A Linear Exponential Comonad in s-finite Transition Kernels and Probabilistic Coherent Spaces

This paper concerns a stochastic construction of probabilistic coherent spaces by employing novel ingredients (i) linear exponential comonads arising properly in the measure-theory (ii) continuous orthogonality between measures and measurable functions. A linear exponential comonad is constructed over a symmetric monoidal category of transition kernels, relaxing Markov kernels of Panangaden's stochastic relations into s-finite kernels. The model supports an orthogonality in terms of an integral between measures and measurable functions, which can be seen as a continuous extension of Girard-Danos-Ehrhard's linear duality for probabilistic coherent spaces. The orthogonality is formulated by a Hyland-Schalk double glueing construction, into which our measure theoretic monoidal comonad structure is accommodated. As an application to countable measurable spaces, a dagger compact closed category is obtained, whose double glueing gives rise to the familiar category of probabilistic coherent spaces.


Introduction
Coherent spaces [19], the original model in which Girard discovered linear logic, provide a denotational semantics of functional programming languages as well as logical systems.Each space is a set endowed with a graph structure, called a web, in which a proof (hence a program) is interpreted by a certain subset, called a clique.The distinctive feature of this model is the linear duality, stating that a clique x ⊆ X and an anti-clique x ′ ⊆ X ′ intersect in at most a singleton #(x ∩ x ′ ) ≤ 1.The linear duality arising intrinsically to the coherent spaces goes along with constructive modelling of logical connectives.There arises a dual pair of multiplicative connectives and of additive ones, together with linear implication for multiplicative closed structure (i.e., *-autonomy of denotational semantics).
Category theoretically (freely from the web-based method), coherent spaces are realised by Hyland-Schalk's double glueing construction [27] G(Rel) over the category of relations Rel, which is the most primary self dual denotational semantics with the tensor (the cartesian product of sets) and the biproduct (the disjoint union of sets).The double glueing lifts the degenerate duality of Rel into a nondegenarate one, called orthogonality, which in turn gives rise to the linear duality so that the coherent spaces reside as an orthogonal subcategory.
Developing the web method, Ehrhard investigates the linear duality in the mathematically richer structures of Köthe spaces [10] and finiteness spaces [11].His investigation of duality leads Danos-Ehrhard [7] to formulate a probabilistic (fuzzy) version of duality in their probabilistic coherent spaces Pcoh.Their construction starts with giving non-negative real valued functions on a web I, reminiscent of probabilistic distributions (but not necessarily to the interval [0, 1]) on the web, so that a clique becomes a subset of R I + .Then, in their probabilistic setting, the linear duality becomes formulated x, x ′ = i∈I x i x ′ i ≤ 1 for x, x ′ ∈ R I + .The precursor of the formulation is addressed earlier in Girard [22].The probabilistic linear duality is accommodated into the linear exponential ! by generalising the original finite multiset functor construction of Girard [19] with careful analysis of permutations and combinations on enumerating members of multisets.The canonicity of their exponential construction is ensured in [6].
The recent trend of probabilistic semantics is more widely applied to transition systems with continuous state spaces for concurrent systems such as stochastic process calculi.The stochastic relation SRel, explored by Panangaden [33,32], provides a fundamental categorical ingredient to the study, analogous to how the category Rel of the relations has been to deterministic discrete systems.Recalling that Rel is the Kleisli category of the powerset monad, SRel is a probabilistic analog of the Giry monad [23], whereby powerset is replaced by a probability measure on a set, giving random choice of points, hence collections of fuzzy subsets are obtained.SRel also provides coalgebraic reasoning for continuous time branching logics [9].Despite the lack of cartesian closed structure, Markov kernels provide a measure theoretic foundation of recent development of various denotational semantics for higher-order probabilistic computations [16,39], theoretically with adequacy and practically with continuous distributions for Monte Carlo simulation.We also remark an intermediate approach on the weighted relational model [29] confining the discrete probability but acquiring *-autonomy and exponential structure for Linear Logic.
This paper aims to give general machinery, inspired from SRel, combining category and measure theories to provide two constructions (i) linear exponential comonad for stochastic processes and (ii) linear duality and probabilistic orthogonality in continuous spaces.The two parts involve comonads, widely used in computer science, but exploration in its continuous stochastic aspect is initiated just recently by [13,17,34] in higher order probabilistic programming.
Our results for each are: (i) The counting process [4] known in the theory of stochastic processes provides a new categorical instance of linear exponential comonad, modelling the exponential modality of linear logic.In particular for countable measurable spaces this gives a simple account of the exponential of Pcoh.Our linear exponential comonad of transition kernels is also seen a continuous version of the weighted relational model in [29] using R + -weighted Rel for the analytic exponential.
(ii) A general continuous framework of the transition kernels provides a new instance of Hyland-Schalk orthogonality and an explanation of how the linear duality arises continuously in terms of measure and measurable functions.
Notice that both (i) and (ii) lack any closed structure for monoidal products in the continuous framework, hence are inconclusive as a model of linear logic.
The paper starts with presenting that a stochastic framework of transition kernels [1] forms a category TKer with biproducts.This in particular explains how kernels need to take the infinite real value ∞.The transition kernels, between measurable spaces and measurable functions, are a relaxation of sub-Markov kernels of Panangaden's SRel. of sub-probability measures.The monoidal product is straightforwardly given by the measure theoretic direct product, same as in SRel.However under the relaxation in our framework, the functoriality of the product needs to be examined carefully.It is well known from the measure theoretic perspective that the functorial monoidal product is ensured by a fundamental Fubini-Tonelli Theorem, for which a standard measure-theoretic restrain is σ-finiteness [33], including finiteness, hence Markov.However from category theoretic perspective, the σ-finiteness has inadequacies in that it is not preserved under the categorical composition.Although a smaller class of the finiteness [3] retains both categorical composition and functorial monoidalness, the class is too small to accommodate the exponential modality for our purpose in this paper.The s-finiteness, recently studied in Staton [38], widens the class of the traditional σ-finiteness in order to gain the composition, but still conserving Fubini-Tonelli for the functoriality of monoidal products.We show that the s-finiteness also provides an exponential construction for us, not only measure theoretically, but also category theoretically.The exponential is shown to be characterised in terms of "counting measures" [4] in the theory of stochastic processes, in which the countable functions for multisets become measurable.We construct an exponential endofunctor on a monoidal category TsKer of the s-finite transition kernels.Furthermore, we construct a linear exponential comonad in TsKer op , modelling the exponential modality in linear logic category theoretically [2,27,30].
Second, observing a contravariant equivalence of TKer to M E of measurable functions and linear positive maps preserving monotone convergence, we formulate an orthogonality between a measurable map f and a measure µ in terms of f dµ ≤ 1, where f ∈ TKer(X , I) and µ ∈ TKer(I, X ).This is a continuous version of Danos-Ehrhard's linear duality for Pcoh.Using an inner product in terms of Lebesgue integral, our orthogonality has an adjunction between operators κ * and κ * of a kernel κ respectively on measurable functions and on measures.The adjunction provides a coherence condition of the orthogonality for the exponential comonad, which is shown to drastically simplify the general method of Hyland-Schalk.The orthogonality allows us to construct a double glueing G(TsKer op ) à la Hyland-Schalk, in which the coproduct and product become different as well as the monoidal tensor and cotensor.We follow Hyland-Schalk to formulate the two subcategories, slack and tight orthogonality categories, of G(TsKer op ).Our main interest afterward is the tight category.
Finally, the full subcategory TsKer ω of the countable measurable spaces is investigated, whose morphisms collapse to the transition matrices from the kernels.TsKer ω has a dagger functor internalising the contravariant equivalence in the restriction to the subcategory.This brings a monoidal closed structure in TsKer ω , making the category compact closed.The double gluing instance G(TsKer op ω ) becomes * -autonomous and its subcategory T(TsKer op ω ) of the tight orthogonality is shown equivalent to the category Pcoh of probabilistic coherent spaces.The equivalence for the first time gives a precise formulation of the folklore among the linear logic commmunity (cf.[31]).In turn this particularly subsumes the coincidence of our TsKer ω with R + -weighted Rel ( [29]) known earlier in the community.
The paper is organised as follows: Section 2 presents various categories of transition kernels and measurable spaces.Section 3 starts with a measure theoretic study on exponential measurable spaces and exponential transition kernels.The goal of Section 3 is to construct a linear exponential comonad over a monoidal category TsKer op .Section 4 is an application of Hyland-Schalk double glueing to our measure theoretic construction.Section 5 restricts to the countable measurable spaces in particular for obtaining Pcoh.This section recalls some basic definitions and a theorem from measure theory, necessary in this paper.
(Terminology) N denotes the set of non negative integers.R + denotes the set of non negative reals.R + denotes R + ∪ {∞}.S n denotes the symmetric group over {1, . . ., n}. δ x,y is the Kronecker delta.For a subset A, χ A denotes the characteristic function of A. The Dirac delta δ x (A) is χ A (x). ⊎ denotes the disjoint union of sets.
Definition 1.1 (σ-field X and measurable space (X, X )).A σ-field over a set X is a family X of subsets of X containing ∅, closed under the complement and countable union.A pair (X, X ) is called a measurable space.The members of X are called measurable sets.The measurable space is often written simply by X , as X is the largest element in X .For a measurable set Y ∈ X , the measurable subspace X ∩ Y , called the restriction on Y , is defined by In this paper, a measurable function unless otherwise mentioned is to the Borel set B + over R + from some measurable space (X, X ).
Definition 1.4 (measure).A measure µ on a measurable space (X, X ) is a function from X to R + satisfying (σ-additivity): Definition 1.5 (integration).For a measure µ on (X, X ), and a (X , B + ) -measurable function f , the integral of f over X wrt the measure µ is defined by X f (x)µ(dx), which is simply written X f dµ.It is also written X dµf .Theorem 1.6 (monotone convergence).Let µ be a measure on a measurable space (X, X ).For an monotonic sequence {f n } of (X , B + )-measurable functions, if f = sup n f n , then f is measurable and sup X f n dµ = X f dµ.
Definition 1.7 (push forward measure µ • F −1 along a measurable function F ).For a measure µ on (Y, Y) and a measurable function ) becomes a measure on (Y ′ , Y ′ ), called push forward measure of µ along F .The push forward measure has a following property, called "variable change of integral along push forward F ": The push forward measure µ ′ is often denoted by µ • F −1 by abuse of notation.
2 Category TKer, its Dual M E and Monoidal Subcategory TsKer of s-finite Transition Kernels This section starts with introducing a category TKer of transition kernels with convolution (i.e., an integral transform on the product) as categorical composition.Measures and measurable function on a measurable space both arise as certain morphisms in the category.A contravariant equivalence is shown to a category M E of measurable functions.When imposing s-finiteness to kernels, a monoidal subcategory TsKer with (countable) biproducts is obtained.
• For a measure µ on X , is a measure on Y, where B ∈ Y.
In particular, for a Dirac measure δ a with any a ∈ X, • For a measurable function f on Y, is measurable on X , where x ∈ X.
In particular, for a characteristic function χ B for any B ∈ Y, It is direct to check, by the monotone convergence theorem 1.6, that κ * f is measurable.
A characterization is known in (1) for which general mappings α in place of κ * in turn define transition kernels as follows: Proposition 2.3 (Lemma 36.2 [1]).Let E(X ) denote the set of all R + -valued measurable functions on a measurable space X .If a function α : becomes a transition kernel from (X, X ) to (Y, Y).Moreover κ is the unique transition kernel satisfying κ * f = α(f ) for all f ∈ E(X ).-TKer denotes the category where each object is a measurable space (X, X ) and a morphism is a transition kernel κ from (X, X ) to (Y, Y).The composition is the convolution of two kernels κ : (X, X ) −→ (Y, Y) and ι : (Y, Y) −→ (Z, Z): id (X,X ) is the unit kernel δ : (X, X ) −→ (X, X ), defined for x ∈ X and A ∈ X by; That is, for each x ∈ X, δ(x, −) is the Dirac measure on (X, X ).
-M E denotes the category whose objects are measurable spaces, same as TKer, but whose morphisms M E (X , Y) consists of any linear positive map α : E(X ) −→ E(Y) preserving monotone convergence.The composition is simply that of the functions.
It is now well known that the composition (2) for sub-Markov kernels (cf.below Remark 2.6) comes from Giry's probabilistic monad, resembling the power set monad of the relational composition (cf.[23,33]).Instead of using the monad applied to our general setting for the transition kernels, we give a simpler intuition how the composition (2) arises via the simpler composition of M E by assuming the expected functoriality (ι In particular, taking which by (1) imposes the definition of the composition of the two kernels.
The morphisms of SRel are called sub-Markov kernels.They are called Markov kernels when κ(x, B) = 1 for any x ∈ X. Note: The bounded condition of (ii) is derivable from (i), thus the condition (ii) is redundant when defining SRel as a subcategory of TKer.
We also remark here a crucial reason seen immediately in the next Subsection 2.2 why SRel needs to be extended to TKer in this paper: The coprpduct of SRel given in [33,32] is not a biproduct in SRel, but it is in TKer (cf.Proposition 2.9).The biproduct existing in TKer will be crucial to the main purpose of the paper constructing exponential structure in the s-finite subcategory to be introduced in Section 2.3.Proposition 2.3 says category theoretically; Proposition 2.7.TKer and M E are contravariantly equivalent.The equivalence is given by the contravariant functor ( ) * : On the objects, ( ) * acts as the identity.On the morphisms, the functoriality The contravariant equivalence ( ) * in particular gives a direct account on the measurable functions as the homset in Remark 2.5 by TKer(X , I) ∼ = M E (I, X ).
Remark 2.8.The contravariant functor ( ) * when restricted to the Markov kernels SRel gives a contravariant equivalence to Vec E b , where each object is the subspace E b (X ) of bounded measurable functions, which forms a vector space.The boundedness makes the space not only a vector space but moreover a Banach space with the uniform norm f = sup x∈X |f (x)|.The opposite category is studied in [33] as the category of the predicate transformers, stemming from Kozen's precursory work on probabilistic programming.Taking measurable functions as predicates and measures as states, the ordinary satisfaction relation, say µ |= f , is generalised into integrals, say f dµ giving a value in the interval [0, 1].In the present paper in Section 4.2, this satisfaction relation will be explored in terms of the orthogonality relation.
Proof.Given a countable family {(X i , X i )} i of measurable spaces, we define i where i X i := { i {i} × A i | A i ∈ X i } is the σ-field generated by the measurable sets of each summands.
(Coproduct): (3) defines a coproduct for TKer.The injection in j : (X j , X j ) −→ ( {i}×X i , X i ) is defined by in j (x j , i A i ) := δ(x j , A i ).The mediating morphism Note (3) is the same instance as the known coproduct in SRel.However, in the relaxed structure of TKer, we have moreover; (Product): (3) becomes a product for TKer.The projection pr given by pr i ((j, x), A i ) := δ i,j • δ(x, A i ).The mediating morphism & i g i for given morphisms Note the construction for the meditating morphism is not closed in Markov kernels, but is so in transition kernels.This construction shows how values of measurable functions include the infinite real ∞ when I becomes infinite.We check the uniqueness of the mediating morphism, say m: The required commutativity for m is pr j • m = g j , which holds by (4) if and only if m(b, A j ) = g j (b, A j ) for all j.Since m(b, −) is a measure and {A j } j∈J are disjoint, this yields the definition & i g i of the mediating morphism.
The unit of the biproduct is the null measurable space T = (∅, {∅}).This subsection ends with the following remark, which though is not required to proceed to read the paper.
Remark 2.10 (TKer is traced wrt the biproduct).TKer is a unique decomposition category [24,25], which is a generalisation of Arbib-Manes partially additive category studied in [33] for SRel.A countable family of TKer morphisms {κ i : (X, X ) −→ (Y, Y) | i ∈ I} is summable so that i∈I κ i (x, B) is a transition kernel.Then (TKer, ) is traced so that any κ : (X, ) which is the standard trace formula corresponding to Girard's execution formula for Geometry of Interaction [20].Refer also to [26] for the definition of the Tr using the execution.The trace provides to model feed back as well as iteration on a given morphism.In particular, Int construction by Joyal-Street-Verity [28] yields a compact closed completion of TKer with taken as tensor.Be aware that the monoidal product addressed in this paper is not this one but the measure theoretic direct product of Definitions 2.11 and 2.19 introduced below.

Monoidal Product and Countable Biproducts in TsKer
This subsection introduces a subcategory TsKer of s-finite transition kernels.The s-finiteness is a relaxation of a standard measure theoretic class of the σ-finiteness so that the σ-finiteness resides intermediately between finiteness and s-finiteness.The relaxed class of the s-finite kernels is closed under composing kernels, which is not the case in the class of σ-finite kernels.Inside the subcategory TsKer, monoidal products of morphisms are functorially defined to accommodate Fubini-Tonelli Theorem for the unique integration over product measures.TsKer is also shown to retain the countable biproducts in TKer of the previous subsection.Definition 2.11 (product of measurable spaces).The product of measurable spaces (X 1 , X 1 ) and (X 2 , X 2 ) is the measurable space (X 1 × X 2 , X 1 ⊗ X 2 ), where X 1 ⊗ X 2 denotes the σ-field over the cartesian product X 1 × X 2 generated by measurable rectangles In order to accommodate measures into the product of measurable spaces, each measures µ i on (X i , X i ) need to be extended uniquely to that on the product.The condition of σ-finiteness ensures this, yielding the unique product measure over the product measurable space: Definition 2.12 (σ-finiteness).A measure µ on (X, X ) is σ-finite when the set X is written as a countable union of sets of finite measures.That is, ∃A 1 , A 2 , . . .∈ X such that µ(A i ) < ∞ and X = ∪ ∞ i=1 A i .Definition 2.13 (product measure).For a σ-finite measures µ i on (X i , X i ) with i = 1, 2, there exists a unique measure µ on ( . µ is written µ 1 ⊗ µ 2 and called the product measure of µ 1 and µ 2 . The product measure derived from σ-finite measures guarantees a basic theorem in measure theory, stating double integration is treated as iterated integrations.Theorem 2.14 (Fubini-Tonelli).For σ-finite measures µ i on The measure theoretical basic Fubini-Tonelli Theorem will become crucial also to the categorical study of the present paper, not only dealing with functoriality of morphisms on the product measurable spaces (cf.Proposition 3.11 below), but also giving a new instance of the orthogonality using the measure theory in Section 4.
Although one can impose σ-finiteness for the transition kernels κ(x, −) (uniformly or nonuniformly in x), this class of kernel is not closed in general under the composition in the category TKer.For the sake of category theory, one remedy for ensuring the compositionality is to tighten the class into the finite kernels.This class confined to the measures is used in finite measure transformer semantics [3] for probabilistic programs.However the class of the finite kernels is not closed under our exponential construction (Definition 3.9) later seen in Section 3.2.Thus, we need another remedy to loosen the condition contrarily, which is how s-finiteness arises below.While its notion was earlier established in [37,18], the s-finiteness is recently studied by Staton [38] in modelling programming semantics.In addition to the compositionality in our categorical setting, the relaxed class of the s-finite kernels is shown to retain the Fubini-Tonelli Theorem (Proposition 2.18) working with the uniquely defined product measure.Definition 2.15 (s-finite kernels [37,38]).Let κ be a transition kernel from (X, X ) to (Y, Y).
-κ is called finite when sup x∈X κ(x, Y ) < ∞; i.e., the condition says that up to the scalar 0 < a < ∞ factor determined by the sup, κ is Markovian.
-κ is called s-finite when κ = i∈N κ i where each κ i is a finite kernel from (X, X ) to (Y, Y) and the sum is defined by ).This is well-defined because any countable sum of kernels from (X, X ) to (Y, Y) becomes a kernel of the same type.
In the definition of s-finiteness, note that ( i∈N κ i ) * = i∈N κ * i and ( i∈N κ i ) * = i∈N (κ i ) * for the operations of Definition 2.2: That is, the preservation of the operation ( ) * (resp. of ( ) * ) means the commutativity of integral over countable sum of measures (resp. of measurable functions).Remark 2.16.Both classes of the finite kernels and of the s-finite kernels are closed under the categorical composition of TKer.This is directly calculated for the finite kernels, to which the s-finite ones are reduced by virtue of the note in the above paragraph.We refer to the proof of Lemma 3 of [38] for the calculation.In particular, the class of s-finite kernels is closed under push forwards along measurable functions.The both classes form wide subcategories of TKer introduced below Definition 2.19.
The definition subsumes that of s-finite measures when (X, X ) is in particular taken the singleton measurable space (I, I).Note that every σ-finite measure is s-finite, but not vice versa: E.g., the infinite measure ∞ • δ a for the Dirac δ a with a ∈ X is not σ-finite, but s-finite.
A characterization of s-finite kernels is directly derived: Proposition 2.17 (Proposition 7 of [38]).A kernel is s-finite if and only if it is a push forward of a σ-finite kernel.
Proof.We prove "only if" part as "if part" is direct because of the inclusion of σ-finiteness into s-finiteness and of the closedness of s-finiteness under push forward.Given a s-finite kernel κ = i∈N κ i with finite kernels κ The original Fubini-Tonelli (Theorem 2.14) for the σ-finite measures extends to the s-finite measures: Proposition 2.18 (Fubini-Tonelli extending for s-finite measures (cf.Proposition 5 of Staton [38])).For the same f as Theorem 2.14 but µ 1 and µ 2 are s-finite measures, it holds; 1 and µ 2 = j∈N µ j 2 with finite kernels µ i 1 s and µ j 2 s, then the following is from (LHS) to (RHS): The first and the last (resp.the second and the second last) equations are by the commutativity of integral over countable sum of measurable functions (resp. of measures) (cf. the note in Definition 2.15).The middle equation is the original Fubini-Tonelli for the σ-finite measures, hence here in particular for the finite ones.
Finally it is derived that the s-finite transition kernels form a monoidal category.Definition 2.19 (monoidal subcategories TsKer of s-finite kernels and Tker of finite ones).TsKer is a wide subcategory of TKer, whose morphisms are the s-finite transition kernels.The s-finiteness of kernels is preserved under the composition of TKer.TsKer has a symmetric monoidal product ⊗: On objects is by Definition 2.13.Given morphisms Alternatively, thanks to Fubini-Tonelli (Proposition 2.18), the monoidal product is implicitly defined as the unique transition kernel The unit of the monoidal product is the singleton measurable space (I, I).
Tker is a monoidal wide subcategory of TsKer whose morphisms are finite transition kernels.Employing s-finiteness to yield the monoidal product is due to Staton [38].Our further interest in s-finiteness in this paper is its retaining the countable biproducts of TKer defined Proposition 2.9.
Proposition 2.20 (The subcategory TsKer retains the countable biproducts of TKer).TsKer has countable biproducts which are those in TKer residing inside the subcategory.
Proof.The coproduct construction of Proposition 2.9 all works under the additional constraint of the s-finiteness of kernels.For the product construction, the only construction necessary to be checked is that of the mediating morphism & i g i , employing the sum over i ∈ I for a countable infinite I: If given g i 's of the product construction in Proposition 2.9 are s-finite, then each is written g i = j∈N g (i,j) , where each In what follows, the index set I is identified with N. A transition kernel h n is defined for each n ∈ N: where in i : (X i , X i ) −→ j∈I (X j , X j ) is the coproduct injection.Note h n is a finite kernel, as the sum specified by the subscript i + j = n is finite.Then in terms of the finite kernels, the mediating morphism & i g i constructed in Proposition 2.9 is represented as follows to be s-finite: Remark 2.21 (infinite biproducts as colimit in TsKer).The countable infinite biproducts in TsKer is characterised by the colimits inside the subcategory: Given the direct system TsKer.Hence, the colimit is closed in the subcategory TsKer, but not necessarily in Tker.
3 A Linear Exponential Comonad over TsKer op 3.1 Exponential Measurable Space (X e , X e ) This subsection concerns a measure theoretic study on exponential measurable spaces.[4] is a good reference for the subsection.Definition 3.1 (exponential monoid X e ).X e denotes the free abelian monoid (the free semi group with identity) generated by a set X: The members of X e are the formal products x 1 x 2 • • • x n where x i ∈ X and n ∈ AE so that order of factor is irrelevant.The monoid operation for members of X e is obviously the free product.When n = 0, under the convention x 1 x 2 • • • x n = 0, this is the monoid identity (in spite of the multiplicative notation), which is equated with the empty sequence.The monoid operator is written by a product (x, y) → xy.Each member Ü ∈ X e is identified with a finite multiset of elements in X and vice versa.Hence Ü is seen as an integer valued function on X, which vanishes to zero outside the finite sets; Ü(t) = multiplicity of t ∈ X as a factor of Ü.That is, Ü represents the unique multiset of elements X, and vice versa.
For A ⊆ X, we define Then Ü(A) represents the number of elements in A.
The counting function n A on X e is defined for each A ⊆ X, The members of X e can be seen as equivalence classes of ordered sequences in X • e defined below under rearrangement (permutations of factors): Definition 3.2 (non-abelian monoid X • e ).Using • for ordered sequences, X • e denotes the nonabelian monoid generated by X, consisting of ordered sequences where x i ∈ X and n ≥ 0. The monoid operation for members of X • e is obviously the operation • joining sequences in order.Then the abelian monoid X e is the image of the monoid homomorphism F forgetting the order of the factors: where the set (Notation) For any family F of subsets of X, P • (F ) denotes the class of all finite ordered the order of factors is irrelevant.
induced by (X, X )).Every measurable space X on X induces a corresponding measurable space on the set X • e defined by: whose σ-field X • e is the disjoint union of the measure theoretic n-ary direct product of X , on the set e is the σ-field generated by P • (X ) and the subspace of X • e restricted to X •n coincides with the n-ary direct products of the measurable space X : I.e., In terms of category theory, Definition 3.3 says Proposition 3.4.In TKer, the measurable space (X Note that the injection factors through the colimit inclusion in l of the direct system of Remark 2.21 such that in ∞ m = in l • in l m for any l ≥ m.The infinite coproduct moreover becomes infinite biprducts, whose projection pr ∞ m to the m-th component is Because of Remark 2.21, the construction of Proposition 3.4 is closed inside the subcategory TsKer of s-finite kernels (but not in Tker of finite kernels).
Finally, the exponential measurable space (X e , X e ) is obtained by the following equivalent characterisations of a σ-field X e .Proposition 3.5 (σ-field X e over X e (cf.Theorem 4.1 [4])).For a measurable space (X, X ), the following families of subsets of X e all coincide with the σ-field σ(P(X )), which is denoted by X e .
e by F : I.e., the σ-field {F ( ) | ∈ X • e }. (iii) The smallest σ-field wrt which the counting functions n A are measurable for all A ∈ X .(iv) The largest σ-field Y for X e having X as a subspace and such that the monoid product is measurable from Y × Y to Y. (v) The smallest σ-field for X e containing X and for which the monoid product preserves measurability.(vi) The smallest σ-field for X e containing X and closed under the symmetric product.Definition 3.6 (exponential measurable space (X e , X e )).The measurable space (X e , X e ), whose X e is defined by Proposition 3.5 such that X e = σ(P(X )), is called the exponential measurable space of (X, X ).This section ends with a measure theoretic proposition on isomorphisms relating the biproduct and the tensor via the exponential: Proposition 3.7.The following holds for any measurable spaces (X, X ) and (Y, Y): (ii) T e = ({∅}, {∅, {∅}}), which is isomorphic to the monoid unit (I, I).
Proof.We prove (i) since (ii) is direct, as the monoid identity of the exponential monoid of Definition 3.1 is given by the empty sequence.First, the monoid isomorphism (X ⊎ Y ) e ∼ = X e × Y e between the largest measurable sets of each side is given as follows; For any Note Þ ′ and Þ ′′ are unique independently of the choice of the rearrangement, thus mapping Ü to (Þ ′ , Þ ′′ ) gives a monoid isomorphism.Second, the monoid iso is shown to induce the set theoretical isomorphism of the σ-fields of both sides (X ⊎ Y) e ∼ = X e ⊗ Y e .
By the definition of the product of two measurable spaces and Proposition 3.5 (iii), X e ⊗ Y e is the smallest σ-field in which the product of counting functions becomes measurable for all A ∈ X and B ∈ Y.As the exponential function is one to one, Proposition 3.5 (iii) holds with n A replaced by 2 n(A) .Since n A⊎B (ÜÝ) = n A (Ü) + n B (Ý), the following commutes so that the isomorphism becomes that between the two σ-fields.
That is, (X Y) e and X e ⊗ Y e are the smallest σ-fields making the respective functions 2 nA⊎B and 2 nA 2 nB measurable for all A ∈ X and B ∈ Y.
Remark 3.8 (Seely isomorphism).The isomorphism (i) of Proposition 3.7 is a Seely isomorphism [35] in an appropriate category theoretical model of linear logic, as our binary biproduct models the logical connective &.The Seely isomorphism is known derivable [2,30] using category theoretic abstraction from any linear exponential comonad structure with product, which structure will be obtained for a certain class of transition kernels in the next Section 3.3 (cf.Theorem 3.25).

Exponential Kernel κ e in s-finiteness
This subsection concerns a categorical investigation in TsKer on the exponential measurable spaces of Section 3.1.This section starts with seeing the exponential acts not only on objects as defined in Section 3.1 but on the morphisms on TsKer, hence becomes an endofunctor.
(Notation) For a measurable space (X, X ) and m ∈ N, This divides the set X e into the following disjoint union: For any A ∈ X , A (m) is defined same for the subspace X ∩ A.
In what follows, κ n denotes the n-ary cartesian product κ n : (X, X ) n −→ (Y, Y) n , for which the n-ary cartesian product of an object is given by (X, X See the following commutative diagram for the definition ( 6): for any (x 1 , . . ., x m ) ∈ X • e and any ∈ Y e ∩ Y (m) with any m ∈ N.
Directly from the definition, for any permutation σ ∈ S m , (9) is implied using ( 7) by the following (10) for any It is sufficient to check (10) for any rectangle ) by the definition of the product measurable space.
Thus, finally we define κ e : X e × Y e −→ R + for any which definition does not depend on the ordering of We need to check κ e defined above is a transition kernel: The second argument of κ e giving a measure over Y e is direct by the definition (11) because so does the second argument of κ • e .For measurability in X e for the first argument of κ e , by virtue of Proposition 3.5 (ii), it suffices to show that (κ e ) -1 (F -1 (−), ) is measurable in X • e .But this is derived from the measurability of the first argument κ • e in X • e because by the commutative diagram below, yielding .
The so constructed κ e is s-finite, as κ • e resides in TsKer (cf.Proposition 3.4) and s-finiteness is closed under the push forward along F (cf. Remark 2.16).
In order to show the functoriality of the exponential over kernels of Definition 3.9, we prepare the following lemma on π-system and Dynkin system.Lemma 3.10 (for Proposition 3.11: a π-system for X • e ).Let (X, X ) be a measurable space.
(a) For any n ≥ 0, the following family consisting of subsets of is both (i) a π-system and (ii) a Dynkin system.
Recall that a nonempty family of subsets of a universal set is a π-system if the family is closed under finite intersections.It is a Dynkin system if the family contains ∅ and is closed both under complements and under countable disjoint unions.
(b) D n becomes a σ-field, hence coincides with the measurable space X •n for any n ≥ 0. Thus we characterise Proof.As (b) is a consequence of (a) by Dynkin Theorem (cf.[1] for the theorem) stating that any Dynkin system which is also a π-system is a σ-field, we prove (a): (ii) As the empty set is contained in (12), we check the other two conditions: (Closedness under countable disjoint unions) Immediate from the definition (12), observing , where [n] := {1, . . ., n} and X 1 = X.Then using De Morgan and distribution of intersection over union: where The equation ( 13) is that of Definition 1.7 when the push forward measure µ ′ = µ • F -1 is defined for µ( ) := κ • e (−, ) with any fixed − (cf.( 8)), and the measurable function g on Y e is given by ι e (Ý, ∼) with any fixed ∼.
For any x 1 • • • x n ∈ X e and any ∈ Z e ∩ Z (n) such that, by Lemma 3.10 (b), by (13) of variable change by the def of ι e = Y •n κ n ((x 1 , . . ., x n ), d(y 1 , . . ., y n )) ι n ((y 1 , . . ., y n ), F -1 ( )) by ( 7) by the def of (ι • κ) e Remark 3.12 (The exponential construction (−) e preserves s-finiteness, but not finiteness.).In addition that the class retains Fubini-Tonelli for the functorial monoidal product in Section 2.3, the class of s-finiteness is employed in this paper because it makes (−) e an endofunctor as shown above.E.g., its restriction on Tker of the finite kernels is no more an endofunctor but from Tker to TsKer.

A Linear Exponential Comonad over TsKer op
The exponential presented in Section 3.1 and Section 3.2 is shown to provide a linear exponential comonad over the monoidal category TsKer op with countable biproducts, hence a categorical model of the exponential modality of linear logic [2,27,30].
Due to the asymmetry between the first (measures) and the second (measurable functions) arguments of transition kernels in continuous measurable spaces, the exponential comonad considered in Subsection 3.3 is for the opposite category TsKer op2 .
Notation for morphisms in the opposite TsKer op : The category considered in this section is the opposite category TsKer op so that the composition is converse to TsKer: In TsKer op , a morphism κ : (X, X ) −→ (Y, Y) is a transition kernel from (Y, Y) to (X, X ).Accordingly a morphism κ is denoted by κ(A, y) meaning that its left (resp.right) argument determines a measure (resp.a measurable function).In particular, the Dirac delta measure which is the identity morphism on (X, X ) is written by δ(A, x).Hence, the composition of two morphisms κ(A, y) : (X,

Typographic Convention:
In what follows, the following typography is used to discriminate levels of the exponential measurable spaces: x, y, z, . . .∈ X and A, B, C, . . .∈ X for (X, X ).
We recall the definition of linear exponential comonad.Definition 3.13 (linear exponential comonad [27,30]).Let (C, ⊗, I) be a symmetric monoidal category.A linear exponential comonad on C is a monoidal comonad equipped with two monoidal natural transformations c : !−→ ∆ • !(with ∆ denoting the diagonal functor for tensor) and w : !−→ I such that the following holds for each X: We start to construct the structure maps for linear exponential comonad in TsKer op .Proposition 3.14 (Dereliction).d X : (X e , X e ) −→ (X, X ) is defined for ∈ X e and x ∈ X d X ( , x) := δ( ∩ X (1) , x) Recall that ∩ X (1) ⊂ X.Then, this gives a natural transformation d : ( ) e −→ Id TsKer op .
In order to introduce the storage in Proposition 3.17, we prepare; Definition 3.15 (| |: X ee −→ X e ).For a set X, the mapping | | is defined by Note: -| | on X ee ∩ (X e ) (1) is the identity.That is, when n = 1 so that 1 ∈ X ee , it holds | 1 |= 1 .
-| | on X ee ∩ (X e ∩ X (1) ) (n) is the identity.That is, when k i = 1 for all i = 1, . . ., n so that For any ∈ X e , its inverse image along | − | is defined by The following lemma 3.16 ensures that the inverse image | | -1 belongs to X ee .
Proof.For any ∈ X e ∩ X (n) with an arbitrary n, we show that | | -1 belongs to X ee .In the following, Ḟ : (X e ) • e −→ X ee and F : X • e −→ X e denote the forgetful maps defined in Definition 3.2 for the respective appropriate types.
becomes a subset of X (n1) × • • • × X (n k ) , whose union ranges over (n 1 , . . ., n k )s such that In what follows in the proof all the is the same as this.Consider the k-folding cartesian product × of F , that is On the other hand, (15) coincides with F -1 ( ) which belongs to X • e ∩ X •n by the choice .This means (14) belongs to (X e ∩ X (n1) ) × • • • × (X e ∩ X (n k ) ), which yields the assertion, as Proposition 3.17 (Storage).Storage, also called digging, s X : (X e , X e ) −→ (X ee , X ee ) is defined for ∈ X e and y ∈ X ee s X ( , y) Then, this gives a natural transformation s : ( ) e −→ ( ) ee .
Proof.We show that κ ee The both HSs coincide thanks to the following Lemma 3.18.
Lemma 3.18.For any κ : X −→ Y, the following holds for any ∈ X e and y ∈ X ee : ) and ∈ X e ∩ X (m) with m = k i=1 n i .In the following Ḟ and F are the same as in the proof of Lemma 3.16.• m X ,Y : (X e , X e ) ⊗ (Y e , Y e ) −→ ((X × Y ) e , (X ⊗ Y) e ) is defined for every rectangle × with ∈ X e , ∈ Y e and every (x 1 , y 1 )

RHS( ,
Note by the definition, it follows so that finite rectangles with same dimensions suffice.
Proposition 3.20.The dereliction d is a monoidal natural transformation with respect to the monoidalness m of Definition 3.19.

RHS( ,
The last equation is by the functoriality of (−) e preserving the identity.
The both HSs coincide because of the following equality in X e : In terms of the monoidalness, Proposition 3.17 is strengthened by Proposition 3.22 (Monoidality of s).The natural transformation storage s is monoidal.That is Note that the monoidality on the functor ( ) ee is given by (m X ,Y ) e • m Xe,Ye .
Proof.In the proof, it is sufficient to consider an instantiation at any rectangle × ∈ X e ⊗ Y e such that ∈ X e ∩ X (n) and ∈ Y e ∩ Y (n ′ ) for any n, n ′ ≥ 0.
For LHS, by virtue of the note on m X ,Y below Definition 3.19, we calculate the case n = n ′ , as the other case n = n ′ directly makes LHS zero.
under the same condition.)Let − be instantiated with any element in (X ⊗ Y) ee , hence with (x 1 , y 1 ) by ( 5) and commuting integral over countable sum then by (7) this m solely contributes to the sum by the definition of (m X ,Y ) e by variable change (13) wrt the push forward measure along the restriction of by the def of (m X ,Y ) • e using product measure RHS coincides with LHS on any instance × when n and n ′ are given both equal to m i=1 n i , while the other instance when n = n ′ directly makes RHS zero.Thus both HSs coincide.Proposition 3.23 (weakening and contraction).Monoidal natural transformations w X and c X are defined: -(Weakening) w X : (X e , X e ) −→ (I, I) is defined for ∈ X e : w X ( , * ) := δ ,0 , where 0 is the monoid identity in X e .
-(Contraction) c X : (X e , X e ) −→ (X e , X e ) ⊗ (X e , X e ) is defined for Ü 1 , Ü 2 ∈ X e and ∈ X e c X ( , ) by the following composition: Proof.The commutative comonoid conditions are the following (a), (b) and (c): (a) sy Xe,Xe • c X = c X , where sy is the symmetry of monoidal products.This is by where ac is an associativity of ⊗.By Fubini-Tonelli, the condition amounts to the equality (c) (w X ⊗ Id Xe ) • c X coincides with the canonical morphism (X e , X e ) −→ (I, I) ⊗ (X e , X e ) for the monoidal unit.The condition is checked as follows: The last equation holds because | 0Ü |= Ü.
The conditions for the coalgebra morphisms are the following (i) and (ii) respectively for the weakening and for the contraction: For RHS, first we calculate: On the other hand, The second last equation is by the functoriality of (−) e on preserving the identity.
The both HSs coincide because of the following equality in X e .
We will end with summarising this section after Lemma 3.24: Lemma 3.24 (comonoidality of s).s X is a comonoid morphism from (X e , c X , w X ) to (X ee , c Xe , w Xe ).
Proof.The two conditions need to be checked:

Double Glueing and Orthogonality over TsKer op
This section constructs the double glueing over TsKer op in accordance with Hyland-Schalk's general categorical framework [27] for constructing the structure of linear logic, but without the assumption of any closed structure of the base category.In Section 4.1, a crude but non degenerate opposite pair is obtained between product and coproduct as well as between tensor and cotensor, lifting those collapsed in the monoidal category TsKer op .Furthermore an exponential comonad is constructed for the glueing G(TsKer op ) over TsKer op .In Section 4.2, a new instance of Hyland-Schalk orthogonality is given in terms of Lebesgue integral between measures and measurable functions, owing to the measure theoretic study in the preceding sections.The instance in TsKer has an adjunction property, called reciprocal, in terms of an inner product using the integral.The reciprocal orthogonality enables us to retain the exponential comonads to the slack subcategory S(TsKer op ).Following the framework [27], the double gluing considered in this paper is along hom-functors to the category of sets.An object is a tuple (X , U, R) so that X is an object of TKer op , and U and R are sets U ⊆ TKer op (I, X ) and R ⊆ TKer op (X , I).
The forgetful functor exists G(TKer op ) −→ TKer op forgetting the second and the third components of the objects.
Similarly by starting from a subcategory TsKer op of the opposite of the s-finite kernels, the double glueing category G(TsKer op ) is defined as a subcategory of G(TKer op ).As a general result of Hyland-Schalk [27] applying to our TsKer op , we have: Proposition 4.2.G(TsKer op ) is a monoidal category with products and coproducts, which is collapsed to the corresponding structures of TsKer op by the forgetful functor.
Given objects X = (X , U, R) and The tensor unit is given I = (I, {Id I }, TsKer op (I, I)).For a subset U of a homset and a morphism f of appropriate type, U • f and f • U denote the respective subsets composed and precomposed with f element-wisely to U .Product s denotes the mediating morphism for ∐ as the coproduct in TsKer op .
The unit for the coproduct is (T , ∅, {∅}).Remark 4.3 (product/coproduct and tensor/cotensor).The product and the coproduct of G(TsKer op ) do not coincide, despite that the forgetful functor makes them collapse into the biproduct in TsKer op .Similarly, another tensor product is defined, say the cotensor `, owing to the nonsymmetricity of the second and the third components for the tensor object: Remind that in what follows * denotes the unique element of the singleton measurable space I of I.
Our linear exponential comonad over TsKer op in Section 3.3 lifts to that for G(TsKer op ), along directly with Hyland-Schalk exponential construction in double glueing (cf.Section 4.2.2 of [27]).Hyland-Schalk give an exponential structure in a double gluing category G(C) by a natural transformation k : C(I, −) −→ C(I, (−) e ) making C(I, −) linear distributive.In our concrete framework TsKer op , the natural transformation k is given as follows: Definition 4.4.A natural transformation k : TsKer op (I, −) −→ TsKer op (I, (−) e ) is defined by the following instance k X (u) : I −→ X e for every u : I −→ X : is shown as follows with any Proof.
The third equation is because d X (dy, x) = δ(dy, x) as y ∈ X e ∩ X (1) .
In order to define certain exponential comonads in G(TsKer op ), the following linear distributivity is crucial on respecting the comonoid structure TsKer op .Lemma 4.6 (linear distributity of k X (−)).The natural transformation k X (−) : TsKer op (I, −) −→ TsKer op (I, (−) e ) meets the following criteria (i), (ii) and (iii) of Hyland-Schalk (cf.pg.209 [27]) in order to make TsKer op (I, −) linear distributive: (Remind the notation below that C(I, −) is a functor so instantiated both by object and by morphism.)(i) well-behavior wrt the comonad structure Proof.(ii) and (iii) are direct as so is the following equations stipulating the commutativity diagrams: where ∃ ∅ is the unique morphism to the empty measurable space (emptyproduct as terminal object) and Id ∅ : ∅ −→ TsKer op (I, I).
Hence, we need to prove (i) having two equalities: With Lemma 4.6, the following is an instance of C = TsKer op of Hyland-Schalk's Proposition 36 of [27] for G(C).Proposition 4.7 (Hyland-Schalk exponential comonad on glueing [27].).There are two kinds (I) and (II) of linear exponential comonad on G(TsKer op ) as follows so that the forgetful functor to TsKer op preserves the structure.For an object X = (X , U, R) in G(TsKer op ), (I) X e = (X e , k X (U ), TsKer op (X e , I)), where k X (U ) = {k X (u) : where k X (U ) as above, but ?R is the smallest subset of TsKer op (X e , I) Containing the weakening w X : X e −→ I (c) Closed under the following for the contraction c X :

Orthogonality as Relation between Measures and Measurable Functions
The Hyland-Schalk orthogonality relation [27], when applied concretely to the measure theoretic framework in the present paper, becomes a relation between measures and measurable functions over a common measurable space.The relation is shown to satisfy a property "reciprocity", which is derivable from the adjunction of the inner product in terms of integral of a measurable function over a measure.Our reciprocal orthogonality is strong enough to guarantee a certain relevant structure maps Hyland-Schalk employed in [27] to obtain product and exponential structures for the certain subcategory, called the slack orthogonality category S(C), of the double glueing G(C).
Note in this subsection, we do not assume any closed structure (i.e., the linear implication).The categories considered in this subsection are either TsKer op or TKer op .
Definition 4.8 (orthogonality on a monoidal category C).An orthogonality on a monoidal category C is a family of relation ⊥ R between maps I −→ R and those R −→ I satisfying the following conditions on isomorphism, identity and on tensor for a monoidal category by Hyland-Schalk (cf.Definition 45 [27]).Note: Although the original orthogonality is for a monoidal closed category, we consider a general monoidal one without the implication.(isomorphism) If ι : R −→ S is an isomorphism then for any u : I −→ R and x : R −→ I, For U ⊆ C(I, R), its orthogonal U • ⊆ C(R, J) is defined by This gives a Galois connection so that then this becomes an orthogonality relation.An orthogonality satisfying ( 16) is called reciprocal Proof.We show the condition ( 16) entails the three conditions (isomorphism), (identity) and (tensor) of Definition 4.8.Moreover, the (tensor condition) is strengthened into the (precise tensor) obtained by replacing "imply" with "iff".The derivation is direct for the isomorphism and the identity conditions.For the precise tensor condition, observing u⊗v = (Id R ⊗v)•u = (u⊗Id R )•v, one antecedent implies the descendant by reciprocity either on Id R ⊗v or on u ⊗ Id S , and vice versa.
In order to construct an orthogonality on TsKer op , we define an inner product of TsKer op , which has the adjunction property: Definition 4.10 (inner product).For a measure µ ∈ TKer op (X , I) and a measurable function f ∈ TKer op (I, X ), we define Then the two operators in Definition 2.2 become characterised as follows: Lemma 4.11 (adjunction between κ * and κ * ).In TsKer op , for any measure µ : X −→ I, any measurable function f : I −→ Y and any transition kernel κ : Y −→ X , Proof.The following starts from LHS and ends with RHS of the assertion, using Fubini-Tonelli: Using the inner product, an orthogonality relation on TsKer op is defined.Definition 4.12 (orthogonality in terms of integral).For a measurable function f ∈ TsKer op (I, X ) and a measure µ ∈ TsKer op (X , I), the relation Proposition 4.13 (⊥ X is a reciprocal orthogonality in TsKer op ).The relation ⊥ X defined in Definition 4.12 is an orthogonality in TsKer op , and moreover is reciprocal.
Proof.By Lemma 4.9, it suffices to show that the relation is reciprocal to satisfy the condition (16), which is derived by Lemma 4.11.
The orthogonality of Definition 4.12 gives rise to the following full subcategory of G(TsKer op ).Definition 4.14 (slack category S(TsKer op ) (cf. [27] for the general definition)).
The slack orthogonality category S(TsKer op ) is the full subcategory of G(TsKer op ) on those objects (X , U, R) Consider when X = B(R) and G is generated by the Borel subsets of an interval (a, b).Then the conditional expectation of ( 18) is This determines the action κ * on functions, and correspondingly the action κ * on measures by means of κ * µ(B) = X κ * χ B (x)µ(dx) (cf.( 1)).In particular, U f and R µ of (17) contain the following respective uncountable subsets: The above construction of the conditional expectation is generalised when a given measure µ is s-finite.We may write µ = Σ i µ i with each µ i being a probability measure.For each probability space ((X, X ), µ i ), the conditional expectation E i [∼| G] yields an endomap (κ i ) * on E(X ) for certain contraction kernel κ i .Then s-finite κ := i κ i becomes a contraction s-finite kernel by virtue of the property on the mixture of measures X g d( i µ i ) = i ( X g dµ i ).Proof.By virtue of the tensor condition for orthogonality, the monoidal product of Proposition 4.2 is shown closed in the subcategory S(TsKer op ): It suffices to show that the third component of (X , U, R) ⊗ (Y, V, S) is perpendicular to U ⊗ V .Take ν from the third component, then but which implies ν ⊥ X ⊗Y f ⊗ g by the tensor condition of the orthogonality of Definition 4.8.
In [27], to obtain an exponential structure as well as an additive one for the slack category, Hyland-Schalk employ certain relevant structure maps for a general category C. We remark that C = TsKer op in this paper, automatically validates their structure maps: Remark 4.17 (Hyland-Schalk's positive and negative maps are implicated by reciprocity).Hyland-Schalk (in Definition 51 of [27]) call a map f : R → S is positive (resp.negative) with respect to U ⊆ C(I, R) and Y ⊆ C(S, I) when f • u ⊥ S y implies (resp. is implied by) u ⊥ R y • f for all u ∈ U, y ∈ Y .When U and Y are the whole homsets, they say positive and negative outright.When a map is both positive and negative, it is called focused.We remark that our reciprocity (16) in TsKer op ensures these property on maps: That is, if an orthogonality is reciprocal, then any map is automatically both positive and negative hence focused with respect to any U and Y .
By the remark, the slack category S(TsKer op ) over TsKer op , has product and coproduct, and moreover an exponential comonad.Proposition 4.18 (product and coproduct in S(TsKer op )).The slack category S(TsKer op ) is closed under products in G(TsKer op ) of Proposition 4.2.
Proof.By Hyland-Schalk's Propositions 52 in [27] because their presupposition positivity (res.negativity) of the projection (resp.injection) of finite products (resp.coproducts) is entailed from our reciprocal orthogonality condition by Remark 4.17.This section starts by considering a discrete (i.e., countable) restriction of the transition kernels within the transition matrices.The restriction makes the integration for the categorical composition into simpler algebraic sum, and turns out to give an involution in the full subcategory TKer ω of the countable measurable spaces.The involution is directly shown to imply dagger compact closedness of the subcategory TsKer ω .Second, the double glueing is constructed over the compact closed category, so that a * -autonomous structure is obtained.Finally, the orthogonality of the previous section is extended over the involution, and the tight orthogonality subcategory of the double glueing is shown to coincide with Danos-Ehrhard's category of probabilistic coherent spaces [7].

Involution in
TKer ω and Closed Structure of TsKer ω When the set Y of (Y, Y) is countable, the integral of the composition ( 2) is replaced by the cruder sum: In the countable case, κ(x, {y}) is written simply by κ(x, y), and the collection (κ(x, y)) x∈X,y∈Y is called a transition matrix, as the composition (19) becomes the matrix multiplication, under the same countable condition making ι(y, C) into ι(y, {c}).This yields the full subcategory TKer ω consisting of the countable measurable spaces in TKer.Definition 5.1 (TKer ω ).A measurable space (X, X ) is countable when the set X is countable.
TKer ω is the full subcategory whose objects are the countable measurable spaces in TKer.Then the morphisms of TKer ω are chatacterised as the transition matrices between two countable measurable spaces.
The involution of Proposition 5.2 with the monoidal product directly yields the compact closed structure of the subcategory TsKer ω of TKer ω .Proposition 5.4 (dagger compact closed category TsKer ω ).Let TsKer ω be the full subcategory of TsKer consisting of the countable measurable spaces.Then the dagger TsKer ω of Proposition 5.2 becomes compact closed whose dual object (and its extension to the contravariant functor) is given by the involution ( ) * .In particular, the monoidal closed structure is by one to one correspondence between the transition matrices κ((x, y), z) and κ(x, (y, z)); Proof.The unit φ X : I → X * ⊗X for the compact closedness is given by the matrix whose element for each ( * , (x 1 , x 2 )) ∈ I × (X × X) is φ X ( * , (x 1 , x 2 )) = δ x1,x2 .The counit ψ X : X ⊗ X * → I is the transpose matrix of the unit.This directly yields the dagger compact closedness φ X = σ X ,X * • (ψ X ) * .Remark 5.5 (TsKer ω = TKer ω ).TsKer ω coincides with TKer ω .As the former is a wide subcategory of the latter, we need to check the fullness for the subcategory: Every morphism in TKer ω is a transition matrix (κ(x, y)) x∈X,y∈Y , whose each element κ(x, y) is approximated as a countable sum i∈N κ i (x, y) with finite κ i (x, y)'s.Thus κ = i∈N (x,y)∈X×Y κ i (x, y), where each κ i (x, y) determines the transition matrix (δ z,(x,y) κ i (x, y)) z∈X×Y .The coincidence of the two categories means that the discretisation makes the s-finiteness redundant so that the well behaved monoidal composition is freely obtained in TKer ω with respect to Fubini-Tonelli.The well behavior is a direct consequence that the monoidal product and composition become algebraic when the morphisms of continuous kernels collapse into transition matrices in TKer .
In spite of the remark, in what follows in Subsection 5.2, we continue to use TsKer ω , whereby the connection to the continuous case studied in the previous sections is seen direct.

Double Glueing and
via the identification under the isomorphisms by ( ) * : TsKer op ω (I, X ) ∼ = TsKer op ω (X , I).Moreover, G(TsKer op ω ) becomes *-autonomous, whose monoidal closedness is given by the following implication in terms of the involution and the cotensor `: A direct corollary is obtained when the slack category studied in Section 4.2 is restricted to the discrete measurable spaces: Corollary 5.7 (The slack S(TsKer op ω ) is a model of linear logic).The slack subcategory S(TsKer op ω ) is defined by Definition 4.14 over TsKer op ω , which becomes the full subcategory of S(TsKer op ) consisting of the objects whose first components are countable measurable spaces.Then S(TsKer op ω ) is *-autonomous with finite products (hence coproducts), and equipped with linear exponential comonads.
Proof.It suffices to show the following (i),(ii) and (iii), but whose latter two are direct and (i) is by Theorem 54 of [27], whose supposition on the positive (resp.negative) projections (resp.injections) and the three positive structure maps d, w and c (for k ) is by Remark 4.17

Exponential Comonad for Tight Category T(TsKer op
ω ) This parts starts with introducing the tight orthogonality subcategory T(TsKer op ω ) of G(TsKer op ω ), and shows that the subcategory is also a categorical model of linear logic.
Proof.Apply the symmetry condition to the descendant of (20), then y * ⊗ u ⊥ S * ⊗R ( f ) * .This happens to be a descendant of (tensor) whose two antecedents are y Hence the two orthogonality are the antecedent of the tensor condition.The second assertion is by the reversibility of the above argument using precise tensor.The third assertion is by Lemma 5.10.Lemma 5.13.If an orthogonality is symmetric, the stable tensor implies the stable implication; That is, for all U ⊆ C(I, R) and Y ⊆ C(S, I), (stable implication) Hence the orthogonality in TsKer op ω stabilises the implication.Proof.The *-autonomy X ⊸ Z = (X ⊗ Z * ) * makes the stable implication into a stable tensor via (U • ) * = (U * ) • , which equality is obtained directly by the symmetry orthogonality.The second assertion is by Lemma 5.10 and Proposition 5.11.
An orthogonality is called stable when it satisfies both stable tensor and stable implication.
The stable orthogonality of TsKer op ω ensures an exponential comonad on T(TsKer op ω ) as well as both monoidal product and product and coproduct.Proposition 5.14 (monoidal product in the tight orthogonality category).The tight category T(TsKer op ω ) has the following monoidal product so that the forgetful to TsKer op ω preserves the structure: with the tensor unit (I, {Id Proof.By Proposition 61 of Hyland-Schalk [27] for a general C with stable orthogonality, which proposition is applicable to C = TsKer op ω thanks to Lemma 5.12. Proposition 5.15 (product and coproduct in T(TsKer op ω )).The tight category T(TsKer op ω ) has the following products and coproducts so that the forgetful functor to TsKer op ω preserves the structures: Proof.By Hyland-Schalk's Proposition 63 in [27] for a general C with stable orthogonality because their presupposition positivity (res.negativity) of the projection (resp.injection) of finite products (resp.coproducts) is entailed from our reciprocal orthogonality condition by Remark 4.17.
Finally, exponential structure for the tight category T(TsKer op ω ) is obtained as an instance of Hyland-Schalk general construction for T(C) in [27].Proposition 5.16 (exponential comonad on T(TsKer op ω )).T(TsKer op ω ) has the following exponential comonad so that the forgetful to TsKer op ω preserves the structure: Proof.By Theorem 65 of [27] for a general monoidal category with a stable orthogonality because their presupposition for the theorem (described below) is automatically derived by Remark 4.17: All the structure maps d, s, w, c for linear exponential comonad and all maps of the form ? are positive for k (i.e., with respect to ∀U k X (U ) and C(I, I)) and the product projections are focused.

Probabilistic Coherent Spaces
Definition 5.17 (Pcoh [7,6]).The definition of the category Pcoh of probabilistic coherent spaces starts with the inner product and the polar: Pcoh has a tensor ⊗ and a product & as follows: in which for ∈| X | e and ∈| Y | e : -L( , That is, when and are given explicitly by in which S denotes the stabiliser subgroup of S n at := (a 1 , . . ., a n ) defined by S := {σ ∈ S n | a i = a σ(i) ∀i = 1, . . ., n} Note that the definition (23) does not depend on the ordering of for the stabiliser subgroup S as S n /S σ( ) = σ(S n /S ) for any permutation σ ∈ S n , hence its action on is well defined.
The *-autonomy of Pcoh with products (hence, coproducts) as well as the monoidal comonad in [7] is in particular derived by the following proposition.
Proof.The key property for the equivalence is that the measures (i.e., the homset TKer op ω (X , I)) and the measurable functions (i.e., the homset TKer op ω (I, X )) becomes isomorphic in TKer op ω by virtue of the involution ( ) * , and furthermore in TsKer op ω they both collapse to bounded functions from X to R + , hence both reside in R X + in Pcoh.The orthogonality < , > in Pcoh coincides with < | > in TsKer op ω , as the integral of Definition 4.12 collapses to the sum in the subcategory of discrete measurable spaces.An object X = (|X |, PX) in Pcoh corresponds one to one to the object X = (X, PX, (PX) • ) in TsKer op ω , preserving the involution ( ) ⊥ .Every morphism from X to Y in Pcoh is by definition an element P(X ⊥ `Y), which is the second component of X ⊥ `Y in TsKer op ω .Composition of Pcoh is the product of matrices, same as TsKer op ω .E.g., in particular their map fun(u) : PX −→ PY for u ∈ Pcoh(X, Y) (cf.Section 1.2.2 [7]) is written in TsKer op ω simply by fun(x) = u * x ∈ PY.Since the tensor and the additive structures are direct, only the exponential structure is checked on (i) objects and on (ii) morphisms.In the both levels, Danos-Ehrhard's exponential construction in Pcoh turns out to coincide with that of Hyland-Schalk for double glueing applied to our TsKer This has shown that t e ( , ) = (!t) , .
Proof.The dereliction and the storage become directly the discretisation of the corresponding maps in TsKer op (hence of T(TsKer op ω )), defined respectively in Propositions 3.14 and 3.17.Compare the first and second formulas respectively with (22) and (23) to see the opposite enumeration.It holds (♮t) , = !!(!t) , .The two exponentials become isomorphic [12], by the following natural isomorphism in terms of the multinomial coefficient m : !−→ ♮, defined by

Conclusion
This paper offers four main contributions: (i) Presenting a monoidal category TsKer of s-finite transition kernels between measurable spaces after Staton [38], with countable biproducts.Showing a construction of exponential kernels in TsKer by accommodating the exponential measurable spaces for counting process into the category.
(ii) Constructing a linear exponential comonad over TsKer op , modelling the exponential modality in linear logic.Although this initiates a continuous linear exponential comonad employing a general measure theory, though we leave it a future work on any monoidal closed structure inside TsKer required for modelling the multiplicative fragment of the logic.
(iii) Giving a measure theoretic instance of Hyland-Schalk orthogonality in terms of an integral between measures and measurable functions.The instance is inspired by the contravariant equivalence between TKer of the transition kernels and M E of measurable functions, and realised by adjunction of a kernel acting on both sides.We examine a monoidal comonad in the double glueing G(TsKer op ) inheriting from TsKer op of (i) and also that in the slack orthogonality subcategory S(TsKer op ).
(iv) (Discretisation of (i), (ii) and (iii)): Obtaining a dagger compact closed category TsKer ω when restricting TsKer of (i) to the countable measurable spaces.We show an equivalence of the tight orthogonality category T(TsKer op ω ) to Pcoh of probabilistic coherent spaces by virtue of the discrete collapse of the orthogonality of (ii) into the linear duality of Pcoh.
We now discuss some future directions.Our categories TKer with countable biproducts and TsKer with tensor are inspired from the standard measure-theoretic formalisation of probability theory, and similarly the linear exponential comonad over TsKer op from the counting process for exponential measurable spaces.We believe our semantics of transition kernels will provide a general tool for semantics of higher order probabilistic programming languages such as probabilistic PCF [7,15], of which Pcoh is a denotational semantics.We need to examine a concrete example making continuous Markov kernels indispensable (rather than discrete Markov matrices) for interpreting probabilistic computational reductions as a stochastic process.For this, any monoidal closed structure fundamental to denotational semantics needs to be explored in continuous measure spaces.Recent development [15,5] on CCC extension induced by Pcoh for continuous probabilities may be seen as a mutual construction of our construction because ours starts with the continuity to obtain Pcoh as a discretisation.
An important future work is to connection to Staton' s denotational semantics [38] for commutativity of first-order probabilistic functional programming, in which s-finiteness of kernels characterises commutativity of programming languages.We are interested in how our trace structure for feedback and probabilistic iteration (cf.Remark 2.10) may play any role in his probabilistic data flow analysis using categorical arrows.That is, a direction towards a probabilistic Geometry of Interaction employing the continuous categories of the present paper.Another promising further study is association with point process monad in [8] using distribution between Giry monad and multisets.This may provide a general categorical understanding how our measure theoretic commutative monoid, seen as counting process, yields the exponential comonad for linear logic.
Relating our model to Girard's coherent Banach spaces [21] on one hand involves analysing the contravariant equivalence of Proposition 2.7 under certain constraints required from logical and type systems.On the other hand, the double glueing in Section 4 will give a direct bridge to the duality of coherent Banach spaces, employing a (variant of) a Chu construction, which is known as another instance of Hyland-Schalk orthogonality.
After the submission of the earlier version of the paper, series of novel literature are published [13,17,34] on continuous exponentials for higher ordered programming.Their approach to measure-theoretic continuity would warrant our future work mentioned above.Especially, Ehrhard's measurable exponential [13] in cones, giving a continuous extension of discrete probability, could be quite beneficial.

Definition 1 . 2
(σ(F ) and Borel σ-field B + ).For a family F of subsets of X, σ(F ) denotes the σ-field generated by F , i.e., the smallest σ-field containing F .When X is R + and F is the family O R + of the open sets in R + (with the topology whose basis consists of the open intervals in R + together with (a, ∞) := {x | a < x} for all a ∈ R + ), the σ-field is denoted by B + , whose members are called Borel sets over R + .Definition 1.3 (measurable function).For measurable spaces(X, X ) and (Y, Y), a function f :
|).The equality to RHS holds because | 0 |= 0. Theorem 3.25.(( ) e , s X , d X , m X ,Y , m I ) equipped with c X and w X is a linear exponential comonad on TsKer op .
well defined so that the naturality of k ι e • k X (u) = k Y (ι • u) : I −→ Y e for any ι : X → Y and u : I → X in TsKer op

Lemma 4 . 9 (
reciprocal orthogonality).If a family of relation satisfies the following for every u : I −→ R, x : S −→ J, and f : R −→ S,

Proposition 4 .
19 (exponential comonad on the slack category).S(TsKer op ) has the following exponential comonad:!(X, U, R) = (!X , k X (U ), ?R)where ?R is defined as in Proposition 4.7, but the clause (b) is replaced by;{χ • w X | Id I ⊥ I χ} ⊆?RProof.By Proposition 53 of[27], which supposition on the positivity of the three structure maps d, w and c (for k ) is a direct consequence in TsKer op by Remark 4.17

5
Dagger TKer ω , Monoidal Closed TsKer ω and Pcoh as Tight T(TsKer op ω ) : (i) The *-autonomy of the slack category is inherited from G(TsKer op ω ) of Proposition 5.6 to the slack monoidal product in Lemma 4.16.(ii) The finite products of Proposition 4.18 are closed in the discrete subcategory.(iii) The linear exponential comonad in Proposition 4.19 is closed in the discrete subcategory.
which can be seen as a matrix (u) a∈|X|,b∈|Y| of columns from |X| and of rows from |Y |.Composition is the product of two matrices such that (uv) a,c = b∈|Y| u a,b v b,c for u : X −→ Y and v : Y −→ Z.(dual) X ⊥ = (|X|, PX ⊥ ) and u ⊥ ∈ Pcoh(Y ⊥ , X ⊥ ) is the transpose of a matrix u ∈ Pcoh(X, Y).

Remark 5 . 20 (
The opposite TsKer op coincides with Pcoh's left enumeration).Our choice taking the opposite of TsKer starting from Section 3.3 turns out to yield Danos-Ehrhard's choice of left enumeration in formalising the exponential in(23).When the right enumeration is chosen oppositely, there arises another exponential, say ♮, (♮t) , := ∈L( , ) ! !t ρ = σ∈Sn/S n i=1 t ai,b σ(i) (m X ) , = m( ) δ , Remind the multinomial coefficient of ∈ X e ∩ X (n) is defined by m( ) := n! a∈X (n) (a)! , which number is equal to the cardinality | S /S | of the quotient independently of the ordering of .