2.2 Linear Logic
2.2.2 Categorical semantics
We refer to a survey by Melliès [59] for an exhaustive study of the categorical semantics of Linear Logic. We choose here to detail the axiomatization by Seely, known as Seely categories, as it will fit well our study of Differential Linear Logic.
Monoidal and˚-autonomous categories We first describe the structure of monoidal closed categories, which are the good axiomatization for models ofMLL. Although this is one the simplest categorical structures, find-ing examples of those in functional analysis is not at all straightforward, as it requires to solve"Grothendieck’s problème des topologies"(see Section3.6).
Definition 2.2.2. A symmetric monoidal categorypC,b,1qis a categoryCendowed with a bifunctorband a unit 1with the following isomorphisms:
αA,B,C :pAbBq bC»Ab pBbCq ρA:Ab1»A, λA:IbA»A
symA,B:AbB»BbA
with the following triangle and pentagon commutative diagrams assuring the coherence of the associativity and the symmetry.
The Identity rule Figure 2.2:The inferences rules for Linear Logic
$?Γ, A
Figure 2.3:The logical cut-elimination rules for the exponential rules ofLL
AbB
pAb1q bB
ρAb1B
88
αA,1,B
//Ab p1bBq
1AbλB
ff
pAbBq b pCbDq
αA,B,CbD
**
ppAbBq bCq bD
αAbB,C,D 55
αA,B,CbD
pAb pBb pCbDqqq
pAb pBbCqq bD αA,BbC,D //Ab ppBbCq bDq
OO
Definition 2.2.3. A monoidal closed category is a symmetric monoidal categorypC,b,1qsuch that for each object Awe have a adjunction
_bA% rA,_s which is natural inA. The objectrA, Bsis called an internal-hom inC.
Definition 2.2.4. A˚-autonomous category[1] is a symmetric monoidal closed category pC,C,1C,p¨(¨qCq
with an objectKsuch that the transposeA //pA(Kq(Kto the natural transformationevA,K:Ab pA( Kq //Kis an isomorphism for everyA.
Notation 2.2.5. We writeδA “ evA,K.
Example2.2.6. The category offinite-dimensionalreal vector spaces and linear functions between them, endowed with the algebraic tensor product is˚-autonomous.
The previous example allows us to introduce informally notations which will be recalled and formally defined in Chapter3and widely used throughout this thesis: whenE is a real locally convex and Haussdorf topological vector space (for example, any finite-dimensional or any normed space will do), then we denote byE1the vector space consisting of all the linear continuous scalar maps`:E //R. Several topologies can be put on this vector space, although one has a canonical one which corresponds to the topology of uniform convergence on bounded subsets ofE. This topology is called the strong topology. WhenEis normed, this corresponds to endowingE1 with the following norm :
k¨k1:f ÞÑsupkxkď1|fpxq|
Thus a space is said to bereflexivewhen it is linearly homeomorphic to its double dual.
Example2.2.7. The category of Hilbert spaces isnot˚-autonomous: if indeed any Hilbert space is reflexive, that is ismorphic to its double dual, Hilbert spaces and linear continuous functions between them is not a monoidal closed category. Indeed, the space of linear continuous functions from a Hilbert space to itself is not a Hilbert, nor it is reflexive in general. More generaly, the category of reflexive spaces and linear continuous maps between them is not˚-autonomous, as reflexive spaces are not stable by topological tensor products nor by linear hom-sets.
Remark2.2.8. We havepK(Kq »1C.
A particular degenerate example of˚-autonomous categories are those where the duality is a strong monoidal endofunctor (see Definition2.2.11) onpC,C,1Cq:
Definition 2.2.9. A compact-closed category is a˚-autonomous category where for each objectsAandBthere is an isomorphism natural inAandB:
A˚bCB˚ » pAbCBq˚.
Example2.2.10. The category offinite-dimensionalreal vector spaces is compact closed, as any finite dimensional vector space is linearly isomorphic to its dual.
Interpreting MLL ConsiderA1, ..., An formulas ofLLand$ A1, ..., An a sequent of formulas ofLL. In a monoidal closed categoryC, we interpretAias an objectJAiK:
• We interpret a formulaAbBbyJAKCJBKans thus the formula1by1C. Therefore, there is a natural isomorphism between the set of morphisms
f : 1C //JA1K
˚ (p....(JAnKqC
ofCinterpreting$A1, ..., Anand the set of morphismsf :A1b...An´1 //An. This being said, one constructs by induction on their proof tree the interpretationJπKKof a proofπofMLL. We interpret the sequent$A1, ..., An as a morphism1 //JA1`¨ ¨ ¨`AnK.
• The interpretation of the axiom rule is$AK, Ais the morphism1 //p1
JAKPKAJ(JAK.
• Iff :JΓKK //JAKinterprets proofπof$Γ, Aandg:JAK //J∆Kinterprets a proofπ1of$AK,∆, then the proof of the sequentΓ,∆resulting from the cut between theπandπ1isg˝f :JΓKK //J∆K.
• The interpretation of the introductions of1andKcorrespond respectively to the identity map11C : 1C //1C an to the post-composition by the isomorphismrΓK»JΓK
˚ (K.
• The interpretation of$Γ, A, Band of$Γ, A`Bare the same.
• From mapsf : JΓKK //
JAKandg :J∆KK //
JBK, one constructs the image off andgby the bifunctor b:fbg:JΓKKbJ∆KK //JAKbJBK.
Interpreting MALL The interpretation of the additive connectives and rules are done via a cartesian structure pˆ,J, tqonC, as detailed in Section2.1.1. Recall thattA:A //Kdescribe the fact thatKis a terminal object.
In a ˚-autonomous category with a cartesian product, the dual pA˚ ˆB˚q˚ of a product is a co-product, interpreting the connective‘.
Strong monoidal co-monads Once thelinearpart ofLLis interpreted, one needs to interpret non-linear proofs.
This is done through linear/non-linear adjunctions [4], or equivalently through strong-monoidal co-monads [71].
The second point of view is the one developed here.
Definition 2.2.11. A strong monoidal functor between two monoidal categories pC,C,1Cq andpD,D,1Dqis a functor equipped with natural isomorphisms:
1DbDFpxq
Definition 2.2.12. A comonad on a categoryCis an endofunctorT : C //C with natural transformationsµ : T //T˝Tandd:T //Idsatisfying the following commutative diagrams for each objectAofC.
T3pAq T2pAq
With the above notations, the co-unit is dD : F ˝ GpDq //D is the image of 1GpDq via the isomorphism DpF˝GpDq, Dq »CpGpDq,GpDqq, and the comultiplication isµ:F˝GÑ pF˝Gq2.
Definition 2.2.13. The coKleisli category of a co-monad T is the category CT whose objects are objects of C, and such thatCTpA, Bq “ CpT A, Bq.
Then the identity inCT of an objectAcorresponds inCtodA : T A //A, and the composition of two arrows f :T A //Bandg:T B //Ccorresponds inCT to the arrow:
g˝Tf “ g˝T f ˝µA.
Then from every co-monad one constructs an adjunction between a category and its co-Kleisli:
CT CT
T
U
%
in which the functorsTandUare deduced fromT:
T:
Interpreting the structural rules ofLL ConsiderpC,b,1,p.q˚qaSeely Category, that is a˚-autonomous cat-egory with a cartesian productpˆ,Jqand endowed with a strong monoidal comonad! : pC,ˆ,Jq //pC,b,1q.
Then we interpret the formulas ofLLas previously andJ!AK “ !JAKandJ?AK “ p!pJAK
˚qq˚.
Remark2.2.14. This strong monoidal functor is said to satisfySeely’s isomorphism:
!pAˆBq »!Ab!B (2.1)
The strong monoidal functor!provides natural isomorphisms:
mA,B: !pAˆBq »!Ab!B, (2.2)
m0: !J »1. (2.3)
Then one defines the natural transformations:
cA:!A!∇A//!pAˆAqm»A,A!Ab!A (2.4)
wA:!A!nA//!Jm»01 (2.5)
The natural transformationcmodels the contraction rule by pre-composition. Likewise,wgives us the interpreta-tion for the weakening rule. The co-unitdgives us the interpretation of the dereliction rule by pre-composition.
Proposition 2.2.15. [59] The morphismswAanddAdefine natural transformationswandd.
If moreover! is a co-monad, then one gets the interpretation of the promotion rule: from the interpretation f : !JΓKK //Aof a sequent$?Γ, A, one constructs:
prompfq: !JΓKK
µJΓK
K//!!JΓKK
!f //!A.
Remark 2.2.16. Notice that the categorical interpretation of the promotion rule is the only one using the co-multiplicationµ.
Remark 2.2.17. [59, 5.17.14] A co-monad which is a strong monoidal endofunctor on a categoryL leads to a monoidal adjunction betweenLand its co-Kleisli categoryL!:
L! K L U
!
A smooth classical semantics, exponentials as distributions. In this paragraph we introduce informally the intuition which will guide our understanding of Differential Linear Logic. Indeed, the interpretation of a differen-tiation operator imposes a smoothness condition of the maps of the co-Kleisli category of the exponential.
Let us observe from a functional analysis point of view what has been defined previously. We interpret formulas A, BbyR-vector spaceEandF with some topology allowing us to speak about limits, continuity and differen-tiability. The categoryLis then the category of these vector spaces and linear continuous maps between them.
Let us denote by C the co-Kleisli category L!. The maps f : E //F of this category are linear maps f : !E (F, and can be described as power series betweenEandF in classic vectorial models ofLL[18,19].
The multiplicative conjunctionAb_ is then interpreted by a (topological) tensor product, whose right adjoint is the hom-setLpA,_q.
The interpretation for1, neutral forb, is thus the fieldR. Following remark2.2.8, the interpretation forKis such thatLpK,Kq “ R, thusKis one-dimensional and:
K »1»R (2.6)
The dualityp_q1corresponds to some topological dualE˚ “ LpE,Rq, and thus via the definition ofCand the properties of a˚-autonomous category we get:
!E» p!Eq2»Lp!E,Rq1»CpE,Rq1 (2.7) Thus the exponential!Emust be understood as the dual of the space of non-linear scalar functions defined on E, as a topological vector space of linear continuous scalar functions (also calledlinear forms) acting on non-linear
continuous scalar functions. Likewise, we get an intuition of the dual of the exponential as a space of non-linear functions.
?pE˚q »CpE,Rq (2.8)
IfCis the category of vector spaces and smooth functions, then the exponential is interpreted as a space of distri-butions [67], see Section7.3.2and Chapter7.