Lineales Martin Hyland andValeria Paiva1 The firstaim of thisnote is todescribe an algebraic structure, more primitive than lattices and quantales, which corresponds to the intuitionistic flavour of Linear Logic we prefer. This part of the note is a total trivialisation of ideas fromcategory theory and we play with a toy-structure a not distant cousin of a toy-language. The second goal of the note is to show a generic categorical construction, which builds models for Linear Logic, similar to categorical models GC of [de P1, but more general. The ultimate aim is to relate different categorical models of linear logic. The first part of thenote consistsof twosections. The first section introduces lineales; the second adds some structure to lineales, compares our work to other approaches and show the main result of this part. The second part of the noteconsists of four sections, whichrun along similar lines to part I. In section 3 we defineour basic categorical construction, section 4adds the extra structure corresponding to section 2 and shows the main result of part II. Section 5, adding the modalities ! >> and <? >>, has no corresponding section in part I, as we have not even tried to find the right notion of << >> in the restricted set-up of lineales. Section 6 describes some preliminary conclusions and further work. 1. Introducing lineales We start by considering a very familiar structure, a commutative monoid ~n the category of posets. We are thinking of posets as a restriction of the From DPMMS andComputer Laboratory, University of Cambridge. o que nos fnz p0050r, n' 4, Abril de 2997 108 Martin Hyland and Valerin Pair general notion of categories. That is the opposite of what people nornially d in CS when theyexplain the notion of acategory as a generalization of a posel We call a commutative (or symmetric) monoid in the category Posets pre-lineale. [In the more general set-up we're thinking of a monoid object ii the category of categories.] t~~(~cf ~o5~5~,à fl~ C~C~',~Definition 1 A p~'e-Iiñ~aIeis a poset (L, ) with a given compatible symmetric nzonoida, structure (L, o, e). That is, a set L equipped witha binary relation > satisfying: • a a forall a in L (reflexivity) • a b and b c =~a c (transitivity) • a bandb a=~a=b(antisymetry) together with a monoid structure (o, e) consisting of a multiplication ' o : L x L -p L and a distinguishedobject e of L, such that the following hold: • (a o b) o c = a o (b o c) (associativity) • aoe=eoa=a(identity) • aob-boa (symmetry) The structures are compatible in the sense that, if a b, we have a o c ~ b o c, forall c in L. We write a quadruple (L, ,o, e) for a pre-lineale. Note that, even ifwe want to think of <o>> as a form of conjunction, we do not havea a a = a (idempotency) nor a e for all a in L. Thus the relation between the order structure and the multiplication is not as tight as in a sup-lattice. But a pre-lineale is not the toy-structure wewant to play with. Apre-lineale corresponds, in the more general set-up of categories, to a symmetric monoidal category and we are interested in symmetric monoidal closed categories. To trivialise this notion we first define: Definition 2 Suppose L is a pre-lineale and a, b e L. If there exists a largest x E I. such that a a x b then this element is denoted a -a b and it is called the relative pseudocomplementof a wrt b. Thus, by definition, if a -a b exists in a pre-lineale L then • ao(a-ob) b • if a a y b for some y, then y a -o b Lineales Definition 3 A lineale isa pre-lineale (L,~ ,o, e) such that a -o b exists for all a and b in L. Since we defined a lineale to be a simplification of the notion of a symmetric monoidal closed category, we have an obvious proposition: Proposition I A lineale (L, , o, e, -o) has the following properties: 1. If a b, for any c in L, c-a a c -o b and b -o c a -o 2. a o b c ~ a b -o c The proof is very easy, it only uses the definition of -a and L. Observe that the item 1 in the proposition says, in the more general set-up of categories, that -a: L x L -~ L is a << bifunctor >>, contravariant in its first coordinate arid covariant in the second coordinate, while item 2 says there is an adjunction between functors 0 o b: L -f L and b -a 0 : L -* L. Another observation is that as e o a a for any a E L, we know e a -o a anda a-oeforanyaE L. Note that if we denote by I any element of L and write (a)1 for (a -a I) we have: (i)a b ~ b-'a1by prop 1.1. (ii)aa(a-oI) I-~=~a o a1 Iimpliesa a'~-oJ~a°-by prop 1.2. Properties (i) and (ii) are called by Dunn the Intuitionistic Contraposition. Definition 4 A }-e~ting~1inealeis a lineale (L, , o, e, -a) equipped with a given compatible symmetric monoidal structure (0,1) weakly de Morgan-dual to a x.. That means that • the given structure (0, 1) satisfies - (associativity) a 0 (b 0 c) = (a 0 b) 0 c - (syminetry)aob=boa • the structure (0, 1) is compatible with (L, ~, a, e) means that, as before, if a b then for anycin L,aoc boc • the object 1. is the identity for0 aDI=IDãa 110 Martin Hyland and ValerEa Pal • we have associative (or absorptive) laws (a 0 b) a c a 0 (b a c) a a (b 0 c) (a a b) Dc Note that, if we write (a)1 for (a -a 1), 1 the identity for 0, we ca show a-'0 b a -a b simply using symmetry of < o and the distributive law above, as follows: ao(a1Db) (aaa-'-)Ob = (aoa-oJjOb IDb=b With definition 4 we are trying to capture the (intuitionistic !) notion tha conjunction and disjunction are not de Morgan dual - as they are in Classica Logic, but instead, we have: • ~ • a10b1 (aob)' We can prove, Proposition 2 A Heyting lineale L satisfies (a) a1 a b1 (a 0 b)-'-, (b) a1 0 b' (a o To show (b), as (a o b)1 = (a a b)-a 1, it is enough to show (a1 0 Li~)o (a o b) 1, easy as (a10 b1) a (b o a) = a10 (b' ob o a) a-'-D(Ioa) = (a'DI) oa = a1 a a I To show (a) a1 o b' (a 0 b)' we use the same kind of reasoning, as it is enough to show (a1 a b1) a (a 0 b) I. Lineales in Note that as e is the identity for o and I the identity for 0, a1 oI-~- (a 01)1 = a1 = a-ia e =~I-'e By proposition 1.2, as e ol I we have e 1-ol = I', thus I~= e. But in the weakly-dual case we cannot guarantee that e-1- = I as we only knõw that a10e1 (a oe)1 =a1= a-1-DI=~e1 e Note that the condition of compatibility says in the more general set-up of categories that 0 is a covariant bifunctor. We would call the symmetric monoidal structure (0, I) de Morgan-dual to if we had equality in condition (a) and (b). In that case we would call the neale a strong Heyting lineale. One may think that names were badly chosen as a lineale already satisfies ~-hat maybe be called a Heyting condition, namely a 0 b c ~ a b-ac ut lineales have no notion of disjunction whatsoever, while Heyting lineales an be restricted to Heyting algebras if 0 satisfies a universal property (cf. elow in def. 5). • Additive lineales A (Heyting) lineale is characterized by its << multiplicative>> structure given y ( , o, -o, e) (perhaps also (0, I)). But we can have another << layer>> of ructure, called its additive structure. efinition 5 A semi-additive (Heyting) lineale is a (Heyting) lineale equipped ith an extra symmetric monoidal structure, notation (x, 1) such that given a and b L, a x b satisfies • axb aandaxb b • If m is such that vi a and m b then m a x b Note that a x b is defined as a binary greatest lower bound; that having iary gib's we can easily define finite rt-ary ones and that 1 is the empty-set , which means that for all a E L, a 1. In particular e I (and I ~ 1, if it is ~sent). Also (x, 1) being a symmetric monoidal structure means 112 Martin Hyland and Vi~leriaPi • (axh)xc=~-ax(hxc) • axb=bxa • axl=Ixa=a A semi-additive lineale corresponds to a symmetric monoktal clõ category with products in the more general framework. Definition 6 An additive (Heyting) lineale is a semi-additive (Heytiiig) lin equipped with an another symmetric monoidal structure, notation (~,0> such t given a and b in L, a ~ b satisfies: • a a®bandb ae3b • ifa nandb nthenaeb n Dually, 0 a for anya e L, in particular, 0 I, 0 e and 0 I. Observe that the conditions in the definition 5 and 6 above are restrictions to the poset set-up of the conditions on the existence of produ~ and coproducts. They could be described in terms of adjunctions, in. this c~ Galois connections, to a diagonal functor, i~: L -, L x L. Note that they determine a lattice structure in L. If the four constants I, e, 0 and I are distinct we have a picture like ~0z but they may coincide. Trivial examples of additive Heyting lineales are Heyting algebras (whe a and x and 0 and + coincide and 0 = I and I = e) and Boolean algebras ( before plus a-~-~-= a). Proposition 3 In an additive Heyting lineale we have the distributive laws: • ao(b~c)=(aob)~(aoc) • aD(bxc) (aob)x(aoc) Notice that the first law is a direct consequence of the fact that ti Lineales :113 <~category>> L is a symmetric monoidal closed one, as ~ is a coproduct kind coproducts are preserved by functors which have right-adjoints. The semi-law is a consequence of x being a categorical product, as b x c b and b x c c impliesao(bxc) aobandao(ln<c) aoc, so aD(bxc) (aO(bxc))x(aD(bxc)) (aob)x(aoc) Comparison with other approaches It seems reasonable to compare the approach taken here with the one by Hesselink using Girard monoids. Quite apart from the fact that Girard mono ids are based on/par the linear connective less amenable to intuitive explanations, Hesselink's approach is based on the classical equivalence between A -* B ~nd -~ A v B. It seems to us that one should strive for the more general set-up - in this case the intuitionistic one - as that allows us to restrict ourselves to the classical case, when (and if) wanted. A strong Heyting lineale can be seen as a Girard monoid wrt 0 and a Girard monoid restricts to a phase structure, the model for linear logic provided by Girard himself in [tcs60]. Also a Girard monoid is ageneralization of the de Morgan monoids in Dunn, the semantical model for relevance logic. The definition of a Heyting additive lineale is also very similar to some work done by Ginsberg and also Fitting on bilattices. Again the difference is that the structure on the horizontal direction need not be a lattice. The conditions forced on us by the (categorical) adjunction are not strong enough for that, but of course a bilattice is a rather special case of an additive Heyting lineale., Rules and axioms of Linear Logic Axioms: A F-A (identity) I--I IFITF-1,A r,oF-~ Structural Rules: r ~ (permutation) F I- A, ~ A, F F- ~ (cut) F,F'F-~',I Logical Rules: (var1) ±_~Lil~L-g?~F B1 F-~\ 114 Martin Hyland and Valeria P~ Multiplicatives: (unit,) r F-A (unit,.) F F- AFF-I,A F,A,BF-A Ff-A,A F'F-B,A' (®,) F, A ~ B F-A F, F' F A ® B, A, A' ~ F, A I- A F', B FA ~ F F-A, B, A r',r",AoBF-A,A' I~F-AoB,A F F-A,A F',B F-A' r,A F-Br, r', A B F-~, ~ (r) F F A B * Additives: I~F-A,A FFB,A r,AF-A F,BI-~ F F-A&B,A F,A&B F-A F,A&B F F,AF-A F,BF-A ~ FF-A,A FF-B,A F,A$B F-A F FÃB,A F F-AEDB, *Observe that in rule (-or) weonly deal with one formula on the right-ha side of the turustyle, according to our intuitionistic flavour of Linear Logic Then we have another obvious proposition Proposition 4 An additive Heyting Lineale (L, , o, -a, 0, e, I, ±, x, 1, 0) is algebraic model of Linear Logic, as described above. Just read atomic propositions in LL as elements of L, Fas leq, ® as 0 ai the other connectives and constants for their homonimous. Note that the poset reflection of GC isa lineale, the simplest non-collaps one (see figure above). 3. A categorical construction Suppose C is a concrete linear category with products, by that we mean concrete symmetric monoidal closed category withproducts. And suppose th L is an object of C endowed with a (Heyting) lineale structure ( , 0, -a, (perhaps also (0, 1)). To make notation manageablewewrite: • [U, V] for the internal horn in C, Lineales II 5 • U ® V for the tensor product in C, with identity I; • U x V for the categorical product in C, with identity I. Then wecan construct the categoryMLC, which has as objectsmorphisms of Cof the form U® X-~-*L. One such object is written as (U ~-+--X)andcalled A. Given two objects, says A = (U ÷~+-X)and B = (V ÷ ~-1--Y),the morphisrr*s of MLC are pairs of morphisms of C, f: U -4 V and F: Y - X such that the following diagram is satisfied, 1J®F U®Y ~LI®X L where the diagram being satisfied means that given u 0 y in U 0 Y, the composite morphism cx . (U® F) applied to (u ®y) as an element of L is smaller than ~. (f® Y) applied to (u 0 y). Simplifying, morphisms are pairs of maps in C(f,F),f: U-i VandF: Y->Xsuch that a(u,Fy) f~(fu,y) It is easy to verify thatMLC is a category with an abundance of symmetric monoidal structures. Proposition 5 The construction above really defines a category MLC. Clearly identities are pairs of identities of C, composition is composition in each coordinate and associativity is an immediate consequence of the associativity in C. Linear structure of MLC One of the possible symmetric monoidal structures of MLC is: Definition 7 Given two objects A = (U ~-f--X)and B = (V ~-~-~--Y)in MLC we define A 0 B their tensor product as follows: A® B =(U® V~~LJV, X]x[U, Y]) The morphism a ® ~3 intuitively says a ® ~3(a, v, f, g) = a (a, fv) a 1~(v, gu). 116 Martin Hyland and Vrderin Paiva To define the morphism a ® ~ consider the following map, which we call a: (U® V)®([V,Xjx[U, VI) ~ V®[V,X] ~®~'> U~X-~-÷L Similarly we define (U® V) ® ([V. X] x [U, YD-~--~L.Then to get a® ~ we pair a and ~ and use the multiplication < o >> of L, as follows: (U®V)®([V,XJxLLI,YJ) <°'~LxL -2---~L Proposition 6 The construction above induces a bifunctor, covariant in both coordinates, with identity IM given by (I ~-~---~--I), where the morphism I ® I I -~--~Ljust picks up the object e > from L. Note that ® is not a categorical product, for instance we have no projections, even if C is a Cartesian closed category. Definition 8 Given two objects A = (U ~±--X) and B = (V #~-Y)in McC we dcfinr [A, B] their internal horn as follows [A, BI = ([U, VI x IV, XI ~ U ® Y) The morphism cx -o f~>intuitively says (a ~) (f, F, a, y) = a (u, Fy) --a J3 (fu, y). The definition of the morphism a -a ~ is similar to the definition of ® above. First consider maps ~ and ~: (LU, VI x IV, XI) 0 (U 0 Y) ~ ®~®~>[U,VI 0 U 0 V --~-~ V 0 V ~ Then, to obtain a -o we pair ~ and ~ and compose the result with -a, considered as a map from L x L to L: ([U,V]x[Y,X])®(U®Y)~~> LxL~ L Note that if we consider the internal horn [A, A]=([U, UIx[X, Xl ÷~-~--~--U® X), there is always a morphism from IM to it, I I [U,U]x[X,XJ ~--~-~---U®X as C is symmetric monoidal closed with products and e a (a, x) -a a (a, x). Lineales 117 Proposition 7 The construction above induces a bifunctor, contravariãztin its first coordinate and covariant in its second coordinate. Having defined both a tensor product and an internal hom, we want to prove that they provide us with a symmetric monoidal closed cate~gory. Proposition 8 The category MLC is a symmetric monoidal closed category. The proof is simple, one has to verify the natural isomorphisn-i: Horn (A ® B, C) Horn (A, [B, C]) This can be done by looking at the diagram U®V ~ [V,XJx[U,YJ U X f <fz,f2> <f,f2> fi W ~ z [V,W1x[Z,Y]<~i~ v®z If the morphism (f, <ft' 12>) is a Hom (A 0 B, C), then given (u, v) in U 0 V and z in Z, we know (a 0 ~)(u, v,f1z,f2z) y (f(u, v), z). That means, by definition of tensor, that a(u, f1zv) 0 ~Mv,f zu) ~y(f (u, v), z). But as Lis a lineale, a (a, fiw) 0 ~ (v, f2zu) y (f (ii, v), z) ~ a (u, fzzu) ~ (a, f2zv) -o ? (f (u, v), z) Now to show that kf,f2>, f~)is in Hom (A, [B, C]) we have to show a (a, fi (v, z)) (f3 -0 y~~'fu,f2u, v, z) But (~--a y) (fri. f2u, v, z) = ~3(v, f2uz) -a 'y (fuv, z) which we know, if ransposing is allowed. If we have a Heyting lineale we can also define another bifunctor < D> of bjects in MLC. )efinition9 Given two objects A = (U ~-~-i-----X)and B = (V ~-+-Y)in MLC we define o B their /par operator as follows: ADB=([X,VJx[Y, LJ]~X®V) Martin Hyland and Vc~leriaPaiva The morphism a 0 ~3 intuitively says (a 0 ~3)(f, g, x, y) = a (x, gy) ~ f3 (fx, y). The definition of the morphism cx 0 [~is similar to the definitions of ~ and 1-. -] above. First consider maps a and ~: (IX, VI x IV, U]) 0 (X 0 Y) ~ ®~')[X, VI 0 X 0 Y) ~' ~ V ®~'-~--, L ([X,V]x[Y,U])®(X®Y) ~øX~Y)[yU]®x®y) Xøeval)U® X-~-~L Then to obtain cx 0 ~we pair ~and 13 and compose the result with 0, considered as a map from LxL to L ([U,V]x[Y,X])®(U®Y)<>LxL~9~~~L Proposition 9The operationADB defines a bifunctor 0 : MLC x MLC -4 )VILC with identity given by the object IM=(< 111), wherethemapl: 101=1 -4 L picks up the object I from L. 4. Additive structure ofMLC Now we want to define products and coproducts in MLC. To do that we need at least • a semi-additive (Heyting) lineale • (disjoint?) coproducts in C. Note that it is not necessary to add products and coproducts to M~Cat the same time. Suppose C is a linear category with coproducts. Then a form of distributivity holds, namely: U®(V± W)=~U® V+ U® W As C is symmetric monoidal closed, the functor U 0 (-) has a right-adjoint, [U, -1, hence it preserves colirnits and, in particular, initial objects and coproducts. Definition 10 Given two objects A = (U -5-f----X) and B (V ~-i----Y)in M~Cwe define A & B their categorical product as follows: A & B = (U x V ~X + Y) Lineales 119 The morphism a & ~3 intuitively says (a & ~3)(u, v, (~))= a (u~x)~'< ~3(a, y). But note that, despite the similarity with previous definitions, the multiplication < x is not used, what is used is the structure on C, as an element of X + V is either (x, 0) or (y, 1) but not both. Proposition 10 The operation<< & above defines a bifunctor& : MLC xMLC-3 MLC, with identitygiven by IM (I ~ 0) and A &B is really a categorical produ ct inMLC. To define the morphism (U x V) 0 (X + Y) & ~ L in C, which corresponds to the object A & B inMLC, we do: ~t~®1+~t 2 ®1 (") (UxV)®(X+Y)~(UxV)®X+(UxV)®Y >U®X+V~V m-~ Projections are trivially given by projections in C in the first coordinate and !anonical injections in the second coordinate. UxV X+V 1ti U Li < I X ë have a diagonal functor A : MLC -f MmC xMLC ___ I UxU X+X ~n by the diagonal in C in the first coordinate and the canonical folding ~in the second coordinate. b show the universal property of products we consider an object C = (W ~-~+---Z) that thereare maps inMLCof the form w T z ~ JF gf JG U ~- X V ~-Y 120 . Martin Hyland anal Valeria PaiV' Then there is a unique map in MLC from C to A & B, W ( Z <f, g> (~) UxV I X+Y Dually we can define Definition 11 Given two objects A = (U +~-~--X)and B = (V -~-t--Y)in MLC we define A 0 B their categorical coproduct as follows: AE?B=(U+V< aE~f3 XxV) The morphism a 0 >> intuitively says (a® ~)((~),x, y) = a (x, g-y) 0 ~3(fx, y). It is another easy proposition to show that A 0 B is abifunctor with identity = (0 4-~-1-1) and A 0 B is a categorical coproduct. Note that as morphisms of C °M and 1M are isomorphic, but not as objects of MLC. Note also that the additive structure of the lineale L is not used at all. The category MLC was defined following the pattern of CC, so it is no surprise that Proposition 11 The category MLC is a model of Linear Logic as described before. The last observation in this section is that we can describe another useful monoidal structure inMLC. Definition 12 Given two objects A (U ~-i--X)and B = (V k-~t---Y)in MLC we define A 0 B another tensor product as follows: A a B = (U 0 V ~-~-~-~-X0 Y) The mnorphismn << a 0 intuitively says (cx 0 ~3)(u 0 v, x 0 y) = cx (u, x) 0 ~ (a, y). Its usefulness will become apparent in the next section. 5. Modalities inMLC Now the intention is to define a comonad inMLC to provide an interpretation of the modality or exponential ~! >> of Linear Logic. Lineales 121 We start by recalling the rules for the modality <<1 >'. These arc~: _____ rE-B F, ! A F- B (dereliction) r, I A F-B (weakening) F',!A,!AF-B . !FF-A F, I A F- B (contraction) IV. F I-! A (!) But as observed by several people, the four rules for the modality << ! >> fall neatly into two pairs. The pair (II, III) has to do with putting back into the logic, in a controlled way, contraction and weakening and thepair (I, IV) make5 <<!>>look like the0 modal operator of S4. Suppose C is a linear category which has countable coproducts (instead of finite ones as in the last section). Then using the well-known construction of MacLane ([CWM] p. 168 theorem 2) we can show that C has free (commutative?) monoids, as C being symmetric monoidal closed the other condition in MacLane's theorem is automatically satisfied. Having free monoids means that there exists a functorF: C -3Mon C, which is left-ad joint to the forgetful functor U:Mon C -9 C. In other words, there is an adjunction <F, U, ~, c> : C -4 Mon C, which wewrite simply as F -1 U. The adjunction says that every map on C of the form, X_~~L,U(V, Ily, FLy) corresponds, by a natural isomorphism, to a monoid homomorphism fof the form (X, 1lx~,~tx)~(V, i~y,~y) Wewrite ( )~for the composite functor U • F: C -f C, recall fromMacLane that X = II je~X' and denote by (*, 1. FL) the corresponding monad in C. Note that the unit of the adjunction F -I U, the natural transformation ii: C -f C takes any object X of C to the carrier of the free monoid X~.Also the co-unit of the adjunction e : Mon C -4 Mon C takes any free monoid (Xe, -yr, ~f) arising from an arbitrary monoid (X, ~, Ft) to itself. Thus e:FU(M,1,~t) = (M,11*,1L*) -~(M,i,ti) where the morphism e corresponds to <<iteration >> of the original multiplication Ft. Now, in this stronger version of the existence ofmonoids, the monad (*, i~,~t) is easily proved a strong monad, so there are morphisms st(X,Y)[X,VJc->[ ,iIc 122 Martin Hyland and LJa!eria Paiva and using these we can define the endofunctor below. Definition 13 The endofunctor S : MLC -* MLC takes an object (U +~-I----X)ofMLC to the object (U~-~f-~--XD, where intuitively uSa (x1, X2, ..., xn) means UeLXI and uax2 and ... and UaX~. The object Sa of MLC isdefined by the sequence of morphisms U®X -~-~L U -+ IX, LI -~--* [IC, L~] U®X-3L -~---*L So far so good and very similar to what happens in GC. But if we try to make another definition Definition 14 The endofunctor T : MLC -9 MLC takes an object (U *~'-f---X)of MLC to the object (U 4~-[U, K]), where intuitively uTaf means uafu. But to give themorphism in U® [U,X1-1-~9Lwe would necd to <<duplicate U, sothat U®[U,X] ~ Also to obtain comonoids in MLC, which would satisfy rules (contriiction) and (weakening), for instance 1<' 1 U®U < I [U,X']x(U,X'J we need U's with some kind of structure. Thus the proposal at the moment is to take C with free comonoids, having 'ree comonoids means that there exists a functor F1 : C -m'Common C, which s left-adjoint to the forgetful functor U1 : Common C -* C. In other words, here is an adjunction <P1. U1, 'q, c> : C -4 Mon C, which we write simply as-H U1. The adjunction says that every map on C of the form, X-1----->U (V. !y, 6y). Irresponds by a natural isomorphism, toa comonoid homomorphism7of the form Lineales 123 (X*, 11x, 6x) ~ (V. i~y,~y) We write Õfor the composite functor U • F: C -9 C, and denote by' (( ),~,-q, i-il the corresponding monad in C. Definition 15 The endofunctor F : MLC -* Mi.C takes an object (U i~-)~) ofMLC to the object (LL ~~-~--XD,where intuitively uFa (xi, X2, ..., X~)means that u c~mnbe shared out betweenuj, U2, as many times as necessary so that uiaxi and U2c1x2 and.. . and u~cxx~. Butthis definitionofF has tobe shown to workand this is work in progress. 6. Further work Apart from making sure that the definition of the modality << >~ works properly, which seems to be clear from previous work on Hopf Algebras by Sweedler and others, it seems that the main work that remains to be done is to get things at the right level of generality. The oneadopted here seemsclearly inadequate, as one would like to <<change basis '> on doing the ccnstruction ofMLC, i. e. one would like to have constructionsMLC, with different L's. It is worth mentioning that there is some joint work in progress with Carolyn -Brown from LFCS, Edinburgh connecting the quantales n-iodels for Linear Logic arising from Petri Nets to the dialectica-like ones proposed in Brown/Gurr, Lics'90, see [H&dP] for the extension that allows Petri Nets with multiplicities >2. References [Bar] M. Barr. Autonomous Categories, LNM 752, Springer-Verlag, 1979. [B&G] C. Brown and D. Gurr. A Categorical LinearFramework for Petri Nets, LICS'90. [CWM] S. Mac Lane. Categories for the WorkingMathematician, Springer-Verlag, '71. [Dun] M. Dunn. Handbook of Philosophical Logic. [H&dPl M. Hyland and V. C. V. de Paiva. ACategory ofMultirelations, preprint June '90. [Law] F. W. Lawvere. Metric Spaces, Generalized Logic, and Closed Categories, Rendiconti del Seminario Matematico e Fisico di Milano, 43 (1973). [Laf] Y. Lafont. From Linear Algebra to Linear Logic,manuscript, November '88. [dePl V. C. V. de Paiva. ADialectica-like Model of Linear Logic, LNCS 389, [Wins] C. Winskel. A Category of Labelled Petri Nets andCompositional ProofSystem, LICS' 88. [Stri T.Streicher.Adding modalities to CAMIEic personal communication at PSSL,March '90. [Swe] M. Sweedler. Hopf Algebras, W. A. Benjamin, Inc., '69.