A Dialectica-like Model of Linear Logic

Valeria C. V. de PaivaDept. Informatica PUC-RJ

R.M.S.Vicente,225 - Rio de Janeiro 22453BRAZIL

The aim of this work is to define the categoriesGC, describetheir categoricalstructureand show they area model of Linear Logic. The secondgoal is to relatethosecategoriesto the DialecticacategoriesDC, cf.[DCJ, using different functorsforthe exponential“of course”. It is hopedthat this categoricalmodel of Linear Logicshouldhelpus to get a betterunderstandingof the logic, which is, perhaps,the firstnon-intuitionisticconstructivelogic.

This work is divided in two parts,eachonewith 3 sections.Thefirst sectionshowsthat GC is amonoidalclosedcategoryanddescribesbifunctorsfor tensor “0”, internalhorn “[—, —]“, par “u”, cartesianproducts“&“ andcoproducts“s”. Thesecondsectiondefineslinear negationasa contravariantfunctor obtainedevaluatingthe internalhornbifunctorat a “dualisingobject”. Thethird sectionmakesexplicit theconnectionswithLinear Logic, while the fourth introducesthe comonadsusedto model the connective“of course”. Section5 discussessomepropertiesof thesecornonadsand finally section6 makesthe logical connectionsoncemore.

This work grewout of suggestionsof J.Y. Girardat theAMS-Conferenceon Cate-gories,Logic andComputerSciencein Boulder1987, whereI presentedmy earlierworkon the Dialecticacategories,hencethe title. Still on the lines of given credit whereitis due, I would like to saythat Martin Hyland, underwhosesupervisionthis work waswritten,hasbeena continuoussourceof ideasand inspiration. Many heartfelt thanksto him.

1. The main definitions

We startwith a finitely completecategoryC. Thento describeGC say that itsobjectsarerelationson objectsof C, that is monicsA ~ U x X, which we usuallywrite as(U ~ X).

Given two suchobjects,(U ~- X) and (V L Y), which wecall simply A andB, amorphismfrom A to B consistsof a pair of mapsin C, f: U —* V andF4Y —+ X, suchthat a pullback conditionis satisfied,namelythat

(U x F)1(o~)~ (f x Y)1(/3), (1)

where (~~)_1 representspuilbacks.

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Using diagrams,wesay (f,F) is a morphismin GC if there is a (unique)map in~, k: A’ —~B’ makingthetrianglecommute:

a~I Ia/3’ UxF

B’>— ~UxY ~UxX

I xY

B>— -~VxY

rhereA’ is thepullbackof a alongU x F andB’ thepullback of /3 alongf x Y. Note

iat we refer to the object (U ~- X) as “a”, meaningthe (equivalenceclassof the)ionic, aswell asA.

The intuition hereis that, if we considera and /3 set-theoreticrelations,thereis aLorphism from a to /3

a

Y

wheneveru a F(y) then f(u) /3 y.It is easyto seeGC is a category, sincecompositionis just compositionin each

Dordinate’.If thebasecategoryC is cartesianclosed,aswell asfinitely complete,thecategory

C hasa symmetricmonoidalstructure(tensorproduct) denotedby “0” that canbeadeclosed. This tensorbifunctor0 seemssomewhatinvolved andnot very intuitive,tt it is exactlywhat is neededto showthat GC is symmetricmonoidalclosed,givene internalhorn, [—, ~}Gc definedbelow.

efinition 1. Thereisan internalhorn bifunctor in GC, [—, —]Gc: GC°~xGC—* GCyenby

[A,B1GC=(VUXXY~+UXY) (2)

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where intuitively the relation J3a readsas (f,F)/3cx(u,y) 1ff wheneveruaF(y) thenf(u)/3y.

Formally, wedefine/3a asthegreatestsubobjectE of VU x X~’x U x Y suchthatE A A’ � B’, whereA’ is thepullbackof A alongthemap

V~1xX1’xUxY~L-~UxX,

B’ is thepullbackof B alongVUx X’1’ x U x Y ~ V x Y and “A” meanspullbackagain.

To guaranteetheexistenceof suchgreatestsubobject,we insist on C beinglocallycartesian closea By that wemeanthat for anyobject A of C, the slicecategoryC/Ais cartesianclosed,cf. [See] 1984.

Definition 2. AssumingC locally cartesiandosed,considerthe tensorproductin GCgivenby theoperation 0: GC x GC -.+ GC which takesthe pair of objects(A, B) to

~ (3)

Intuitively, (u,v)a0 /3(f,g) 1ff uaf(v) and v/3g(u).

Thefunctor “0” is not a categoricalproduct,for example,projectionsdo not existnecessarily,but it is associativeand symmetric.The object I = (1 4 1) is theunit forthis tensorproduct.

Anothertensorproduct,similar to the tensorbifunctorin DC canbedefined,butit is not left-adjoint to theinternalhorn.

Definition 3. Thebifunctor 0: GC —~ GC, which takes(A,B) to

AOB=(UxV?~?XxY) (4)

is associativeand symmetric. It has the sameunit I = (1 4 1) as the bifunctor “0”andintuitively therelation (u,v)a 0 /3(x,y) holds 1ffuax andv[3y.

Proposition 1. ThecategoryGC is a symmetricmonoidalclosedcategory.

Proof: It’s enoughto seethenatural isomorphism

HOInGC(A0 B,C) ~ HomGc(A,[B, CIGC). (5)

Usingdiagrams,

a

UxV I XVXYU U I X

f (F1,F2) (7,~2)

‘1w ( Z W1TxY~ i VxZ

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For symmetryreasonsthat will be apparentlater, we want to introduceyet an-otherbifunctor, to be called “Ii”, which is, in somesense,dual to the tensorproduct“0” bifunctor. To definethe bifunctor “If” categoricallywe needC with stable(underpuilbacks)and disjoint coproductscf. [M/R] 1977.

Definition 4. Considerthe bifunctor “If” that takes(A, B) to

AIfB = (Us’ x VS” ~ X x Y). (6)

Therelation defining AfjB saysthat (f, g)aIff3(x,y) if f(y)ax or g(x)/3y.

Notice that the object I = (1 ~+- 1), where0 is theemptyrelationon 1 x 1, is theunit for the operation“If” and that thereis a naturalmapI —~I, but not conversely.

Notice as well that tensor “0” and its dual “If” have very similar “carriers”, butduality hereis transformingthemetalanguage“and” into “or”.

Now we definecartesianproductsand coproductsin GC.

Proposition2. If C is a finitely completecategorywith disjoint and stablecoproductsthenGC hascategoricalproductsand coproducts.

Proof: Categoricalproductsaregivenby the bifunctor&: GC x GC —~GC, whichtakesthe pair of objects(A,B) to

A&B=(UxV~X+Y) (7)

and therelation “a&/3” is given, intuitively, by (u, v) a&/3 (‘~)if either uax or v/3v.

Categorically,we takethe coproductmap inducedby the morphismsA x V U x~xU

X x V andB x U >—* V x Y x U. The object A&B is a cartesianproduct,as caneasilybe checkedandthe object 1 = (1 4 0) is the unit for the cartesianproduct andsoa terminalobject in GC.

The constructionabovecan be dualised. Thus, if we takethe coproductmapofaxY ~9xX

AxY>-~UxXxYandBxX>--*VxYxXthatgivesus

AEDB=(U+V~XxY) (8)

wherethenaturalrelationreadsas (‘f) a~ /3(x, y) if either uax or v/3y . Clearly

the endofunctorED definescoproductsin GC. The object 0 = (0 4 1) is the unit forthis construction.

A remarkis that theintuitive “or” in thedefinitions of & and ED is givenby takingcoproducts,while theone in the definition of If is areal “or”.

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2. Linear negation in GC

We definein GC a strongcontravariantfunctor,which inducesan involution on asubcategoryof GC.

Recall that, givenasymmetricmonoidalclosedcategoryC, a contravariant strongf~nctorT: C —÷ C is afunctorsuchthat, for ~leverypairofobjects(A,B) in C, thereis afamily of mapssi(A,B): [A, BJc —~[TB, TAlc makingthefollowing diagramscommute.

MI I [X,Y]®[Y,Z] ) [X,Z]

I st®stl 1st[X, X] [TX, TX] [TI’, TX] o [TZ, TY] ) [TZ, TX]

st M

Definition 5. Considertheinternal horn bifunctor evaluatedat I = (1 ~ 1) in thesecondcoordinate, that is consider [—, I]GC. This obviouslydefinesa contravariantfunctor(—)~:GC°~—+ GC.

More precisely to eachobject (U i~- X), the functor (—)~associatesthe object

(X +~i- U) where the relation ~ intuitively saysx J u if wheneveruax then I.As “~L”is the empty relation, it is neverthe case,so if we aredealingwith decidablerelationsin Sets,xJ-’~uif it is not the casethat uax. Hencethe namelinear negation.

Proposition 3. Thefunctor(—)~:GC°~—~GC is a strongcontravariant functor.

Now wewant to considerthesubcategory“DecGC”, whoseobjectsarethe decid-ableobjectsin GC, that is decidablerelationson C.

Definition 6. By a decidableobject on GC wemeanthat (U i~- X) is suchthat thea .L±° .

canonicalmapfrom (U I-f- X) to (U i-i- X) is an isornorphism.

Our nextpropositionis to givenamesto structures.Following Barr, cf. [Bar] page13, we saythat a *-autonomouscategorycomprises:

1. A symmetricmonoidalclosedcategoryC.2. A strong(contravariant)functor *: C°” —~C, thus the functor * anda family of

mapsst*:[A,B]~c~ ~3. An equivalenced = dA:A —f A** suchthat thefollowing diagramcommutes

[A,B]c at [B*,A*]c

___________ jst

[d~,d][A,B}c~ I

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Proposition 4. ThesubcategoryDecGC is an *-autonornouscategory,for * = (_)J.

It is easyto verify that thetensorproduct“0” of GC distributesoverthecoproduciED anddually, that the bifunctorpar “If” distributesover the cartesianproduct “&“

Thefollowing isomorphismshold in GC

A 0 (B ED C) ~ (A 0 B) e (A 0 C) and Alt(B&C) ~ (AfIB)&(AIfC).

Notice that ‘multiplicatives’ distributeover ‘additives’. But therearealsonatura

morphismsof the form (A 0 A’) 0 (BIf C) —~-~(A 0 B)If(A’ 0 C) and symmetricall~

(AIfB) o(C 0 C’) —L (A 0 C)fj(B0 C’), which reduceto themorphismsA0 (BIfC) _!_

(A 0 B)IfC and (AffB) 0 C’ -~ AIf(B 0 C’).

3. Classical Linear Logic and GC

The categoryGC cameinto existenceaiming to be a categoricalmodel of ClassicaLinear Logic. It stemsfrom a suggestionof Girard in Boulder 87, to whom I am verbgrateful,andto agreatextentit fulfils its promise. In particular,thecategoryGC is ~very interestingmodel of ClassicalLinear Logic, sinceit doesnot collapsethe units o“tensor” and “par” into a singleobject.

Thereareat leasttwo equivalentpresentationsof ClassicalLinearLogic with slighivariationsin notation.

Theoriginal one, cf. [Gir] 1986 page22, is very sleekandelegant,but it is handt

readof a categoricalmodel from it.Seelyin [See] 1987,on theotherhand,givesapresentation,which is gearedtowar&

the symmetriesand thus morehelpful. In his presentationa sequenthastheform

Gi, C2, . .. , Gn F D1, D2,... , Dm,

wherethe commason theleft shouldbe thoughtassomekind of conjunctionarid thoson theright, somekind of disjunction.

A (propositional)ClassicalLinear Logic consistsof formulae and sequents.For~mulaeare generatedby the binary connectives0, If, &, ED and —o and by the unar~operation(_)~1.,from a setof constantsincluding I, I, 1 and 0 and from variables.

Thesequentsaregeneratedby thefollowing rules, from initial sequentsor axioms

Axioms:A F A (identity)F-I IFrF1,~ F,~F-~A F A-1-’- A-’--’- F A (negation)

StructuralRules:

FFL~ rFA,z~ A,r’F~’(permutation) (cut)

r,r’F~’,L~

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Logical Rules:

Multiplicatives:

r,AFB,~

~,B-’-F A-’-,~(var)

rF~(unit,)

r,.rF ~

r,A,B F ~(oi)

r,AeBF L~

r,AFz~ r’,BF~’

~,~‘,AffBF ~

rF~(unit,.)

rFA,~ T’,BFL~’(-oj)

r,r’,A —o B F i~’,L~

Additives:

r,AFB,~(-or)

rFA-oB,~

(&,)r,A&B F ~

r F A,~(EDr)

rF AEDB,Z~

r,B F L~

F,A&B F /~

r F B,z~

rF ~

A remarkon notation. Seely writes in his paper“x” for “&“, “+“ for “ED”, 0 forIf and -‘ for (—)~,but wewant to keep,asmuchaspossible,theoriginal notationfrom[Gir].

We would like GC with all thestructuredefinedbefore,to be a categoricalmodelof ClassicalLinear Logic. But it is clear that we do not have morphismsof the formA~1

—f A for all objectsA in GC. So, not all theobjectsareequivalentto their doublelinear negations,A ~ A-’-1.

Thus, weomit from the systemjust presentedthe negationaxiom A’-’~F A. It isinterestingto notethat, apartfrom theaxiom,weonly haveto changethenegationrule(var). Actually wetransformthe rule (var) into two rules, the rules (var,) and (var,.)asbelow,

(If,)

rFA,~ r’FB,/.~’(0,.)

r,r’ F A®B,L~,Li’

rF A,B,z~(If,.)

r F AIfB,L~

rFA,z~ rFB,~(&r)

rFA&B,~

r,AFz~ r,BF~(ei)

T,AEDB F ~

r,AF~.(var,.)

r F A-1-,L~

r F B,z~(var,)

~,B1 F L~

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Thenonly the rule (var,) is satisfiedin GC. That happensbecausethe loaredealingwith is really intuitionistic, at the bottom level. Thus, for example,model, the objects(A 0 B)-’- and (A1IfB-’-) “look” exactly the same;they arethe form (Xv x ~-~- U x V). But taking in considerationthe relations, whave a morphismin one direction: (A.I.IfB~t~)F (A 0 B)1. This is just as itcasein Intuitionistic Logic, thinking of “If” as “or” and “0” as “and”. Lookthe rule (var,.) weseethat it is not satisfiedin general,sinceif we havea morGØA —f D, that is equivalentto amorphismG ~ (A —o D). But from A1IfD sproveintuitionistically, A —oD, but not conversely.Thus wehaveto leaveout (~

Let the newlogical system,without A11 F A and (var,.), be calledL.L~ortimesjust L.L.

Theorem. ThesymmetricmonoidalclosedcategoryGC, with bifunctorstensoruct 0; “par” If; internal horn [—, —lGc; cartesianproduct &; coproduct ED anltravariant functor (_)1 for linear negation, is a modelof L.L~.Thus to eachment r FL.L~A correspondsthe existenceof a morphismin GC,(f, F): IFI —4

jrj F-GC JAJ.

Theproof is merely to checkeachof theaxiomsand rules.Notice that rules Oi and hr arefundamental,since they indicatehow we ~

interpretthe sequentsin thecategoryGC. They show that

Gi,G2,...,GnFD1,D2,...,Dm

shouldbereadasthereexists amorphismin GC,

lG~I0...01G11 —4 IDkIIf...IfID1I

Corollary. Thesubcategory‘Dec GC” is a modelof ClassicalLinear Logic.

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we 4. Modalities in GC~he Interpretationsof themodal,or exponential,operators“!“ and “?“ of Linear Logic,of in a categoricalset-up, should correspondto a comonadand a monad, respectively,

satisfyingcertainconditions. Wediscussendofunctorson GC, whichcouldplaytherole;he of theconnective“!“ in ClassicalLinearLogic. Thefirst ideawas,following the modelat of DC, to look at free monoidsin C and seewhetherthey would induceappropriate

comonoidsin GC.

) SomedefinthonsConsiderthemonadgivenby theconstructionoffree-monoidsin C. Thus, suppose

we are given an adjunctionF H U: C ~ Mon C and call S0, or alternatively*, thecompositionFU: C — C.

Intuitively X* standsfor “finite sequencesof elementsof X” and f* for “f appliedto eachelementof the sequence”.Also So hasclearly a monadstructureand it does

u~l- not preserveproducts. Despitethat So inducesan endofunctorS:GC —~ GC whichor hasanaturalstructureasa comonad.

a Sa *Definition 1. TheendofunctorS is givenby S(U i-i- X) = (U +÷ X ) on objects. Therelation “Sa “is givenby thepullback below

SA ______

UxX* )(UXX)*Cu,x

wheretheauxiliary mapsC(_,) aregivenby theserieofequivalences:

V x I’ ~ (V x

Y~2~9(VxY)*V

Y* -~-~ (V x y)*V

v Y~C~y)(V x

Therelation “Sa” readsintuitively as

“u(Sa)(x1,. ..,xk) if uax1 and ... anduaxk”

and S appliedto a morphism(f,F) in GC is (f,F*).Thefunctor5: GC —* GC hasanaturalcomonadstructure,inducedby themonad

structureof S0. Alas, this comonadhasnot the nice categoricalpropertiesit hadwithrespectto the categoriesDC, due to thefact that the tensorproductin GC is muchmore complicatedthan the. one in DC. There are other very natural monadstoconsiderin C, if C is cartesianclosed.

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Definition 2. For eachU in C, let Tu: C —~ C be the endofunctorwhich takes~ tofgEY~.

That is clearly a monadin C with unit (i~TU)X:X —* Xu given by the “cormap”, andmultiplication (/2T~)x:XUXU ...+ Xu, simply “precompositionwithnal”. We now turn ourattentionto definingacomonadin GC “induced by the m~

Definition 3. ConsiderthefunctorT: GC ~ GC which takes(U ~- X) to the

(U ~ Xu) and the object (V L Y) to (V ~ yV) The relation “TcE” is definthe pullback below

TA fa

UXXU fUxX(iri,ev)

Intuitively, it saysthat “u(Ta)g if uag(u)”, whereg € Xu. To completethedeflisay that T applied to a map(f,F): A -~ B is (f, F(—)f):TA -~ TB.

It is easyto show that T hasacomonadstructureinducedby themonadstru(of thefunctorsT~.

Moreover, the monadsT~relateto So in a very specialway, describedby[69] asa “distributive law”. More interestingis the fact that A above,inducesdistributivity law A, this timebetweenthe comonadsT andS in GC.

Proposition 1. For eachU and X in C, wehavea natural transformation

Ax: SoT~X-4 T~SoX

correspondingto (Xu)* ~ (X*)U, satisfyingtheappropriatediagramsfor a distriLlaw. Thereis also a natural transformationA: TS—~ ST, at eachobjectA, AA:T.~STA is given by (1~,(A)~)where (A)x:(XU)* .4 (X*)u is the distributive LC. This natural transformationA satisfies the conditionsfor a “distributive 1.comonads”.

Using Distributive Laws

It is widely known that the compositionof monadsis not always a monadgiven a distributivelaw A, we candefinethe compositemonaddefinedby A, cf. [IWe can alsodefinethe “lifting” of one of the monadsandseveralrelationshipsathe categoriesof algebrasand Kleisli categoriesinvolved.

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Definition 4. The compositemonad(T,LjSo)A in C, takesX ‘—p (X*)L~.Similarly, wehavethecompositecomonad,inducedbyA andgivenby (TS):GC -~ GC, which takes

(U ~- X) to (U TSa

Besidesthe “compositemonad”,a distributive law providesa “lifting” of oneofthe monadsto thecategoryof algebrasfor theothermonad.

~ct Proposition 2. ThemonadTu:C C lifts to the categoryof So-algebras.Dually,the comonadT in GC lifts to thecategoryofS-coalgebras.

byWe-candescribeanothermonadTu:CS0 ~ CS~.The endofunctorT~applied to

an So-algebra(X,j: X~-~ X), givesths So-algebra(Xu,h). The newstructuralmap

h: (XL~)* XU is given by the composition(Xu)* -~ (X*)u ~ Xu. The endo-functor ~: (GC)S ~ (GC)S hasa comonadstructuregivenby themonadstructureofT~.

Clearly composingthe functor Tu with the natural transformationr~:I -~ S~wehavea monadmorphisma:T~- TuSowhich takesXu f-* (X*)(~.We alsowrite /3 forSo —+ T~So,which is (~lT~) appliedto thefunctor S~,thustakingX~~+ (X*)u.

Similarly, therearecomonadmorphisms6: TS T and ~:TS —~S, where foreachA in GC, 6A: TSA -~ TA is given by 6A = (‘u~ax) and lcA: TSA —~ SA byIcA = (lu,/3x).

Proposition 3. Themonadand comonadmorphismsaboveinduce:

mapsin the categoriesof algebras,~: CTUSO~ C~’~’and~: CTUSO ~ C50;mapsin the categoriesof coalgebras6: GCT -.4 GCTS and7~:GC5

—+ GCTS.

Our next aim is to relatethecategories(CS0)TU andC50T~- dually (GCS)T andGCST.

Proposition 4. Thereis an equivalenceof categoriesof algebras,ye ~o:(CS0)T~~ ~

of and respectively,of categoriesof coalgebra.s~: (GCS)T~4 GCST.

The proofsof Propositions2, 3 and4 given in Beck’s paperfor algebrastranslateexactly to the coalgebrascase,thuswe omit them.

Clearly themonadSo doesnot lift to thecategoryof Tu-algebras,sincewe cannotdefine the Tu-structural mapfor S0X using A, but it seemsto lift to the Tu-Kleislicategory,CT~.Cleanly we are talking about duality once more, but that is a moresubtlecase.

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Proposition 5. ThemonadSo “lifts” to theKleisli categoryCT~.Dually, thecornonadS lifts to theKleisli categoryGCT.

TheendofunctorS0: CT~—+ C~takesX to X~anda morphismin CT~,X —* Y -

S0(f0) Acorrespondingto a mapf: X —fTUYin C - to thecompositionS0(X) —* S0Tu(Y)—4

TuS0(Y)whichcorrespondsto So(f):S0X—* S0Yin CT~.ThefunctorSohasanaturalcornonadstructure.

This is a generalconsequenceof the existenceof the distributive law. The pointhereis that all thepropositionsabovecouldbereadoff from Street’spaper “The formaltheori, of momads“, by a clever 2-categoricallyminded reader.But we will not go intothe 2-categoricalaspectsof thetheoryhere.

Using thepropositionsabovewe cansumup the resultsof this sectionin the four“squares”below. Eachsquarehasthreesidesconsistingof adjoint-pairsand the lastsidegiven by a naturalmorphism.In C, relatingalgebrasandKleisli categoriesandinGC relatingcoalgebrasand Kleisli categories.

CTU S0 ) CTU CT~So C~,

Ii Ii I IICSo

GCTS GCT GCTS I GCT

•11 _____ TI 1 _____ 11GCS( )GC GC5’ GC

Note that if C hasequalisersthen, the two top squaresare totally composedofadjoint-pairs,but we do not pursueit here, sinceit is not clear that equalisersin Cwould imply equalisersin GC.

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5. Properties of the comonadsT and!In the last sectiontheendofunctorT: GC -~ GC wasdefinedandshownto havea

naturalcomonadstructure.Thisendofunctorseemsareasonablecandidateto representtheconnective“!“. For a startit hasa “dual” endofunctor,to bedenotedR, describedin thenext paragraph.

Definition 5. TheendofunctorR takestheobject(U ~- X) to (Ux ~ X), theobject

(VLYJtO(V1’ RJY) and thernap(f,F):A—f B tothemap(f(—)F,F):RA--fRB.

Therelation “Ra” is definedusingthepullback ofA >~.+ U x X along the evaluation(evirj)

morphismU x X U x X and intuitively it says “g(Ra)x zffg(x)ax”.

Thefunctor R is thefunctorpart of a monad,with unit i~: A —+ RAgiven by theconstantmap 1/u: U —+ Ux in the first coordinateand identity on X. Multiplicationj~:WA -~ RA is given by “restriction to the diagonal” Pu: UXXX Ux in the firstcoordinateandidentity on X.

We would like to havein GC resultsfor “T” analogousto the onesfor “!“ in DC.For example,we would like the isomorphismrelatingcategoricalproducts to tensorproducts!(A&B) ~=!A®!B.But there is no obvious relationshipbetweenT(A&B) =

(U xV~(X+Y)~”)andT(A)OT(B)=(U xV4-+-X~~~’xY~1’). What wedohaveis arelationbetweenthetensorproductsin GC.

Proposition 6. Thereis a natural isomorphisrnin GC, T(A0 B) ~ TA0 TB.

For a far more interestingresult, ~recall that the T-Kleisli categoryGCT hasasobjectsthe objectsof GC but asmapsfrom A to B, mapsin GC from TA to B.

Proposition 7. The maps from A to B in the T-Kleisli categoryGCT, are in 1-1correspondencewith mapsfrom A to B in thecategoryDC.

Proof: We want to check

HomGcT(A,B) = HomGc(TA,B) ~ HomDc(A,B).

Thesecondequivalenceholds, sinceamap(f, F): TA —~ B in GC, correspondstof: U -~ V andF: Y —4 XU, satisfyingthe condition

(U x F)~(a)<(f x Y)~(/3). (1)

That correspondsto f: U —.~ V and, by exponentialtranspose,to F: U x Y —‘ X,satisfyinga correspondingcondition(*) that is amap (f,F): A - B in DC.

Sinceobjectsarethe samein both categoriesGC andDC, Proposition7 impliesthat thereis an equivalencebetweencategoriesGCT andDC.

The cornonad “!“

Consider now the compositecomonadTS defined in the last section,with thedifferencethat now So denotesfree commutativemonoids in C. Thus S(U ~- X) =

(U ~ X*) and S takesa morphism(f,F): A -~ B to (f, F*): SA -4 SB. Let the

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comonadTS be called “!“. The functor part of “!“ acts on objects as !(U ~- X) =

(U ~ (X*)u) andon maps!(f,F) = (f,F*(_)f).As wehaveshown “!“ is thefunctorpart of thecompositecomonad

e~:!A —~ A, 8!: !A —~!!A)

andwe canconsiderthecategoriesGC! of !-coalgebrasandGC~,the !-Kleisli categoryMoreover,theobjects“!A” haveanaturalcomonoid-likestructure,with respectto “0”

Proposition 8. Therearenatural morphisrnsin GC as follows1. From theobject !A to I, given by the terminalmapon U and the natural ma1

1 —4 (X*)u.2. From !A to !A0!A, which is given by the diagonalmapin C, A.: U —4 U x U an

themap8: (X*)~~~x (X~)~~~’~ (X~.

The map 8 is given, intuitively, by taking a pair of functions(~,cr), eachof thenof theform U x U —4 X’~,to the productmap~5x c precomposingit with the diagonain U and post-composingit with themultiplication on X~,asfollows,

* * p. *U—÷UxU —~ X x X —iX.

Proposition 9. Wehavethefollowingnatural isomorphismsfor eachA and B in GC!(A&B) —~!AØ!B.

Proof: Look at thefollowing seriesof equivalences:

!(A&B) = TS(A&B)~ T(SA® SB)~ TSA0 TSR=!AØ!B.

Proposition 10. TheKleisli categoryGC is cartesianclosed.

That is aneasycorollary of theabove,since

HomGc.(A0 B, C) ~ HomGc(TS(A0 B), C) ~ HomGc(!A0!B, C) ~

~ HOmGC(!A, [!B, C]GC) ~ HomGc,(A,[B, CIGO,).

Corollary. Themorphismsfrom A to B in the categoryGC~,correspondnaturally t

morphismsin the categoryDCs from A to B.

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6. Linear Logic with modalities

The compositecomonad“!“ definedin the last sectionsatisfiesthe rules for themodality “!“, but we would like also a monad“?“ satisfying the rules for the dualconnective,called by Girard “why not ?“.

We recall the rules for themodality “!“. Theseare:

r,Al—B ri-BI. (dereliction) II. (weakening)

r,!AI-B r,!AI-B

r,!A,!AFB !rFAIII. (contraction) W. (!)

r,!AI-B !rHA

Theorem. The categoryGC with the compositecornonad ‘1” definedin Section5 isa modelfor Linear Logic enrichedwith modality ‘9”.

Theproof is againto checktherules andit is straightforward.Moreover,usingRthe dual endofunctorto T we canget a monadto model ?. We just haveto composeRwith themonadin GC inducedby themonadU ~ U’ in C. The compositemonadsatisfiesall necessaryconditions.

Concludingremarks

To concludeit is perhapsworth mentioningsomeof the severalquestionsthat thework on the categoriesDC and GC prompts,apartfrom the onesalreadymentionedin theintroduction.

1. Is thereaninterestingconnectionbetweenthecategoricalmodelsDC andGC andGirard’s newwork on the Geometryof Interactions?

2. Sincewe think of mapsin DC andGC as“linearmorphisms”,in oppositionto themoreusualmorphismsin the Kleisli categories,canwe characterizebilinearmapsin this context?Thereis someinterestingwork of Kock, but theobviousapproachdoesnot work, dueto thefact that thecomonads“!“, or rather,their functorparts,arenot strongfunctors.

3. We haveshunnedawayfrom the2-categoricalaspectsof everythingdiscussedpre-viously, but that is not, probably, the best policy, as was indicatedby the needof distributive laws. More to the point, thereis a very interestingquestionof us-ing “spans” insteadof relationsin the constructionof DC and GC, which wassuggestedby Aurelio Carboni.

4. We have workedonly with commutativeversionsof the connectives,that is withsymmetric tensorproducts, “par” bifunctors etc. There is a interestingcasetolook at, if this comrnutativityconditionis dropped. On thoselines thereseemstobe someconnectionwith Joyal and Street’swork on braidedmonoidalcategories.In particularthereis also a preprint by D. N. Yetter on “Quantalesand (Non-commutative)Linear Logic”.

5. Finally, thereis theverypromising,but asyet very vagueideaof connectingLinearLogic with ConcurrencyandParallelism. The ideabeing that Linear Logic mayprovidean integratedlogic, whereonewouldhopeto modelcomputationalprocesses

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in a lessad hoc fashionthan it hasbeenup to now. In particular,Petri Netshavebeenproposedasa model for LinearLogic, cf. [Girl 1987.

References

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