Convention 1.7. Hereafter, the unqualified term “theory” shall always mean a geometric theory. If we wish to refer to theories from classical first-order logic, we shall signpost this explicitly.

Convention 1.13 (Geometricity).

The term “geometry” shows up in two very different settings—first, as in non-Archimedean geometry, and second as in geometric mathematics, the regime of constructive mathematics described in Remark 1.12 (2). To reduce confusion, we say that a proof satisfies geometricity if it adheres to the constraints of geometric mathematics in the second sense.

Discussion 1.15 (Points of $\mathbb {T}_{\mathbb {R}}$ ).

Unlike point-set topology, where one first specifies the set of real numbers and then equips it with a topology, the localic approach begins with a lattice of opens. Here, the symbols $P_{q,r}$ generate a frame behaving like the lattice of rational open intervals. The usual Dedekind real numbers are then recovered as the completely prime filters of this frame.

Convention 1.19. We denote the space of Dedekinds as $\mathbb {R}$ . By Remark 1.18, the points of $\mathbb {R}$ may be regarded as models of $\mathbb {T}_{\mathbb {R}}$ or the theory described by Definition 1.17. For suggestiveness, whenever we want to work with a generic Dedekind x, we write $x\in \mathbb {R}$ .

Convention 1.20 (Topology on upper reals).

Denote $\overleftarrow {[-\infty ,\infty )}$ to be the space of upper reals excluding $\infty $ .Footnote ¹²

• • The points of $\overleftarrow {[-\infty ,\infty )}$ are subsets of $\mathbb {Q}$ , ordered by subset inclusion as below:

  $$ \begin{align*}x\sqsubseteq y \iff x\geq y.\end{align*} $$

  We call $\sqsubseteq $ the specialisation order, which reverses numerical order.

• • The topology on $\overleftarrow {[-\infty ,\infty )}$ is the Scott topology, generated by the basic opens

  $$ \begin{align*}[-\infty, q):=\{x \mid x< q\}\qquad q\in\mathbb{Q} ,\end{align*} $$

  i.e., the rational rays (“the upper reals strictly less than q”).Footnote ¹³

• • To reflect the reversal of order, we decorate one-sided intervals with arrows indicating the direction of refinement under $\sqsubseteq $ , e.g., $\overleftarrow {[0,\infty )}$ or $\overleftarrow {[-\infty ,\infty )}$ .Footnote ¹⁴

Fact 1.21 (One-sided vs. Dedekind Reals).

1. (i) There is a natural map

   $$\begin{align*}\mathbb{R} \longrightarrow \overleftarrow{[-\infty,\infty)}, \end{align*}$$

   sending a Dedekind real $x=(L_x,R_x)$ to its right cut $R_x$ . The map $x\mapsto R_x$ is monic [Reference Ng and Vickers26, Corollary 1.27], but not a topological embedding: the real line $\mathbb {R}$ is Hausdorff, whereas the Scott topology on $\overleftarrow {[-\infty ,\infty )}$ is non-Hausdorff. Thus $\mathbb {R}$ cannot be realised as a subspace of the upper reals. The case for lower reals is analogous.

2. (ii) Given a non-empty set of reals $\{x_i\}_{i\in I}$ , define

   $$\begin{align*}\underset{i\in I}{\mathrm{inf}}\,x_i := \bigcup_{i\in I} \{ q\in \mathbb{Q} \,| \, x_i<q \}\qquad \text{and}\qquad \sup_{i\in I}x_i := \bigcup_{i\in I} \{ q\in \mathbb{Q} \,| \, q<x_i \}.\end{align*}$$

   Classically, when $\{x_i\}_{i\in I}$ is bounded appropriately, these again define reals. In the point-free setting, however, one must specify which real.

   As defined, a set-indexed infimum of Dedekind/upper reals only determines an upper real, and there is in general no constructive way to recover a Dedekind real from one-sided data. The analogous statement holds for suprema of Dedekind/lower reals. Nonetheless, working classically remains compatible with a point-free interpretation of spaces, even if this lies outside the strict regime of geometric mathematics.Footnote ¹⁵ This tension between geometric and point-free mathematics reappears in Theorem 2.21, and is unpacked more fully in Discussion 2.23.

Discussion 1.22 (Types vs. Models as Filters).

Informally, a complete type p over a model M corresponds to an ultrafilter of the Boolean algebra of definable subsets of M, which we denote as $\mathcal {B}_M$ . By contrast, models of a propositional theory $\mathbb {T}$ correspond to the completely prime filters of its Lindenbaum algebra $\Omega _{\mathbb {T}}$ . The appearance of filters in both contexts is suggestive, but there is a subtlety. Because geometric syntax lacks negation (cf. Remark 1.6), $\Omega _{\mathbb {T}}$ is typically not Boolean, and so its completely prime filters need not be ultrafilters. However, when $\Omega _{\mathbb {T}}$ is Boolean, its prime filters are exactly its ultrafilters, and so the two notions converge.Footnote ¹⁶

Discussion 1.23 (Logical Complexity).

The example of Dedekind reals highlights a basic contrast between two ways of assessing logical complexity. In classical first-order model theory, logical complexity is typically tied to combinatorial invariants (e.g., number of types, number of non-isomorphic models, etc.). From this perspective, $\mathrm {Th}(\mathbb {Q},<)$ is complex since it is unstable. By contrast, in categorical logic, complexity is more naturally tied to syntactic expressiveness (e.g., coherent vs. geometric, propositional vs. predicate etc.—see [Reference Johnstone21, Remark D1.4.14]). From this perspective, the theory presenting the Dedekind reals is well-behaved since it is essentially propositional.

The upshot is that logical complexity is not absolute: although linear orders exhibit the strict order property and hence instability in classical first-order logic, the same data may admit a useful description when formulated in a different framework, in our case geometric logic.

Warning 1.24. Since both Stone spaces of types and localic spaces are built from filters, it is tempting to identify the Stone space of 1-types $\mathrm {S}_1(\mathbb {Q},<)$ with the space of Dedekind reals. While Discussion 1.22 shows a meaningful point of contact between the two, the identification is clearly incorrect. Unlike $\mathrm {S}_1(\mathbb {Q},<)$ , the space of Dedekind reals does not contain infinities or infinitesimals—its models are the honest real numbers belonging to the interval $(-\infty ,\infty )$ . The distinction is also visible topologically: $\mathrm {S}_1(\mathbb {Q},<)$ is totally disconnected and compact, whereas the space $\mathbb {R}$ is connected and non-compact.

Discussion 1.25. An upper real only records the rational strictly larger than itself and nothing else. Consequently, a generic upper real does not determine a complete type over $(\mathbb {Q},<)$ : the associated set of formulae is consistent but not maximal (except in the degenerate cases $\pm \infty $ ). Moreover, since the upper reals cannot determine rationals smaller than itself, they do not carry a well-defined strict order.Footnote ¹⁷ In point-free terms, this reflects the fact that the upper reals correspond to prime filters which are not, in general, ultrafilters.

Discussion 1.26. This article is concerned with multiplicative seminorms on certain rings, as they arise in Berkovich geometry. It is therefore natural to ask why the upper reals, rather than the Dedekind reals, provide the appropriate codomain for these seminorms.

The guiding reason is that multiplicative seminorms are naturally built by successive upper approximations. As Summary Theorem 0.1 shows, the multiplicative seminorms on $K[T]$ correspond to sequences of nested discs

$$\begin{align*}D_{r_1}(k_1)\supseteq D_{r_2}(k_2)\supseteq \cdots, \end{align*}$$

which are refined monotonically from above. From a topological viewpoint, this behaviour aligns more closely with the Scott topology on the upper reals than with the Hausdorff topology of the Dedekind reals (cf. Fact 1.21).

Discussion 1.30. Regarding the topological aspects of Definition 1.27:

1. (i) Despite its point-set formulation, the Berkovich analytification $X^{\mathrm {an}}$ is suggestive from the point-free perspective. By axiomatising the geometric theory of multiplicative seminorms, the topos theory automatically produces a generalised space whose points are precisely such seminorms.

   This perspective already played a role in [Reference Ng and Vickers27], where it led to a natural distinction between Dedekind- and upper-real-valued seminorms. As summarised in [Reference Ng and Vickers27, Discussion 5.9], if multiplicative seminorms are also required to be positive-definite, then they must be valued in the Dedekind reals; otherwise, it is more natural for them to be valued in the upper reals.

2. (ii) The Berkovich topology can be more explicitly characterised as the weakest topology such that for all $f\in K[X]$ and for all $\alpha \in \mathbb {R}$ , the sets

   (6) $$ \begin{align} U(f,\alpha):=\{ |\cdot|\in X^{\mathrm{an}} \,\big|\, |f|<\alpha \}\\ V(f,\alpha):=\{ |\cdot|\in X^{\mathrm{an}} \,\big|\, |f|>\alpha \} \nonumber \end{align} $$

   are open in $X^{\mathrm {an}}$ . Hence, notice the Berkovich topology (as defined) crucially depends on the fact that $|\cdot |$ is valued in the Dedekinds as opposed to say, the upper reals.Footnote ¹⁹

3. (iii) Why should we regard the space of multiplicative seminorms as filling in the “missing points” of the base field? Notice that Definition 1.27 is defined for any valued field K, not necessarily non-Archimedean. By the Gelfand–Mazur Theorem, every multiplicative seminorm on $\mathbb {C}[T]$ corresponds to an evaluative seminorm, and so one deduces $\mathbb {A}_{\mathrm {Berk}}^1\cong \mathbb {C}$ when $K=\mathbb {C}$ . However, when K is non-Archimedean, then there exist more multiplicative seminorms than just the evaluative seminorms.

Summary Theorem 1.31 [Reference Berkovich7, Sections 3.4–3.5].

Let K be a non-Archimedean field (possibly with the trivial valuation), and let X be a K-scheme of finite type. Then the following GAGA-type results hold:

1. (i) X is connected if and only if $X^{\mathrm {an}}$ is (arcwise) connected;

2. (ii) X is separated if and only if $X^{\mathrm {an}}$ is Hausdorff;

3. (iii) X is proper if and only if $X^{\mathrm {an}}$ is compact.

Convention 1.34 (On the Berkovich Spectrum).

1. (i) Unless stated otherwise, all seminorms in this section are multiplicative and bounded by the given Banach norm.

2. (ii) The given norm on the Banach ring $\mathcal {A}$ is denoted as $||\cdot ||$ . By contrast, to emphasise that $\mathcal {M}(\mathcal {A})$ is a topological space, its points will be represented as $|\cdot |_x$ , or even $x\in \mathcal {M}(\mathcal {A})$ when the context is clear.

Proof. Let $|\cdot |_x\in \mathcal {M}(\mathcal {A})$ . Restricting $|\cdot |_x$ to linear polynomials defines a bounded K-seminorm

(42) $$ \begin{align} |\cdot|_x^{\mathrm{lin}} \colon \mathcal{A}_{\mathrm{Lin}} \to \overleftarrow{[0,\infty)}. \end{align} $$

By Fact 1.21, the map $x\mapsto R_x$ sending a Dedekind to its right Dedekind section is monic. Consequently, (i) follows immediately from Preparation Lemma 2.6 and Theorem 2.20.

We now prove (ii). Throughout the remainder of the proof we make the following classical assumption:

• (⋆) Any upper real $\gamma _U$ besides $\pm \infty $ is canonically identified with a Dedekind real $\gamma $ whose right Dedekind section is $\gamma _U$ , and similarly for (bounded, non-empty) lower reals.

In the forward direction, following Equation (42), the restriction map

(43) $$ \begin{align} x \longmapsto |\cdot|_x^{\mathrm{lin}} \end{align} $$

defines a map $\mathcal {M}(\mathcal {A}) \to \overleftarrow {\mathcal {M}(\mathcal {A}_{\mathrm {Lin}})}$ .

Conversely, by Theorem 2.20, any bounded K-seminorm on $\mathcal {A}_{\mathrm {Lin}}$ is of the form $|\cdot |_{\mathcal {F}}$ for a unique R-good filter $\mathcal {F}$ . We extend $|\cdot |_{\mathcal {F}}$ to a multiplicative seminorm on $\mathcal {A}$ in two stages.

Discussion 2.22. Let us sketch the original argument from classical Berkovich geometry.

• • Given any rigid disc $D_{r}(k)$ such that $0<r\leq R$ , define a multiplicative seminorm $ |\cdot |_{D_{r}(k)}$ on $\mathcal {A}$ .

• • Next, given any nested sequence of discs $\mathsf {D}:= D_{r_1}(k_1)\supseteq D_{r_2}(k_2)\supseteq \dots $ , define the multiplicative seminorm $|\cdot |_{\mathsf {D}}:=\mathrm {inf}_{\mathsf {D}} |\cdot |_{D_{r_i}(k_i)}$ .

• • Finally, given $|\cdot |_x\in \mathcal {M}(\mathcal {A})$ , define the nested sequence $\mathsf {D}_{x}:=\{D_{|T-k|_x}(k)\,|\, k\in K \, \text {and}\, |k|\leq R\}$ . Check that $|\cdot |_x$ and $|\cdot |_{\mathsf {D}_x}$ agree on linear polynomials, and conclude $|\cdot |_x=|\cdot |_{\mathsf {D}_x}$ .

The parallels with the proof for Theorem 2.20 are clear. However, the argument must be adjusted and finitised appropriately in order to work in our context. Some important differences:

1. (i) “Rational” discs. Both R-good filters and the nested sequence of discs $\mathsf {D}_x$ give rise to approximation arguments, but their approximants differ in important ways. In particular, whereas the radius of a formal ball $B_{q}(k)$ is rational in the usual sense that $q\in Q_{+}$ is a positive rational number, in non-Archimedean geometry a rigid disc $D_r(k)\in \mathsf {D}_x$ is said to have rational radius when

   $$\begin{align*}r\in\Gamma:=\{|k|\in [0,\infty) \,|\, k\in K\},\end{align*}$$

   i.e., r belongs to the value group of K. The terminology is standard, and indicates an analogy between $Q_{+}$ and $\Gamma $ , but it is important to remember that they are not the same—particularly when K is trivially-valued.

2. (ii) K-seminorms. Whereas the original argument starts by defining a multiplicative seminorm on $\mathcal {A}$ , before restricting it to the linear polynomials to perform certain checks, we instead defined a new algebraic structure (which we call K-seminorms) on the space of linear polynomials $\mathcal {A}_{\mathrm {Lin}}$ .

3. (iii) Use of filters. While Berkovich’s original argument shows that every $|\cdot |_x\in \mathcal {M}(\mathcal {A})$ corresponds to a nested descending sequence of discs, this representation is not unique. In particular, two different sequences of discs may define the same multiplicative seminorm on $\mathcal {A}$ . We resolve this issue by appealing to the more natural language of filters, which allows us to obtain a representation result: every $|\cdot |_x\in \mathcal {M}(\mathcal {A})$ is uniquely associated with an R-good filter $\mathcal {F}_x$ .

4. (iv) Use of formal balls. As already pointed out in Discussion 2.10, our result holds for both trivially and non-trivially valued K. This is in contrast to the original argument, which only works for non-trivially valued K.

Items (i) and (ii) reflect our decision to work with the upper reals as opposed to the Dedekinds, and strike a careful balance: whilst the upper reals are particularly suited to analysing the filters of formal balls (cf. Observation 2.14), they also impose strong (constructive) restrictions on the algebra (cf. Remark 2.18, but see also Discussion 2.23). Items (iii) and (iv) give evidence that filters (as opposed to nested sequences of discs) and formal balls (as opposed to the classical rigid discs) are the correct language for studying $\mathcal {M}(\mathcal {A})$ .

Discussion 2.23 (On Assumption $(\star )$ ).

Theorem 2.21 is classical because its proof relies on Assumption $(\star )$ , namely, that any bounded upper (or lower) real admits a canonical extension to a Dedekind real. Some natural points of discussion:

1. (i) Why is Assumption $(\star )$ classical? Consider the obvious argument: given an upper real R defined as the set of rationals strictly greater than 1, we define a lower real L as the set of rationals strictly less than 1. Hence, conclude that $(L,R)$ is the Dedekind real canonically associated with R.

   This argument will strike most as reasonable, so where does it fail constructively? Discussion 1.25 reminds us that an upper real is blind to the rationals less than itself, suggesting it may only have the same knowledge as a Dedekind real by way of classical reasoning. Reformulated more precisely, we claim that if any upper real R can be associated with a lower real L such that $(L,R)$ defines a Dedekind real, then every proposition p has a Boolean complement $p'$ —which holds classically, but not constructively (cf. Remark 1.4).

   To prove our claim, let p be any proposition, and define the subset of rationals:

   $$\begin{align*}R:=\{ q\in\mathbb{Q} \,|\, \text{either } "q>1\text{" }or\ "p \text{ holds and } q>0\text{"}\}.\end{align*}$$

   In other words, R is a kind of schizophrenic upper real: since p holds if and only if $R<1$ , R may define the upper real $1$ or the upper real $0$ , depending on the truth value of p. Now, suppose we have some lower real L such that $(L,R)$ is Dedekind. Define a new proposition $p' \leftrightarrow \frac {1}{2}<L$ . If $(L,R)$ indeed define a Dedekind, one deduces

   1. (a) $\top \to p\vee p'$ ;

      [Why? Locatedness of $(L,R)$ gives $\frac {1}{2}<L\vee R<1$ .]

   2. (b) $p\land p'\to \bot $ .

      [Why? Since $p\to R<\frac {1}{2}$ , this gives $p\land p'\to (\frac {1}{2}<L)\land (R<\frac {1}{2})$ , contradicting separatedness of $(L,R)$ .]

   This shows $p'$ is a Boolean complement of p, as claimed.

2. (ii) Is Assumption $(\star )$ necessary? One might hope to constructivise Theorem 2.21 by replacing Dedekind-valued seminorms on ${\mathcal {A}}$ with upper-valued ones. However, our current proof proceeds by extending a multiplicative seminorm on $K[T]$ to ${\mathcal {A}}$ by taking limits, a step which does not interact well with upper reals constructively (cf. Remark 2.18). It is not clear how to modify this argument without relying on Assumption $(\star )$ . In any case, $(\star )$ is unavoidable if one wishes to recover the classical Berkovich spectrum.

Summary 2.24 (Classical vs. Geometric Mathematics).

In the discussion below, the term “geometric” is used in the sense of geometric mathematics, i.e., a specific regime of constructive mathematics natural in point-free topology, dealing with constructions arising from colimits and finite limits (cf. Remark 1.12). Since our present work is influenced by point-free ideas, the question arises: to what extent do the results of this section adhere to these constraints?

1. (i) Theorem 2.20 is a fully geometric result. We only require K to be a non-Archimedean field, a structure which can be expressed geometrically. The only potential subtlety is our use of Preparation Lemma 2.6 in the proof; however, we only invoke item (i) of the Lemma, which satisfies geometricity.

2. (ii) Theorem 2.21 is both non-constructive and non-geometric on several levels. As already mentioned in Discussion 2.23, it invokes Assumption ( $\star $ ), which is classical since it implies the LEM.Footnote ³² In addition, it assumes Hypothesis 2.1, which requires K to be a complete field, a non-geometric property. Moreover, as pointed out by Johnstone [Reference Johnstone18, Section 2], any non-trivial ring R satisfying the geometric formulation of the field axiom

   $$ \begin{align*}"\,\forall a\in R .\,\,a=0 \,\,\text{or}\,\, a\,\,\text{is a unit}\text{"},\end{align*} $$

   must be decidable, and thus a discrete field. In other words, adhering to geometric mathematics would rule out many (topological) fields of interest in Berkovich geometry, e.g., the p-adic complex numbers $\mathbb {C}_p$ —which is certainly against the spirit of the theorem.

3. (iii) Theorem 2.20 can be read as extracting the geometric content of Theorem 2.21. However, as presently stated, Theorem 2.20’s connections to Berkovich spectra are somewhat indirect, not least because of the geometricity issues underlying Hypothesis 2.1. Can we do any better?

   Discussion 2.23 already pointed out the difficulties in removing Assumption $(\star )$ from Theorem 2.21. Moreover, although the polynomial ring $K[T]$ and $\mathcal {A}_{\mathrm {Lin}}$ can be defined geometrically for any field K, it is presently unclear how to do the same for the convergent power series ring $K\{R^{-1}T\}$ .Footnote ³³ In other words, in the context of geometric mathematics, even defining “multiplicative seminorm on ${\mathcal {A}}$ ” is a challenge. Alternatively, one may wish to modify the definition of a K-seminorm to be Dedekind-valued, but this creates new issues for Theorem 2.20. In particular, $|\cdot |_{\mathcal {F}}$ from Construction 2.17 is defined by taking $\mathrm {inf}$ ’s and so, at least constructively, $|\cdot |_{\mathcal {F}}$ has to be upper-valued (cf. Fact 1.21).