Interpreting LL pol

In Section2.2.2 we interpreted! as a co-monad on L. Here, we take the point of view of a strong monoidal adjunction between!and the forgetful functorU.

Usually one requires Seely’s isomorphism :

!pNˆMq »!Nb!M. (2.12)

We are going to require the strong monoidality of?instead of!: this is justified in a polarized setting by the fact that the above isomorphisms take place in the category interpreting the negative formulas, while the strong monoidiality of!would be interpreted in the category interpreting the positives formulas. Indeed, as it will ap-pear in Chapters6and7, the negatives formulas are the one interpreted by some complete spaces (completeness in understood in this particular exeample in a wide sense: spaces may be Mackey-complete, or quasi-complete).

Complete spaces play the role of co-domainsF of smooth functionf PC⁸pE, Fq. Positive one the contrary may not be complete, but may verify other properties preserved by inductive limits : in particular, they may be inter-preted by barreled spaces3.4.22or bornological spaces6.2.13. Thus in a setting rich enough to interpret differenti-ation, and thus with some notion of completeness, negative formulas are interpret as complete spaces, and positive ones as the formulas which need not be complete.

However, in the theory of topological vector spaces, the Seely isomorphism is not true with a non-completed tensor product b. More specifically, when interpreting!E as a space of distribution, the Kernel theorem7.3.5 states this isomorphism for a completed tensor product. It does state the density of⁴ C⁸pE,Rq bC⁸pF,Rqin C⁸pEˆF,Rqand the fact that the topology induced byC⁸pEˆF,RqonC⁸pE,Rq bC⁸pF,Rqis the injective topology. It is by completing the tensor product that we obtain thus an isomorphism, which is dualized to be stated in the above form2.12.

With these arguments in mind, we want to have an interpretation for !and?such that!N “ ?^KLpN^KRq, satisfying:

?pPbQq »_N ?P`?Q (2.13)

When the composition of the negations is not the identity on the positive (as it is the case in our negative chiralities), we have thus:

p!pNˆMqq^KLKR»_Pp!N^KLb!M^KRq^KR (2.14) Thus, while Seely’s isomorphism2.12is most of the time described as a linear/non-linear monoidal adjunc-tion [59, Def. 21]:

pN ⁸,ˆq pP,bq

!

U

%

we ask here for astrongmonoidal adjunction:

pP^8,op,‘q pN ^op,`q

?

U

%

Remark2.3.24. In an unpolarized setting,!is a co-monad: L //L, andN⁸ is the co-Kleisli categoryL_!. For any objectN PN ⁸we have an isomorphism betweenN andUp!Nq “ !N inN ⁸: the morphismN //!N corresponds tof “1!N inL, while the morphism!N //N corresponds tog : !!N ÝÝÑ^d!N !N Ý^dÝ^NÑN. They are indeed inverse from one another inN⁸:

f˝!g “ f˝!g˝µ_!Nby definition of˝! (2.15)

“ !dN ˝!d!N˝µ!N (2.16)

“ !d_Nby the second commutative diagram for comonads, see2.2.12 (2.17) However, inN⁸ the second commutative diagram for comonads say that!d_N andd_!N are the same arrow: they both act as a unit for the composition˝!, and thus are equal by the unicity of units. Moreover, we have:

g˝!f “ d_N˝d_!Ng˝µ_N “ d_N.

Thusgandf are inverse one another, andN »_N⁸ Up!Nqthe adjunction between!andU results in fact in a closure onN ⁸. Likewise, the adjunction between?andU (another functor denoted byU, as it is thought as a forgetful functor) is a closure onP⁸.

Definition 2.3.25. A classical model ofLL_polconsists in

• A negative chiralitypP,b,1qandpN ,`,Kqwith a strong monoidal left closurep´q^KL :P //N^op % p´q^KR :N ^op //P,with a polarized closure´:N //P %ˆ:P //N suchˆ˝´ “ Id_N,

• A cartesian structure onN pN ,ˆ,Jqsuch thatˆis distributive over`

• A co-cartesian categorypP⁸,‘₈,0₈qand a co-cartesian categorypN ⁸,ˆ₈,J₈qwith a strong monoidal left closure

p´q^KL,8:P⁸ //N ^8,op% p´q^KR,8:N ^8,op //P⁸.

4In the context of topological vector spaces we have a biproduct and in particularEˆF»E‘F

• A strong monoidal right closure adjunction

? :pP^8,op,‘,0q //pN ^op,`,1q %U :pN<sup>op</sup>,`,1q //pP^8,op,‘,Jq.

• A family of isomorphisms inP:

closP :´P^KL » pˆPq^KR natural inP.

Then asˆ´»Id_N » p´^KLq^KRone has the isomorphisms´ˆP »P^KLKR. Interpreting the exponential connectives

Proposition 2.3.26. We define a strong monoidal functor frompN⁸,ˆqtopP,bqby

!N “ ?˝UpN^KRq^KR

!` “ ?˝Up`^KRq^KR. We interpret the formula?PforPPPas⁵?UˆP.

Proof. It follows from the fact that,we have a closure betweenN ⁸andP^8,op, that isp´q^KL,8˝p´q^KR,8 “ Id_N8. Remark2.3.27. The previously defined model is calledclassicalas the double-negation is asked to be the identity on negatives. However, this definition would also suits an intuitionist setting by not asking the negation to define a closure. Then!as defined above would not necessarily be a strong monoidal functor betweenpN⁸,ˆqandpP,bq.

The algebra structure of? Let us detail how the exponential rules are interpreted in this context, following the pattern detailed in Definition2.3.21. From the strong monoidality of!, one has natural isomorphisms inN :

m^K_P,Q: ?P`?Q»?pP‘Qq m0: ?0» K

If we denote by∇P :P‘P //Pthe co-diagonal, then we obtain as previously, see2.4:

cP : ?P`?P //?P wP :K //?P

These define natural transformations which interprets respectively the contraction and weakening rules, by precomposition.

Interpreting the dereliction rule By definition, the co-unit of the adjunction

? :pP^8,op,‘,0q //pN ^op,`,1q %U :pN <sup>op</sup>,`,1q //pP^8,op,‘,Jq is a natural transformationd^K_N PN ^opp?UpNq, Nqwhich interprets the dereliction. We denote by

dN :N //?UpNq

the morphism corresponding tod^K_N inN via the isomorphismN ^opp?UpNq, Nq »N pN,?UpNqq.

Thus consider a morphismf PNpP^KR, Mqinterpreting the proof of$M, P. One constructs the interpreta-tion of the proof of$M,?P as the morphismf˜PN pˆ1, M`?UpˆPqq »N pˆp1b p?UpˆPqq^KL:

ˆp1b p?UpˆPqq^{KL ˆpdˆPq}

KL

ÝÝÝÝÝÝшppˆPq^KLq »ˆ´P^KL »P^KRÝÑ^f M Promotion is interpreted as before by functoriality of!andU.

5We need to apply a shift toPbefore constructing!Pbecause of our categorical definitions. In a interpretation where?N^Krepresents the space of smooth scalar functionsC⁸pN,Rq, it amounts of saying that one needs a complete domain to define smoothness.