2. Foundations

In this section, we introduce the type-theoretical setting in which we work and then present the type-theoretical formulations of some of the preliminary notions that form the basis of our work. Our type-theoretical conventions follow those of de Jong and Escardó (Reference de Jong, Escardó and Kobayashi2021; Reference de Jong2023) and The Univalent Foundations Program (2013).

We work in Martin-Löf Type Theory with binary sums $(-) + (-)$ , dependent products $\Pi$ , dependent sums $\Sigma$ , the identity type $(-) = (-)$ , and inductive types including the empty type $\mathbf{0}$ , the unit type $\mathbf{1}$ , and the type $\mathsf{List}( {A})$ of lists, or words, over any type $A$ . We denote by $\pi _1$ and $\pi _2$ the first and the second projections of a $\Sigma$ type. We use the abbreviation $\mathbf{2}_{}$ for the sum type $\mathbf{1}_{} + \mathbf{1}_{}$ , and refer to this as the type of Booleans. We adhere to the convention of the HoTT Book (2013) of using $(-) \equiv (-)$ for judgmental equality and $(-) = (-)$ for the identity type.

We work explicitly with universes, for which we adopt the convention of using the variables $\mathscr{U}, \mathscr{V}, \mathscr{W}, \ldots$ The ground universe is denoted $\mathscr{U}_0$ and the successor of a given universe $\mathscr{U}$ is denoted ${ {\mathscr{U}} }^+$ . The least upper bound of two universes is given by the operator $(-) \sqcup (-)$ which is assumed to be associative, commutative, and idempotent. We do not assume that the universes are, or are not, cumulative. Furthermore, ${ {(-)} }^+$ is assumed to distribute over $(-) \sqcup (-)$ . Universes are computed for the given type formers as follows:

• • Given types $X: \mathscr{U}$ and $Y: \mathscr{V}$ , the type $X + Y$ inhabits universe $\mathscr{U} \sqcup \mathscr{V}$ .

• • Given a type $X: \mathscr{U}$ and an $X$ -indexed family $Y: X \to \mathscr{V}$ , both $\sum _{x: X}Y(x)$ and $\prod _{x: X}Y(x)$ inhabit the universe $\mathscr{U} \sqcup \mathscr{V}$ .

• • Given a type $X: \mathscr{U}$ and inhabitants $x, y: X$ , the identity type $x = y$ inhabits universe $\mathscr{U}$ .

• • The type $\mathbb{N}$ of natural numbers inhabits $\mathscr{U}_0$ .

• • The empty type $\mathbf{0}$ and the unit type $\mathbf{1}$ have copies in every universe $\mathscr{U}$ , which we occasionally make explicit using the notations $\mathbf{0}_{\mathscr{U}}$ and $\mathbf{1}_{\mathscr{U}}$ .

• • Given a type $A: \mathscr{U}$ , the type $\mathsf{List}( {A})$ inhabits $\mathscr{U}$ .

We assume the univalence axiom and therefore function extensionality and propositional extensionality. We maintain a careful distinction between structure and property, and reserve logical connectives for propositional types, that is, types $A$ satisfying $\mathsf{isProp}\left ( {A} \right):\equiv \Pi _{({x, y} \,:\, {A})}{x = y}$ . We denote by $\Omega _{\mathscr{U}}$ the type of propositional types in universe $\mathscr{U}$ , that is, $\Omega _{\mathscr{U}}:\equiv \Sigma _{({A} \,:\, {\mathscr{U}})}{\mathsf{isProp}\left ( {A} \right)}$ . A type $A$ is called a set if its identity type is always a proposition, that is, ${x}={y}$ is a propositional type, for every pair of inhabitants $x, y: A$ .

We assume the existence of propositional truncation, given by a type former $\left \| {-} \right \|: \mathscr{U} \to \mathscr{U}$ and a unit operation $\left | {-} \right |: A \to \left \| {A} \right \|$ . It is given by the recursion principle:

\begin{equation*} \Pi _{({P} \,:\, {\mathscr{V}})}{\mathsf{isProp}\left ( {P} \right) \to (X \to P) \to (\left \| {X} \right \| \to P)}\text{.} \end{equation*}

The existential quantification operator is defined using propositional truncation as

\begin{equation*} \exists _{(x \,:\, A)}\ B(x):\equiv \left \| {\Sigma _{({x} \,:\, {A})}{B(x)}} \right \|\text{.} \end{equation*}

When presenting proofs informally, we adopt the following conventions for avoiding ambiguity between propositional and non-propositional types.

• • For the anonymous inhabitation $\left \| {A} \right \|$ of a type $A$ , we say that $A$ is inhabited.

• • For truncated $\Sigma$ types, we use the terminologies there is and there exists.

• • We say specified inhabitant of type $A$ to contrast it with the anonymous inhabitation $\left \| {A} \right \|$ . Similarly, we say there is a specified or chosen element to emphasize that we are using $\Sigma$ instead of $\exists$ .

• • When we use the phrase has a, we take it to mean a specified inhabitant. In contrast, we say has some when talking about an unspecified inhabitant. If we want to completely avoid ambiguity, we prefer to use the more explicit terminology of has specified or has unspecified.

If a given type $A$ is a proposition, it is clear that it is logically equivalent to its own propositional truncation. The converse, however, is not always true: there are types that are logically equivalent to their own propositional truncations, despite not being propositions themselves. Types that satisfy this more general condition of being logically equivalent to their own truncations are said to have split support and have previously been investigated by Kraus et al. (Reference Kraus, Escardó, Coquand and Altenkirch2017).

We also have the notion of local smallness that refers to the equality type always being small with respect to some universe.

We have mentioned the propositional resizing axiom in Section 1. Having formally defined the notion of $\mathscr{V}$ -smallness, we can now give the precise definition of this axiom.

By impredicative mathematics in the context of univalent foundations, we mean the use of the above axiom. Our work here is concerned with the development of locale theory without the use of the above axiom.

We will use the following axiom of set replacement:

Definition 5 (Axiom: set replacement (cf. (Rijke, Reference Rijke2022, Axiom 18.1.18))). Let $f: X \to Y$ be a function from a type $X: \mathscr{U}$ into a set $Y: \mathscr{V}$ . The set replacement axiom says that if $X$ is $ {\mathscr{U}^{\prime}}$ -small and the type $Y$ is locally $\mathscr{W}$ -small then the type ${\mathsf{image}( {f})}$ is $ {(\mathscr{U}^{\prime} \sqcup \mathscr{W})}$ -small, where

\begin{equation*} {\mathsf{image}( {f})}:\equiv \Sigma _{({y} \,:\, {Y})}{\exists _{(x \,:\, X)} {f(x)}={y}}. \end{equation*}

An inhabitant of the set replacement principle can be constructed as a higher inductive type and is thus a theorem in a foundational setting where all higher inductive types are available (such as cubical type theory (Cohen et al. Reference Cohen, Coquand, Huber, Mörtberg and Uustalu2018)). We call this an axiom here since we would like to carefully keep track of which higher inductive types are used in our work. Moreover, de Jong (Reference de Jong2023, Theorem 2.11.24) showed that the set replacement axiom is logically equivalent to the existence of small set quotients, which means that this principle does not require the use of an additional HIT if small set quotients are already available.

We now proceed to define, in the presented type-theoretical setting, some preliminary notions that are fundamental to our development of locale theory.

We will also be talking about families $(I, \alpha)$ where the function $\alpha$ is an embedding. We refer to such families as embedding families.

3. Locales

A locale is a notion of space characterized solely by its lattice of opens. The lattice-theoretic notion abstracting the behavior of a lattice of open subsets is a frame: a lattice with finite meets and arbitrary joins in which the binary meets distribute over arbitrary joins.

Our type-theoretic definition of a frame is parameterized by three universes: (1) for the carrier set, (2) for the order, and (3) for the join-completeness, that is, the index types of families on which the join operation is defined. We adopt the convention of using the universe variables $\mathscr{U}$ , $\mathscr{V}$ , and $\mathscr{W}$ for these, respectively.

As we have done in the definition above, we adopt the shorthand notation $\bigvee _{i: I} x_i$ for the join of a family $( {x_i})_{{i}: {I}}$ . We also adopt the usual abuse of notation and write $L$ instead of $| L |$ .

The reader might have noticed that we have not imposed a sethood condition on the underlying type of a frame in the above definition. The reason for this is that it follows automatically from the antisymmetry condition for partial orders that the underlying type of a frame is a set.

We also note that, in the work of de Jong and Escardó (Reference de Jong, Escardó, Baier and Goubault-Larrecq2021; Reference de Jong, Escardó and Kobayashi2021; Reference de Jong2023), the term $\mathscr{V}$ -dcpo is used for a directed-complete partially ordered set whose directed joins are over $\mathscr{V}$ -families. This terminology leaves the carrier and the order universes implicit, as the completeness universe is the only one relevant to the discussion in most cases.

We gave a highly general definition of the notion of frame in Definition 11: all three universes involved in the definition are permitted to live in separate universes. This generality, however, is never needed in our work since it is known that complete and small lattices cannot be constructed predicatively. This was first shown by Curi (Reference Curi2010a; Reference Curi2010b), who proved that the existence of complete, small lattices is not provable in Aczel and Rathjen’s ConstructiveFootnote ¹ ZF set theory (Reference Aczel and Rathjen2010), which is a predicative system.

In our setting, an analogous result was proved by de Jong and Escardó (Reference de Jong, Escardó and Kobayashi2021), who showed that the existence of a nontrivial complete and small lattice is equivalent to a form of propositional resizing that is known to be independent of type theory—see (de Jong and Escardó Reference de Jong, Escardó and Kobayashi2021, Theorem 35) for details. Their result is more direct and is in the style of reverse constructive mathematics (Ishihara Reference Ishihara2006): they show directly that the existence of a nontrivial complete and small lattice implies a form of propositional resizing. This formally shows that, when we adopt a predicative approach to locale theory, we are forced to work with large and small-complete lattices.

In light of this, we restrict attention to $(\mathscr{U}^+, \mathscr{U}, \mathscr{U})$ frames, which we refer to as large, locally small, and small-complete frames.

We adopt the notational conventions of Mac Lane and Moerdijk (Reference Mac Lane and Moerdijk1994). A locale is a frame considered in the opposite category, denoted $\mathbf{Loc}:\equiv {\mathbf{Frm}}^{\mathsf{op}}$ . To highlight this, we adopt the standard conventions of (1) using the letters $X, Y, Z, \ldots$ (or sometimes $A, B, C, \ldots$ ) for locales and (2) denoting the frame corresponding to a locale $X$ by $\mathscr{O}( X)$ . For variables that range over the frame of opens of a locale $X$ , we use the letters $U, V, W, \ldots$ We use the letters $f$ and $g$ for continuous maps $X \to Y$ of locales. A continuous map $f: X \to Y$ is given by a frame homomorphism $f^*: \mathscr{O}( Y) \rightarrow \mathscr{O}( X)$ .

In point-set topology, a basis for a space $X$ is a collection $\mathscr{B}$ of subsets of $X$ such that every open $U \in \mathscr{O}( X)$ can be decomposed into a union of subsets contained in $\mathscr{B}$ . These subsets are called the basic opens. In the context of point-free topology, the notion of a basis is captured by the lattice-theoretic notion of a generating lattice. In (Johnstone Reference Johnstone2002, p. 548), for example, a basis for a locale $\mathscr{O}( X)$ is defined as a subset $\mathscr{B} \subseteq \mathscr{O}( X)$ such that every element of $\mathscr{O}( X)$ is expressible as a join of members of $\mathscr{B}$ . We now give the formal definition of this notion in our type-theoretical setting.

Notice that the embedding requirement says that the family in consideration does not have repetitions. However, it will often be convenient to work with families that might have repetitions. Such repetitions can be removed (constructively) if necessary, by factoring the family through its image. In the case of a spectral locale, for example, there can be different intensional bases but the extensional basis is unique.

Even though we do not require a basis to be given by a small family, we always work with small bases in our work.

Although the directedness requirement is not essential, as explained in the above remark, we prefer to include it in our definition of basis for technical convenience. Finally, we also note that the directification of a small basis gives a small basis.

4. Spectral and Stone Locales

The standard impredicative definition of a spectral locale is as one in which the compact opens form a basis closed under finite meets (see II.3.2 from (Johnstone Reference Johnstone1982)). To talk about compactness, we define the way-below relation.

The statement $U \ll V$ is thought of as expressing that open $U$ is compact relative to open $V$ . An open is said to be compact if it is compact relative to itself:

4.1 Spectral locales

Definition 23 (Compact locale). A locale $X$ is called compact if the top open $\mathbf{1}_{X}$ is compact.

We denote by $\mathsf{K}( {X})$ the type of compact opens of a locale $X$ . In other words, we define

\begin{equation*} \mathsf{K}( {X}) \quad:\equiv \quad \Sigma _{({U} \,:\, {\mathscr{O}( X)})}{\,{U} \text{ is compact}}. \end{equation*}

Since the locales we consider are large, we have that the type $\mathsf{K}( {X})$ lives in universe $\mathscr{U}^+$ . A spectral locale, also sometimes called coherent in the literature (Johnstone Reference Johnstone1982, II.3.2), is defined in impredicative locale theory as one in which the compact opens (1) are closed under finite meets and (2) form a basis. In our formulation of this notion, we capture the same idea but impose the additional requirement that the type $\mathsf{K}( {X})$ be small.

Definition 24 (Spectral locale). A locale $X$ is called spectral if it satisfies the following conditions.

1. (SP1) It is a compact locale (i.e., the empty meet $\mathbf{1}_{X}$ is compact).

2. (SP2) Compact opens are closed under binary meets.

3. (SP3) For any $U: \mathscr{O}( X)$ , there exists some small directed family $( {K_i})_{{i}: {I}}$ , with each $K_i$ compact, such that $U = \bigvee _{i: I} K_i$ .

4. (SP4) The type $\mathsf{K}( {X})$ is small.

Notice that in the impredicative notion of spectral locale, everything is small. One way of understanding the above definition is that we only make the carrier large, but everything else, including the basis, remains small. This is important for a variety of reasons. Without the smallness condition, it does not seem possible to (1) define Heyting implication, and hence open nuclei, (2) show that frame homomorphisms have right adjoints, and so define the notion of perfect frame homomorphism. Additionally, a natural Stone-type duality to consider is that between small distributive lattices and spectral locales, for which it becomes absolutely necessary that we work with small bases as in the above definition. Notice that the same kind of phenomenon occurs when we work with locally presentable categories, which are large, locally small categories with a small set of objects that suitably generate all objects by taking filtered colimits (Adámek and Rosicky (Reference Adámek and Rosicky1994); Johnstone (Reference Johnstone2002)).

Lemma 25. For any locale $X$ , the statement that $X$ is spectral is a proposition.

Proof. Immediate from Lemma 20 and the fact that the $\Pi$ type over a family of propositions is a proposition.

The natural notion of morphism between spectral locales is that of a spectral map.

Definition 26 (Spectral map). A continuous map $f: X \to Y$ of spectral locales $X$ and $Y$ is called spectral if it reflects compact opens, that is, the open $f^*(V): \mathscr{O}( X)$ is compact whenever the open $V: \mathscr{O}( Y)$ is compact.

We denote by $\mathbf{Spec}$ the category of spectral locales with spectral maps as the morphisms. We recall the following useful fact (see e.g. Escardó (Reference Escardó1999)).

Lemma 27. Let $X$ be a spectral locale and let $U, V: \mathscr{O}( X)$ be two opens.

\begin{equation*}U \le V \quad \text{if and only if} \quad \Pi _{({K} \,:\, {\mathscr{O}( X)})}{\, K \text{ is compact} \to K \le U \to K \le V}.\end{equation*}

In the context of the definition of spectrality that we gave in Definition 24, where we have conditions ensuring that the compact opens form a small basis, it is natural to ask whether the basis in consideration is unique. This is indeed the case, but it is subtler than one might expect and is proved in Theorem 36 below. We first need some preparation.

Given that the conditions in Definition 24 capture the idea of the compact opens behaving like a small basis closed under finite meets, one might wonder if the same notion can be formulated by starting with a small basis closed under finite meets and requiring it to consist of compact opens.

Definition 28 (Intensional spectral basis). An intensional spectral basis for a locale $X$ is a small intensional basis $( {B_i})_{{i}: {I}}$ for $X$ satisfying the following three conditions.

1. (1) For every $i: I$ , the open $B_i$ is compact.

2. (2) There is some index $t: I$ such that $B_t = \mathbf{1}_{X}$ .

3. (3) For any two $i, j: I$ , there exists some $k: I$ such that $B_k = B_i \wedge B_j$ .

We say that the basis is extensional if the map $B: I \to \mathscr{O}( X)$ is an embedding. When we say spectral basis without any qualification, we mean an extensional spectral basis.

In Remark 17, we mentioned that a basis can always be directified by closing it under finite joins. The same applies to spectral bases as well since the join of a finite family of compact opens is compact.

Lemma 29. Given any locale $X$ with an intensional basis $( {B_i})_{{i}: {I}}$ , every compact open of $X$ falls in the basis, that is, for every compact open $K$ , there exists an index $i: I$ such that $B_i = K$ .

Proof. Let $K: \mathscr{O}( X)$ be a compact open. As $( {B_i})_{{i}: {I}}$ is an intensional basis, there must be a specified directed family $( {i_j})_{{j}: {J}}$ on $I$ such that $K = \bigvee _{j: J} B_{i_j}$ . By the compactness of $K$ , there must be some $k: J$ such that $K \le B_{i_k}$ . Clearly, $B_{i_k} \le K$ is also the case, since $K$ is an upper bound of the family $( {B_{i_j}})_{{j}: {J}}$ , which means we have that ${K}={B_{i_k}}$ .

Corollary 30. Given any locale $X$ with an intensional basis $( {B_i})_{{i}: {I}}$ consisting of compact opens, we have an equivalence of types ${{\mathsf{image}( {B})}}\simeq {\mathsf{K}( {X})}$ .

Lemma 31. If a locale $X$ has an unspecified, intensional, spectral, and small basis, then $X$ is spectral.

Proof. First, notice that the conclusion is a proposition since being spectral is a proposition (by Lemma 25). This means that we may appeal to the induction principle of propositional truncation and assume we have a specified intensional spectral basis $( {B_i})_{{i}: {I}}$ . We know that ${{\mathsf{image}( {B})}}\simeq {\mathsf{K}( {X})}$ by Corollary 30, which is to say that the type $\mathsf{K}( {X})$ is small, by the smallness of ${\mathsf{image}( {B})}$ (given by Lemma 18) meaning (SP4) is satisfied.

The top element falls in the basis and is thus compact so (SP1) holds. Given two compact opens $K_1$ and $K_2$ , they must be basic meaning there exist $k_1, k_2$ such that $K_1 = B_{k_1}$ and $K_2 = B_{k_2}$ . Because the basis is closed under binary meets there must be some $k_3$ with $B_{k_3} = K_1 \wedge K_2$ which means condition (SP2) also holds.

For (SP3), consider an open $U: \mathscr{O}( X)$ . We know that there is a specified a small family $( {i_j})_{{j}: {J}}$ of indices such that $U = \bigvee _{j: J} B_{i_j}$ . The subfamily $( {B_{i_j}})_{{j}: {J}}$ is then clearly a small directed family with each $B_{i_j}$ compact which is what we needed.

Lemma 32. If a locale $X$ is spectral, then it has a specified, extensional, and spectral small basis.

Proof. Let $X$ be a spectral locale. We claim that the inclusion $\mathsf{K}( {X}) \hookrightarrow \mathscr{O}( X)$ , which is formally given by the first projection, is an extensional and spectral small basis. The fact that it is small is given by (SP4). By (SP1) and (SP2), we know that this basis contains $\mathbf{1}_{X}$ and is closed under binary meets. It remains to show that it forms a basis. To define a covering family for an open $U: \mathscr{O}( X)$ , we let the index type be the subtype of compact opens below $U$ , which is again small by (SP4), and the family is again given by the first projection. It is clear that $U$ is an upper bound of this family so it remains to show that it is its least upper bound. Consider some $V$ that is an upper bound of this family. By Lemma 27, it suffices to show $K \le U$ implies $ K \le V$ , for every compact open $K$ . Any such compact open $K \le U$ is below $U$ by construction, which implies $K \le V$ since it is an upper bound.

Definition 33 (Extensionalization of an intensional basis). For an intensional basis $( {B_i})_{{i}: {I}}$ on a locale $X$ , its extensionalization is defined by taking the index set to be ${\mathsf{image}( {B})}$ , and the family to be the corestriction of $B$ to its image, which is given by the first projection, and hence is an embedding.

Lemma 34. If $( {B_i})_{{i}: {I}}$ is an intensional spectral basis, then so is its extensionalization.

Lemma 35. From an unspecified intensional spectral basis on a locale $X$ , we can obtain a specified extensional spectral basis.

Proof. Let $X$ be a locale with an unspecified intensional spectral basis. By Lemma 31, we know that $X$ is a spectral locale and therefore that it has a specified extensional basis by Lemma 32.

Recall that a type $X$ is said to have split support if $\left \| {X} \right \| \to X$ (Kraus et al. Reference Kraus, Escardó, Coquand and Altenkirch2017). With this terminology, the above lemma says that the type of intensional spectral bases on a locale $X$ has split support.

Theorem 36. The following are logically equivalent for any locale $X$ .

1. (1) $X$ is spectral.

2. (2) $X$ has an unspecified intensional small spectral basis.

3. (3) $X$ has an unspecified extensional small spectral basis.

4. (4) The inclusion $\mathsf{K}( {X}) \hookrightarrow X$ is an extensional spectral basis, where to an open $U$ we assign the directed family of compact opens below it as in the construction of Lemma. 32

5. (5) $X$ has a specified intensional small spectral basis.

6. (6) $X$ has a specified extensional small spectral basis.

Notice that the first four conditions are propositions, but the last two are not in general.

Proof. We have already established one direction of the logical equivalence (2) $\leftrightarrow$ (5), since (6) $\rightarrow$ (5) and Lemma 35 gives (2) $\rightarrow$ (6). The converse is simply the unit of propositional truncation. Every extensional basis is intensional, and so (6) $\to$ (5) is clear. The implication (1)  $\rightarrow$  (6) is Lemma 32. By Corollary 30, we know that ${\mathsf{image}( {B})}\simeq {\mathsf{K}( {X})}$ for any intensional spectral basis $( {B_i})_{{i}: {I}}$ . This implies that any extensional spectral basis is equivalent to the basis $\mathsf{K}( {X}) \hookrightarrow X$ . Conversely, the inclusion $\mathsf{K}( {X}) \hookrightarrow X$ is always a small extensional basis, so that we get (6) $\leftrightarrow$ (4). We also clearly have (6) $\rightarrow$ (3) $\rightarrow$ (2), which concludes our proof.

4.1.1 Examples of spectral locales

The terminal locale (denoted $\mathbf{1}_{\mathscr{U}}$ ) is the locale defined by the initial frame $\Omega _{\mathscr{U}}$ .

Example 37 (The initial frame). For every universe $\mathscr{U}$ , the (large) type $\Omega _{\mathscr{U}}$ forms a poset under the order defined by implication. This poset is locally small since each proposition $P \Rightarrow Q$ is small. Note that the antisymmetry of this order is exactly propositional extensionality. Furthermore, this poset forms a frame: the top element is the true proposition $\top _{\mathscr{U}}$ , the meet operation is given by the conjunction of two propositions, and the join of a family of propositions $( {P_i})_{{i}: {I}}$ is defined as

\begin{equation*} \bigvee _{i: I} P_i \quad:\equiv \quad \exists _{i: I}{P_i}\text{,} \end{equation*}

which is a small proposition for every small family of propositions.

The fact that these define meet and join operations is easy to see. For the distributivity law, it follows directly from the recursion principle of propositional truncation that

\begin{equation*}P \wedge \exists _{i: I} Q_i \quad \leftrightarrow \quad \exists _{i: I} P \wedge Q_i\text{,}\end{equation*}

for every $P: \Omega _{\mathscr{U}}$ and every small family of propositions $( {Q_i})_{{i}: {I}}$ .

Before proving the universal property of the initial frame, we first establish the following lemma:

Lemma 38. Let $\beta: \mathbf{2}_{\mathscr{U}} \to \Omega _{\mathscr{U}}$ be the function defined as $\beta (\mathsf{0}):\equiv \bot$ , $\beta (\mathsf{1}):\equiv \top$ . Every $P: \Omega _{\mathscr{U}}$ is equal to

1. (1) the join $\bigvee _{p: P} \top _{\mathscr{U}}$ , and

2. (2) the directed join $\bigvee \left \{ { \beta (b) \mid b: \mathbf{2}_{\mathscr{U}}, \beta (b) \le P } \right \}$ .

Therefore, $\beta: \mathbf{2}_{\mathscr{U}} \to \Omega _{\mathscr{U}}$ forms an intensional basis for the terminal locale.

Proof. (1) is easy to see since $P = \top$ if and only if $P$ holds. For (2), we show that ${P}={\bigvee \left \{ { \beta (b) \mid b: \mathbf{2}, b \le P } \right \}}$ by antisymmetry. If $P$ holds, then $P = \top$ and hence $\top \le P$ . For the other direction, observe that $P$ is an upper bound of the family since $\bot$ is trivially below $P$ and $\top \le P$ implies that $P$ holds.

The family $\left \{ { \beta (b) \mid b: \mathbf{2}, \beta (b) \le P } \right \}$ is directed since it is always inhabited by $\bot = \beta (\mathsf{0})$ , and for every pair of Booleans $b_1, b_2: \mathbf{2}$ with $\beta (b_1) \le P$ and $\beta (b_2) \le P$ , we have that

\begin{equation*}\beta (b_1 \vee b_2) = \beta (b_1) \vee \beta (b_2)\text{,}\end{equation*}

and also $\beta (b_1 \vee b_2) \le P$ , meaning the join $\beta (b_1) \vee \beta (b_2)$ falls in the family.

Proposition 39. The frame $\Omega _{\mathscr{U}}$ is the initial object in the category $\mathbf{Frm}_{\mathscr{U}}$ .

Proof. Let $L$ be a frame. By Lemma 38, $P: \Omega _{\mathscr{U}}$ can be expressed as the small join $\bigvee _{p: P} \top$ . This implies that every frame homomorphism $h: \Omega _{\mathscr{U}} \to L$ is necessarily unique as it must satisfy the equality:

\begin{equation*} h(P) = h\left ( {\bigvee _{p: P} \top } \right) = \bigvee _{p: P} h(\top) = \bigvee _{p: P} \mathbf{1}_{L}. \end{equation*}

We accordingly define the unique map $\Omega _{\mathscr{U}} \to L$ as above. It is easy to see that this defines a frame homomorphism.

Lemma 40. The terminal locale $\mathbf{1}_{\mathscr{U}}$ is compact.

Proof. It follows directly from the definition of the join that, for every family $( {P_i})_{{i}: {I}}$ of $\mathscr{U}$ -propositions, we have that $\top \le \bigvee _{i: I} P_i$ if and only if some $P_i$ holds, that is, $\top \le P_i$ for some $i: I$ .

Lemma 41. The terminal locale $\mathbf{1}_{\mathscr{U}}$ is spectral.

Proof. Using Theorem 36, it suffices to show that the basis constructed in Lemma 38 satisfies the conditions from Definition 28.

1. (1) We need to show that both $\beta (\mathsf{0})$ and $\beta (\mathsf{1})$ are compact. We have already established in Lemma 40 that $\beta (\mathsf{1}) = \top$ is compact. The bottom open $\beta (\mathsf{0}) = \bot$ is compact in any frame as shown in Example 22.

2. (2) The top proposition $\top _{\mathscr{U}}$ is obviously contained in the basis since we have $\beta (\mathsf{1}) = \top _{\mathscr{U}}$ .

3. (3) Let $b_1, b_2: \mathbf{2}$ be a pair of Booleans. It is easy to see that the meet $\beta (b_1) \wedge \beta (b_2)$ falls in the basis, since we have $\beta (b_1 \wedge b_2) = \beta (b_1) \wedge \beta (b_2)$ .

5. Predicative Form of the Posetal Adjoint Functor Theorem

We start with the definition of the notion of an adjunction in the simplified context of lattices.

Definition 55. Let $(X, \le _X)$ and $(Y, \le _Y)$ be two preordered sets. An adjunction between $X$ and $Y$ consists of a pair of monotone maps $f: X \to Y$ and $g: Y \to X$ satisfying

\begin{equation*} f \dashv g \quad:\equiv \quad f(x) \le y \leftrightarrow x \le g(y)\ \text{for every $x: X$, $y: Y$.} \end{equation*}

In locale theory, it is standard convention to denote by $f_*: \mathscr{O}( X) \to \mathscr{O}( Y)$ the right adjoint of a frame homomorphism $f^*\,:\,\mathscr{O}( Y)\,\to \,\mathscr{O}( X)$ corresponding to a continuous map of locales $f: X \to Y$ . The right adjoint here is known to always exist thanks to a simple application of the Adjoint Functor Theorem which amounts to the definition:

\begin{equation*} f_*(U) \quad:\equiv \quad \bigvee \{ V: \mathscr{O}( Y) \mid f^*(V) \le U \}. \end{equation*}

In the predicative setting of type theory, however, it is not clear how the right adjoint of a frame homomorphism would be defined as the family $\{ V: \mathscr{O}( Y) \mid f^*(V) \le U \}$ might be large in general. This gives rise to the problem that it is not a priori clear that its join in $\mathscr{O}( Y)$ exists. To resolve this problem, we use the assumption of a small basis. The use of a small basis for the posetal Adjoint Functor Theorem in a predicative setting was independently observed by Tom de Jong (personal communication).

Proof. Let $h: \mathscr{O}( Y) \to \mathscr{O}( X)$ be a monotone map of frames and assume that $Y$ has some small basis.

The forward direction is easy: suppose $h$ has a right adjoint $g: \mathscr{O}( X) \to \mathscr{O}( Y)$ and let $( {U_i})_{{i}: {I}}$ be a family in $\mathscr{O}( Y)$ . By the uniqueness of joins, it is sufficient to show that $h(\bigvee _i U_i)$ is the join of the family $({h(U_i)})_{{i}: {I}}$ . It is clearly an upper bound by the fact that $h$ is monotone. Given any other upper bound $V$ of $({h(U_i)})_{{i}: {I}}$ , we have that $h(\bigvee _i U_i) \le V$ since $h(\bigvee _i U_i) \le V \leftrightarrow \left (\bigvee _i U_i\right) \le g(V)$ meaning it is sufficient to show $U_i \le g(V)$ for each $U_i$ . Since $U_i \le g(V)$ if and only if $h(U_i) \le V$ , we are done as the latter can be seen to hold directly from the fact that $V$ is an upper bound of the family in consideration.

For the converse, suppose ${h(\bigvee _i U_i)}={\bigvee _{i: I} h(U_i)}$ for every small family of opens $( {U_i})_{{i}: {I}}$ . Since right adjoints are unique, we may appeal to the induction principle of propositional truncation and assume we have a specified basis $( {B_i})_{{i}: {I}}$ . We define the right adjoint of $h$ as

\begin{equation*} g(U) \quad:\equiv \quad \bigvee \left \{ B_i \mid i: I, h(B_i) \le U \right \}. \end{equation*}

We need to show that $g$ is the right adjoint of $h$ , hat is,

\begin{equation*}h(V) \le U \leftrightarrow V \le g(U)\end{equation*}

for any two $V: \mathscr{O}( Y)$ , $U: \mathscr{O}( X)$ .

For the $(\!\rightarrow\!)$ direction, assume $h(V) \le U$ . It must be the case that $V = \bigvee _{j: J} B_{i_j}$ for some specified basic covering $( {i_j})_{{j}: {J}}$ . This means that we just have to show $B_{i_j} \le g(U)$ for every $j: J$ , which is the case since $h(B_{i_j}) \le h(V) \le U$ .

For the $(\!\leftarrow\!)$ direction, assume $V \le g(U)$ . This means that we have

\begin{align*} h(V) &\quad \le \quad h(g(U))\\ &\quad =\quad h\left (\bigvee \left \{ B_i \mid i: I, h(B_i) \le U \right \}\right)\\ &\quad =\quad \bigvee \left \{ h(B_i) \mid i: I, h(B_i) \le U \right \}\\ &\quad \le \quad U \end{align*}

so that $h(V) \le U$ , as required.

Our primary use case for the Adjoint Functor Theorem is the construction of Heyting implications.

Definition 58. By Theorem 56, for any continuous map $f: X \to Y$ of locales where $Y$ has an unspecified small basis, the frame homomorphism $f^*: \mathscr{O}( Y) \to \mathscr{O}( X)$ has a right adjoint, denoted by

\begin{equation*} f_*: \mathscr{O}( X) \to \mathscr{O}( Y). \end{equation*}

Let us also record the following fact about perfect maps, which we will need later.

A proof of this fact can be found in (Escardó Reference Escardó1999), and it works in our predicative setting.

In fact, the converse is also true in the case of spectral locales. The proof given in (Escardó Reference Escardó1999) works, once we know that the required adjoints are available, as established above. We include it in order to illustrate this point.

6. Meet-Semilattice of Scott Continuous Nuclei

In this section, we construct the defining frame of the patch locale. It is an observation due to Escardó (Reference Escardó1999, Reference Escardó2001) that the patch locale of a locale $X$ can be defined by the frame of Scott continuous nuclei on $X$ . We start by constructing the meet-semilattice of all nuclei on a frame.

In Section 7, we will work with inflationary and binary-meet-preserving functions that are not necessarily idempotent. Such functions are called prenuclei. We also note that, to show a prenucleus $j$ to be idempotent, it suffices to show $j(j(U)) \le j(U)$ as the other direction follows from inflationarity. In fact, the notion of a nucleus could be defined as a prenucleus satisfying the inequality $j(j(U)) \le j(U)$ , but we define it as in Definition 63 for the sake of simplicity and make implicit use of this fact in our proofs of idempotency.

Proof. The top nucleus is defined as the constant map with value $\mathbf{1}_{X}$ and the meet of two nuclei as $j \wedge k:\equiv U \mapsto j(U) \wedge k(U)$ . It is easy to see that $j \wedge k$ is the greatest lower bound of $j$ and $k$ so it remains to show that $j \wedge k$ satisfies the nucleus laws.

Inflationarity can be seen to be satisfied from the inflationarity of $j$ and $k$ combined with the fact that $j(U) \wedge k(U)$ is the greatest lower bound of $j(U)$ and $k(U)$ . To see that meet preservation holds, let $U, V: \mathscr{O}( X)$ .

\begin{align*} (j \wedge k)(U \wedge V) &\quad =\quad j (U \wedge V) \wedge k (U \wedge V) \\ &\quad =\quad j(U) \wedge j(V) \wedge k(U) \wedge k(V) \\ &\quad =\quad (j(U) \wedge k(U)) \wedge (j(V) \wedge k(V)) \\ &\quad =\quad (j \wedge k)(U) \wedge (j \wedge k)(V) \end{align*}

For idempotency, let $U: \mathscr{O}( X)$ . We have

\begin{align*} (j \wedge k)((j \wedge k)(U)) &\quad =\quad j (j(U) \wedge k(U)) \wedge k(j(U) \wedge k(U)) \\ &\quad =\quad j(j(U)) \wedge j(k(U)) \wedge k(j(U)) \wedge k(k(U)) \\ &\quad \le \quad j(j(U)) \wedge k(k(U)) \\ &\quad =\quad j(U) \wedge k(U) \\ &\quad =\quad (j \wedge k)(U) \end{align*}

We now show that this meet-semilattice can be refined to the Scott continuous nuclei (i.e., the perfect nuclei).

Proof. Let $X$ be a locale. The construction is the same as the one from Lemma 61; the top element is the constant map with value $\mathbf{1}_{X}$ , which is trivially Scott continuous so it remains to show that the meet of two Scott continuous nuclei is Scott continuous. Consider two Scott continuous nuclei $j$ and $k$ on $\mathscr{O}( X)$ , and let $( {U_n})_{{n}: {N}}$ be a directed, small family of opens.

\begin{align*} (j \wedge k) \left ( {\bigvee _{n: N} U_n} \right) &\quad \equiv \quad j \left ( {\bigvee _{m: N} U_m} \right) \wedge k \left ( {\bigvee _{n: N} U_n} \right) && \\ &\quad =\quad \left ( {\bigvee _{m: N} j(U_m)} \right) \wedge \left ( {\bigvee _{n: N} k(U_n)} \right) && [\text{Scott continuity of }j\text{ and }k]\\ &\quad =\quad \bigvee _{(m, n): N \times N} j(U_m) \wedge k(U_n) && [\text{distributivity}]\\ &\quad =\quad \bigvee _{n: N} j(U_n) \wedge k(U_n) && [\dagger] \\ &\quad \equiv \quad \bigvee _{n: N} (j \wedge k)(U_n) && [\text{meet preservation}] \end{align*}

where, for the $(\!\dagger\!)$ step, we use antisymmetry. The $(\!\ge\!)$ direction is immediate. For the $(\!\le\!)$ direction, we need to show that $\bigvee _{(m, n): N \times N} j(U_m) \wedge k(U_n) \le \bigvee _{n: N} j(U_n) \wedge k(U_n)$ , for which it suffices to show that $\bigvee _{n: N} j(U_n) \wedge k(U_n)$ is an upper bound of $\{j(U_m) \wedge k(U_n)\}_{(m, n): N \times N}$ . Let $m, n: N$ be two indices. As $( {U_n})_{{n}: {N}}$ is directed, there must exist some $o$ such that $U_o$ is an upper bound of $\{U_m, U_n\}$ . Using the monotonicity of $j$ and $k$ , we get $j(U_m) \wedge k(U_n) \le j(U_o) \wedge k(U_o) \le \bigvee _{n: N} j(U_n) \wedge k(U_n)$ as desired.

7. Joins in the Frame of Scott Continuous Nuclei

The nontrivial component of the patch frame construction is the join of a family $( {k_i})_{{i}: {I}}$ of Scott continuous nuclei, as the pointwise join fails to be idempotent in general, and not inflationary when the family in consideration is empty. When the family in consideration is directed, however, the pointwise join is idempotent, and so it is the join in the poset of Scott continuous nuclei.

Regarding arbitrary joins, the situation is more complicated. A construction of the join, due to Escardó (Reference Escardó1998), is based on the idea that, if we start with a family $( {k_i})_{{i}: {I}}$ of nuclei, we can consider the family with index type $\mathsf{List}( {I})$ of words over $I$ , defined by

\begin{equation*} (i_0i_1 \cdots i_{n-1}) \mapsto k_{i_{n-1}} \circ \cdots \circ k_{i_1} \circ k_{i_0}. \end{equation*}

This family is easily seen to be directed.

To talk about such families of finite compositions over a given family of (pre)nuclei, we adopt the following notation. Recall that a word over type $I$ is either empty or consists of the insertion of some $i: I$ onto some other word $s$ . We denote the latter by $is$ . Similarly, we write $s t$ for the concatenation of two words $s, t: \mathsf{List}( {I})$ .

To define the join operation, we will use the iterated composition operator $(-)^{*}$ which we define below.

Definition 67 (Family of finite compositions). Given a small family $( {k_i})_{{i}: {I}}$ of nuclei on a given locale $X$ , its family of finite compositions is the family defined by

\begin{equation*} k^*(i_0\cdots i_{n-1}) \quad:\equiv \quad k_{i_{n-1}} \circ \cdots \circ k_{i_0}. \end{equation*}

By an abuse of notation, we omit the superscript “ $*$ ” when there is no possibility of confusion.

The order of composition is actually not important. A finite composition $k_s$ is, in general, not a nucleus. It is, however, always a prenucleus which we prove below.

Proof. If $s$ is the empty list, we are done as it is immediate that the identity function $\mathsf{id}$ is a prenucleus. If $s$ is of the form $i s^{\prime}$ , with $i: I$ , we need to show that $k_{s^{\prime}} \circ k_i$ is a prenucleus. For meet preservation, let $U, V: \mathscr{O}( X)$ .

\begin{align*} (k_{s^{\prime}} \circ k_i)(U \wedge V) &\quad =\quad k_{s^{\prime}}(k_i(U \wedge V)) \\ &\quad =\quad k_{s^{\prime}}(k_i(U) \wedge k_i(V)) && [\text{$k_i$ is a nucleus}] \\ &\quad =\quad k_{s^{\prime}}(k_i(U)) \wedge k_{s^{\prime}}(k_i(V)) && [\text{inductive hypothesis}] \\ &\quad =\quad (k_{s^{\prime}} \circ k_i)(U) \wedge (k_{s^{\prime}} \circ k_i)(V) \end{align*}

For inflationarity, consider some $U: \mathscr{O}( X)$ . We have that $U \le k_i(U) \le k_{s^{\prime}}(k_i(U))$ , by the inflationarity properties of $k_i$ and $k_{s^{\prime}}$ (the latter following from the inductive hypothesis).

Proof. Let $j$ and $( {k_i})_{{i}: {I}}$ be, respectively, a nucleus and a family of nuclei on a locale and assume that $j$ is an upper bound of the family $( {k_i})_{{i}: {I}}$ . We proceed by list induction. Base case: the empty composition is the identity function, and it is easy to see that we have $\mathsf{id}(U) \equiv U \le j(U)$ . Inductive step: consider a list of the form $i s$ , with $i: I$ , and assume inductively that $k_{s} \le j$ .

\begin{align*} k_{s}(k_i(U)) \quad &\le \quad k_{s}(j(U)) && [\text{monotonicity of $k_{s}$ (Lemma 68 and monotonicity of prenuclei)}] \\ \quad &\le \quad j(j(U)) && [\text{inductive hypothesis}] \\ \quad &\le \quad j(U) && [\text{idempotency of $j$}] \end{align*}

We are now ready to construct the join operation in the meet-semilattice of Scott continuous nuclei, hence constructing the defining frame of the patch locale of a locale $X$ .

Theorem 73 (Join of Scott continuous nuclei). Let $( {k_i})_{{i}: {I}}$ be a family of Scott continuous nuclei. The join of $K$ can be calculated as

\begin{equation*} \left (\bigvee _{i: I} k_i\right) (U):\equiv \bigvee _{s: \mathsf{List}( {I})} k_{s}(U). \end{equation*}

Proof. It must be checked that this is (1) indeed the join, (2) is a Scott continuous nucleus, that is, it is inflationary, binary-meet-preserving, idempotent, and Scott continuous. The inflationarity property is direct. For meet preservation, consider some $U, V: \mathscr{O}( X)$ . We have:

\begin{align*} \left (\bigvee _{i: I} k_i\right)(U \wedge V) &\quad =\quad \bigvee _{s: \mathsf{List}( {I})} k_{s}(U \wedge V) \\ &\quad =\quad \bigvee _{s: \mathsf{List}( {I})} k_{s}(U) \wedge k_{s}(V) && [\text{Lemma 70}] \\ &\quad =\quad \bigvee _{s,t: \mathsf{List}( {I})} k_{s}(U) \wedge k_{t}(V) && [\dagger] \\ &\quad =\quad \left ( {\bigvee _{s: \mathsf{List}( {I})} k_{s}(U)} \right) \wedge \left ( {\bigvee _{t: \mathsf{List}( {I})} k_{t}(V)} \right) && [\text{distributivity}] \\ &\quad =\quad \left ( {\bigvee _{i: I} k_i} \right)(U) \wedge \left ( {\bigvee _{i: I} k_i} \right)(V) \end{align*}

where step ( $\dagger$ ) uses antisymmetry. The $(\le)$ direction is direct whereas for the $(\ge)$ direction, we show that $\bigvee _{s: \mathsf{List}( {I})} k_{s}(U) \wedge k_{s}(V)$ is an upper bound of the family $( {k_{s}(U) \wedge k_{t}(V)})_{{s,t}: {\mathsf{List}( {I})}}.$ Consider arbitrary $s, t: \mathsf{List}( {I})$ . By the directedness of the family of finite compositions, we know that there exists some $u: \mathsf{List}( {I})$ such that $k_{u}$ is an upper bound of $\{k_{s}, k_{t}\}$ . We then have

\begin{equation*} k_{s}(U) \wedge k_{t}(V) \le k_{u}(U) \wedge k_{u}(V) \le \bigvee _{s: \mathsf{List}( {I})} k_{s}(U) \wedge k_{s}(V). \end{equation*}

For idempotency, let $U: \mathscr{O}( X)$ .

[\begin{align*} \left (\bigvee _{i} k_i\right)\left (\left (\bigvee _{i} k_i\right)(U)\right) &\quad \equiv \quad \bigvee _{s: \mathsf{List}( {I})} k_{s}\left ( \bigvee _{t: \mathsf{List}( {I})} k_{t}(U) \right) \\ &\quad =\quad \bigvee _{s: \mathsf{List}( {I})} \bigvee _{t: \mathsf{List}( {I})} k_{s}(k_{t}(U)) & [\text{Lemma}\ {70}]\\ &\quad \le \quad \bigvee _{s,t: \mathsf{List}( {I})} k_{s}(k_{t}(U)) & \, \\ &\quad \le \quad \bigvee _{s: \mathsf{List}( {I})} k_{s}(U) & [\dagger]\\ &\quad \equiv \quad \left ( {\bigvee _i k_i} \right)(U), \end{align*}

where for the step ( $\dagger$ ) it suffices to show that $\bigvee _{s: \mathsf{List}( {I})} k_{s}(U)$ is an upper bound of the family $( {k_{s}(k_{t}(U))})_{{s,t}: {\mathsf{List}( {I})}}.$ The prenucleus $k_{s t}$ is an upper bound of $k_{s}$ and $k_{t}$ (as in Lemma 71). We have that

\begin{equation*}k_{t}(k_{s}(U)) = k_{s t}(U) \le \bigvee _{u: \mathsf{List}( {I})}k_{u}(U)\text{.}\end{equation*}

For Scott continuity, let $( {U_j})_{{j}: {J}}$ be a directed family over $\mathscr{O}( X)$ . We then have:

\begin{align*} \left (\bigvee _{i: I} k_i\right)\left (\bigvee _{j: J} U_j\right) &\quad \equiv \quad \bigvee _{s: \mathsf{List}( {I})} k_{s}\left (\bigvee _{j: J} U_j\right) \\ &\quad =\quad \bigvee _{s: \mathsf{List}( {I})} \bigvee _{j: J} k_{s}(U_j) && [\text{Lemma}\ {70}] \\ &\quad =\quad \bigvee _{j: J} \bigvee _{s: \mathsf{List}( {I})} k_{s}(U_j) && [\text{joins commute}] \\ &\quad \equiv \quad \bigvee _{j: J} \left (\bigvee _{i: I} k_i\right)(U_j) \end{align*}

as required.

The fact that $\bigvee _i k_i$ is an upper bound of $( {k_i})_{{i}: {I}}$ is easy to verify. To see that it is the least upper bound, consider a Scott continuous nucleus $j$ that is an upper bound of $( {k_i})_{{i}: {I}}$ . Let $U: \mathscr{O}( X)$ . We need to show that $(\bigvee _i k_i)(U) \le j(U)$ . We know by Lemma 69 that $j$ is an upper bound of the family of finite compositions, since it is an upper bound of $( {k_i})_{{i}: {I}}$ , which is to say $k_{s}(U) \le j(U)$ for every $s: \mathsf{List}( {I})$ , that is, $j(U)$ is an upper bound of the family $( {k_s(U)})_{{s}: {\mathsf{List}( {I})}}$ . Since $\left (\bigvee _i k_i\right)(U)$ is the least upper bound of this family by definition, it follows that it is below $j(U)$ .

We use Lemma 72 to prove the distributivity law.

Lemma 74 (Distributivity). For any Scott continuous nucleus $j$ and any family $( {k_i})_{{i}: {I}}$ of Scott continuous nuclei, we have the equality:

\begin{equation*} j \wedge \left (\bigvee _{i: I} k_i\right) = \bigvee _{i: I} j \wedge k_i. \end{equation*}

It follows that the Scott continuous nuclei form a frame.

Notice that the truth value of the relation $j \le k$ between two Scott continuous nuclei lives by default in universe ${ {\mathscr{U}} }^+$ . However, it always has an equivalent copy in the universe $\mathscr{U}$ if the locale in consideration is spectral.

Lemma 76. For any spectral locale $X$ and any two Scott continuous nuclei $j, k: \mathscr{O}( X) \to \mathscr{O}( X)$ , we have that

\begin{equation*} j \le k \quad \text{if and only if}\quad j(K) \le k(K)\,\,\text{for all }\,\, K:\mathsf{K}( {X}), \end{equation*}

and hence $\mathsf{Patch}(X)$ is locally small.

Proof. The usual pointwise ordering obviously implies the basic ordering so we address the other direction. Let $j$ and $k$ be two Scott continuous nuclei on a spectral locale $X$ and assume $j(K) \le k(K)$ , for all $K: \mathsf{K}( {X})$ . We need to show that $j(U) \le k(U)$ for every open $U$ so let $U: \mathscr{O}( X)$ . By (SP3), $U$ can be decomposed as $U = \bigvee _{i: I} K_i$ for some directed covering family $( {K_i})_{{i}: {I}}$ consisting of compact opens. We then have $j(\bigvee _{i: I} K_i) = \bigvee _{i: I} j(K_i)$ by Scott continuity and

\begin{equation*}\bigvee _{i: I} j(K_i) \le \bigvee _{i: I} k(K_i)\end{equation*}

since $j(K_i) \le k(K_i)$ for every $i: I$ . Finally, because the quantification is over the small type $\mathsf{K}( {X})$ , the proposition “ $j(K) \le k(K)\text{ for all }K:\mathsf{K}( {X})$ ” is small.

8. The Coreflection Property of $\mathsf{Patch}$

We prove in this section that our construction of $\mathsf{Patch}$ has the desired universal property: it exhibits $\mathbf{Stone}$ as a coreflective subcategory of $\mathbf{Spec}$ . The notions of closed and open nuclei are crucial for proving this universal property. We first give the definitions of these. Let $U$ be an open of a locale $X$ .

1. (1) The closed nucleus induced by $U$ is the map $V \mapsto U \vee V$ .

2. (2) The open nucleus induced by $U$ is the map $V \mapsto U \Rightarrow V$ .

We denote the closed nucleus induced by open $U$ by $\mathbf{c}(U)$ and the open nucleus induced by $U$ by $\mathbf{o}( U)$ .

Consider the closed nucleus formation operation $U \mapsto \mathbf{c}(U)$ . This defines a frame homomorphism $\mathscr{O}( X) \to \mathscr{O}( \mathsf{Patch}(X))$ , whose corresponding continuous map is denoted

\begin{equation*}\varepsilon: \mathsf{Patch}(X) \to X\text{.}\end{equation*}

In Lemma 80, we will show that this map is perfect. Before Lemma 80, however, we record an auxiliary result:

The proof of Lemma 79 can be found in Escardó (Reference Escardó1999). It is omitted here as it is unchanged in our predicative setting.

Proof. We have to show that the right adjoint $\varepsilon _*$ of $\mathbf{c}(-)$ is Scott continuous. Let $( {k_i})_{{i}: {I}}$ be a directed family of Scott continuous nuclei. Thanks to Lemma 79, it suffices to show

\begin{equation*}\left (\bigvee _{i: I} k_i\right)(\mathbf{0}) = \bigvee _{i: I} \varepsilon _*(k_i).\end{equation*}

By Lemma 66, we have that $\left (\bigvee _{i: I} k_i\right)(\mathbf{0}) = \bigvee _{i: I} k_i(\mathbf{0})$ .

We will later show that the perfect map $\varepsilon: \mathsf{Patch}(X) \to X$ is the counit of the coreflection of interest.

8.1 ${\mathsf{\rm Patch}}$ is Stone

Before we proceed to showing that the patch locale has the desired universal property, we first need to show that $\mathsf{Patch}(X)$ is Stone for any spectral locale $X$ . For this purpose, we need to (1) show that it is compact and (2) construct a basis for it consisting of clopens. We start with compactness.

Lemma 81. For every spectral locale $X$ , the locale $\mathsf{Patch}(X)$ is compact.

Proof. Recall that the top element $\mathbf{1}$ of $\mathsf{Patch}(X)$ is defined as $\mathbf{1}_{\mathsf{Patch}}:\equiv U \mapsto \mathbf{1}_{X}$ . Because $\varepsilon ^*$ is a frame homomorphism, it must be the case that $\mathbf{1}_{\mathsf{Patch}} = \varepsilon ^*(\mathbf{1}_{X})$ , meaning it suffices to show $\varepsilon ^*(\mathbf{1}_{X}) \ll \varepsilon ^*(\mathbf{1}_{X})$ . By Lemma 60, it suffices to show $\mathbf{1}_{X} \ll \mathbf{1}_{X}$ which is immediate by (SP1).

To construct a basis consisting of clopens, we will use the following fact, which was already mentioned above:

Lemma 82. The open nucleus $\mathbf{o}( U)$ is the Boolean complement of the closed nucleus $\mathbf{c}(U)$ .

Lemma 83. Let $X$ be a spectral locale. Given any Scott continuous nucleus $j: \mathscr{O}( \mathsf{Patch}(X))$ , we have that

\begin{equation*} j \quad =\quad \bigvee _{K: \mathsf{K}( {X})} \mathbf{c}(j(K)) \wedge \mathbf{o}( K) \quad =\quad \bigvee _{\stackrel {K_1,\mkern +2mu K_2: \mathsf{K}( {X})}{K_1 \le j(K_2)}} \mathbf{c}(K_1) \wedge \mathbf{o}( K_2). \end{equation*}

Proof. The second equality in the statement is clear, so let us show the first one. We use the fact, proved in (Johnstone Reference Johnstone1982, Lemma II.2.7), that for any nucleus $j$ on any locale,

\begin{equation*} j = \bigvee _{U: \mathscr{O}( X)} \mathbf{c}(j(U)) \wedge \mathbf{o}( U). \end{equation*}

Suppose now additionally that $X$ is spectral and the nucleus $j$ is Scott continuous. The inequality $\bigvee _{K: \mathsf{K}( {X})} \mathbf{c}(j(K)) \wedge \mathbf{o}( K) \leq \bigvee _{U: \mathscr{O}( X)} \mathbf{c}(j(U)) \wedge \mathbf{o}( U) = j$ is trivial, so let us show the reverse one. Let $K: \mathsf{K}( {X})$ and notice that $ \left (\mathbf{c}(j(K)) \wedge \mathbf{o}( K)\right)(K)= (j(K)\vee K)\wedge (K\Rightarrow K)= j(K)$ . Therefore

\begin{equation*}j(K)= \left (\mathbf{c}(j(K)) \wedge \mathbf{o}( K)\right)(K) \leq \left ( \bigvee _{K^{\prime}: \mathsf{K}( {X})} \mathbf{c}(j(K^{\prime})) \wedge \mathbf{o}( K^{\prime}) \right) (K),\end{equation*}

so the required relation follows from Lemma 76.

Theorem 84. $\mathsf{Patch}(X)$ is a Stone locale, for every spectral locale $X$ .

Proof. Compactness (ST1) was given in Lemma 81. For (ST2), let $j: \mathscr{O}( \mathsf{Patch}(X))$ . We take the intensional basis given by the family $\left ( \mathbf{c}(K_1) \wedge \mathbf{o}( K_2) \right)_{(K_1, K_2): \mathsf{K}( {X}) \times \mathsf{K}( {X})}\text{.}$ For each Scott continuous nucleus $j$ , we define its basic covering family to be $\gamma: I_j \to \mathscr{O}( \mathsf{Patch}(X))$ , defined by

\begin{align*} I_j \quad &:\equiv \quad \Sigma _{({K_1,K_2} \,:\, {\mathsf{K}( {X})})}{(K_1 \le j(K_2))}\text{, and}\\ \gamma (K_1, K_2)\quad &:\equiv \quad \mathbf{c}(K_1) \wedge \mathbf{o}( K_2)\text{.} \end{align*}

Even though this basic covering family is not a priori directed, we know that it can be directified as explained in Remark 17. We therefore obtain a specified, small, intensional basis of clopens by Lemma 83. We denote the directed form of the basis by $B^{\uparrow }$ which is small by Remark 17 since $\mathsf{K}( {X})$ is small by (SP4). Since clopens are compact in compact locales by Lemma 47, the clopens fall in the basis by Lemma 29. Therefore, we get an equivalence of types

\begin{equation*}\mathsf{C}( X) \simeq {\mathsf{image}( {B^{\uparrow }})}\text{,}\end{equation*}

where $B^{\uparrow }$ denotes the closure of the above basis under finite joins (as explained in Remark 17). This concludes (ST3) by Lemma 18.

8.2 The universal property of ${\mathsf{\rm Patch}}$

We now show that $\mathsf{Patch}$ is the right adjoint to the inclusion $\mathbf{Stone} \hookrightarrow \mathbf{Spec}$ .

Lemma 85. Given any spectral map $f: X \to A$ from a Stone locale into a spectral locale, define a map $\bar {f}^*: \mathscr{O}( {\mathsf{Patch}(A)}) \to \mathscr{O}( X)$ by

\begin{equation*} \bar {f}^*(j) \quad:\equiv \quad \bigvee _{K: \mathsf{K}( {A})} f^*(j(K)) \wedge \neg f^*(K). \end{equation*}

Then, the map $\bar {f}_*: \mathscr{O}( X) \to \mathscr{O}( {\mathsf{Patch}(A)})$ defined by

\begin{equation*} \bar {f}_* (V) \quad:\equiv \quad f_* \circ \mathbf{c}(V) \circ f^* \end{equation*}

is the right adjoint of $\bar {f}^*$ .

Proof. First, note that $f_* \circ \mathbf{c}(V) \circ f^*$ is indeed a Scott continuous nucleus, and both $\bar {f}^*$ and $\bar {f}_*$ are clearly monotone. Let us first prove the forward implication. Assume that for $j: \mathscr{O}( {\mathsf{Patch}(A)})$ and $V: \mathscr{O}( X)$ , the relation $\bar {f}^*(j)\leq V$ holds. By Lemma 76, in order to show that $j \leq \bar {f}_*(V)$ , it suffices to show that for any compact open $K: \mathsf{K}( {A})$ , the inequality $j(K) \leq \bar {f}_*(V)(K)$ holds. Hence, let $K: \mathsf{K}( {A})$ , and note that

\begin{equation*} f^*(j(K))\wedge \neg f^*(K) \leq \bar {f}^*(j)\leq V\text{.} \end{equation*}

Notice that $f^*(K)$ is clopen, by the fact that $f^*$ is a spectral map and Lemma 48. It is therefore complemented in the lattice $\mathscr{O}( X)$ , and so we have $f^*(j(K))\leq V\vee f^*(K) = \mathbf{c}(V)(f^*(K))$ , which by adjunction yields $j(K)\leq f_*(\mathbf{c}(V)(f^*(K)))= \bar {f}_{*}(K)$ , as required.

Let us now show the reverse implication. Let $ j: \mathscr{O}( {\mathsf{Patch}(A)})$ and $V: \mathscr{O}( X)$ and assume that $j\leq \bar {f}_*(V)$ . Once again, by the definition of the ordering on $\mathscr{O}( {\mathsf{Patch}(A)})$ , for all $K: \mathsf{K}( {A})$ we have $j(K) \leq f_*(V \vee f^*(K))$ , which by adjunction equivalently yields $f^*(j(K))\leq V \vee f^*(K)$ . Since $f^*(K)$ is clopen, and hence complemented in the lattice $\mathscr{O}( X)$ it follows that $f^*(j(K)) \wedge \neg f^*(K) \leq V$ . Hence, $\bar {f}^*(j)\leq V$ .

Theorem 86. For every spectral map $f: X \to A$ from a Stone locale into a spectral locale, there exists a unique continuous map $\bar {f}: X \to \mathsf{Patch}(A)$ satisfying $\varepsilon \circ \bar {f} = f$ , as illustrated in the following diagram in $\mathbf{Spec}$ :

Proof. Assume that a locale map $\bar {f}: X \to \mathsf{Patch}(A)$ satisfies the condition in the theorem. Then, for any $j: \mathscr{O}( {\mathsf{Patch}(A)})$ , one has

\begin{align*} \bar {f}^*(j) &\quad =\quad \bar {f}^* \left ( \bigvee _{K: \mathsf{K}( {A})}\mathbf{c}(j(K)) \wedge \mathbf{o}( K)\right) && [\text{Lemma}\ {83}]\\ &\quad =\quad \bigvee _{K: \mathsf{K}( {A})} \bar {f}^*\left ( \mathbf{c}(j(K)) \wedge \mathbf{o}( K)\right) && [\bar {f}^*\text{ preserves small joins}] \\ &\quad =\quad \bigvee _{K: \mathsf{K}( {A})} \bar {f}^*\left ( \mathbf{c}(j(K))\right) \wedge \neg \bar {f}^*\left (\mathbf{c}(K)\right) && [\bar {f}^*\text{ preserves binary meets and complements}] \\ & \quad =\quad \bigvee _{K: \mathsf{K}( {A})} f^*\left ( j(K)\right) \wedge \neg f^*\left ( K\right) && [\text{commutativity of the diagram}] \end{align*}

and hence $\bar {f}$ is uniquely determined. If we now define a monotone map $\bar {f}^*: \mathscr{O}( {\mathsf{Patch}(A)}) \to \mathscr{O}( X)$ by

\begin{equation*} \bar {f}^*(j) \quad:\equiv \quad \bigvee _{K: \mathsf{K}( {A})} f^*(j(K)) \wedge \neg f^*(K), \end{equation*}

it is easy to show it preserves the top element (namely the top nucleus with constant value $\mathbf{1}_{A}$ ) because $\mathbf{0}_A$ is compact. It also preserves binary (pointwise) meets as

\begin{align*} & \bar {f}^*(j_1)\wedge \bar {f}^*(j_2)\\ &= \bigvee _{K_1,K_2: \mathsf{K}( {A})} f^*\left ( j_1(K_1)\right) \wedge \neg f^*\left ( K_1\right) \wedge f^*\left ( j_2(K_2)\right) \wedge \neg f^*\left ( K_2\right) && [\text{distributivity}] \\ &= \bigvee _{K_1,K_2: \mathsf{K}( {A})} f^*\left ( j_1(K_1) \wedge j_2(K_2)\right) \wedge \neg f^*\left ( K_1\right) \wedge \neg f^*\left ( K_2\right) && [\text{$f^*$ preserves binary meets}] \\ &\leq\! \bigvee _{K_1,K_2: \mathsf{K}( {A})}\! f^* \!\left ( j_1(K_1\vee K_2) \wedge j_2(K_1\vee K_2)\right) \wedge \neg f^*\!\left ( K_1\right) \wedge \neg f^*\!\left ( K_2\right)\!\! && [\text{monotonicity}] \\ &= \bigvee _{K_1,K_2: \mathsf{K}( {A})} f^*\left ( (j_1\wedge j_2)(K_1\vee K_2)\right) \wedge \neg f^*\left ( K_1\right) \wedge \neg f^*\left ( K_2\right) && \, \\ &= \bigvee _{K_1,K_2: \mathsf{K}( {A})} f^*\left ( (j_1\wedge j_2)(K_1\vee K_2)\right) \wedge \neg f^*\left ( K_1 \vee K_2\right) && [\text{De Morgan law}] \\ &= \bar {f}^*(j_1\wedge j_2). && [\text{$\mathsf{K}( {X})$ closed under $(-) \vee (-)$}] \end{align*}

Moreover, Lemma 85 and Theorem 56 ensure that $\bar {f}^*$ preserves small joins and so it is a frame homomorphism.

Let us finally show that $\bar {f}$ makes the diagram commute. Since compact opens from a small basis for $A$ , it suffices to show that $\bar {f}^*(\mathbf{c}(K))=f^*(K)$ for any $K: \mathsf{K}( {A})$ . Let $K: \mathsf{K}( {A})$ and note that

\begin{align*} \bar {f}^*(\mathbf{c}(K)) &\quad \equiv \quad \bigvee _{K^{\prime}: \mathsf{K}( {A})} f^*\left ( K\vee K^{\prime}\right) \wedge \neg f^*\left ( K^{\prime}\right) \\ &\quad =\quad \bigvee _{K^{\prime}: \mathsf{K}( {A})} f^*\left ( K\right) \wedge \neg f^*\left ( K^{\prime}\right) && [\text{$f^*$ preserves binary joins}] \\ &\quad = \quad f^*(K)\wedge \bigvee _{K^{\prime}: \mathsf{K}( {A})} \neg f^*\left ( K^{\prime}\right) && [\text{distributivity}] \\ & \quad =\quad f^*(K)\wedge \mathbf{1}_{X}&& [\mathbf{0}_{X}\ \text{is compact}] \\ & \quad =\quad f^*(K), \end{align*}

as required.

9. Summary and Discussion

We have constructed the patch locale of a spectral locale in the predicative and constructive setting of univalent type theory. Furthermore, we have shown that the patch functor

\begin{equation*}\mathsf{Patch}: \mathbf{Spec} \rightarrow \mathbf{Stone}\end{equation*}

is the right adjoint to the inclusion $\mathbf{Stone} \hookrightarrow \mathbf{Spec}$ , which is to say that $\mathsf{Patch}$ exhibits the category $\mathbf{Stone}$ as a coreflective subcategory of $\mathbf{Spec}$ . As we have elaborated on in Section 4, answering this question in a predicative setting involves several new ingredients, compared to Escardó (Reference Escardó1999, Reference Escardó2001):

1. (1) We have reformulated the notions of spectrality and Stone-ness in our predicative type-theoretic setting and have shown that crucial topological facts about these notions remain valid under these reformulations.

2. (2) We have developed several predicative formulations of the notion of spectral (resp. Stone) locale and have shown that they are equivalent in Theorem 36 (resp. Theorem 54).

3. (3) In Escardó’s (Reference Escardó1999) construction of the patch locale, the proof of the universal property relies on the existence of the frame of all nuclei. As it is not clear that the poset of all nuclei can be shown to form a frame predicatively, we developed a new proof of the universal property using Lemma 85, which is completely independent of the existence of the frame of all nuclei.

We have formalized all of our development, most importantly Theorem 84 and Theorem 86. The formalization has been carried out by the third-named author (Tosun Reference Tosun2023) as part of the TypeTopology library (Escardó and contributors, 2018).

In previous work (Escardó Reference Escardó1999, Reference Escardó2001), which forms the basis of the present work, the patch construction was used to

1. (1) exhibit $\mathbf{Stone}$ as a coreflective subcategory of $\mathbf{Spec}$ , and

2. (2) exhibit the category of compact regular locales and continuous maps as a coreflective subcategory of stably compact locales and perfect maps.

In our work, we have focused on item (1). The question of taking a predicative approach to item (2) was previously tackled by Coquand and Zhang (Reference Coquand and Zhang2003) using formal topology. We conjecture that it is possible to instead use the approach we have presented here, namely, working with locales with small bases and constructing the patch as the frame of Scott continuous nuclei. It should also be possible to show predicatively that the category of small distributive lattices is dually equivalent to the category of spectral locales as defined in this paper.