A positive model of MALL with bornological tensor

6.3 A positive model of MALL with bornological tensor

6.3.1 Co-products of bornological spaces

By Proposition6.2.15, we have that Mackey tensor and the co-product of lcs preserves the bornologicality of spaces. That is, bornological spaces are well behaved with respect to thepositive connectivesof Linear Logic. It is not the case however for the negative connectives of linear logic:

Proposition 6.3.1. [44, 13.5.4] The productś

iPIEiof any familypEiqI of bornological spaces is bornological if and only ifK^I is bornological, if and only if the cardinal ofIdoes not admit a Ulam measure [77].

Remark6.3.2. The question of whether there are sets which carry a Ulam measure is an open question. InZF C however, if the cardinal of a set is accessible, then it does not admits a Ulam measure. There are models ofZF C in which every cardinal is accessible.

6.3.2 A tensor product preserving the Mackey-topology

Spaces of linear functions. The space of linear maps endowed with the topology of uniform convergence on bounded subsets is not necessarily bornological: we need to bornologize it.

Remark 6.3.3. The topological dual does not necessarily preserve the bornological condition. As recalled by Erhhard [18], one can construct on the vector space`<sub>8</sub>a non-null bounded linear continuous function which sends c<sub>0</sub>(i.e. all sequences converging to0) to the scalar0. This function cannot be continuous on`₈“`<sub>1</sub><sup>K</sup>, endowed with its normal topology, asc<sub>0</sub>is dense in`₁^K. Thus`₈is not bornological.

Remember from Section2.2.3that the normal topology of`<sub>8</sub> “`₁^Kis the topology of uniform convergence on equicontinuous subsets of`1.

More generally, one cannot prove without further hypothesis on a lcsE that ifF is bornological, then so is LβpE, Fq.

Definition 6.3.4. We denote byLbornpE, Fqthe lcs of all linear continuous functions between two lcsEandF, endowed with the bornologification of the topologyLβpE, Fq.

The bounded tensor product. Let us recall more precisely the monoidal structure of BORNVECas explained by Kriegl and Michor [53, 5.7].

Proposition 6.3.5. The bounded tensor product onBORNVECis symmetric and associative, andEbβ_ is left adjoint to the hom-set functor which mapsFtoLpF,_q.

From this monoidal structure on BORNVEC, we deduce the monoidal structure ofBTOPVEC:

Proposition 6.3.6. Theβtensor product onBTOPVECis symmetric ans associative, andEbβ_ is left adjoint to the hom-set functor which mapsF toLbornpF,_q.

Definition 6.3.7. ConsiderEandF two lcs. According to Section3.6, theβ-tensorEbβF is the vector space EbF endowed with the finest lcs topology such that the canonical bilinear maphβ : EˆF //Ebβ F is β-hypocontinuous.

Neighbourhoods of0inEbβF are then generated by the prebasis consisting of product of bounded sets inE (resp. F) and0-Neighbourhoods inF (resp. E). This tensor product enjoys a universal property with respect to β-hypocontinuous functions: these are the bilinear function which, restricted to a bounded set ofE(resp.F) are continuous onF(resp.E). Then we recall from Section3.6:

Proposition 6.3.8. If f : E ˆF //Gis β-hypocontinuous, then there is a unique linear continuous map f_β :EbβF //Gsuch thatf “f_β˝h_β.

IfE andF are bornological, thenβ-hypocontinuous bilinear functionsf : EˆF //Gare exactly those which send a product of bounded setsBEˆBF PEˆF to a bounded sets ofG. As the bounded sets of a product are exactly product of bounded sets (this is straightforward from the definition of the product topology, see Section 3.1.4), we have then the following fact:

Lemma 6.3.9. IfEandF are bornological, then theβ-hypocontinuous bilinear functions fromEˆF to any lcs are exactly the bilinear bounded ones.

Thus on bornological lcs theβ-tensor product is the bounded tensor product described by Kriegl and Michor [53, I.5.7]:EbβFis the algebraic tensor product with the finest locally convex topology such thatEˆFÑEbF is bounded.

Proposition 6.3.10. ConsiderEandF two bornological lcs. ThenEbβFis bornological.

Proof. ConsiderGa lcs. By definition of theβ-tensor product, we have the algebraic equality:

LpEbβF, Gq „BβpEbβF, Gq.

By the previous lemma, we have thatBbetapEˆF, Gq „ BpEˆF, Gq, the space of bilinear bounded maps betweenEˆFandG.

Consider now a bounded linear mapf PLpEbβF, Gq. This function coincide with a bounded bilinear map f˜onEˆF: products of bounded sets inB_EandB_F respectively are bounded inEbβF, and thus f˜sends a BEˆBF on a bounded set inG. Thusf˜PBbetapEbβF, Gq, and thusf is continuous onEbβF. Therefore EbβF is bornological.

We have thus a monoidal structure onBTOPVEC, where thetensor product need not be submitted to any closure operation.

6.3.3 A bornological ` for C

A model in MACKEY We proved in Section6.3.2thatBTOPVECendowed withbβis a symmetric monoidal category. Following our quest for classical models ofDiLL, the natural idea then would be to define an interpre-tation of`as the dual ofbβ in MACKEY: E`MACKEYF :“ ppEqĘ¹_µ^bornbβpFĘq¹_µ^bornq¹_µ. This would result in a model ofMALLin the category MACKEY, in whichpositive connectives are bornological lcs. We try now to formalize this idea that positives are interpreted by bornological lcs in a model made of Mackey lcs, through a chirality between bornological lcs and Chu pairs. The following developments must then be read in analogy with Section5.3on Weak spaces.

Preconvenient dual pairs To obtain dual pairs wich result on bornological lcs, and not just Mackey lcs, a little more material is needed. Indeed, considering a dual pairpE1, E2q, the dual ofEs_1,µpE^born

1,E2qcontains but is not restricted toE2. To have an adjunction, one would need to consider bornological dual pairs , which are the one called preconvenient by Frolicher and Kriegl in [26, 2.4.1].

Remember from Section4that we have the following right polarized closures, whereP is the functor which maps a lcsEto the pairpE, E¹qandMmaps a pairpE1, E2qtoE_1,µpE1,E2q:

TOPVEC CHU P

Mp_q

%

TOPVEC CHU^op P^K

pMp_qq¹_µ

%

Definition 6.3.11. We denote by PRECONVthe category of dual pairspE1, E2qwhich are invariant the compo-sition of the functorsP˝Born˝Top˝M. That is, the dual of the bornologification ofE1,µpE₁,E₂qis stillE2. According to [26, 2.4.1] these are exactly the dual pairs such thatE2contains all the bounded linear forms onE1

endowed with the bornology ofσpE₁, E₂qweakly bounded subsets. We denote byDPRECONVthe category of dual pairspE₂, E₁qsuch thatpE₁, E₂q PPRECONV.

Remark6.3.12. Following Frölicher and Michor, we remark that a dual pairpE₁, E₂qis bornological if and only if the bornologification of the Mackey-topology onE₁is exactly the Mackey topology onE₁.

Proposition 6.3.13. The following diagrams define a right polarized chirality:

BTOPVEC PRECONV P

Mp_q¯ ^born

%

pBTOPVEC,bβ,Rq pPRECONV^op,`,pR,Rqq

P^K

pMp_qqй_µ^born

%

Proof. Remember that morphisms in CHUbetweenpE1, E2qandpF1, F2qare pairspf, f¹qin pLpE1, E2q, LpF2, F1qqsuch that such that the following diagram commute:

E1 E2

F₁^˚ F₂^˚

f

p¨q^˚ p¨q^˚ f^1˚

.

Let us show then that we have natural isomorphisms, forFa bornological lcs andpE1, E2qa bornological pair:

LpF,E¯^born_1,µpE

1,E₂qq »CHUppE1, E2q,pF, F¹qq.

By definition, aspE1, E2qis a preconvenient dual pair, we haveE¯^born_1,µpE

1,E2q » E_1,µpE1,E2q, and the adjunction follows from Section4. The adjunction for the second diagram goes likewise, because we reversed the order of the vector spaces in the dual pair.

The second diagrams features strong monoidal adjunctions: by the universal property ofbβin BORNVECand the fact that whenE is bornological we haveLpE, Fq “ LpE, Fq. Moreover, whenEandF are bornological thenpEbβFq¹“LpEbβF,Rq “LpE, F^ˆq “LpE, F¹q, and thus

P^KpEbβFq “ pEbF,LpE, F¹qq^K“ pLpE, F¹q, EbFq “P^KpEq`P^KpFq.

Likewise, we have that whenE andF are bornologicalEbβF is bornological, and thus already endowed with its Mackey-topology. ThusMpEй, Eq^bornbβMpFй, Fq^born “MpEbF,LpE, F¹qq

Notice that because the category PRECONVis not symmetric when it concerns its dual pairs, we are not here in the setting of the negative chiralities defined in Section2.3.2. Indeed, the second adjunction is betweenBTOPVEC

andDPRECONVwhen it should be betweenBTOPVECand PRECONV^op

Theorem 6.3.14. These adjunctions between BTOPVEC and DPRECONV define a positive interpretation of MALL.

Proof. We showed the adjunctions in Proposition6.3.13. We need now to show that the two closures define the same action on the negatives. With the categorical notations of Section2.5.2.3, this would amount to:

clos_N :ˆN^KR» p´Nq^KL. Thus we must prove that for any pairpN₁, N₂q PDPRECONVwe have:

PppMppPĞ1, P2qqq¹_µ^born“P^KpMppN¯1, N2qq^born

This equation is straighforward, aspMppP1, P2qqq¹_µ “ pP2qµpP1q, which is by definition bornological thusPppMppPĞ1, P2qqq¹_µ^born“ pP2, P1q, and likewiseP^KpMppN¯1, N2qq^born“ pP2, P1q

Remark6.3.15. In this proof, we showed that the closure operation is the identity even on the interpretation of the negatives. Thus one cannot argue that any computation is done here.