Dirichlet is Natural

Giry and Lawvere’s categorical treatment of probabilities, based on the probabilistic monad G , oﬀer an elegant and hitherto unexploited treatment of higher-order probabilities. The goal of this paper is to follow this formulation to reconstruct a family of higher-order probabilities known as the Dirichlet process . This family is widely used in non-parametric Bayesian learning. Given a Polish space X , we build a family of higher-order probabilities in G ( G ( X )) indexed by M ∗ ( X ) the set of non-zero ﬁnite measures over X . The construction relies on two ingredients. First, we develop a method to map a zero-dimensional Polish space X to a projective system of ﬁnite approximations, the limit of which is a zero-dimensional compactiﬁcation of X . Second, we use a functorial version of Bochner’s probability extension theorem adapted to Polish spaces, where consistent systems of probabilities over a projective system give rise to an actual probability on the limit. These ingredients are combined with known combinatorial properties of Dirichlet processes on ﬁnite spaces to obtain the Dirichlet family D X on X . We prove that the family D X is a natural transformation from the monad M ∗ to G ◦ G over Polish spaces, which in particular is continuous in its parameters. This is an improvement on extant constructions of D X [17,26].


Introduction
It has been argued that exact bisimulations between Markovian systems are better conceptualized using the more general notion of bisimulation metrics [29].This is because there are frequent situations where one can only estimate the transition probabilities of a Markov chain (MC). 3Such uncertainties lead one naturally to using a metric-based notion of approximate equivalence as a more robust way of comparing processes than exact bisimulations.Here, we wish to take a new look at this issue of uncertainty in the model and suggest a novel and richer framework to deal with it.We keep the idea of using a robust means of comparison (typically the Kantorovich or Prohorov metrics lifted to MCs), but we add a second idea: namely to introduce a way of quantifying the uncertainty in the chains being compared.
To quantify uncertainty in the Markov chains, we propose to explore in the longer term concepts of "uncertain Markov chains" as elements of type X → G 2 (X), where X is an object of Pol, the category of Polish spaces (separable and completely metrisable spaces) and G is the Giry probability functor.This is to say that the chain takes values in "random probabilities" (ie probabilities of probabilities). 4his natural treatment of behavioural uncertainty in probabilistic models will allow one to formulate a notion of (Bayesian) learning and therefore to obtain notions of 1) models which can learn under observations and 2) of behavioural comparisons which can incorporate data and reduce uncertainty.Bisimulation metrics between processes become random variables and learning should decrease their variability.
One needs to set up a sufficiently general framework for learning under observation within the coalgebraic approach.Learning a probability in a Bayesian framework is naturally described as a (stochastic) process of type G 2 (X) → G 2 (X) (so G 2 (X) → G 3 (X) really!) driven by observations.For finite Xs this setup poses no difficulty, but for more general spaces, one needs to construct a computational handle on G 2 (X) -the space of uncertain or higher-order probabilities.This is what we do in this paper.
To this effect, we build a theory of Dirichlet-like processes in Pol.Dirichlet processes [1,16] form a family of elements in G 2 (X) indexed by finite measures over X [1, p.17]5 and which is closed under Bayesian learning.
Integral to our construction is a method of "decomposition/recomposition" which allows us to build higher-probabilities via finite approximations of the underlying space (the limit of which lead to a compactification of the original space).In order to lift finite higher-probabilities we use a bespoke extension theorem of the Kolmogorov-Bochner type in Pol (Sec.2.3).Kolmogorov consistent assignments of probabilities on finite partitions of measurable spaces (or finite joint distributions of stochastic processes) can be seen systematically as points in the image under G of projective (countable co-directed) diagrams in Pol.
Using the above we show that Dirichlet-like processes in Pol can be seen as natural transformations from M * (the monad of non-zero finite measures on Pol) to G 2 built up from finite discrete spaces.The finite version of naturality goes under the name of "aggregation laws" in the statistical literature and can be traced back to the "infinite divisibility" of the one building block, namely the Γ distribution.(This opens up the possibility of an axiomatic version of the construction presented here, see conclusion.)

Notations & basic facts
We provide a primer of general topology as used in the paper in Appendix A. A useful reference on the matter is [11].Weak convergence of probability measures is treated in [7,27].

Weak topology
A measure P on a topological space X is a positive countably additive set function defined on the Borel σ-algebra B(X) verifying P (∅) = 0. We will only consider finite measures on Polish spaces, i.e.P (X) < ∞.When P (X) = 1, P is a probability measure.We write G(X) for the space of all probability measures over X with the weak topology [7,27], the initial topology for the family of evaluation maps EV f = P → X fdP where f ranges in C b (X) and where (C b (X), • ∞ ) is the Banach space of real-valued continuous bounded functions over X with the sup norm.A neighbourhood base for a measure P ∈ G(X) is given by the sets where f i ∈ C b (X), i > 0. One can restrict w.l.o.g. to the subset of real-valued bounded uniformly continuous functions, noted U b (X).Importantly, if X is Polish the weak topology on G(X) is also Polish (see e.g.Parthasarathy, [27] Chap.2.6) and metrisable by the Wasserstein-Monge-Kantorovich distance [31].We denote the convergence of a sequence (P n ∈ G(X)) n∈N to P ∈ G(X) in the weak sense by P n P .The "Portmanteau" theorem ( [7], Theorem 2.1) asserts that P n P is equivalent to P n (B) → P (B) for all P -continuity sets B, i.e.Borel sets s.t.P (∂B) = 0. P -continuity sets form a Boolean algebra ( [27], Lemma 6.4).The support of a probability P ∈ G(X) is noted supp(P ) and is defined as the smallest closed set such that P (supp(P )) = 1.For X, Y Polish and P ∈ G(X), Q ∈ G(Y ), we write P ⊗ Q ∈ G(X × Y ) the product probability, so that

Giry monad
The operation G can be extended to a functor G : Pol → Pol compatible with the Giry monad structure (G, δ, μ) [19].For any continuous map f : X → Y we set G(f )(P ) = B ∈ B(Y ) → P (f −1 (B)), i.e.G(f )(P ) is the pushforward measure.For a given X, δ X : evaluates a probability on the Borel set B. We have the "change of variables" formula: for all P ∈ G(X), f : X → Y and g : Y → R bounded measurable, Y gdG(f )(P ) = X g•fdP .Finally, G preserves surjectivity, injectivity and openness: Proof.We recall that elements of G(X) for X Polish verify the Radon property: for all Borel set B ∈ B(X) and all P ∈ G(X), P (B) = sup {P (K) | K ⊆ B, K compact} (see [9], Chap.7).(i) Let f be an injective continuous map.Let P, Q ∈ G(X) be such that P (B) = Q(B) for some Borel set B. Then, there must exist a compact be some probability.By the measurable selection theorem [32], there exists a measurable function g : Y → X such that g(y) ∈ f −1 (y), which implies f • g = id Y .Let P be the pushforward measure of Q through g, i.e.P (B) . ., g n , 1 , . . ., n ) be some basic neighbourhood of some P ∈ G(X), and let P ∈ N P be in the neighbourhood of

Finite measures
The set of all finite non-negative Borel measures on a Polish space, noted M (X), is a Polish space when endowed with the weak topology ([9] Theorem 8.9.4).M : Pol → Pol is a functor extending G, mapping continuous functions to the corresponding pushforward morphism.The monad multiplication μ X can be conservatively extended to a morphism from M 2 (X) to M (X) by defining μ X (P ) = B ∈ B(X) → M (X) EV B dP .The everywhere zero measure, noted 0, is an element of M (X) that we might want to exclude: M (X) being Hausdorff implies that the set of nonzero measures M * (X) Lemma 2.2 Strictly positive finite measures on a Polish space X form (when they exist) a Polish subspace of M (X).We denote this subspace by M + (X).
and forms a Polish subspace of M (X). 2 Summing up, we have for X Polish the following inclusions of Polish spaces of finite measures: Note also that M and M * are endofunctors on Pol but M + is not, unless one restricts to the subcategory of epimorphisms.

Normalisation of measures
We note ν X : is the total mass of the measure.ν X verifies an useful property:

Densities and convolution
The Radon-Nikodym theorem asserts that measures in G(R) absolutely continuous with respect to the Lebesgue measure admit integral representations such that P (A) = A fdx.In this case P is said to have density f with respect to the Lebesgue measure.f is sometimes noted dP dλ , where λ denotes Lebesgue.For P, Q ∈ G(R) with respective densities w.r.t.Lebesgue f P , f Q , the convolution product of P and Q is defined to be the measure P * Q having density Kallenberg [22], Lemma 1.28).

Finitely supported measures
When X is a finite, discrete space such that V. Danos, I. Garnier / Electronic Notes in Theoretical Computer Science 319 (2015) 137-164

Projective limits of topological spaces
Many of our theorems will deal with spaces obtained as projective limits (also known as inverse limits or cofiltered limits) of topological spaces.These topological projective limits are defined as adequate topologisations of projective limits in Set, the usual category of sets and functions.
Let (I, ≤) be a directed partially ordered set seen as a category and let D : I op → Set be a cofiltered Set diagram.The projective limit of D is a terminal cone (lim D, π i ) over D where lim D is the set and the π i : j D(j) → D(i) are the canonical projections.Notice that D is contravariant from I to Set.As emphasised in the definition, lim D is the subset of the cartesian product i D(i) containing all sequences of elements that respect the constraints imposed by the diagram D. The elements of lim D are called threads and the maps D(i ≤ j) : D(j) → D(i) are the bonding maps.Of course, lim D can be empty (see [34] for a short example).A sufficient condition to ensure non-emptiness of the limit is to consider functors D where I is countable and the bonding maps are surjective [6].As a convenience, we will note those bonding maps as π ij D(i ≤ j), and we write countable cofiltered surjective diagrams ccd for short.
Writing U : Top → Set for the underlying set functor, cofiltered limits in Top for diagrams D : I op → Top are obtained by endowing the Set limit of U • D with the initial topology for the canonical projections {π i } i∈I .The following useful additional fact follows by considering lim D as the intersection of the (closed) subsets of i D(i) satisfying D(i ≤ j)(π j (x)) = π i (x) for all pairs (i, j) s.t.i ≤ j.Lemma 2.4 ([11], Ch. 1, §8.2, Corollaire 2) lim D is a closed subset of i D(i).

The Bochner extension theorem
The construction of a stochastic process given a system of consistent finitedimensional marginals is an important tool in probability theory, a classical example being the construction of the Brownian motion using the Kolmogorov extension theorem [25].Besides Kolmogorov's there are many other variants, collectively called Bochner extension theorem [24].They differ in the amount of structure of the space over which probabilities are considered (measurable, topological or vector spaces) -and we will make crucial use of the Bochner extension theorem for Polish spaces, which admits a particularly elegant presentation.In words, the Bochner extension theorem states that any projective family of probabilities that satisfy the diagram constraints (elements of lim G • D) can be uniquely lifted to a probability over the limit space (elements of G(lim D)) -and what's more, this extension is a homeomorphism!This presentation of the Bochner extension seems not to be well-known: a similar statement is given in Metivier ( [24], Theorem 5.5) in the case of locally compact spaces, which intersects but does not include Polish spaces; Fedorchuk proves the continuity of G on the class of compact Hausdorff spaces in [15] while more recently Banakh [4] provides an extension theorem in the more general setting of Tychonoff spaces, using properties of the Stone-Čech compactification.

Zero-dimensional Polish spaces and their properties
It is natural in applications to consider finitary approximations of stochastic processes.Accordingly, the correctness of such approximations should correspond to some kind of limiting argument, stating that increasingly finer approximations yield in some suitable sense the original object.In view of the Bochner extension theorem, it suffices to consider as input a projective family of probabilities supported by the finitary approximants of the underlying space.However, the very same theorem tells us that we can only obtain by this means probabilities on a projective limit of finite spaces (also called profinite spaces), a rather restrictive class: Proposition 3.1 A space is a countable projective limit of finite discrete spaces if and only if it is a compact, zero-dimensional Polish space.
The proof can be found under a slightly different terminology in Borceux & Janelidze [10], where it is shown that these spaces correspond to Stone spacesindeed, profinite spaces are exactly the spaces homeomorphic to the Stone dual of their Boolean algebra of clopen sets!As the proof of this proposition is quite enlightening for the developments to come, we provide it here.
Proof.Let D : I op → Pol fin be a ccd of finite spaces.Polishness of lim D comes from the closure of Pol under countable limits.Finite spaces are compact and by Tychonoff's theorem so is i D(i).Lemma 2.4 asserts that lim D is closed in this compact product, hence lim D is itself compact.Recall that lim D has the initial topology for the canonical projections maps π i : lim D → D(i), therefore a base of lim D is constituted of finite intersections of prebase opens π −1 i (X i ), for X i ⊆ D(i).Since the D(i) are discrete, any of their subsets is clopen and so are the prebase opens; we conclude by noticing that a finite intersection of clopen sets is again clopen.
Conversely, let Z be a compact zero-dimensional Polish space.As Z is zerodimensional Polish, its topology is generated by a countable base of clopen sets.Since Z is compact, each clopen can be written as a finite union of base clopens.Therefore its Boolean algebra of clopens Clo(Z) is also generated by the same countable base, and is itself countable.Note that Clo(Z) does not depend on the choice of the base!Let us consider the set I(Z) of all finite clopen partitions of Z.For any i ∈ I(Z), there exist by assumption a continuous surjective quotient map f i : Z → i.Since i is discrete, the fibres of f i are clopen.Note that I(Z) is also countable.I(Z) is partially ordered by partition refinement: for all i, j ∈ I(Z), we write i ≤ j if there exists a surjective "bonding" map f ji : j → i such that f ji • f j = f i (any such map, if it exists, is unique).I(Z) is also directed by considering pairwise intersections of the cells of any two partitions.The system of finite discrete quotients of Z together with the bonding maps f ji clearly defines a ccd that we write D : I(Z) op → Pol fin , mapping each element of I(Z) to itself and the partial order of refinement to the bonding maps.Therefore, there exists a limit cone (lim D, π i ).By universality of this cone, there exists a unique continuous map η : Let us show that η is an homeomorphism.As lim D and Z are both compact, it is enough to show that η is a bijection.Recall that Clo(Z) separates points (it contains a base for a Hausdorff topology) therefore for any x = y ∈ Z we can exhibit two clopen cells separating them, implying that η is injective.Surjectivity of η is a consequence of that of the quotient and bonding maps. 2 We denote by Pol cz the full subcategory of Pol where objects are compact and zero-dimensional -by the previous proposition, these spaces are exactly the profinite Polish spaces.From the data of a projective system of finitely supported probabilities, Prop.3.1 together with Bochner's extension theorem (Thm.2.5) only allow us to obtain probabilities supported by such profinite spaces.Our extension of Dirichlet as a natural transformation from the setting of finite spaces to that of arbitrary Polish spaces must therefore imperatively bridge the gap from profinite spaces to arbitrary Polish spaces.
The solution we propose is mediated by zero-dimensional Polish spaces in a decisive way.More precisely, our construction can be framed as the iterative reduction of the extension problem to increasingly smaller subcategories of Pol (depicted below): the (full) subcategory of zero-dimensional spaces Pol z , that of compact zero-dimensional spaces Pol cz and finally the subcategory Pol fin of finite Polish spaces.The categorical setting is informally sketched in the following figure: The two essential operations, highlighted in the figure above, are: • the zero-dimensionalisation Z, which yields a zero-dimensional refinement of a Polish space for which a countable base of the topology has been chosen, and • the zero-dimensional Wallman compactification ω, which yields a compact zerodimensional Polish space from a zero-dimensional one along, again, a choice of a base of clopens sets.
We attract the attention of the reader on the fact that these operations are a priori not functorial.However, as we shall see in the rest of this section, these operations exhibit powerful properties which are sufficient to proceed to the extension.

Zero-dimensionalisation
Zero-dimensionalisation takes as input a Polish space X along a choice of some countable base F for X.It produces a Polish zero-dimensional topology on the same underlying set as X, that we denote by z F (X).
Proposition 3.2 Let (X, T X ) be Polish and let F be a countable base for X.Let Boole(F) be the Boolean algebra generated by F. Let z F (X) be the space which admits Boole(F) as a base of its topology.z F (X) verifies the following properties: (iii) the Borel sets are preserved: In order to prove Prop.3.2 we need some classical facts from descriptive set theory, taken verbatim from Kechris [23], Sec.13: Lemma 3.3 For any Polish space (X, T X ) and any closed set A, there exists a Polish topology Lemma 3.4 Let (X, T X ) be Polish and let {T Xn } n∈N be a family of Polish topologies on X, then the topology T X∞ generated by ∪ n T Xn is Polish.Moreover if ∀n, T Xn ⊆ B(T X ), then B(T X∞ ) = B(T X ).

Proof. (Proposition 3.2)
Consider the family of Polish topologies T X An n∈N , as obtained using Lemma 3.3.Lemma 3.4 entails that the topology generated by Since F is a base of T X and F ⊆ Boole(F), an equivalent base of the topology generated by ∪ n T X An is Boole(F).By definition, we deduce that the topology of z F (X) is generated by ∪ n T X An .
(i) Lemma 3.4 entails that the resulting space is indeed Polish.An equivalent base to δ and the elements of this base are clopen, hence the resulting space is also zero-dimensional.
(ii) Zero-dimensionality is a trivial consequence of taking a Boolean algebra as a base.
(iii) Preservation of Borel sets is a further consequence of Lemma 3.4.
(iv) Continuity of the identity is a trivial consequence of the fact that z F (X) is finer than X. 2 To the best of our knowledge, we can't do away with the dependency on F: one can exhibit a Polish space X with two distinct bases Despite this apparent lack of canonicity, any Polish topology is entirely determined by its collection of zero-dimensional refinements6 : Theorem 3.5 Any Polish space X has the final topology for the family {id F : z F (X) → X} F of all the (continuous) identity maps from its zerodimensionalisations, where F ranges over all the countable bases of X.
The proof of this theorem relies on the following lemma.Lemma 3.6 Let X be a Polish space and (x n ) n∈N → x a convergent sequence in X.Let F be a countable base for X.
then it is trivial to exhibit the convergence property by referring to the topology of X only.If not, we have that This family still constitutes a neighbourhood base.The countable union of countable sets is countable, therefore it constitutes a countable base of X. 2 Notice that the topologies of G(X) and G(z F (X)) might be different, and there is in general no continuous map from G(X) to G(z F (X)).It should also be emphasised that the "zero-dimensionalisation" of a Polish space is not an innocent operation: for instance if X is compact and non-zero-dimensional then z F (X) will never be compact!However, we have the following powerful analogue to Thm. 3.5: Theorem 3.7 For X Polish, G(X) has the final topology for the family of identity maps {G(id F ) : G(z F (X)) → G(X)} F where F ranges over countable bases of X.
Proof.As before, it is sufficient to prove that a map f : G(X) → Y is continuous if and only if all precompositions f • G(id F ) are continuous.If f is continuous then the composites clearly also are.Let us consider the reverse implication and suppose that all composites are continuous.Let (P n ) n∈N G(X) P be a sequence of probabilities converging weakly to P in G(X).It is sufficient to exhibit one F s.t.P n P in G(z F (X)).Let us recall the following theorem ([7], Theorem 2.2): For any Y Polish, let U be a subset of B(Y ) such that (i) U is closed under finite intersections (ii) each open set in X is a finite or countable union of elements in U.If P n (A) → P (A) for all A in U, then P n Y P .Recall that Boole(F) is a base of z F (X).This base trivially verifies condition (i) of the previous theorem.It is therefore sufficient to build a base F of X such that condition (ii) is verified, i.e.P n (A) → P (A) for all A ∈ Boole(F).Observe that the P -continuity sets in X form a Boolean algebra ( [27], Lemma 6.4).It then suffices to form a base of X included in the Boolean algebra of continuity sets of X, which is always possible: for any point x ∈ X, there can at most be countably many radii s.t. the open ball B(x, ) has a boundary with strictly positive mass. 2

Zero-dimensional Wallman compactifications
Compactifications are topological operations embedding topological spaces into compact spaces.Common examples are the Alexandrov one-point compactification (for locally compact spaces) of the Stone-Čech compactification for Tychonoff spaces.In most settings, this embedding is also required to be dense.By choosing the compactification carefuly, one can preserve some relevant properties of the starting space -in our case, Polishness and zero-dimensionality.A well-behaved class of compactifications (that includes Alexandrov and Stone-Čech as special cases) is that of Wallman compactifications.The general method by which one obtains such a compactification from a given topological space X can be decomposed in two steps: (i) one first selects a suitable sublattice of the lattice of open sets of X (a Wallman base); (ii) then, one topologises (in a standard way) the space of maximal ideals of that particular sublattice.
These compactifications are surveyed in Johnstone [21] and (less abstractly) in Beckenstein et al. [5].Van Mill [30] provides some facts on Wallman compactifications of separable metric spaces.An extensive topos-theoretic perspective is also given by Caramello [12].In the remainder of this section, we present this compactification method and apply it to the case of Polish zero-dimensional space, yielding a zerodimensional compactification that we denote by ω.We then highlight its connection with Prop.3.1 and study those of its properties that are relevant to our goal.

Spaces of maximal ideals
All the material here is standard from the litterature on Stone duality for distributive lattices.See e.g.Johnstone [21] for more details.For the proof of (i) and (ii), see [21], II resp.3.5 and 3.6.2

Wallman bases and compactifications
Wallman compactifications are defined as spaces of maximal ideals over Wallman bases, which are particular lattices that are also bases in the topological sense.Here, we will follow the definition given in [21]: Definition 3.9 ([21], IV 2.4) Let X be a topological space and let T X be its lattice of open sets.A Wallman base is a sublattice of T X that is a base for X and which verifies: For all U ∈ T X and x ∈ U , there exists a V ∈ T X such that X = U ∪ V and x ∈ V .
The following lemma is key in considering a space of maximal ideals over a Wallman base as a compactification (see ([21], IV 2.4) for a proof): Lemma 3.10 Let X be a topological space and let L be a Wallman base for X.
We are now in position to define Wallman compactifications: Definition 3.11 Let X be a T 0 space and L a Wallman base.We denote ω L (X) = max(L) the Wallman compactification of X for L.

Zero-dimensional compactifications
We will now show that taking the inverse limit of the finite partitions of a Polish zero-dimensional space (as in the proof of Prop.3.1) corresponds -when applied to a non-compact Polish zero-dimensional space -to a Wallman compactification of that space, which exhibits very good properties.
Consider a zero-dimensional Polish space Z.In opposition to the compact case, the Boolean algebra Clo(Z) of clopens of Z is not necessarily countably generated: we therefore consider partitions of Z taken in some countable Boolean sub-algebra C ⊆ Clo(X) such that C is a (topological) base for Z. Observe that such a base is always trivially a normal Wallman base.In the following, we call such countable Boolean sub-algebras that generate the topology "Boolean bases".We define:

C(X) {C | C is a countable Boolean base of X}
We write I C (Z) for the directed partial order of clopen partitions of Z taken in C ∈ C(X).Since C is countable, so is I C (Z).We recall that the construction of I C (Z) is described in the proof of Prop. 2 As ω C (Z) is always a profinite space, Prop.3.1 ensures there always exists a cofiltered diagram D in Pol fin such that lim D ∼ = ω C (Z).We will switch from one point of view to the other freely.We should insist on the fact that our compactification is not the Stone-Čech compactification, as these are in general not metrisable (except when compactifying an already metrisable compact space, obviously).Take for instance the discrete (hence zero-dimensional) Polish space N: βN has cardinality 2 2 ℵ 0 ([33], Theorem 3.2) while Polish spaces have cardinality at most 2 ℵ 0 .We would obtain Stone-Čech if we were to take the Wallman compactification over the full lattice of open sets, however.ω C (Z) enjoys a property reminiscent of Stone-Čech: Proposition 3. 13 Let Z be a Polish zero-dimensional space.For each continuous map f : Z → K to a compact zero-dimensional space K, there exists a Boolean base C and a continuous map Proof.Prop.3.1 entails that there exists a ccd D K : I op → Pol fin s.t.K ∼ = lim D K , with limit cone (lim D K , {π i : lim D k → D k (i)} i∈I ).Note that by continuity of π i •f , each finite clopen partition of K induces a finite clopen partition of X.By choosing a Boolean base of clopens C of Z that contains f −1 (Clo(K)), we can exhibit a compactification ω C (Z) with an associated cone (ω Corollary 3.14 For any continuous f : Z → Z between zero-dimensional spaces, there exists Boolean bases C, C of respectively Z and Z such that there exists a map Zero-dimensional Polish Wallman compactifications were considered in [2], which however does not state Prop.3.13.

Projective limit measures on zero-dimensional compactification
For Z Polish zero-dimensional, the developments of Sec.3.2 allow us to map any measure in G(Z) to G(ω C (Z)) (for any choice of a Boolean base C) through G(η C ). Crucially, thanks to Lemma 2.1 this is a faithful operation.
Therefore any measure on Z can be obtained, up to isomorphism, as a projective limit of finitely supported measures.However, as pointed out before, the converse operation is the difficult one.Let D be a diagram such that ω C (Z) ∼ = lim D and {P i } i ∈ lim G • D a projective family of finitely supported probabilities.There is in general no way to assert that the corresponding projective limit probability P ∈ G(ω C (Z)) obtained through the Bochner extension theorem restricts to G(Z).We delineate the conditions under which a probability can be restricted to a subspace and propose a simplification of previous arguments (see [26]), based on the properties of the Giry monad.Note that the results to follow are not specific to zero-dimensional spaces.
Polish subspaces of Polish spaces are always G δ sets (and conversely, see [23], 3.11), hence Borel sets.This allows for a simple restriction criterion.Proposition 3. 15 Let P ∈ M (Y ) be a finite measure on a Polish space Y and let X ⊆ Y be a Polish subspace (hence a G δ in Y ).The restriction of P to X, defined as the set function Proof.P | X is trivially a finite measure on the trace σ-algebra.We observe that B(X) = B(Y ) ∩ X: this is a consequence of Theorem 15.1 in [23] (essentially, this follows from the Borel isomorphism theorem for Polish spaces), therefore P | X ∈ M (X).Since P (X) = 1 and X ∈ B(Y ), P | X (X) = 1 and P | X ∈ G(X).The converse is easy. 2 This criterion lifts to "higher-order" probabilities, that is probabilities over spaces of probabilities, thanks to the multiplication of the Giry monad.The following theorem states that such a higher order probability measure restricts to a subspace if and only if it restricts in the mean.This is essentially Theorem 1.1 in [26].

Theorem 3.16 For all X ⊆ Y Polish spaces and all
Proof.The forward implication is trivial.By Lemma 2.1, G 2 (X) is a subspace of G 2 (Y ).By Prop.3.15, it is sufficient to prove that P (G(X)) = 1.By assumption that μ(P )| X ∈ G(X) and Prop.3.15, we have that μ(P )(X) = 1, which unfolds as G(Y ) EV X dP = 1.So it suffices to prove that G(Y ) EV X dP = 1 ⇒ P (G(X)) = 1.Assume P (G(X)) < 1, then there must exist a Borel set A ⊆ G(Y ) \ G(X) with P (A) > 0. Any probability p ∈ A assigns positive measure to some Borel set B ⊆ Y \ X, therefore A EV X dP < 1. 2

The Dirichlet process
The Dirichlet process stands out among other Bayesian methods in that the prior and posterior distributions are second order probabilities, that is elements of G 2 (X).
Learning becomes an operation of type X → G 2 (X) → G 2 (X), mapping some evidence in X and a prior in G 2 (X) to a posterior in G 2 (X), and it can be proved that the second-order stochastic process induced by sampling from identically and independently distributed random variable will converge (in Kullback-Leibler divergence, hence in the weak topology [18]) to a singular distribution over the law of the target.

The Dirichlet distribution
For a fixed finite discrete space X, Dirichlet is a function D X : M + (X) → G 2 (X), the parameter in M + (X) representing the initial prior as well as the degree of certainty about this prior (encoded in its total mass).As we highlight below, D X is continuous and verifies other properties, among which naturality and normalisation.Some of the material on the finitary Dirichlet distribution contained in this section can be found (presented differently) in e.g.[17].In the following we take X = {x 1 , . . ., x n } to be a finite discrete space of cardinality n.

Definition of D
for all (p i ) i<n ∈ Δ n ∩ (0, 1) n−1 , with d X (Q) = 0 elsewhere.In Equation 1, Γ(q) is the continuous gamma function, verifying Γ(n) = (n − 1)! for n integer and defined as Γ(t) = ∞ 0 x t−1 e −x dx.It is clear that for any bounded continuous function

Proposition 4.1 (Γ representation)
For all , where dP dλ denotes the density of P with respect to Lebesgue.
Proof.We write x n 1− i<n x i .For any Q ≡ (q 1 , . . ., q n ) ∈ M + (X), the function ν X maps the half-line {tQ | t ∈ R >0 } to its unique point of intersection with Δ n .This yields a change of variables T (t, q 1 / |Q| , . . ., q n−1 / |Q|) = Q with determinant t n−1 .Expressing the density of ⊗ i Γ(q i , 1) in these new coordinates and integrating over t to obtain the density at the point of intersection, one obtains: The integral identity A more detailed proof can be found in [1], Sec.2.3 or [3], Chap.27.For an extensive survey of other representations of Dirichlet, see [17].

Extension of D to non-zero measures
Both the definition in Eq. 1 and the Γ representation have the drawback of being confined to strictly positive measures M + (X) (since Γ is not defined at 0). M + is not a functor unless one restricts to the category of Polish spaces and maps with dense ranges.This is a serious drawback, as strict positivity of a measure is a topological property: in particular, it is not preserved by zero-dimensionalisation of the underlying space!We therefore continuously extend D X to the better-behaved nonzero measures, so that it verifies for Q ∈ M + (X \ {x i }): where e i : X \ {x i } → X is the inclusion (observe that M * (e i )(Q)(x i ) = 0).The following proposition asserts that this extension is well-defined as a continuous map.

Naturality of D
As we are going to show, D is a natural transformation from M * to G 2 .This property, usually called aggregation in the literature, is an indirect consequence of the closure under convolution of the Γ distribution, a property that we assume without proof: Γ(q 1 , r) * Γ(q 2 , r) = Γ(q 1 + q 2 , r) ( A corollary of Eq. 3 is that the pushforward of a product of Γ distributions is a product of Γ distributions. First of all, Prop.4.2 allows us to restrict our attention to f surjective and Q = x i → q i ∈ M + (X) strictly positive.Then: Example 4.5 Let us verify naturality on a simple case: take The last integral can be carried out using the generalised binomial theorem on the rightmost term: the integral identity holds ( [20], 3.191.1).On the other hand, we have: So naturality holds.

Normalisation
The Dirichlet distribution Let us verify this identity for a parameter Q = (q 1 , . . ., q n ) ∈ R n >0 ⊆ M + (X).Observe that Eq. 4 holds when |X| = 2, as D X (q 1 , q 2 ) degenerates to a Beta X (q 1 , q 2 ) distribution which is known to have mean ( q 1 q 1 +q 2 , q 2 q 1 +q 2 ) (see eg [3], Sec.16.5 for a definition of the Beta distribution and the proof of this property).
In the case of X an arbitrary finite discrete space, let

Extension to zero-dimensional Polish spaces
The finite support case is instructive but lacks generality.We proceed to the extension of finitely supported Dirichlet distributions to Dirichlet processes supported by arbitrary zero-dimensional Polish spaces.Our construction preserves both naturality and continuity -in fact, it can be framed as the extension of the natural transformation Proof.We prove existence, naturality and uniqueness.Existence.For any given choice of a Boolean base C, let η C : Z → ω C (Z) be the embedding of a Polish zero-dimensional space Z into its compactification ω C (Z) (Lemma 3.10).ω C (Z) is compact zero-dimensional so by Prop.3.1 there exists a ccd of finite spaces D such that ω C (Z) ∼ = lim D. Let us construct Dlim D , the extension of Dirichlet to lim D (see Fig. 1).Applying the functor Applying the finitary Dirichlet D on the base of this cone yields a ccd in

. Commutation of normalisation and Dirichlet averaging
G 2 • D, of which we take the limit, obtaining a terminal cone T = (lim By universality of T , there exists a unique morphism u :

1). This yields a morphism
Dlim D : M * (lim D) → G 2 (lim D) that trivially coincides with D when lim D happens to be finite.In order to conclude the existence part of the extension, we need to show that Dlim . By Theorem 3.16, it suffices to check that for any which by Prop.3.15 amounts to checking that this measure attributes full measure to η C (Z.We take advantage of the normalisation property (Eq.4) of D. Thanks to this property and to the naturality of μ, the diagram in Fig. 2 commutes.The Bochner extension theorem entails the universality of the cone (G(lim D), {G(π i )} i ) at the top of the diagram, therefore commutation of the diagram in Fig. 2 entails the existence of a unique morphism from the cone (M * (lim D), ν D(i)•M * (π i ) i ) to (G(lim D), {G(π i )} i ) (morphism represented as a dashed line in Fig. 2).This morphism is no other than the normalisation ν lim D : M * (lim D) → G(lim D).Therefore, (a) Reducing naturality to the case of compact zero-dim.Polish spaces.
M * (ω C (Z)) M * (D Z (i)) Trivially, Hence, up to isomorphism, DZ restricts to a morphism DZ : M * (Z) → G 2 (Z).This concludes the proof of existence.Naturality.For any map f : Z → Z between zero-dimensional Polish spaces, we must prove DZ • M * (f ) = G 2 (f ) • DZ .By Corollary 3.14, we can reduce the task to the case of a morphism ω CC (f ) : ω C (Z) → ω C (Z ) between compact zerodimensional spaces (see Fig. 3a).It remains to prove Dω where D Z and D Z are their respective finite discrete quotient ccd s.Let us write (ω C (Z ), {π i : ω C (Z ) → D Z (i)} i ) the terminal cone corresponding to D Z .The universal property of this limit cone allows to reduce the problem to the commutation of the diagram in Fig. 3b: • DZ for all i.Therefore, Q induces through D and D the same projective family of finite-dimensional Dirichlet distributions which yields (by unicity of extensions, see Theorem 2.5) a contradiction. 2

Extension to arbitrary Polish spaces
Let X be an arbitrary Polish space.As shown in Theorem 3.7, G(X) has the final topology for the family of identity maps {G(id F ) : G(z F (X)) → G(X)} F where F ranges over countable bases of X.In order to harness this theorem, we need the following fact: Lemma 4.7 Let X be Polish and z F (X), z G (X) be two zero-dimensional refinements as constructed in Prop.3.2.Then D z F (X) and D z G (X) are equal in Set.
Proof.The set of countable bases of X is directed by union.Let us write H ≡ F ∪G.The (continuous) identity functions id FH : z H (X) → z F (X) and id GH : z H (X) → z G (X) lift to identity functions G 2 (id FH ), G 2 (id GH ), and similarly for the functor M * .Therefore, the commutation relation boils down in Set to the equality of D z F (X) and D z G (X) (and similarly for G). 2 Finally, we have: Proof.Let X be a Polish space.Consider the family {z F (X)} F of its zerodimensional refinements, as constructed in Prop.3.2.For each z F (X), Theorem 4.6 asserts the existence of a continuous Dirichlet map Dz F (X) : M * (z F (X)) → G 2 (z F (X)), which extends by continuity of the identity and functoriality to a continuous map By Lemma 4.7, all these maps coincide in Set.Theorem 3.7 allows to conclude.2

Conclusion
Our construction of the Dirichlet process in categorical style subsumes existing ones [17,26] while establishing continuity and naturality.However, further work, which we intend to pursue right away, is required to consolidate our understanding of the finitary approximation framework we have built for higher-order probabilities.The Giry monad can be generalised from Pol to the category of Tychonoff spaces, however our construction relies heavily on the properties of Polish spaces: for instance we use the fact that zero-dimensional Polish spaces are Borel sets of their compactifications (Prop.3.15); the measurable selection theorem used in Lemma 2.1 also requires the spaces considered to be Polish.The process by which we rebuild Dirichlet relies on some simple properties of Γ distributions.Naturality is a consequence of closure of Γ under convolution (a particular case of infinite divisibility also exhibited by e.g.normal distributions), and the fact that Dirichlet restricts to X, which is only a subset of its compactification wX, follows from the normalisation property (see §4.1).By axiomatising these properties, we can generalise our main result.However, it remains to be seen whether other interesting distributions on R >0 fit the conditions and generate Dirichlet-like processes.
Beyond the immediate questions above, we can return to the less immediate goals expounded on in the introduction, namely higher-order learning using uncertain chains of Dirichlet type.Any uncertain Markov chain τ , meaning a morphism X → G 2 (X) in Pol, can be post-composed with the multiplication of G to obtain the "mean" Markov chain of type X → G(X).We will investigate the case where τ takes values in Dirichlet processes -focusing on the tractable "uncertain chains of Dirichlet type".Such chains can be decomposed as α : X → G(X) followed by the Dirichlet natural transformation D X : M * (X) → G 2 (X).The first component α is τ 's parameterising chain.As μ • D X • α is the normalised version of α, α is again up to normalisation the mean chain of the uncertain τ .Our construction ensures that τ is continuous by construction.At this stage, it is already possible given τ : X → G 2 (X) to quantify the uncertainty at each point by considering moments of the "Kantorovich" random variable K , where K X is defined over the probability triple (G(X), τ(x)) and d K is metrises G(X).The next step is to adapt the Bayesian learning scheme which in the discrete case maps the prior D X (Q) to the posterior D X (Q + s), for Q in M * (X) the current parameter, given s a multiset of observed values in X (seen as a counting measure).Via the projective limit construction, learning can be led at the level of behavioural approximants [13] and a subsidiary goal is to understand how the two levels relate.The second goal consists of in extending the probabilistic Kantorovich metric to uncertain chains (of this specific type) and understand its evolution under learning.Until now we assumed that the state of the system is fully observable, but the above questions should be developed as well in a broader context where the state is only partially and noisily so.In this setting, naturality of D might allow to compare uncertain chains defined on distinct state spaces by embedding them in some universal Polish space -giving a quantitative account of both the differences in their respective state spaces and their dynamics.

A Topological and measurable spaces
We recall some basic facts about topological and measurable spaces.

Basic definitions
A topological space (X, T X ) is given by a set X and a set T X of open subsets of X such that X ∈ T X , ∅ ∈ T X and T X is closed under arbitrary unions and finite intersections.A set is closed if its complement is open.It is clopen if it is both closed and open.A set of subsets S ⊆ T X is a base of T X whenever it is closed under finite intersections and its closure under arbitrary unions yields T X .A set of subsets S ⊆ T X is a pre-base of T X when its closure under finite intersections is a base of T X .The closure of a subset A ⊆ X is noted A, it is the smallest closed set containing A. Conversely, the interior of A ⊆ X is noted int (A), it is the largest open set contained in A. The boundary of A is noted ∂A and is defined as ∂A A \ int (A).A space (X, T X ) is separable if there exists a countable subset D ⊆ X that is dense in X, i.e.D = X.Except where it might lead to ambiguities, we will omit T X and write the space simply X.A map f : X → Y between two topological spaces is continuous if and only if for all Topological spaces with continuous maps form a category, noted Top.

Initial and final topologies
Let I = {f i : X → (X i , T X i )} i be a family of functions f i from a set X into topological spaces (X i , T X i ).The initial topology induced by I is the coarsest topology on X making the f i continuous.If is defined as the topology T I generated by the sets The final topology is defined dually, as the finest topology on X making a family of functions It is straightforward to check that this defines a topology.
Limits and colimits in Top are defined by endowing them with resp. the initial and final topologies on the Set limits and colimits.In particular, the topological product is defined as the initial topology on the Set product w.r.t. the canonical projections.

Metric and metrisable spaces
A distance function on a set X is a function d : X × X → [0, ∞] that obeys the following axioms, ∀x, y, z ∈ X: (i) symmetry: d(x, y) = d(y, x), (ii) d(x, y) ≥ 0, d(x, y) = 0 iff x = y, (iii) d(x, y) ≤ d(x, z)+d(z, y).Any distance d on X induces the topology of a metric space, with base the open balls B(x, ) = {y | d(x, y) < }, for all x ∈ X, > 0. A metric space is noted (X, d).A sequence of points (x n ∈ X) n∈N converges to a point x ∈ X if for all > 0, there exists a N ∈ N such that for all n ≥ N , d(x n , x) < .A sequence is Cauchy if for all > 0, there exists an N ∈ N such that for all m, n ≥ N , d(x m , x n ) < .A metric space is complete if all Cauchy sequences converge.A space is metrisable if its topology is generated by some distance.A space is completely metrisable if its topology is generated by some distance that makes it complete.A Polish space is a separable, completely metrisable space.Polish spaces form a full subcategory of Top, noted Pol.Pol has all countable limits and all countable disjoint unions [11].It includes the category of finite, discrete topological spaces Pol fin as a full subcategory (a space X is discrete if T X = ℘(X)).

Separation conditions
A topological space X is Hausdorff if for any two points x, y ∈ X there exists disjoint open sets O x , O y such that x ∈ O x , y ∈ O y .In a Hausdorff space, all singletons are closed.X is completely regular if for any closed set F ⊆ X and any point x ∈ X \ F , there exists a continuous function f : X → R such that f (x) = 0 and f (y) = 1 for all y ∈ F .A space is Tychonoff if it is completely regular and Hausdorff.All metrisable spaces are Tychonoff.

Compactness
An open cover of a space X is a family {O i ∈ T X } i of open subsets such that ∪ i O i = X.A topological space is compact if any open cover of the space has a finite sub-cover.A subset of X is compact if it verifies this property.All finite subsets are compact.The continuous image of a compact set is compact.All spaces we consider will be Hausdorff, accordingly all compact spaces will be implicitly Hausdorff.All compact subspaces of Hausdorff spaces are closed.Tychonoff 's theorem asserts that an arbitrary product of compact spaces is compact.A continuous bijection between compact (Hausdorff!) spaces is always a homeomorphism.

Zero-dimensional spaces
A topological space is zero-dimensional if it has a base of clopen sets.The set of clopen sets of a space X is noted C(X).It is a Boolean algebra.One easily deduces that zero-dimensional Polish spaces have a countable base of clopen sets.Zero-dimensionality is a hereditary property and is preserved by subspaces.

Compactifications
A compactification of a (Tychonoff) topological space X is a compact space Y into which X embeds homeomorphically and such that the closure of X in Y is Y itself (a non-Tychonoff spaces need not embed in its compactification).

A.2 Measurable spaces
A measurable space (X, Σ X ) is a set X along a σ-algebra Σ X , that is a set of subsets of X closed under complements and countable unions that contain X.If X is a topological space we note B(X) the Borel σ-algebra generated from its topology.A map f : (X, Σ X ) → (Y, Σ Y ) between measurable spaces is measurable if f −1 (A) ∈ Σ X for all A ∈ Σ Y .If f : X → Y is a continuous map, f is also measurable between the corresponding Borel measure spaces.Borel measure spaces arising from Polish spaces verify the "Isomorphism theorem" [23]: Theorem A.1 For all X and Y Polish spaces, B(X) ∼ = B(Y ) if and only if X and Y have the same cardinality.

B Proof of Theorem 2.5
This proof is adapted from Metivier [24].
Let D : I op → Pol be our ccd, with canonical projections π i : lim D → D(i).Let {P i } i ∈ lim G • D be given.We proceed to continuously extend this family to an element P ∈ G(lim D).
. By directedness, A is an algebra of lim D-Borel sets.We define the set function P 0 : A → [0, 1] by P 0 (π −1 i (B)) = P i (B).Codirectedness of the family {P i } i ensures that (i) P 0 is consistent as a function and that (ii) P 0 is finitely additive, therefore P 0 is a charge.As P 0 is finite, hence σ-finite, it is sufficient to exhibit that P 0 is σ-additive on A and the Carathéodory extension theorem ( [35], Theorem 1.7) will yield the sought unique projective limit Borel measure.σ-additivity is equivalent to the implication ∀n, P 0 (A n ) ≥ δ ⇒ ∩ n A n = ∅ for all δ > 0 and all decreasing sequence of Borel sets (A n ) n∈N ([8], Prop.1.3.3).
Let (A n ) n be such a sequence.Each A n is by construction of the form A n = π −1 i (B * i,n ) for some i ∈ I, where B * i,n ∈ B(D(i)).We map this sequence (A n ) n to a family {B cn ∈ B(D(c n ))} cn of Borel sets indexed by an increasing sequence (c n ) n∈N , cofinal in I, such that for all n, A n = π −1 cn (B cn ) and B c n+1 ⊆ π −1 cnc n+1 (B cn ).The cofinal increasing sequence (c n ) n∈N is constructed by induction on any fixed enumeration of I.By construction, there is some i n ∈ I for which A n = π −1 in (B in ).By cofinality, there exists c n ≥ i n and by measurability, B cn π −1 incn (B in ) is measurable.By directedness, A n = π −1 cn (B cn ).Now consider m ≤ n with A n ⊆ A m .We have For n fixed, this generalises to ).We construct a nonempty (compact!)set K s.t.K ⊆ A n for all n.By cofinality of (c n ) n∈N , it is sufficient to construct of a family of non-empty compact sets {K cn ⊆ B cn } n∈N that is projective, i.e. verifying π cnc n+1 (K c n+1 ) = K cn for all n.Such a projective family of compact sets can in turn be obtained from a sequence of non-empty compact sets K cn n∈N verifying K c n+1 ⊆ π −1 cnc n+1 (K cn ).Indeed, setting, for all m, K cm = ∩ n≥m π cmcn (K cn ), we trivially have that K cm is compact.As an intersection of a decreasing sequence of non-empty compact sets (in a metrisable space), K cm is also non-empty (this is Cantor's intersection theorem).Moreover, K cm = ∩ n≥m π cmcn (K cn ) ⊇ ∩ n≥m+1 (π cmc m+1 • π c m+1 cn )(K cn ) ⊇ π cmc m+1 (∩ n≥m+1 π c m+1 cn (K cn )) = π cmc m+1 (K c m+1 ).To prove the reverse inclusion, it suffices to show that for all x ∈ K cm , π −1 cmc m+1 (x) ∩ K c m+1 = ∅.We have

Theorem 2 . 5
For all D a ccd in Pol, G(lim D) ∼ = lim G • D. We denote by bcn : lim G • D → G(lim D) this homeomorphism.

Proposition 3 . 8
Let X be a set and L be a (distributive) sublattice of the lattice of subsets of X.The space max(L) has the set of maximal ideals of L as points and admits subsets of the form B(O) = {I ∈ max(L) | O ∈ I} as a base.Moreover:

Proof.
It suffices to check that the family B(O) where O ranges in L is indeed a base (i.e.closed under finite intersections).Maximal ideals are by definition proper.Since L is distributive, maximal ideals on L are moreover prime: for all I ∈ max(L), B, B ∈ L, if B ∩ B ∈ I then either B ∈ I or B ∈ I ([21], I 2.4).Let B, B ∈ L be given.We show B(O) ∩ B(O ) = B(O ∩ O ).Consider I ∈ B(O) ∩ B(O ): we have O ∈ I and O ∈ I, therefore (by primality) O ∩ O ∈ I, which implies I ∈ B(O ∩ O ).Conversely, if I ∈ B(O ∩ O ) then O ∩ O ∈ I. Since ideals are downward closed, we must have O ∈ I and O ∈ I.

3 . 1 .
Proposition 3.12 For C ∈ C(Z), let D C : I op C (Z) → Pol fin be the diagram of finite clopen partitions of Z seen as discrete spaces, then lim D C is a zero-dimensional compactification of Z homeomorphic to ω C (Z).Proof.Existence and non-emptiness of lim D C stems from surjectivity of the bonding maps and countability of C. Note that lim D C is Polish.Zero-dimensionality is an hereditary property, so it only remains to exhibit an homeomorphism with ω C (Z).First, observe that since C is a Boolean algebra, maximal C-ideals are in one-to-one correspondence with maximal C-ultrafilters via the complement map: elements of lim D C correspond to C-filters, they are upward closed and codirected by intersection.They are moreover maximal: for anyU ∈ lim D C and all C ∈ C, either C ∈ U or C c ∈ U .A basic clopen in lim D C is of the form π −1 i (C) where C ∈ i ∈ I C , which correspond to the ultrafilter {U ∈ lim D C | C ∈ U }.This in turns, through the negation map, correspond to a basic clopen of ω C (Z) (seeProp.3.8).Every basic clopen of ω C (Z) similarly correspond to a basic clopen in lim D C .Therefore, the spaces are homeomorphic, from which we conclude that lim D C is a Polish zero-dimensional compactification.

Corollary 4 . 3
Consider f : X → Y surjective for X, Y finite and discrete, then:

2 Proposition 4 . 4 D
r) V. Danos, I. Garnier / Electronic Notes in Theoretical Computer Science 319 (2015) 137-164 Proof.It suffices to consider the case of |X| = 2, |Y | = 1, in which case M + (f ) : M + (X) → M + (Y ) is simply the addition and G(M + (f )) is by definition the convolution!Eq. 3 then yields the claim.An induction on |f −1 (y)| for each y ∈ f (X) allows to conclude in the general case.X → D X is a natural transformation D : M * ⇒ G 2 when M * and G 2 are restricted to Pol fin .Proof.It suffices to show that for any f : X → Y a (continuous) function between finite discrete spaces, one has

Theorem 4 . 6
D from Pol fin to Pol z , the full subcategory of zero-dimensional Polish spaces and continuous maps.In what follows, we denote by F | C : C → Pol the restriction of the domain of some endofunctor F : Pol → Pol to a subcategory C of Pol.When unambiguous, we drop this notation.There exists a unique (up to isomorphism) natural transformation D : M | Polz ⇒ G 2 | Polz such that D coincides with D on Pol fin .
• DZ As already argued in the proof of Prop.3.13, any finite discrete clopen partition of ω C (Z ) induces a finite discrete clopen partition of ω C (Z) since the two spaces are related by the continuous function ω CC (), therefore the diagram in Fig. 3b commutes.Uniqueness.Assume there exists two distinct natural transformations D, D : M * (Z) → G 2 (Z) that coincide with D on finite spaces.It is clear that it is enough to exhibit a contradiction in the case of Z compact zero-dimensional Polish.We refer to Fig. 1 for the notations.Let D be a ccd of finite spaces such that Z ∼ = lim D, with canonical projections π i : lim D → D(i).By assumption, there must exist a measure Q ∈ M * (Z) such that D(Q) = D (Q).But both D and D verify (by assumption of naturality and consistency with the finitary case) the equalities

Theorem 4 . 8
There exists a unique (up to isomorphism) natural transformation D : M * ⇒ G 2 such that D coincides with D on Pol.
using the initial assumption, we have x ∈ (X \∪ n i=1 O i )∩O, which is an open set in X.The result follows.
2Proof.(Theorem 3.5) It suffices to prove that for all topological space Y , a function f :X → Y is continuous if and only if f • id F : z F (X) → Y is continuous for all countable base F. The forward implication is trivial.Assume that for all countable base F, f •id B : z F (X) → Y is continuous.Consider a converging sequence (x n ) n∈N → x in X.It is sufficient to exhibit one space z F (X) where this sequence also converges.Lemma 3.6 gives as a sufficient criterion that x does not belong to ∂O for any O ∈ F. Let us build such a base.Consider a dense set D of X.Let d : X 2 → [0, 1] be some metric that completely metrises X.Without loss of generality, assume x ∈ D. Write r n d(x, d n ) for d n ∈ D \ {x}.For all n, take the family of open balls centred on each d n with rational radii strictly below r n , e.g.r n /3.Since diam(B(d n , r n