generated by an axiom-set $(I,C)$ on $S^{\prime}$, then the condition 3 can be replaced by

1. (FTM3a)

$a\leq^{\prime}b\implies r^{-}a\vartriangleleft r^{-}b$, 2. (FTM3b)

$r^{-}a\vartriangleleft r^{-}C(a,i)$ for each $i\in I(a)$.

Remark 26*.*

Let $\mathbf{1}$ denote the formal topology $(\left\{*\right\},=,\in)$. Then, a formal point $\alpha$ of a formal topology $\mathcal{S}$ corresponds to a formal topology map $r_{\alpha}\colon\mathbf{1}\to\mathcal{S}$ given by $*\mathrel{r_{\alpha}}a\overset{\mathrm{def}}{\iff}a\in\alpha.$

Set-presentations of formal topologies provide a stronger notion of inductive generation.

Definition 27**.**

A formal topology $\mathcal{S}=(S,\leq,\vartriangleleft)$ is set-presented if there exists an axiom-set $(I,C)$ on $S$ such that

[TABLE]

In this case, $(I,C)$ is called a set-presentation of $\mathcal{S}$.

Proposition 28**.**

The following are equivalent over $\mathbf{ECST}+\mathrm{FPA}$:

$\mathrm{\mathrm{NID}}_{<\omega}$; 2. 2.

The class of formal topology maps from a set-presented formal topology to an inductively generated formal topology is set-generated.

Proof.

Since $\mathbf{1}$ is set-presented, 2 implies 1 by Proposition 24 and Remark 26. Note that we need $\mathrm{FPA}$ in this direction.

Conversely, assume $\mathrm{\mathrm{NID}}_{<\omega}$. Let $\mathcal{S}=(S,\leq,\vartriangleleft)$ be a formal topology with a set-presentation $(I,C)$, and let $\mathcal{T}=(T,\leq^{\prime},\vartriangleleft^{\prime})$ be a formal topology inductively generated by an axiom-set $(J,D)$. Then, a formal topology map $r\colon\mathcal{S}\to\mathcal{T}$ is a model of the following generalised geometric theory over $S\times T$ (cf. Aczel et al. [2, Proposition 7.8]):

[TABLE]

where (5.5), (5.6), (5.7), and (5.8) are derived from 1, 2, a, and b, respectively, using the fact that $\mathcal{S}$ is set-presented by $(I,C)$. The disjunctions such as $\bigvee_{b\in T}(a^{\prime},b)$ must be read as $\bigvee_{b\in T}\bigwedge\left\{(a^{\prime},b)\right\}$. Then, the required conclusion follows from Remark 6. ∎

5.4 Equalisers in the category of concrete spaces

We show that $\mathrm{\mathrm{NID}}_{<\omega}$ is equivalent to the existence of equalisers in the category of concrete spaces, a predicative notion of point-set topology by Sambin [15]. Ishihara and Kawai [8, Section 4] have already shown that the category is complete and cocomplete using $\mathrm{SGA}$. Hence, the essence of this subsection is that the converse holds.

Definition 29**.**

A concrete space is a basic pair $(X,\Vdash,S)$ such that

$X=\mathop{\mathrm{ext}}\nolimits S$, 2. 2.

$\mathop{\mathrm{ext}}\nolimits a\cap\mathop{\mathrm{ext}}\nolimits b=\mathop{\mathrm{ext}}\nolimits(a\downarrow b)$

for all $a,b\in S$, where

[TABLE]

We also define $U\downarrow V\overset{\mathrm{def}}{\,=\,}\bigcup_{a\in U,b\in V}a\downarrow b$ for $U,V\in\mathrm{Pow}(S)$.

Let $\mathcal{X}_{1}=(X_{1},\Vdash_{1},S_{1})$ and $\mathcal{X}_{2}=(X_{2},\Vdash_{2},S_{2})$ be basic pairs. A relation pair $(r,s)\colon\mathcal{X}_{1}\to\mathcal{X}_{2}$ is said to be convergent if

$\mathop{\mathrm{ext}}\nolimits_{1}S_{1}=r^{-}\mathop{\mathrm{ext}}\nolimits_{2}S_{2}$, 2. 2.

$\mathop{\mathrm{ext}}\nolimits_{1}(s^{-}a\downarrow s^{-}b)=r^{-}\mathop{\mathrm{ext}}\nolimits_{2}(a\downarrow b)$,

where $\mathop{\mathrm{ext}}\nolimits_{i}\;(i=1,2)$ is the operator given by (5.9) associated with $\Vdash_{i}\,(i=1,2)$.

Concrete spaces and convergent relation pairs form a subcategory $\mathbf{CSpa}$ of $\mathbf{BP}$. Specifically, $\mathbf{CSpa}$ is a coreflective subcategory of $\mathbf{BP}$.

Proposition 30** **(Ishihara and Kawai [8, Theorem

7.1]).

For any basic pair $\mathcal{X}$, there exist a concrete space $\widetilde{\mathcal{X}}$ and a relation pair $(r,s)\colon\widetilde{\mathcal{X}}\to\mathcal{X}$ such that for any concrete space $\mathcal{Y}$ and a relation pair $(u,v)\colon\mathcal{Y}\to\mathcal{X}$ there exists a unique convergent relation pair $(\widetilde{u},\hat{v})\colon\mathcal{Y}\to\widetilde{\mathcal{X}}$ which makes the following diagram commute:

[TABLE]

Proof.

See Ishihara and Kawai [8, Theorem 7.1] for the details. Their proof can be carried out in $\mathbf{ECST}+\mathrm{FPA}$. ∎

The construction of an equaliser of $\mathbf{CSpa}$ uses the notion of convergent subset.

Definition 31**.**

Let $\mathcal{X}=(X,\Vdash,S)$ be a basic pair. A subset $D\subseteq X$ is said to be convergent if

$D\between\mathop{\mathrm{ext}}\nolimits S$, 2. 2.

$D\between\mathop{\mathrm{ext}}\nolimits a\mathrel{\&}D\between\mathop{\mathrm{ext}}\nolimits b\implies D\between\mathop{\mathrm{ext}}\nolimits(a\downarrow b)$.

The class of convergent subsets of a basic pair $\mathcal{X}$ is denoted by $\mathrm{Conv}(\mathcal{X})$.

An equaliser in $\mathbf{CSpa}$ is constructed from a generating subset of a certain class. In the lemma below, $\Diamond_{2}U$ denotes the set $\left\{a\in S_{2}\mid\exists x\in U\left(x\Vdash_{2}a\right)\right\}$ for each subset $U\subseteq X_{2}$.

Lemma 32** (Ishihara and Kawai [8, Proposition 6.3]).**

Let $\mathcal{X}_{1}=(X_{1},\Vdash_{1},S_{1})$ and $\mathcal{X}_{2}=(X_{2},\Vdash_{2},S_{2})$ be concrete spaces and $(r_{1},s_{1}),(r_{2},s_{2})\colon\mathcal{X}_{1}\to\mathcal{X}_{2}$ be a parallel pair of morphisms in $\mathbf{CSpa}$. If the class defined by

[TABLE]

is set-generated, then the parallel pair has an equaliser.

Proof.

See the proof of Ishihara and Kawai [8, Proposition 6.3]. The proof can be carried out in $\mathbf{ECST}$. ∎

Proposition 33**.**

The following are equivalent over $\mathbf{ECST}+\mathrm{FPA}$:

$\mathrm{\mathrm{NID}}_{<\omega}$; 2. 2.

$\mathbf{CSpa}$* has equalisers.*

Proof.

(1 $\rightarrow$ 2) This has been proved by Ishihara and Kawai [8, Proposition 6.3] using $\mathrm{SGA}$. We reformulate the proof using $\mathrm{\mathrm{NID}}_{<\omega}$. Assume $\mathrm{\mathrm{NID}}_{<\omega}$, and let $(r_{1},s_{1}),(r_{2},s_{2})\colon\mathcal{X}_{1}\to\mathcal{X}_{2}$ be a parallel pair of morphisms in $\mathbf{CSpa}$. It suffices to show that the class $\mathcal{E}$ given in (5.11) is set-generated. Define a set of finitary rules on $X_{1}$ by

[TABLE]

Then, $\mathcal{E}$ is the class of $R$-closed subsets of $X_{1}$, and thus it is set-generated.

(2 $\rightarrow$ 1) Assume that $\mathbf{CSpa}$ has equalisers. In the following, we write $S_{\Delta}$ for the basic pair $(S,\Delta_{S},S)$ on a set $S$ and $\mathrm{Fin}(S)_{\supseteq}$ for the concrete space $(\mathrm{Fin}(S),\supseteq,\mathrm{Fin}(S))$.

Let $R$ be a set of finitary rules on a set $S$. Define relations $r_{1},r_{2}\subseteq\mathrm{Fin}(S)\times R$ by

[TABLE]

Obviously, $(r_{1},r_{1})$ and $(r_{2},r_{2})$ are relation pairs from $\mathrm{Fin}(S)_{\supseteq}$ to $R_{\Delta}$. By Proposition 30, $(r_{1},r_{1})$ and $(r_{2},r_{2})$ determine unique convergent relation pairs $(\widetilde{r_{1}},\widehat{r_{1}})$ and $(\widetilde{r_{2}},\widehat{r_{2}})$ from $\mathrm{Fin}(S)_{\supseteq}$ to $\widetilde{R_{\Delta}}$, respectively, which make the diagram (5.10) commute. Let $(p,q)\colon\mathcal{X}\to\mathrm{Fin}(S)_{\supseteq}$ be an equaliser of $(\widetilde{r_{1}},\widehat{r_{1}})$ and $(\widetilde{r_{2}},\widehat{r_{2}})$ in $\mathbf{CSpa}$, and write $\mathcal{X}=(X,\Vdash,K)$. Define a set $G$ of subsets of $S$ by

[TABLE]

Let $x\in X$ and $(a,b)\in R$ such that $a\subseteq\alpha_{x}$. Since $(p,q)$ is convergent, there exists a $B\in\mathrm{Fin}(S)$ such that $x\mathrel{p}B$ and $B\supseteq a$. Thus, $x\mathrel{(r_{1}\circ p)}(a,b)$. Since $(r_{1},r_{1})\circ(p,q)\sim(r_{2},r_{2})\circ(p,q)$, we have $x\mathrel{(r_{2}\circ p)}(a,b)$, so there exist $C\in\mathrm{Fin}(S)$ and $y\in b$ such that $x\mathrel{p}C$ and $C\supseteq\left\{y\right\}\cup a$. Then $y\in\alpha_{x}$, and so $\alpha_{x}$ is $R$-closed.

We show that $G$ generates the class of $R$-closed subsets of $S$. Let $\beta$ be an arbitrary $R$-closed subset, and $z\in\beta$. Define a relation pair $(p_{\beta},q_{\beta})\colon\left\{*\right\}_{\Delta}\to\mathrm{Fin}(S)_{\supseteq}$ by

[TABLE]

It is easy to see that $(p_{\beta},q_{\beta})$ is convergent. Then, for any $(a,b)\in R$

[TABLE]

Thus $({r_{1}},{r_{1}})\circ(p_{\beta},q_{\beta})\sim({r_{2}},{r_{2}})\circ(p_{\beta},q_{\beta})$, and so $(\widetilde{r_{1}},\widehat{r_{1}})\circ(p_{\beta},q_{\beta})\sim(\widetilde{r_{2}},\widehat{r_{2}})\circ(p_{\beta},q_{\beta})$ by Proposition 30. Hence, $(p_{\beta},q_{\beta})$ factors uniquely through $(p,q)$ via a convergent relation pair $(u,v)\colon\left\{*\right\}_{\Delta}\to\mathcal{X}$ as follows:

[TABLE]

Since $*\mathrel{p_{\beta}}\left\{z\right\}$, there exist $x\in X$ and $B\in\mathrm{Fin}(S)$ such that $*\mathrel{u}x$, $x\mathrel{p}B$, and $B\supseteq\left\{z\right\}$. Thus $z\in\alpha_{x}$. Moreover, for any $y\in\alpha_{x}$, there exists an $A\in\mathrm{Fin}(S)$ such that $\left\{y\right\}\subseteq A\subseteq\beta$. Hence $\alpha_{x}\subseteq\beta$. Therefore, $G$ generates the class of $R$-closed subsets.

Note that $\mathrm{FPA}$ is needed only in the direction (2 $\rightarrow$ 1). ∎

Remark 34*.*

Ishihara and Kawai [8, Proposition 6.4] showed that $\mathbf{CSpa}$ has small products using $\mathrm{SGA}$. Thus, if $\mathbf{CSpa}$ has equalisers, then $\mathbf{CSpa}$ is complete under $\mathrm{FPA}$. Moreover, coequalisers in $\mathbf{CSpa}$ can be constructed exactly as in $\mathbf{BP}$ [8, Lemma 5.2]. Thus, $\mathbf{CSpa}$ is cocomplete under $\mathrm{\mathrm{NID}}_{1}$ and hence under $\mathrm{\mathrm{NID}}_{<\omega}$ as well. Therefore, the following are equivalent over $\mathbf{ECST}+\mathrm{FPA}$:

$\mathbf{CSpa}$ has equalisers; 2. 2.

$\mathbf{CSpa}$ is complete and cocomplete.

We summarise the equivalents of $\mathrm{\mathrm{NID}}_{<\omega}$.

Theorem 35**.**

The following are equivalent over $\mathbf{ECST}$:

$\mathrm{\mathrm{NID}}_{<\omega}$; 2. 2.

$\mathrm{SGA}$; 3. 3.

$\mathrm{\mathrm{NID}}_{n}\;(n\geq 2)$; 4. 4.

The class of models of any geometric theory is set-generated.

Moreover, each of the following statement together with $\mathrm{FPA}$ is equivalent to $\mathrm{\mathrm{NID}}_{<\omega}$ over $\mathbf{ECST}$:

The class of formal points of any inductively generated formal topology is set-generated; 2. 6.

The class of formal topology maps from a set-presented formal topology to an inductively generated formal topology is set-generated; 3. 7.

$\mathbf{CSpa}$* has equalisers;* 4. 8.

$\mathbf{CSpa}$* is complete and cocomplete.*

Acknowledgements

The authors were supported by the Japan Society for the Promotion of Science (JSPS), Core-to-Core Program (A. Advanced Research Networks). The second author was supported by JSPS KAKENHI Grant Number JP16K05251. The third author was supported by Istituto Nazionale di Alta Matematica as a fellow of INdAM-COFUND-2012.