Robustly Parameterised Higher-Order Probabilistic Models

We present a method for constructing robustly parameterised families of higher-order probabilistic models. Parameter spaces and models are represented by certain classes of functors in the category of Polish spaces. Maps from parameter spaces to models (parameterisations) are continuous and natural transformations between such functors. Naturality ensures that para-meterised models are invariant by change of granularity – ie that parameterisations are intrinsic. Continuity ensures that models are robust with respect to their parameterisation. Our method allows one to build models from a set of basic functors among which the Giry probabilistic functor, spaces of cadlag trajectories (in continuous and discrete time), multisets and compact powersets. These functors can be combined by guarded composition, product and coproduct. Parameter spaces range over the polynomial closure of Giry-like functors. Thus we obtain a class of robust parameterised models which includes the Dirichlet process, point processes and other classical objects of probability theory such as the de Finetti theorem. By extending techniques developed in prior work, we show how to reduce the questions of existence, uniqueness, naturality, and continuity of a parameterised model to combinatorial questions only involving ﬁnite spaces.


Introduction
In a widely read paper [1], Beylin and Dybjer reinterpret coherence for monoidal categories as a result of normalisation on a linear non-commutative lambda-calculus.They prove that the structural arrows of a monoidal category are characterised by their domain and codomain.In this paper we follow a parallel path for probabilistic models.There are several differences with Dybjer's correspondence, however.
First, we work in Pol, the concrete category of Polish spaces Polish spaces form a classic and convenient environment to construct a large variety of stochastic models.We exploit this potential to build up a sufficiently expressive stock of structure arrows.Specifically, we build up structure arrows based on a two-sorted polynomial type theory of 'parameter' functors and 'model' ones.Model functors include as primitives the Vietoris functor of compact non-determinism, the Giry functor of probabilities, and most interestingly the Skorokhod functor of cadlag functions -i.e. with countably many jumps and values in any Polish space -which captures traces of processes both in discrete and continuous time.Model functors can be combined by products, coproducts, and guarded composition; parameter functors are less malleable (perhaps because we do not understand them well yet).Hence, our type theory allows for iterations of 'compact' non-determinism (as captured by the Vietoris functor) and probabilism (the Giry monad) -which have been a fundamental pursuit in quantitative models of concurrency theory since Segala's early work [18,10,19,14].In addition, topology (and the possible recourse to metrics) allows one to talk about continuity and to quantify notions of approximate bisimulation.Our arrows come pre-equipped with metric interpretations which several and in particular van Breugel et al. have convincingly argued are fundamental in probabilistic modeling [6,20].
Thus, our stock of arrows is remarkably expressive.Of course, this would amount to little, without a characterization of arrows equality.Here again, our result is slightly different from Dybjer's.We do not show that natural transformations are uniquely determined by their 'types'.Instead, using the 'machine' built in earlier work [5,4], we show that structure arrows are completely characterised by their behaviours on finite spaces.In effect, our result makes the structure arrow equality problem (the equivalent of the normalisation of linear lambda-terms if we follow our analogy) purely combinatorial.For some special choices of parameter and model functors (of the Giry type), we can even show rigidity [4], that is to say, types do characterise arrows.Regarding existence, we provide a converse to the above result, and prove that not only are structure arrows wholly determined by the finite case, but that finite data is enough to define such arrows.
The standpoint presented here departs from the more common "forward semantics" approach where one knows already what one wants to semanticize.Here we embrace a less trodden path (but usually rewarding, as in the Ehrhard-Regnier invention of differential linear logic via the study of spaces of sequences, or Girard's own carving of linear logic via coherence spaces) of "reverse semantics" where the mathematical tractability of the semantic universe is the primary tool for constructing the universe of computational discourse.
The outline of the paper is as follows: we start with a 'slicing' of the category of Polish spaces in convenient layers, and recall the basic points of our existence and unicity 'Machine', spelling out the conditions on our parameter and model functors.With this behind us, we attack the description of the type theory and develop a string of propositions which justify that choice by building its semantics.We conclude with our "normalisation" theorem and an application of our framework to the celebrated de Finetti theorem, a key result in probability theory and statistics.

Preliminaries
We work in the category Pol of completely metrisable and separable topological spaces and continuous maps.There is the obvious Borel functor B : Pol → Meas mapping a Polish space to the measurable space with the Borel σ-algebra and mapping continuous maps to measurable ones, together with the underlying set functor U : Pol → Set with the obvious forgetful action.Pol has all countable limits and all countable coproducts [3, IX].

Pol endofunctors
We introduce some Pol endofunctors which are going to be used as examples in the rest of the paper and form the primitive bricks for our structure arrows.First, there is the Giry functor G [9,17] which maps a space X to the space of Borel probability measures (with the topology induced by the Kantorovich metric) over X. Related to G are the finite nonzero positive measure functor M + ∼ = G × R >0 .A parallel construction is that of the Vietoris functor V which maps X to the "hyperspace" of its compact subsets topologised with the Hausdorff distance [13, 4.F].The finite multiset functor B and the related finite list functor W are also Pol endofunctors [4].Finally, we have a pair of functional functors.The Skorokhod functor D, which maps X to the space of cadlag (right-continuous with left limits) functions from [0, ∞) to X equipped with the J 1 topology [7], which is fundamental to the study of continuous-time stochastic processes.And for any compact Polish set X, the functor C(X, −) which maps any Polish space Y to that of continuous maps from X to Y .Our family of functors covers both probabilistic behaviour through G, compact nondeterminism through V and spaces of trajectories through D and C.Moreover, working in the category of Polish spaces makes the analogies between these functors all the more striking: G is metrised by Kantorovich, V by Hausdorff and the same holds of D using a metric allowing "time transport".Recent work [16] might shed additional light on these similarities.

The structure of Pol
We slice Pol into the following full subcategories: finite Polish spaces Pol f ; compact zerodimensional spaces Pol cz ; zero-dimensional spaces Pol z ; and compact spaces Pol c : Finite spaces are equipped with the discrete topology.Compact zero-dimensional Polish spaces (such as the Cantor set 2 N ) can be characterised as projective limits of finite Polish spaces.Zero-dimensional spaces (such as the Baire space N N ) are those spaces which admit a base of their topology constituted of clopen sets.These subcategories have interesting structures: any zero-dimensional space equipped with a choice of a countable base of clopen sets can be mapped to its compactification, which is compact zero-dimensional [4].In the other direction, any Polish space equipped with a choice of a countable base can be mapped to a zero-dimensional Polish refinement of its topology.We give some more details on these operations in the next section, together with characterisations of Polish and compact zero-dimensional spaces as respectively colimits and limits of particular diagrams.

Characterisations of zero-dimensional spaces
A countable codirected diagram in A (an A ccd for short) is a functor D : I op → A where I is a countable directed partial order and A is a subcategory of Pol.The characterisation of objects of Pol cz can be formulated as follows: Any Polish space can be written as the colimit of a diagram in Pol z : Proposition 2 ([5], Proposition 3.2).Let X be a Polish space and F a countable base of the topology of X.Let Z X (F) (U (X), Bool(F) ) be the space having the same underlying set as X and the topology generated by the Boolean algebra generated by F. The following holds: (i) Z X (F) is zero-dimensional Polish, (ii) B(Z X (F)) = B(X).We call Z X (F) the zero-dimensionalisation of X with respect to F.

23:4 Robustly Parameterised Higher-Order Probabilistic Models
The family of zero-dimensionalisations of a space X indexed by all countable bases of X forms a codirected diagram.This diagram is indexed by Bases(X) the set of all countable bases of X partially ordered by inclusion; Bases(X) is directed by closing the union of two bases under finite intersection [4, Def.2.10].If F, G are such bases then, if F ⊆ G, the identity function is continuous from Z X (G) to Z X (F).This defines a codirected diagram from the directed partial order Bases(X) to Pol z , that we still denote by Z X .The following statement states that any Polish space is the colimit of its diagram of zero-dimensionals: Theorem 3 ([5], Th. 3.5; [4], Proposition 2.11).For every Polish space X, X ∼ = colim Z X .

Converging in G(X)
We recall some standard facts about convergence in G(X) for X Polish.The boundary of a set A ⊆ X is the set-theoretic difference between its closure and its interior, and is denoted by ∂ X A. By the Portmanteau theorem ([2], Th. 2.1), a sequence (p n ) n∈N of probability measures converges to p ∈ G(X) iff p n (A) → p(A) for each Borel set A which is a p-continuity set, i.e. which verifies p(∂ X A) = 0. Note that for all p, p-continuity sets form a Boolean algebra that we denote C X (p) ( [17], Lemma 6.4).We have the following facts: ) is clopen and included in U , and the conclusion follows.
Lemma 5. Let X, Y be Polish, p ∈ G(X) and let f :

Lemma 6. Let X be Polish and uncountable and let {p i } i∈I ⊆ G(X) be a countable family of probability measures. There exists a countable base
Proof.Let B(x, ) be the open ball of radius > 0 centered on x.Observe that for 0 < < , ∂ X B(x, ) ∩ ∂ X B(x, ) = ∅.Therefore, for a given p, there can at most be countably many radiuses k such that the B(x, k ) are not p-continuity sets, as otherwise the total mass of ∪ k ∂ X B(x, k ) would diverge.Using that a countable union of countable sets is countable, there are at most countably many radiuses For our purposes, it is enough to take E such that E does not intersect the forbidden radii { k }, which can always be done.The sought base F is obtained by considering a countable dense subset D of X and taking F to be the closure under finite intersections of {N (x) | x ∈ D}.Since continuity sets form a boolean algebra, we get Boole(F) ⊆ ∩ i C X (p i ).

3
The Machine The parameterised models we are interested in use 'the Machine' [4], a powerful theorem allowing one to extend a class of natural transformations from finite Polish spaces to arbitrary ones.This extension theorem hinges on particular conditions on the domain and codomain functors of the natural transformation, corresponding respectively to constraints on parameters and models.Accordingly, we will call domain functors 'P-functors' and codomain functors 'M-functors'.Below, we list these conditions.In Section 4, we will study closure properties of these conditions, and derive a syntax for parameters and models.

Parameter condition
The Machine applies to natural transformations whose P-functor commutes with colimits of diagrams of zero-dimensionals (Section 2.2.1).We call this property Z-cocontinuity: In order for the Machine to apply, P-functors must also preserve epis.The parameter condition is the conjunction of these two conditions.Definition 8 (Parameter conditions).An endofunctor F : Pol → Pol satisfies the parameter condition (or equivalently, is a P-functor) if (i) F is Z-cocontinuous and (ii) F preserves epis.

Model condition
The Machine also requires M-functors to verify a list of conditions, corresponding to constraints on models.

The model condition
Before defining the model condition, we introduce some terminology related to commutation of functors with some limits.Definition 10.Let A be a subcategory of Pol.An endofunctor G : M-functors are endofunctors that satisfies the following.

Example 12.
The following are M-functors: (i) the Giry functor G, and finite measure functors M + ; (ii) the multiset and list functors B, W; (iii) the Vietoris functor V; (iv) the Skorokhod functor D and the continuous map functor C(X, −) from a compact Polish space X (see [4]).
Observe that in Definition 11, all conditions are preserved by composition of endofunctors except the last one.We will come back to this in Section 4.

The Machine
The Machine states that natural transformations (parameterised models) between M-functors and P-functors are entirely characterised by their components on finite spaces.Theorem 13 ([4]).Let F 1 be a P-functor and F 2 be a M-functor; one has The unite of the Giry monad η : 1 ⇒ G is entirely characterised by its finite components.
We conjecture that the parameter condition is closed under composition by G, which would imply that multiplication µ : G 2 ⇒ G of the Giry monad is also characterised on the whole of Pol by its finite components.The normalisation ν : is also finitely characterised.Moreover, the Machine allows to prove that it is the unique natural transformation between M + and G [4, Th. 5.2].Classical objects of statistics can be framed as natural transformations: the i.i.d.distribution on sequences of samples iid : G ⇒ G(− N ) (Section 5); the Dirichlet process D : M + ⇒ G 2 a cornerstone of Bayesian nonparametrics [8]; the Poisson process P : M + ⇒ GB which is the prototypical point process.Using the Machine, it is enough to define the Poisson process on finite sets, this is done via where Po(λ) is the Poisson measure on N with parameter λ.

A grammar for parameterised models
We turn now to the main question of the paper, which is to find operations on functors under which the parameter and model conditions are closed.For parameters, this result takes the form of a simple grammar over functors.For models, we develop a simple type system over polynomial terms generated from a family of functors, well-typedness implying the model condition.This syntax for parameter and models lifts to natural transformations, giving rise to a language of natural combinators for parameterised models.
As Pol has all countable limits and coproducts, the category of Pol endofunctors is closed under at most countable coproducts and products (recall that if F, G : Pol → Pol are two endofunctors, their coproduct F + G acts on objects by (F + G)(X) = F (X) + G(X) and on morphisms by (F + G)(f ) = F (f ) + G(f ), and similarly for products).Endofunctors are also trivially closed under composition.

Closure properties of the parameter condition
Let us start with parameter conditions.At the time of writing, we do not know whether these are closed under products and/or functor composition.However, we show that they are closed under coproducts.We also derive specific results for particular functors that altogether yield a sufficiently expressive class of parameterisations.

Finite coproducts preserve the parameter condition
The following facts are easily verified: Proposition 15.If G, H preserve epis, then so does G + H. Proposition 16.The parameter condition is preserved by finite coproducts.

Products of Giry-like functors satisfy the parameter condition
Countable products of Giry-like functors (i.e.G, M + ) satisfy the parameter condition.The case of finite products follow trivially from the same result.Proposition 17.Let {F k } k∈N be given with F k ∈ {G, M + }.Then k F k satisfies the parameter condition.
Proof.It is enough to treat the case of G as M + is naturally isomorphic to G × R >0 .Preservation of epis by G lifts to products of G. Let us prove Z-cocontinuity.We reuse the proof method of ( [5], Th. 3.7).It is enough to exhibit, for all X and all countable family of converging sequences {p It follows from Lemma 4 that for countable Polish spaces there is nothing to show, and we can thus assume w.l.o.g. that X is uncountable.Applying Lemma 6, we get a base F s.t.Boole(F) ⊆ ∩ k C X (p (k) ).Noting that Boole(F) is a base of Z X (F), an application of Th. 2.2 in (Billingsley [2]) concludes.

Giry-like functors over products satisfy the parameter condition
Our grammar for parameterisations admits a way of specifying quantitative relations on points of the underlying space, as shown below in Proposition 18 (which generalises to Giry-like functors M + ).In what follows, we treat countable products.The case of finite products follows easily from the same proof.

Proposition 18. G(− N ) satisfies the parameter condition.
Proof.Preservation of epis still follows from the properties of G. To prove Z-cocontinuity, we follow the proof scheme of Proposition 17.We denote π k : X N → X, π 1...k : X N → X k the canonical projections.Let X be Polish and let (p n ) n∈N → p be a converging sequence in G(X N ).By Lemma 6, there exists a base F of X such that for all k > 0, Boole(F) ⊆ ∩ k C X (G(π k )(p)).The sets of the form π −1 k (O) with O ranging in Boole(F) induce a base H of Z X (F) N .Using Lemma 5 plus the fact that continuity sets are closed under finite intersections, we deduce that H ⊆ C X N (p).Therefore, for all V ∈ H, p n (V ) → p(V ).Using Th. 2.2 in (Billingsley [2]), we get that p n → p in G(Z X (F) N ).We conclude that G(− N ) is Z-cocontinuous.

Closure properties of the model condition
We turn now to closure properties of model conditions.As we are going to show, all model conditions (Definition 11) are closed under all polynomial operations with the exception of the last one, namely Pol f -continuity.

Finite coproducts preserve the model condition
We prove preservation of the model condition under finite coproducts.We proceed with the other parts of the model condition, namely preservation of embeddings, intersections and Pol f -continuity.The following proposition is routine.

Proposition 19. (i) If G, H preserve monos, then so does G + H. (ii) If G, H preserve embeddings, then so does G + H. (iii) If G, H preserve embeddings and intersections, then so does G + H.
The following one is a bit more technical and deserves a proof.Proof.Let (X i ) i∈I be a ccd with limit X, we're assuming that GX = lim i GX i and HX = lim i HX i .The diagrams (GX i ) i∈I and (HX i ) i∈I define a diagram (GX i + HX i ) i∈I where for each i < j there exists a unique arrow Gp i,j + Gp i,j : GX i + HX i → GX j + HX j , with p i,j : X i → X j the connecting morphism of the diagram (X i ) i∈I .The coproduct GX + HX is a cone over this diagram via the maps Gp i + Hp i : GX + HX → GX i + HX i constructed from the canonical projections p i : X → X i by the universal property of coproducts.There must therefore exist a unique continuous map φ : GX + HX → lim i (GX i + HX i ) which maps a thread in GX to the obvious thread in lim i (GX i + HX i ) and similarly for threads in HX.It is clear that φ is injective, moreover it is easy to see that threads in lim i (GX i + HX i ) must be the form (x i ) i∈I with every x i ∈ GX i or with every x i ∈ HX i , and in particular φ is surjective.Thus φ is continuous and bijective, and it only remains to show that it is open.
Let U be open in GX + HX.By definition of the coproduct topology, are open, and thus by definition of the topology on the limit, where each V i (resp.W j ) is open in GX i (resp.HX j ).Since I is cofiltered, for every i, j ∈ I there exists a k ∈ I and morphisms p k,i : X k → X i and p k,j : X k → X j .Let us denote by i ∧ j the choice of such a k for the pair i, j and note that Gp where q i is the canonical projection (lim i HX i + GX i ) → HX i + GX i .By construction it is open in lim i (GX i + HX i ) since it is a union of inverse images of sets which are open in GX i∧j + HX i∧j by definition of the coproduct topology.We claim that φ[U ] = V .For any thread (x i ) ∈ U , if the thread is in the HX component then it must belong to U G and thus there must exist an i such that x i ∈ V i , since we can assume that the connecting morphisms are surjective there exists for each j an element x i∧j ∈ Gp −1 i∧j,i (U i ) in the thread and it follows that the thread belongs to V , and similarly if the thread (x i ) is in the GX component.Similarly, starting from (x i ) in V , it is clear that (x i ) belongs to U and it thus φ is open.

Finite products preserve the model condition Proposition 21. (i) If G, H preserve monos, then so does
Proof.(i) Straightforward.(ii) Since embeddings are equalizers in Pol, this result is simply a case of limits commuting with limits (Mac Lane [15] IX).(iii) Similarly, since intersections are finite limits and they commute with finite limits in Pol.(iv) Finally, since ccds are cofiltered limits, they commute with finite products.

Composition
We now consider the operation of functor composition.The following is trivial: Proposition 22.The conditions 1. to 3. of Definition 11 are preserved under functor composition.
Figure 1 Signature for G and V.
The condition of Pol f -continuity (Definition 11, 4.) does not behave as well: if F, G are Pol f -continuous endofunctors and F maps finite spaces to non-finite spaces, GF has no reason to be Pol f -continuous.On the other hand, if F maps finite spaces to a subcategory with respect to which G is continuous, then the composition GF will be Pol f -continuous.In order to make this intuition formal, we need to capture the global behaviour of functors on the subcategories of Pol.To do so, we propose to abstract functors as monotonic functions on the poset of subcategories of Pol.
Definition 23 (Partial order on subcategories).We denote by (P, ≤) the lattice over the subcategories of Pol displayed in Equation 1 and generated by the subcategory relation, i.e.A ≤ B iff A is a subcategory of B. We will denote by ∧ and ∨ the infinimum and the supremum.
Note that P has as maximal element Pol and as minimal element Pol f .The known behaviour of a endofunctor over Pol can be presented as a monotonic function from P to itself.We call such a function a signature assigned to the functor.
Definition 24 (Signatures and signature assignments).We denote by Sign(P) the set of order-preserving functions from P to itself.We say that an endofunctor G admits f ∈ Sign(P) as a signature if for all A ∈ P, there exists a functor G : A → f (A) such that the following diagram commutes in Cat: where I AB denotes the obvious inclusion functor.If G admits f as a signature, we call the pair (G, f ) a signature assignment.
Example 25.It is known that the Giry functor G and the Vietoris functor V preserve compactness (see resp.Parthasarathy [17], Th. 6.4 and Kechris [13], Th. 4.26).Therefore, both G and V admit the signature (in dotted arrows) in Figure 1.However, the fact that V maps finite spaces to finite spaces implies that it admits a finer signature (Figure 2).Note also that the functor M + does not preserve compactness.
The exact signature of a functor might be unknown.However, it is always possible to assign to a functor the signature corresponding to the constant function A ∈ P → Pol.In fact, the lattice structure on P lifts to signatures: Lemma 26.Define the relation on and similarly for ∨.Then (Sign(P ), ≤ * ) is a lattice and the constant function A → Pol is its maximal element.
Proof.Reflexivity and transitivity are trivial.Assume f ≤ * g and g ≤ * f , then by antisymmetry of (P, ≤) we have f = g.Maximality of the constant Pol function is trivial.
Let us now define a criterion for functor composition ensuring preservation of Pol fcontinuity.
Proposition 27.Let F and G be respectively a A-continuous and a B-continuous functor such that F admits signature f and G admits signature g.In the next section, we will leverage Proposition 27 by defining a type system for polynomial composites of endofunctors.

Syntax for parameterisations and models
We capture the results of this section into grammars for parameterisations and models.

A grammar for parameterisations
In what follows, we let G = {G, M + } be the Giry-like functors (respectively Giry, the finite positive measure and finite nonzero measure functors).We recall that ∆ is the diagonal functor.
Definition 28 (Parameterisations generated by a family of functors).Parameterisations, denoted by P, are defined by the following grammar: where F ranges over functors satisfying the parameter condition.
We have the following expected result: Theorem 29.All P ∈ P verify the parameter condition.
Proof.By induction, using the results of Section 4.1.

23:11
Inferences rules for the type system on models.

A grammar for models
Proposition 27 gives a sufficient condition ensuring that composition of functors satisfying the model condition still satisfies the model condition.We integrate this result in a type system for polynomial composites of functors satisfying the model condition.
Definition 30 (Functor types).A functor type is a pair (A, f ) with A ∈ P and f ∈ Sign(P).
The set of types is defined by T ypes(P) P × Sign(P ).
Functor types are assigned to elements of the polynomial closure of the set of functors that satisfy the model condition.
Definition 31 (Typing judgments).We inductively define a relation between functors satisfying the model condition and functor types through the set of inference rules in Figure 3.The fact that a functor F admits the type (A, f ) will be denoted by F :: (A, f ).
Our type system is sound with respect to the model condition.It is only known to be Pol f -continuous ( [4]).Therefore, GB is a valid model functor but BG breaks the third premise of the composition rule in Figure 3: indeed, G maps finite spaces to compact spaces (Figure 1).
Definition 34.The set of models is defined to be that of typeable functors and will be denoted by M.

Natural parameterised models
Theorem 29 and 32 delineate a class of parameters and models to which the Machine (Theorem 13) applies.These combined results can be reframed concisely as follows:

Applications
It is hard to overstate the importance of independently and identically distributed (i.i.d.) sequences of random of variables and their generalisation to exchangeable processes to probability and statistics, as witnessed by the wealth of powerful asymptotic results which apply to them -to name a few, the law of large number, the central limit theorem and the de Finetti theorem [12].We illustrate the usefulness of our framework by recasting i.i.d.processes and the de Finetti theorem as instances of our parameterised models.

The iid natural transformation
Let X be finite Polish.For all integer n > 0, we construct an arrow iid n X : G(X) → G(X n ) as follows: iid n X (p) = (B 1 , B 2 , . . ., B n ) → p(B 1 ) • p(B 2 ) • • • p(B n ).One easily verifies that this map is well-typed and continuous.We have the following result: Proposition 36.For all all positive integer n, the family iid n X defines a natural transformation iid n : G ⇒ G(− n ).
Proof.Let X, Y be finite spaces and let f : X → Y be a function.Let p ∈ G(X) be given and let (y 1 , . . ., y n ) be a sequence in Y n .We have: We have proved that the iid n transformation is well defined on all finite spaces.One easily checks that G is a P-functor and G(− n ) is a M-functor.Applying Theorem 35, we conclude that iid n admits an unique extension to the whole of Pol.

Example 14 .
Let us give some examples of finitely characterised natural transformations.

20 .
If G, H are A-continuous, then so is G + H.

Figure 2 A
Figure 2 A finer signature for V.

Proof.
Since f (Pol f ) ≤ B, we know that C exists and verifies Pol f ≤ C. Let D : I op → C be a C ccd.By assumption of A-continuity and using hat C ≤ A, F (lim D) ∼ = lim F D. Since F admits f as a signature, F D is a f (C)-ccd and since f (C) ≤ B, G(lim F D) ∼ = lim GF D.

Theorem 32 .
If M :: (A, f ) then M is A-continuous, m admits signature f and M satisfies the model condition.Proof.The proof is by induction.The properties of preservation of monos, preservation of embeddings and preservation of intersections are treated in Section 4.2.Sum rule.Both F and G are A ∧ B-continuous, therefore by Proposition 20, F + G is A ∧ B-continuous (and therefore are least Pol f -continuous).It is clear that a coproduct of finite spaces is finite and similarly for compact zero-dimensional spaces, compact spaces and zero-dimensional spaces.Therefore, F + G admits f ∨ g as a signature.The case of the product rule is similar.Composition.C-continuity is by Proposition 27.That F G admits f • g as a signature is trivial.Example 33.It is instructive to consider the the multiset functor B. It maps finite spaces to finite spaces but we ignore its behaviour on other subcategories, hence the signature:

Theorem 35 .
For all parameterisation P ∈ P and all model M ∈ M,N at(P, M ) ∼ = N at(P | Pol f , M | Pol f ).