The ε tensor product

With the injective tensor product comes another operator, which correspond to the completed injective tensor product (Definition3.6.19) for a large class of lcs who have the approximation property (see [44, chapter 18]).

Called theε-tensor product, it gives a concrete representation of the elements of the completed injective tensor product wheneverEεF »EbˆεF.

Definition 3.6.25. ForEandFtwo lcs, we defineEεF “ pE_γ¹ b_E E_γ¹q¹as vector ofE-hypocontinuous bilinear forms on the dualsE¹_γ andF_γ¹ (the definition of hypocontinuity is done in3.6.4, the one of the Arens dualE_γ¹ in 3.4.16). It is endowed with the topology of uniform convergence on products of equicontinuous sets inE¹, F¹. As on equicontinuous subsets ofE¹ (resp.F¹), the weak topology and the compact-open topology coincide (see Proposition3.4.20). The vector spacepE_γ¹ b_EF_γ¹q¹coincides also to theE-hypocontinuous functions onE¹_wˆF_w¹, which contains in particular the space of all continuous bilinear maps onE_w¹ ˆF_w¹.

Proposition 3.6.26. The lcsEεF induces on the tensor productEbFthe injective tensor product topology.

The following proposition is proved by Jarchow in his textbook. It is interesting to notice that it is proved by using the semi-γ-reflexivity of anylcs:

E“ pE_γ¹q¹_γ

and the fact that we have here a closure operator in the category of lcs: E¹_γ » ppE_γ¹q¹_γq¹_γ. This is concretely what is done every day in denotational models ofLL, the equationE »E^KK is used to show thatLpE^K, Fqis commutative and associative.

Proposition 3.6.27. [44, 16.2.6,16.1.3] Theεproduct is commutative and associative on complete lcs, andEεF is complete if and only ifEorFis complete.

Beware that Jarchow in his proof describes theεproduct by using on the dualE¹ the topology of uniform con-vergence on compact subsets of E. As we are dealing here with complete subsets our notations are coherent.˜ Remember also that Jarchow uses the notationE_γ¹ to denote the topology of uniform convergence on absolutely convex compact subsets ofE, which is coherent with our definition for˜ E_γ¹ 3.4.16only when spaces are complete.

Part II

Classical models of DiLL

Chapter 4

Mackey and Weak topologies as left and right adjoint to pairing

In this part, we describe two models of DiLL, each one using a specific topology on the dual allowing for an involutive linear negation. The first Chapter introduces the notion of quantitative versus smooth interpretation of proofs, and the Weak spaces and Mackey spaces through an adjunction with the category of Chu spaces. The second Chapter5is adapted from a published article by the author and details a quantitative model of DiLL with weak spaces [48]. We highlight the fact that this gives a polarized model of DiLL with a linear negation involutive on the negatives formulas. The third Chapter6takes inspiration from [17], so as to adapt the work of convenient space into a polarized model of DiLL with Mackey spaces, with a linear negation involutive on the positive formulas.

Quantitative semantics and Cartesian closed categories

Introduced by Girard [30], quantitative semantics refines the analogy between linear functions and linear programs (consuming exactly once their input). Indeed, programs consuming exactly n-times their resources are seen as monomials of degreen. General programs are seen as the disjunction of their executions consumingn-times their resources. Mathematically, one can apply this idea by interpreting non-linear proofs as sums of n-monomials.

These sums may be converging [18,32,49], finite [19], or formal [48].

Power series are an efficient answer to the issue of finding cartesian closed categories of non-linear functions in a vectorial setting. The isomorphism betweenC⁸pEˆF, GqandC⁸pE,C⁸pF, Gqqconsists of combinatorial manipulations on the monomials (see in particular Section5.2). The convergence of the power series obtained as a result of these manipulations is proved in a second time, and uses the completeness of the spacesE,F andG.

However, one of the two guidelines for this thesis is the search for models of DiLL in which smooth functions are not necessarily power series, and most importantly spaces are general topological vector spaces. The motivation comes from the need to relate DiLL with functional analysis or differential geometry. One of the best settings for a cartesian closed category of smooth functions was developed by Frölicher, Kriegl and Michor [26,53] , by defining smoothness as the preservation of smooth curvesc:R //E(see Section2.4.3). In Chapter6, we adapt the results of [6] to a classical setting.

The Category of Chu spaces and its adjunctions to T

OP

V

EC

.

The category CHUof duals Pairs.

We describe the category CHUas described by Barr [2]. Chu spaces makes use of the monoidal closed structure of the category VECof all vector space with linear maps between them, endowed with the usual tensor product, and transform it into a˚-autonomous category. We recall first the notion of dual pairs as described previously3.3.18.

Definition 4.0.1. A dual pair is a couplepE, Fqof twoR-vector space together with a symetric bilinear and non degenerate form:

x¨,¨y:

#EˆF //R

px, yq ÞÑ xx, yy (4.1)

As in the litterature [44], whenE is a vector space, so in particular whenE is a lcs, we denote byE^˚ the set of (not necessarily continuous) linear forms` :E //K. Then for any dual pairpE, F,x¨,¨yqwe have linear injections:

EãÑF^˚

xÞÑpx^˚:yÞÑ xx, yyq F ãÑ^˚E

Y ÞÑp^˚y:xÞÑ xx, yyq.

The fact that these are injections comes from the non-degeneracy of the pairing. These definitions extend to functions, defining two endofunctorsp¨q^˚and^˚p¨qon the category VECof vector spaces and linear functions.

Definition 4.0.2. The category CHUhas as objects dual pairs of vector spaces and as arrows pairs of linear maps:

pf :E1 //E2, f¹:F2 //F1q:pE1, F1q //pE2, F2q such that the following diagram commute:

E1 E2

F₁^˚ F₂^˚

f

p¨q^˚ p¨q^˚ f^1˚

We denote by CHUppE1, E2q,pF1, F2qqthe vector space of such pairs of linear functions. Thanks to the symmetry ofx., .y, the commutation of this this diagram is equivalent to the one of the following:

F₂ F₁

˚E2 ˚E1

f¹

˚p¨q ^˚p¨q

˚f

Then one defines on CHUa tensor product and internal hom-set which makes it a duality which makes it a

˚-autonomous category.

Definition 4.0.3. ConsiderpE₁, F₁qandpE₂, F₂qtwo dual pairs with pairings denoted respectively byx¨,¨y1and x¨,¨y2. As in Chapter3we writeLpE, Fqfor the vector space of all linear functions between two vector spaces.

• pE1, F1q^K“ pF1, E1qand the pairing ofpE1, F1q^Kis the symmetric to the pairing ofpE1, F1q.

• pE1, F1q b pE2, F2q “ pE1bE2,CHUppE1, F2q,pF1, E2qqqwith a pairing defined by:xx1bx2,p`1, `2qy “ x`1px1q, x2y2and then extended linearly onE1bE2. This definition is symmetric inpE1, F1qandpE2, F2q by the requirements on dual pairings.

• pE1, F1q(pE2, F2q “ pCHUppE1, E2q,pF1, F2qq, E1bF2qwith a pairing defined asxp`1, `2q, x1by2y “ x`1px1q, y2y2.

Proposition 4.0.4. The categoryCHUis monoidal closed with tensor productband neutralpK,Kq.

Proof. The monoidality ofb in CHU follows from the one of the algebraic tensor product in the category of vector spaces and linear maps. By definition, maps in CHUppE₁, F₁q b pE₂, F₂q,pE₃, F₃qqare pairspf : E₁b

E₂ //E₃, f¹:F₃ //LpE₁, F₂qqverifying the naturality condition in definition4.0.2. By the universal property of

the algebraic tensor product on VEC, we have thus natural isomorphisms between CHUppE₁, F₁qbpE₂, F₂q,pE₃, F₃qq and pairspf :E₁ //LpE₂, E₃q, f¹:F₃bE₁ //F₂q, which is exactly the definition of CHUppE₁, F₁q, LpE₂, F₂q,pE₃, F₃qqq.

The commutative diagrams required for monoidality follow from the one of VEC.

Proposition 4.0.5. The categoryCHUis cartesian with productpE1, F1q ˆ pE2, F2q “ pE1ˆE2, F1‘F2q. The neutral for the cartesian product is thenpt0u,t0uq.

Proof. The projections and factorisation of maps is straightforward from the algebraic product and co-product of vector spaces.

Proposition 4.0.6. The categoryCHUis˚-autonomous with duality the functor p¨q^K:pE, Fq //pF, Eq,pf, f¹q //pf¹, fq.

We have thenpE, Fq^K“LppE, Fq,pR,Rqq.

Proof. The functor p¨q^K defines indeed an equivalence of categories between CHU and CHU^op. As we have pE, Fq^K“LppE, Fq,pR,Rqq, this isomorphism is precisely induced by the transpose of the evaluationppE, Fq ˆ LppE, Fq,pR,Rqq //pR,Rq.

A topological adjunction

Barr in [2] states a categorical interpretation for the Mackey-Arens theorem3.5.3. Recall that this theorem says that amongst all topologies on a certain lcsE preserving the dualE¹, the weak topology (Section3.3.2) is the coarsest one and the Mackey*-topology is the finest one (the Mackey topology onEis the topology of uniform convergence on weak*-compact and absolutely convex subsets ofE¹, see Section3.4.3).

Definition 4.0.7. We denote byP the functor from TOPVECto CHUsending a lcsEon the pairpE, E¹q, and a linear continuous functionf :E //Fon the pairpf, f¹q:, wheref¹ :`PF<sup>1</sup>ÞÑ p`˝fq PE¹q.

Recall that, as all polar topologies, the weak and the Mackey topologies are functorial: they define an endo-functor on the category TOPVECwhich is the identity on linear continuous maps. Indeed, iff PLpE, Fqthenf is continuous fromEw //Fwas for`PF<sup>1</sup>we have`˝f PE¹(see Proposition3.3.16). Likewisefis continuous fromEµtoFµas the image byF¹of a weak* compact inF¹is weak* compact inE¹. We thus define two functors from CHUto TOPVEC. WhenEis a lcs,EσpE,FqandEµpE,Fqare indeed locally convex and separated topological vector spaces [44, 8.4].

Definition 4.0.8. The functorWmaps a dual pairpE, Fqto the lcsE_σpE,Fqand acts as the identity on morphisms.

The functorMmaps a dual pairpE, Fqto the lcsE_µpE,Fq.

Theorem 4.0.9. The weak functorWis right adjoint toPwhile the Mackey functorMis left adjoint toP.

MACKEY CHU WEAK

P

M

%

W

P

%

Proof. ConsiderE andF two lcs. By Proposition3.3.16, the linear continuous functions from Ew toFw are exactly the linear functions such that, in CHU pf, f¹q : pE, E¹q //pF, F¹q. Thus ifE is a space already en-dowed with its weak topology and pE2, F2qa dual pair, one has the linear isomorphismsLpE, F_2,σpF2,E2qq “ LppE, E¹q,pE2, F2qq.

Let us show thatMis left adjoint toPby using directly the Mackey-Arens Theorem3.5.3. We know that the Mackey-topology on a lcs E is the finest one preserving the dualE¹. As it preserves the dual, we have that from any functionf inLpEµ, Fqone deduces a linear functionf¹ :F¹ //pE_µq¹. Thus for any dual pairpE₁, F₁qand lcs F one has an injectionLpE1,µpE₁,F₁q, Fq ĂLppE₁, F₁q,pF, F¹qq, by mappingf to the pairpf, `PF<sup>1</sup>ÞÑ`˝fq.

Let us reason by contradiction and consider some arrowpf, f¹qbetween the dual pairspE₁, F₁qandpF, F¹qwhich does not correspond to a linear continuous function E_1,µpE1,F1q //F. Then consider onE₁ the topology τ consisting of the open sets ofµpE₁, F₁qand those generated by the reverse image byfof the open sets ofF. Then by definition we havef¹ :F¹ //E¹_τ, thusE_τ¹ ĂF1. But as the topologyτ is by definition finer thanµpE1, F1q we haveF1 “ pE_1,µpE1,F1qq¹ ĂE_τ¹. Thusτis a topology finer than the Mackey topology with dualF1which is absurd.

Following this adjunction, we could have described two models ofMALL. Each one would consist of lcs with the weak (resp. Mackey) topology inherited from their dual. The dual of a construction ofLLwould be defined alongside this construction. For example, ifAandBare two formulas ofLL, we would define in WEAK:

JAbBK“ pJAKbJBKqσpJAKbJBK,LpJAK,JBK¹σq

JA`BK“ pLpJAK

1

σ,JBKqq_σpLpJAK¹σ,JBKq,JAK¹bJBK¹q

and likewise in the category MACKEYusing the Mackey topology:

JAbBK“ pJAKbJBKqµpJAKbJBK,LpJAK,JBK¹µq

JA`BK“ pLpJAK

1

µ,JBKqq_µpLpJA

K¹µ,JBKq,JAK¹bJBK¹q

As this is easily extended to the additive, we would construct this way two models of MALL. However,what we do in Chapters5and6adds more features and refines the semantics.

The adjunction between weak spaces and Mackey spaces allows to give an equivalent of Proposition3.5.17for Mackey spaces:

Proposition 4.0.10. ConsiderE and F two lcs endowed with their Mackey-topology. Then we have a linear homeomorphismL_µpE¹_µ, Fq »L_µpF_µ¹, Eq.

Proof. A linear functionf :E_µpEq¹ //F is continuous, if and only iff :E¹ //Fwis continuous, if and only if f : F¹ //E²_wis continuous, if and only iff :F_µ¹ //Eis continuous by Section3.5.3. Thus we have a linear isomorphismf //f¹ betweenLpE_µ¹, Fq »LpF_µ¹, Eq. This isomorphism is continuous: letV be an absolutely convex0-neighbourhood inE, andW an absolutely convex weakly compact set inF_µ¹. AsF is MackeyW^˝is a0-neighborhood on F. Likewise, asE is Mackey we can take V “ K^˝, whereK is absolutely convex and weakly compact inE¹_µ. Then iff¹pWq ĂV we havefpV^˝q ĂW^˝, and thusfpKq ĂW^˝and conversely. Thus pW_K,V˝q¹ ĂW_W,V andf ÞÑf¹is continuous.

Remark4.0.11. In particular, any Mackey-dual is endowed with its Mackey-topology. Indeed, as pE_µpEq¹ q¹„E

by proposition3.5.2, we have thatE_µpEq¹ , which is endowed byµpE¹, Eqby definition, is also endowed with its Mackey-topologyµpE¹, E²q. With respect to what it done for weak topologies, the topologyµpE¹, EqonE¹ should be called the Mackey* topology.

Chapter 5

Weak topologies and formal power series

We consider the category WEAK of lcs endowed with their weak* topology σ^˚pE, E¹q and linear continuous functions between them. We show in the two first sections that WEAKis a model of MALL, and in the third Section we construct a co-monad generating formal power series on it, interpretingDiLL. In the last Section we state the fact that we have here a negative model ofDiLL0,pol, according to the categorical definitions in Chapter 2. This chapter is mainly built from the contents and the form of [48]. It is thus more detailed than most of the thesis, and may provide a nice introduction to the interpretation ofDiLLin topological vector spaces.

Notation 5.0.1. We use the notations introduced in Chapter 3, that is E „ F denotes a linear isomorphisms between the vector spacesEandF, whileE »Fdenotes a linear homeomorphism between the lcsEandF(that is, they have the same vectorial and topological structure).

Contents

5.1 Multiplicative and additives connectives . . . 85 5.1.1 Spaces of linear maps . . . 85 5.1.2 Tensor and cotensor . . . 86 5.1.3 A˚-autonomous category . . . 88 5.1.4 Additive connectives . . . 88 5.2 A quantitative model of Linear Logic . . . 88 5.2.1 The exponential . . . 89 5.2.2 The co-Kleisli category . . . 94 5.2.3 Cartesian closedness . . . 96 5.2.4 Derivation and integration. . . 98 5.2.5 An exponential with non-unit sequences . . . 99 5.3 A negative interpretation ofDiLLpol . . . 101

5.1 Multiplicative and additives connectives

ConsiderFpE,Cqsome vector space of functions fromEtoCandev:EÑFpE,Kq¹. WhenFpE,Kqcontains only linear functions,evis linear. WhenE¹ĂFpE,Kq,evis injective, asE¹separates the points ofE.

Notation 5.1.1. We write the following function asev^FpE,Kq: ev:

#EÑFpE,Kq¹ xÞÑ pevx:f ÞÑfpxqq If there is no ambiguity in the context, we will writeevforev^E1 :EÞÑE².

Dans le document manuscript (Page 87-94)