In this section we introduce a toy syntax called Smooth Differential Linear Logic for which Nuclear spaces and distributions form a classical and smooth model. This is a calculus with no-higher order. We distinguish two classes of formulas, the finitary ones on which an exponential can be applied, and the smooth ones, which represents those on which an exponential has already been applied.
Outlook 11. This calculus should be seen as a toy calculus, allowing to play with the mathematical tools of distributions and (F)-spaces. This is useful, as it allows to highlight the traditional mathematical objects attached to Differential Linear Logic. In section7.5, we provide a higher order negative model ofDiLLwhere the exponential is interpreted as a space of higher order distributions.
7.4.1 The categorical structure of Nuclear and (F)-spaces.
Nuclear (F)-spaces gather all the stability properties to be a (polarized) model of LL, except that we do not have an interpretation for higher-order smooth functions. If!Rnis interpreted asE1pRnq, we do not have an immediate way to define!!Rn.
3Beware that the multiplication between distribution is in general not possible [69], and that it is possible here only because the Fourier transform of a convolution with compact support is a generalized function.
Definition 7.4.1. One defines the exponential as a space of Distributions:
!
#
RnÞÑ!Rn“E1pRn,Rq
`:Rn //RmÞÑ p?p`q:φPE1pRnq ÞÑφp_˝`q PE1pRmq (7.1) From the kernel theorem7.3.9and example7.2.22it follows:
Theorem 7.4.2. The exponential! :EUCL //NDFis a strong monoidal functor.
Definition 7.4.3. Dually, one defines the following functor:
?
#
RnÞÑ?Rn“C8pRn,Rq
`:Rn //RmÞÑ p?p`q:f PC8pRn,Rq ÞÑf ˝`1PC8pRm,Rqqwhere`1is the dual of` (7.2) The the Kernel theorem on functions7.3.5states that?is strong monoidal frompEUCL,‘qtopNUCL,bq.ˆ Remark7.4.4. The Kernel Theorem is all about proving thestrongmonoidality of?, thus allowing for an interpre-tation of the co-structural rulesc¯andw. The fact that there is a natural transformation from¯ ?Rnb?Rmis the easy part, which allows for the interpretation of the structural rulescandw.
Thus the structure of Nuclear (F)-space or Nuclear (F)-spaces gives us a strong monoidal closure p_q1β : NF //NDF$ p_q1β :NDF //NF:
pNDF,bˆπ,Rq pNFop,bˆ,Rq
p¨q1β
p¨q1β
%
and a strong monoidal functor:
pEUCL,ˆ,Rq pNDFop,bˆπ,Rq
p¨q1β
p¨q1β
%
but no interpretation for the shift. We must interpret proofs in the category of complete spaces.
7.4.2 Smooth Differential Linear Logic
In this section, we construct a version of DILL for which Nuclear spaces and distributions are a model, by dis-tinguishing several classes of formulas. We introduce nowSDiLL: its grammar, defined in figure7.1, separates formulas into finitary ones and polarized smooth ones.
Definition 7.4.5. We call Smooth DiLL, denotedSDiLL, the sequent calculus whose formulas are constructed according to the grammar of figure7.1, and whose rules are the ones ofDiLL0(without promotion, as usual).
Its rules are those of DiLL. Thus, the cut-elimination procedure is the same as the one defined originally [23].
What makes it different is the grammar of its formulas: the construction of the formula!!Ais not possible. We see this Smooth Differential Linear Logic as a first step towards Chapter8. In fact, we construct in the next section a model of DiLL with higher-order, base on Nuclear and (F)-spaces.
If we forget about the polarisation of SDiLL, a model of it would be a model of DiLL where the object!!A does not need to be defined. It is thus a model of DiLL where!does not need to be an endofunctor, but just a strong monoidal functor! :FIN //SMOOTHbetween two categories. The categories FINand SMOOTHneed to be both a model of MALL.
This distinction is necessary here to account for spaces of distributions and spaces of smooth functions, which cannot be understood as part of the same ˚-autonomous category. Indeed, if both type of space are reflexive,
E, F :“A|N |P
reflexive lcs arenot preservedby topological tensor products nor spaces of linear maps. One must then refine this category into two monoida subcategories of REFL, and construct an adjunction between the negation functors.
We also give a categorical semantics for an unpolarized version of SDiLL. Remember that considering the development by Fiore [25], if the product on FINis a biproduct, the interpretation ofw,c,w,¯ c¯follow from the strong monoidality of!. One then needs to make precise the interpretation ofdandd.¯
7.4.3 A model of Smooth DiLL with distributions
A categorical model of DILL is clearly a model of SDILL. On the other hand, to interpret SDILL one does not need the co-monadic structure on !. But then, in order to interpret the dereliction, one must ask for a natural transformation accounting for the lost co-unit of the co-monad.
Definition 7.4.6. A unpolarized categorical model of SDiLL consists of a model ofMALLEUCL, and a model of MALLwith biproduct NUCL, equipped with wa strong monoidal functor
! :pEUCL,ˆq //pNUCL,bq,
a forgetful functorU :EUCL //NUCLstrong monoidal inb,`,&,‘and two natural transformationd: ! //U andd¯: ! //Usuch thatd˝d¯“IdEUCL.
The model of smooth formulas is the one with biproduct, as the biproduct accounts for the necessity to sum the interpretation of non-linear proofs, interpreted as maps in NUCL. Thus a polarized model of SDiLL would consist into two polarized model of MALL, with a forgetful functor respecting polarities and an exponential inversing the polarities. This would amount to describe4chiralities, and their respective coherence laws. We leave this to future work, and describe a concrete polarized model of Smooth DiLL with Nuclear (F)-space or Nuclear DF spaces.
A concrete model ofSDiLL. Positive formulas ofSDiLLare interpreted as complete nuclearDF spaces. Nega-tive formulas are interpreted as Nuclear (F)-spaces. A sequent$Γ, Aof Smooth DiLL is intepreted as a continuous linear map inLpJΓK
1 β, Aq.
Remark7.4.7. Notice that we are not here in the categorical context of chiralities. Nuclear (F)-space and Nuclear DF lcs are not a chirality, as we don’t have of a (covariant) adjunction between the two interpreting the shifts.
However, as the two are subcategories of TOPVEC, we don’t need these shifts to interpret the sequents, as they can be interpreted as arrows of TOPVEC, that is continuous linear arrows.
Theorem 7.4.8. The categories of Euclidean, and (F)-space Nuclear, and DF Nuclear spaces, defines a model of SmoothDiLL.
Proof. We interpret finitary formulasAas euclidean spaces. Without any ambiguity, we denote also by Athe interpretation of a finitary formula into euclidean spaces. The exponential is interpreted as!A“E1pAq, extended by precomposition to functions. We explain the interpretation for the rules, which follow the intuition of [23]. We define:
Then we have indeed: d˝d¯“IdA2. The interpretation ofw,c,w,¯ ¯cfollows from the biproduct structure on EUCLand from the monoidality of!, as explained in2.4.2.
We now detail the description of the bialgebraic structure on!Rn.
Proposition 7.4.9. The weakening and contraction have the following interpretation inNUCL:
w:
Proof. The contraction rules is interpreted from the cartesian structure on EUCL and the Kernel Theorem: if we write4 : A //AˆA the diagonal morphisms, we have !4 : !A //!pAˆAq » !Ab!A. From theˆ introduction of section7.3, remember that the isomorphism!pAˆAq » !Ab!Aˆ » p?AKb?Aˆ Kq1 coincides to kP!pAˆAq ÞÑ phbh1 P?AKb?Aˆ Kq ÞÑkppx, x1q ÞÑhpxqh1px1qq.
Weakening is then interpreted by the exponentiation of the terminal morphismw¯“!pnRnq, thuswpφ¯ P!Rnq “ consta PEpt0uq ÞÑφpconsta˝nRnq. Thusw, as a constant scalar function, identifies to a scalar in¯ R. This scalar is its constant value. Ifφcoincides with the generalization of a functiongPC8c pRnqthen we would have:
¯
wpφq “aÞÑa ż
Rn
gpxqdx, from which follows the notation used in the proposition.
Proposition 7.4.10. Thus the dual of the contraction is :
c1:
#?Rnb?ˆ Rn //?Rn»!pRnˆRnq
fbgÞÑ pxÞÑfpxqgpxqqand then extended via the universal property of completion (7.5) Proof. The dualc1ofccorresponds to the composition of the diagonalisation morphism and of the isomorphism resulting from the Kernel theorem on functions 7.3.5, which is exactly proved by density of?Rn //?Rn in
?pRnˆRnq.
Remark7.4.11. During the proofs of this Chapter and of Chapter8, we will equally use the interpretationcorc1 when interpreting the contraction.
We show thatw,¯ c¯have a direct interpretation which follows the intuitions of [6].
Proposition 7.4.12. The cocontraction and codereliction defined through the kernel theorem coincide in fact with the convolution product of distributions and the introduction ofδ0.
¯
During the rest of the proof we use Fourier transforms and tamed distributions, as exposed in section7.3.4. The co-contraction is defined categorically asc¯“!O˝m´1A,A. In the categorical setting, addition in hom sets is defined through the biproduct. But here the reasoning is done backward. We know that‘ “ ˆis a biproduct thanks to O:AˆA //A;px, yq ÞÑx`y, and thus!O:φP!pAˆAq ÞÑ pf PEpAq ÞÑφppx, yq ÞÑfpx`yqq. Moreover iff PEpAˆAqis the sequential limit ofpfnbgnqnP pEpAq bEpAqqN(see theorem7.3.9)
m´1A,Apφbψqpfq “lim
n pφpfnqψpgnqq.
If we write byFφthe Fourier transform of a distribution, we have that ofFφ˚ψ“FφFψ. From the details above we deduce moreover:
F¯cpφ, ψqpfq “m´1A,Apφbψqppx, yq ÞÑ{fpx`yqq “m´1A,Apφbψqfˆxfˆy“φpfˆ qψpfˆ q.
As distributions with compact support are temperate, we can apply the inverse Fourier transform toFφ˚ψ and andF¯cpφ, ψq, and thusc¯coincides to the convolution.
Outlook12. Let us notice that we could use Sobolev spaces [76] as a model for exponentials in Smooth DiLL. This would lead to the possibility of greater applications in the theory of Linear partial differential operators. Indeed, Sobolev spaces are reflexive when one considers derivatives inL2. However, we must refine the Kernel theorem to do so. This is work in progress.