POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS. Contents. Introduction

Transcription

1 Theory and Applications o Categories, Vol. 18, No. 2, 2007, pp Contents POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS J.R.B. COCKETT AND R.A.G. SEELY Abstract. Motivated by an analysis o Abramsky-Jagadeesan games, the paper considers a categorical semantics or a polarized notion o two-player games, a semantics which has close connections with the logic o (inite cartesian) sums and products, as well as with the multiplicative structure o linear logic. In each case, the structure is polarized, in the sense that it will be modelled by two categories, one or each o two polarities, with a module structure connecting them. These are studied in considerable detail, and a comparison is made with a dierent notion o polarization due to Olivier Laurent: there is an adjoint connection between the two notions. 1 Basic polarized games 8 2 Basic polarized game logic 12 3 Polarized categories 16 4 The logic o polarized cut and its semantics 24 5 Additive types or polarized polycategories 37 6 Linear polarized categories 55 7 Multiplicative and additive structure on AJ games 80 8 Depolarization 84 9 Exponential structure Laurent polarized games Concluding remarks 99 Introduction The idea o developing an algebraic and proo theoretic approach to game theory has a certain level o irony, since games, viewed as combinatorial structures, are regarded as being endowed with suicient worldliness that they pass muster as a respectable semantics. The temptation to reinvent them as a type theory and thereby turn this notion o semantics on its head was irresistible. Research partially supported by NSERC, Canada. Diagrams in this paper were produced with the help o the XY-pic macros o K. Rose and R. Moore, the diagxy macros o M. Barr, and TEXCAD by G. de Montmollin. Received by the editors and, in revised orm, Transmitted by Martin Hyland. Published on Mathematics Subject Classiication: 18D10,18C50,03F52,68Q55,91A05,94A05. Key words and phrases: polarized categories, polarized linear logic, game semantics, theory o communication. c J.R.B. Cockett and R.A.G. Seely, Permission to copy or private use granted. POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 5 The realization that the logic o (polarized) games is a just a subtle modiication o the logic o products and sums [CS00] suggested to us that there was a rather dierent approach to understanding these games. Considering the logical complexities o the commuting conversions o the logic o products and coproducts, ΣΠ [CS00], and the connections o that system with games, it is not unnatural to ask i one can avoid the conversions by introducing some extra type constraints on the type theory, and i the resulting system has a game-theoretic interpretation. The answer is in act more arreaching even than the example o ΣΠ might lead one to expect. The key lay in providing a categorical semantics or these subtly changed sums and products. This meant that we had to understand the categorical meaning o polarization and the related notion o ocus. O course, once one puts the question in these terms, the answer inevitably is staring one in the ace. The logic o a two player game cries out to be interpreted as a module between two categories. The problem then is to transport standard categorical notions into this polarized world. Central to this was the idea o an inner adjoint which has the universal properties o an adjoint but in a polarized sense. It remained however to ind a voice or this way o telling the story o polarized games amidst the altogether more practical uses o game theory and a community very ocused (and rightly so) on applications o games. This paper has had a long period o gestation, and many o the ideas underlying the story we wished to tell were just beneath the surace in the community anyway, so it was not surprising that as we began to talk openly about our perspective on these games [C00, C02a, C02b], Olivier Laurent published his work on polarized linear logic [L02] 1. Laurent s view o polarization, while being very similar to ours, at the same time was also subtly dierent. His view o polarization was heavily inluenced by Girard s view o and grouping o the connectives o linear logic. Consequently his work struck a amiliar cord with many linear logicians. Furthermore, Laurent used a Hyland-Ong style game theoretic models to provide a semantics. Inevitably, our view o the polarization o the connectives was rather dierent. We had taken as our starting point the games used by Abramsky and Jagadeesan [AJ92] and this had lead us to a rather dierent organization o the same basic material. At the end o the paper we explain the relationship between the two approaches. The main dierence is two-old: we emphasise dierent operators, and we include operators not included in Laurent s presentation. Almost all these operators may be seen in the simple initary game model that serves as our motivation in the irst section. This perspective makes some important aspects o these game models explicit, which were implicit in previous treatments, such as ocalization and the subtly dierent notions o sums and products possible in the polarized setting. The latter can be interpreted as dierent communication strategies which we discuss in sections 4, 5. For example, we describe a depolarization process which can take a polarized model and produce a -autonomous category, i.e. a non-polarized model o (multiplicative) linear logic. The navigation o the polarized additive connectives and their role in depolarization [L02]. 1 Laurent has published several variants o his polarized logic; or deiniteness our comments reer to 4

2 6 J.R.B. COCKETT AND R.A.G. SEELY is suiciently complex that, without a careul treatment o these connectives, it is not easy to see what properties are required o the polarized model to ensure the depolarization has multiplicatives. The choices made, or example, by Laurent do not support depolarization with multiplicatives. On the other hand, the choices implicit in Abramsky and Jagadeesan are precisely suicient to provide a depolarization with multiplicatives. However, they are not suicient to deliver additives, which requires a dierent additive structure, as we shall discuss in section 8.2. Furthermore, we note that our notion o polarization is compatible with the co-kleisli construction in the presence o the exponentials! and? in act, given a polarized game category with a suitable notion o! and?, there is a polarized co-kleisli construction which lits the semantics to include these exponentials. Such a construction doesn t type in the Laurent setting, and cannot work in such a simple ashion. Lest anyone think otherwise, we should make clear that we do not take the view that one approach is superior to the other. There may be many notions o polarization, each with its own virtues and special properties, and we hope adherence to one will not preclude readers rom the delights o others. Laurent polarization provides a series o categorical doctrines which are parallel to ours. In act, they are linked to our doctrines by adjunctions which use the amily construction to reely add non-polarized additives. The publication o Laurent s work did cause us to wonder again whether there was suicient let in the story we wished to tell. Laurent s work had, or example, provided a very compact (one sided) sequent presentation or games. We had elt that our sequent presentation was a highlight indeed a novelty o our work. But although we can no longer claim originality or providing a sequent logic or these polarized games, we do claim our systems have some interesting eatures. One dubious distinction is that our systems have many more rules! However there are some good reasons or this. We take a very basic approach to these logics, making sure that they correspond transparently to their categorical semantics. However, this is not the real source o their size; rather, it is our continued insistence that these systems need have neither negation nor a commutative multiplicative structure. Thus the calculi we consider are more general those presented in Laurent s work; but more important is that ours are very modular (eatures are added only as needed). We think that the real gain is in the explicit nature o the resulting logic. The story o this game theory has been told many times, oten with the intent o getting the reader to the applications in the semantics o programming as ast as possible [H97, A97]. In this context it has become usual to regard games as being combinatorial structures and thus to be imbued with suicient concreteness to be passable as a semantics. This is not the story we wish to tell here: we take (in common with Laurent) a very proo theoretic approach and when we talk o semantics we are thinking o the categorical models o the proo theory which have as little claim to concreteness as the proo theory itsel. To be sure, we regard it as remarkable and ortuitous that the initial models have a concrete combinatorial description. However, our primary interest in them stems rom the act that they are the result o general constructions and that these constructions allow movement, at a general level, rom one categorical doctrine to another. We think that the view that polarized games have a natural categorical semantics in modules is original to our approach to this subject. Furthermore, the use o a new notion o adjunction ( inner adjoints ) to characterize polarized sums and products, which we POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 7 regard as a starting point or understanding the categorical semantics o polarized games, is also original to our particular way o telling the story. The paper is structured as ollows: in Part I, we set out the basic situation or polarized categories, starting with the basic game model. This structure captures the essence o the notions o polarity and o polarized sums and products. In act, Part I is a sel-contained entity, giving the key ideas o the paper. In Part II we extend this structure; in terms o the communication interpretation o games, we include communication along multiple input and output channels, arriving at the notion o a polarized polycategory. Representing this polystructure with appropriate tensors and pars is done in Part III on representability. Finally, in Part IV, we extend the theory, in particular describing depolarization, the polarized exponentials, and inally the connection with Olivier Laurent s approach to polarity. The reader amiliar with his work ought to reer to Table 17 irst, to get at least an idea o how our notions (and notation) compare with his.

3 8 J.R.B. COCKETT AND R.A.G. SEELY Part I The basic game situation 1. Basic polarized games To begin with we shall present a type system which we claim accounts or the basic structure o 2 player input output games, o the sort studied (in the context o semantics o linear logic) by Abramsky and Jagadeesan [A97]. We consider this as an example o a general process; we shall probe this special case as an illustration, but do not regard it as exhausting the techniques or ideas behind this paper. For example, although we do not consider Conway games in this paper, these appear to be susceptible to a similar treatment albeit with rather dierent type theory. 2 We shall start with a simple type theory or games; however it is not suicient to handle game constructors such as tensor and par. In later sections we shall show how those may be handled by a richer system, which may be more easily understood ater the simpler system has been presented. In addition, we shall present a categorical semantics or these type theories in terms o polarized sums and products. The games we wish to abstract have two players: O the opponent and P the player, each o which has associated moves. When the morphisms between these games are viewed as processes, it is natural to think o the moves as messages which are being passed between processes. It is then usual to classiy these messages rom a system centric perspective: those which originate rom the environment and those which are generated by the process or system itsel. In the codomain o a morphism it is possible to identiy the system messages with player moves and environment messages with the opponent moves. However, in the domain these roles are completely reversed: system message are identiied with opponent moves and environment messages with player moves. An important characteristic o a game is whether the opponent or player starts, as this determines the direction o the initial message. Since initially we shall not consider type constructors like internal hom we cannot ollow the more usual approach o coding a morphism up as a strategy or a single game (o type A B). Instead we have to explicitly deine the morphisms between our games. To acilitate this, in the next section we shall think o games as types and morphisms between games as proos, derivations, or terms, in a manner amiliar rom type theory. The act that there are opponent and player games necessitates that the type theory has opponent and player sequents which accommodate the dierent sorts o games which are available. In addition our basic type theory will have two constructions which allow us to build games as trees whose paths consist o alternating sequences o O-moves and P-moves. Given any inite amily {X i } o O-games, we can construct a P-game X i and dually given any inite amily {Y j } j J o P-games, we can construct an O-game j J Y j. To allow a connection between O-sequents and P-sequents we shall need mixed or cross sequents which operate between O-games and P-games. To illustrate the structure we have in mind, we shall start with a variant 3 o a well- 2 At times we shall reer to combinatorial games ; in this paper, by that phrase we shall mean Abramsky-Jagadeesan style games, not Conway games. 3 We shall use the abbreviation AJ games to reer to our initary variant, using Abramsky- POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 9 known model, viz. Abramsky-Jagadeesan games [A97] which actually gives the initial categorical model o the basic type theory we shall introduce. We start by explaining this model rom the present point o view to motivate the abstractions we shall make with our type theories. In case the reader gets the wrong impression, however, we should remark here that the ininitary games, with strategies (and winning strategies), may be presented as a model o the ollowing ramework as well, although we think the initary variant gives a clearer model, and has the additional virtue o being a ree model o the basic logic. The syntax we present below has no explicit reerence to strategies, however. The appropriate level o categorical generalization o strategies is still unclear to us, and will have to await a sequel. An idea o a possible approach may be ound in the glueing example, Polarized (inite) AJ games. Here a game may be regarded as a inite labeled bipartite tree: the nodes are partitioned into player states and opponent states and a labeled edge is required to start in a dierent partition rom where it ends. When the root is a player node we shall call the game a player game and similarly i the root is an opponent node we shall call it a opponent game. We shall use several notations or these games. A player game is denoted P = {a i : O i i I} = a i : O i where each O i is an opponent game. Moreover, supposing I = {1, 2,..., n}, we could represent this by the ollowing graph. An opponent game is denoted a1 a2 an... O 1 O 2 O n O = (b j : P j j J) = j J b j : P j where each P j is a player game. Again, this might be represented as this graph. b1 b2 bn... P 1 P 2 P n Jagadeesan games or the ininitary games o [AJ92, A97], or or the minor variants that appear in other papers by one or both o those authors.

4 10 J.R.B. COCKETT AND R.A.G. SEELY The binary versions o the basic operations are O O, which takes two opponent games and produces a player game, and P P, which takes two player games and produces an opponent game. The atomic games are given when the index sets are empty. We shall denote these by 0 = = { } and 1 = = ( ). Graphically, these are just leaves on a tree. Given a game G there is a dual game G which is obtained by swapping products or sums and overlining the component indicators (where we assume that double overlining is the identity). Thus, we have: P = a i: O i = a i: O i = (a i : O i i I) Q = j J b j: P j = j J b j: P j = {b j : P j j J} Our basic game type theory will abstract just this basic structure, but these games carry some additional structure which we will present in a later section, and which motivates the multiplicative extension o the basic type theory Maps and strategies. The usual way to speciy maps between these games is via strategies and counter-strategies. However we shall adopt a somewhat dierent approach by directly describing the morphisms between games. Strategies can then be recovered as morphisms rom the inal game 1 (and counter-strategies as morphisms to the initial game 0): using the closed structure which we introduce later we can recover the usual deinition o the morphisms (see Proposition 7.2.2). [Opponent maps: ] b 1 h 1 : O (b 1 : P 1,..., b m : P m ) b m h m where each h i : O P i is a mixed map. We shall occasionally use the in-line notation (b i : h i ) : O (b i : P i i I). Note that the displayed notation has the advantage o not needing subscripts, since the tokens may play that role themselves. [Mixed maps: ] These are either o the orm ak g: O {a 1 : O 1,..., a n : O n } where k {1,..., n}, and g: O O k is an opponent map, or bk : (b 1 : P 1,..., b n : P n ) P where k {1,..., n}, and : P k P is a player map. When the subscript is not necessary (being speciied by the token itsel) we may drop it. [Player maps: ] a 1 h 1 : {a 1: O 1,..., a m : O m } P a m h m where each h j : O j P is a mixed map. We shall occasionally use the in-line notation {a j : h j } j J : {a j : O j j J} P. POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS Remark. Note that this notation gives in eect a notational comparison between CCS and the π-calculus on one side, and the categorical notions o product and coproduct on the other. This is intended, and relects a basic intuition behind this work. This is made even more explicit in the thesis o Craig Pastro [P03] Example. Here is a map between two opponent games: a b ( a ) c () b : c g () d g h e Here are our mixed maps between these given games: { 1. c } b () a d b () ( 3. c ) b {} a d b {} b () a b : c b {} d 1.3. Compositions. Next we deine our compositions, via rewriting: [Opponent opponent composition ] b 1 h 1 b 1 g ; h 1 g ; = b m h m b m g ; h m a b c d [Opponent mixed composition ] b 1 h 1 ; b k = h k ; and g ; a g = a (g ; g) b m h m [Mixed player composition ] ak g ; a 1 h 1 a m h m [Player player composition ] a 1 h 1 a m h m = g ; h k and b ; = b ( ; ) ; = a 1 h 1 ; a m h m ;

5 12 J.R.B. COCKETT AND R.A.G. SEELY POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS Example. Here is a reduction o an opponent map composed with a mixed map. a b c d g h e ( a ) c () b ; ( g ) w () b g () h v () v w ( = g ) w () g () ; h v () = () ; w () = w (() ; ()) = w () Table 1: Basic polarized additives A p A atomic identities A o A { } X i o Y X i p Y cotuple X o Y k X o Y injection i {X o Y i } X o Y i tuple X k p Y X i o Y projection where k I, I It is an easy inductive argument to show that this is a conluent and terminating rewriting which eliminates the composition (as we shall shortly see this is a cut-elimination procedure). Furthermore, these rewritings satisy the associative law in all the conigurations which are possible. (See [CS00] or proos or a similar system in act those proos carry over to the present context, and even become simpler since the permuting conversions o [CS00] are absent in the present context.) Lemma. (i) The above rewriting on maps terminates. (ii) The above rewriting on maps is conluent. (iii) The associative law is satisied by all composible triples. To establish categorical structure or games and morphisms, we must exhibit the appropriate identity maps. Given a player object P = {a i : O i i I} we deine the identity map 1 P = { a i 1 Oi } ; given O = (b i : P i i I) we deine its identity map 1 O = ( b i 1 Pi ). We then have: Lemma. In any possible composition with an identity, the identity acts as a neutral element with respect to that composition. As will be seen in section 3, this means that we have two categories, the player and the opponent category, linked by a module (see Deinition 3.0.1). In that section we give a complete characterization o the categorical models which in addition to being a module must possess polarized products and sums. 2. Basic polarized game logic Beore we look at categorical structures, we shall approach this via type theory, presenting the logic as a sequent calculus over a type theory. In eect, we are presenting the graph structure, together with a cut elimination process, which will motivate and justiy the categorical structure presented later. This basic game logic will be a bit peculiar since we shall need three kinds o sequents: Player sequents: These take the orm: where X and Y are player propositions. X p Y Opponent sequents: These are dual to the player sequents, they take the orm: V o W where V and W are opponent propositions. Cross sequents: These are sel-dual and have the orm: V o Y where V is an opponent proposition and Y is a player proposition. The valid inerences are generated rom the rules in Table 1, which are a graded version o ΣΠ [CS00]. Notice that the rules are symmetric: the symmetry is given by swapping the direction o the sequents while at the same time swapping player or opponent and or. This symmetry arises rom an underlying categorical duality. The logic has our cut rules (Table 2) which correspond to those permitted by the types. The irst two arise as cuts respectively in the player and opponent sequents. The last two are the two possible cuts on the cross sequent. These correspond categorically to the compositions (or rather the let and right actions) expected o a module.

6 14 J.R.B. COCKETT AND R.A.G. SEELY POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 15 Table 2: Basic cut rules X p Y Y p Z X p-cut o Y Y o Z o-cut X p Z X o Z X o Y Y p Z X o Y Y o Z cp-cut X o Z X o oc-cut p Z Table 4: Basic rewrites ; 1 = 1 ; = g ; a g = a (g ; g) b ; = b ( ; ) Table 3: Basic terms {h i } ; = {h i ; } g ; (h i ) = (g ; h i ) ak g ; {h i } = g ; h k (h i ) ; b k = h k ; 1 A :: A p A atomic identities 1 A :: A o A { } h i :: X i o Y {a i : h i } :: a i: X i p Y cotuple g:: X o Y k ak g:: X o a i: Y i injection where k I, I :: X Y g:: Y Z cut ; g:: X Z {h i :: X o Y i } (b i : h i ) :: X o b i: Y i tuple :: X k p Y bk :: b i: X i o Y projection where represents the appropriate type o sequent or each o the our cut rules 2.1. A term logic. In act more is true. As or ΣΠ we may assign terms to this logic (Tables 3 and 4): however, where ΣΠ needed commuting conversions this logic does not because the type system makes the conversions impossible. Essentially this means that it is possible to have combinatorial models or these game processes as there are no manipulations once cut has been eliminated. The terms and term rewrites are similar to the ones we listed or polarized AJ games. To reduce the overload strain on colons, we use :: to denote the term-type membership relation, so t:: U V will mean that t is a term o type U V, where U (say) may be o the orm a: X. Then we can assert that cut elimination steps preserve the equivalence on terms induced by these rewrites. We shall see some examples o these rewrites (in a more general context, and using proo circuits) in sections 5, 5.3 and 6.5; as they are in principle similar to those in [CS00], we shall leave urther examples and details to the reader. However, it is now a simple exercise to prove the ollowing Theorem. The basic game logic satisies cut elimination, and urthermore, the cut elimination process satisies the Church-Rosser property. Anticipating the deinitions o section 3, we can see that a categorical model or this logic must consist o two categories, the player category X p and the opponent category X o, and a module X: X o X p. (Such a module behaves much as one would expect rom the ring theory notion o a 2-sided module, but we shall soon make the notion more concrete, in section 3.) Furthermore, or each index set I we have unctors I : XI o X p and I : XI p X o with the ollowing natural correspondences: {X i Y } in X {X Y i } in X X i Y in X p X Y i in X o These correspondences have the ollowing consequences. Let 1 B and 1 A be respectively the unary game sum and game product. Then there are bijections: 1 Y X in X p Y Y X in X 1 X in X o which shows that 1 is let adjoint to 1 and that the module is generated by these unctors. Notice also that we have the correspondence: { 1 Y i X} {Y i X} Y i X which shows that X p has I-indexed coproducts o objects o the orm 1 B. This gives:

7 16 J.R.B. COCKETT AND R.A.G. SEELY Proposition. A model or the basic game logic is equivalent to an adjunction 1 1 : X o X p in which (I-indexed) coproducts o objects o the orm 1 Y exist in X p and (I-indexed) products o objects o the orm 1 A exist in X o. This means there are plenty o models since any adjunction between a category X o with coproducts and X p with products will automatically produce a model. Clearly any category with products and coproducts will be a model (using the identity adjunction). The proo theory, o course, allows us to generate ree models rom arbitrary modules. The initial model (generated rom the module between empty categories) is the polarized inite AJ games described earlier. This may be proved directly, but will be let here as an exercise, since we shall prove it later by another route when we reine our view o these games and can link it to the original approach to the subject used in [AJ92] which uses strategies and counter-strategies. We shall abstract the notion o model outlined above to develop the theory o polarized categories, and more speciically o polarized game categories, which is the correct domain or considering the semantics or (our sort o) polarized games Remark. One might wonder (as we did) whether a useul type theory may be based on cross sequents o the opposite type: X p o Y. There are philosophical reasons or rejecting these (as there may well be reasons or wanting them), but rom the present point o view, we shall merely point out that such a type theory blocks the inductive construction o identity derivations, such as X Y p X Y, and generally will have an unsatisactory categorical semantics (consider that coproducts cannot have injections due to typing conlicts, or example). 3. Polarized categories To arrive at a semantic doctrine or these basic polarized games, we shall need some o the theory o polarized categories. Although not our primary motivation, it also seems that this is a possible doctrine within which to develop a semantics or Girard s original notion o ludics [G01] Deinition. A polarized category X = X o, X p, X consists o a pair o categories X o, X p together with a module X: X o X p. A module X: X o X p is a prounctor X: X o X p, that is to say, a unctor X op o X p Sets. We can regard such a module as a span Obj(X o ) X Obj(X p ) in the category Sets, subject to the usual module closure condition: this may be regarded as a set o (ormal) arrows whose domain is an object o X o and whose codomain is an object o X p. These arrows must be closed under precomposition with arrows o X o and under postcomposition with arrows o X p, and must satisy the evident associativity and identity equations. We shall write module arrows with a small vertical hatch on the shat o the arrow: A B. Given a polarized category X o, X p, X, there is an obvious dual polarized category X o, X p, X op = X op p, Xop o, X op, where the dual X op is the same ormal set o arrows as X, but now regarded as having the opposite direction: B A. POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS Example. The ollowing example (and several sequels throughout the paper) may be particularly o interest to readers amiliar with double glueing. Suppose C is a category, with distinguished objects I and J, and a distinguished set K o morphisms I J. We shall deine a polarized category X = G(C, K) as ollows. An object o X o is a pair (R, X), or X an object o C and R C(I, X). A morphism (R, X) (R, X ) is given by X X in C so that r R r ; R. Dually, an object o X p is a pair (Y, S), Y an object o C, S C(Y, J); a morphism (Y, S) (Y S ) is given by Y Y so that s S g ; s h S. Finally, a module morphism (R, X) (Y, S) is given by X h Y in C so that or all r R, s S, r ; ; s K. It is easy to show that this is indeed a module. This example may be given using a slightly dierent language. For an object X o C, or morphisms C(I, X), g C(X, J), say that is orthogonal to g, X g, i ; g K. Clearly, or any X h Y, X h ; g i and only i ; h Y g. Such a notion o orthogonality is equivalent to the speciication o a distinguished set K; to get K rom, just set K = { ; g g}. Then the deinition o module maps becomes so that r ; Y s or all r, s, equivalently so that r X ; s or all r, s. Anticipating section 3.2, note that there are two constructions taking us between X o and X p : (R, X) = (X, R ), where R = {h: X J r X h, r R} and (Y, S) = (S, Y ), where S = {k: I Y k Y s, s S}. It is easy to show the ollowing natural bijections, establishing that these are adjoint, and moreover, they characterize the module structure. (R, X) h (Y, S) (R, X) (R, X) h (Y, S) h (Y, S) Deinition. A polarized unctor F = F o, F p, F : X o, X p, X X o, X p, X consists o two unctors F o : X o X o, F p: X p X p, and a module morphism F : X X, viz. F : x m y F o (x) bf (m) g F p (y) satisying F o (a) ; F (m) ; F p (b) = F (a ; m ; b) or x a x in X o and y g b y in X p Deinition. A polarized natural transormation α: F o, F p, F F o, F p, F consists o a pair α := α o, α p o natural transormations α o : F o F o, α p : F p F p making the ollowing commute or any module arrow m: A B. F o (A) αo(a) F o(a) bf (m) bf (m) F p (B) αp(b) F p(b)

8 18 J.R.B. COCKETT AND R.A.G. SEELY The collection o polarized categories, unctors, and natural transormations orms a 2-category which we shall call PolCat. Note that this 2-category o polarized categories is (equivalent to) the slice category Cat/2, where 2 is the 2-point lattice regarded as a category Remark. Although we have a 2-category PolCat, it will not be the case that all notions appropriate or the polarized setting will be the usual notions interpreted in PolCat. In the next section, we shall see a central example o this phenomenon: polarized sums and products are not the usual notions interpreted in PolCat, but will require a new universal property. Later, in Section 4.2, we shall see another important instance o this, when we come to interpret the notions o polarized polycategories and polarized modules again, the appropriate notions are not merely interpretations in an appropriate 2-category. Since the pure category theory is somewhat skewed by the polarized notions, keeping the games interpretation in mind is an excellent guide Inner and outer adjoints; polarized products and sums. In considering polarized structure, it turns out that a mixed notion (partially polarized, partially not) is o use. Consider how we ought to add polarized products and sums (especially with the example o AJ games in mind) Deinition. A polarized category X = X o, X p, X is said to have I-indexed polarized products (or a set I) i there is a unctor I : XI p X o (also denoted ) with the ollowing natural correspondences: { } X i Y i in X X (i)i I Y i in X o X is said to have I-indexed polarized sums i the dual polarized category X op has I-indexed polarized products. X is said to have all inite polarized sums and products i it has I- indexed sums and products or all inite sets I. Note that in this deinition, the polarized sums and products are selected, rather than given by a universal property alone. However, we shall see that they do satisy an appropriate universal property, once we have the right notion o adjunction to describe this situation. Although we shall not need extensions o this deinition, it is obvious that we can deine arbitrary (not necessarily inite) polarized sums and products, and indeed, polarized limits and colimits or more general diagrams, in a similar ashion. One act that strikes one immediately about this notion is that it is not polarized in the most natural sense: viz. this is not the natural notion o limit in the 2-category PolCat. To say a polarized category X has polarized products and sums is not to require that the diagonal (polarized) unctor have an adjoint in PolCat, but rather that it has a pair o mixed unctors ( mixed in the sense that they switch polarity) each o which sets up the expected bijection. This notion may be abstracted as ollows. POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS Deinition. Suppose F : X Y is a polarized unctor, and that G := G o, G p is a pair o unctors G o : Y p X o G p : Y o X p (note that G is not polarized). Then we say F has an inner adjoint G, or equivalently that G has an outer adjoint F, i there are natural bijections F o (X) Y in Ŷ Y F p (X ) in Ŷ X G o (Y ) in X o G p (Y ) X in X p It is now a simple matter to veriy that X has I-indexed polarized products and sums i I : X X I has an inner adjoint. It is worth noting that this notion o inner outer adjunction does not compose. Inner adjoints do have a universal property: Proposition. To say that a polarized unctor F : X Y has an inner adjoint is precisely to say that there are object unctions G o : Y p X o, G p : Y o X p, and natural amilies o module maps ɛ Y : F o G o (Y ) Y, η Y : Y F p G p (Y ), or all Y in Y o, Y in Y p, so that or any module maps g: F o (X) Y, : Y F p (X ), there are unique maps g : X G o (Y ) in X o, : G p (Y ) X in X p, making the ollowing diagrams commute. ηy F o (X) Y F p G p (Y ) g Fo(g ) Fp( ) F o G o (Y ) Y ɛ Y F p (X ) We can express this in a dierent manner. Suppose F : X Y is an ordinary unctor; we can deine modules F : X Y and F : Y X as ollows: X Y in F is a triplet X, F (X) Y, Y and Y X in F is a triplet Y, Y F (X), X. Then i F is a polarized unctor, maps F o (X) Y (or X in X o and Y in Y p ) are maps o the composite module F o Ŷ. Likewise maps Y F p (X ) (or Y in Y o, X in X p ) are maps o the composite module Ŷ F p. With this language, we can state the deining property o an inner adjoint as ollows Proposition. A polarized unctor F : X Y has an inner adjoint i and only i there are module equivalences F o Ŷ = G o and Ŷ F p = Gp, or some unctors G o : Y p X o and G p : Y o X p. We are now in a position to state the obvious corollary that inner (and outer) adjoints are unique up to unique isomorphisms, as with ordinary adjoints Corollary. Suppose a polarized unctor F : X Y has inner adjoints given by (G o, G p, ( ), ( ) ) and (G o, G p, ( ), ( ) ). Then G o = G o and G p = G p are natural equivalences satisying the obvious coherence conditions. On objects, these equivalences are given by unique isomorphisms. The proo is straightorward, and is let to the reader.

9 20 J.R.B. COCKETT AND R.A.G. SEELY 3.2. Modules given by adjunction. We can now return to consider polarized coproducts and products. First, note that as these are given by an inner adjoint, they are unique up to a unique isomorphism. Unary polarized coproducts and products play a special role. First, consider the identity (polarized) unctor on a polarized category 1 X : X X; to say that it has an inner adjoint is precisely to say that the module X is given by an (ordinary) adjunction. For suppose 1 X has an inner adjoint, given by ( ) : X p X o and ( ) : X o X p. Then we have the ollowing natural bijections. Q P in X o Q P in X Q P in X p The converse is obvious. Furthermore, it is clear that this adjunction is given by 1 : X p X o and 1 : X o X p, where 1 is a singleton set. So ( ) = 1 and ( ) = 1 are switch polarity unctors, i.e. inner adjoint to the identity. (One is tempted to call these unctors Pierre and Gaston, or i we think o polarized categories as describing games, these correspond to moves o the sort apres vous, Gaston.) Then the adjunction X o 1 1 generates the module structure; it also shows the connection between polarized and nonpolarized sums and products Lemma. A polarized category has inite (I-indexed) polarized sums and products i and only i there is an adjunction ( ) ( ) : X o X p in which (I-indexed) coproducts o objects o the orm Q exist in X p and (I-indexed) products o objects o the orm P exist in X o. X p As we saw with basic game types, the bijections { 1 Q i P } in X p {Q i P } in X Q i P in X p show that we have ordinary (non-polarized) sums (and products) o objects given by singleton (polarized) sums (and products). Notice that this is the polarized categorical restatement o Proposition In particular it allows us to conclude that a polarized category with polarized products and coproducts is precisely a model or our basic game logic Deinition. We shall call a polarized category which is generated by an adjoint in this ashion an inner polarized category. This means an inner polarized category has an inner adjoint to its identity unction. For inner polarized categories we may construct polarized products and coproducts rom ordinary products and coproducts: POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS Lemma. An inner polarized category which has products in X o and coproducts in X p has polarized products and coproducts. The polarized products are constructed as I P i := I P i, where is (ordinary) product in X o, and dually or polarized sums Example. We already know the polarized category G(C, K) o Example is generated by an adjunction; in addition it has polarized sums (respectively products) i C has ordinary sums (respectively products). Let (R i, X i ) = ( i X i, {h: i X i J r i ; b i h r i R i }) Then i {(R i, X i ) (Y, S)} (R {i}i i, X i ) (Y, S) In addition, G(C, K) o also has ordinary sums and products i C does. (R i, X i ) = ( i (R i ; b i ), i X i) (R i, X i ) = ( R 1,..., R n, i X i) Polarized products are handled dually, and G(C, K) p has ordinary sums and products deined dually. I C is distributive, so are G(C, K) o and G(C, K) p The 2-category o polarized games. We now wish to briely consider the 2- category o polarized categories with inite polarized products and coproducts. As we think o an object in this category as a model or the basic polarized game logic we shall call the 2-category PolGam and reer to the objects as polarized game categories. We start by describing the unctors o this 2-category: Deinition. Suppose X, X are polarized categories with polarized sums. A polarized unctor F : X X preserves polarized sums i F p preserves and F preserves cotupling and injections. Explicitly, F p ( I A i) = I F o(a i ) and the ollowing diagrams commute, or i : A i B (i I), and : A A k (k I). I F o(a i ) bk(fo()) F b (i) I F o (A) = F p (B) bf (bk()) Fp( i I ) F p ( I A i) F preserving polarized products is deined dually. Then polarized categories with inite polarized sums and products, polarized unctors that preserve polarized sums and products, and polarized natural transormations orm the 2-category PolGam.

10 22 J.R.B. COCKETT AND R.A.G. SEELY Proposition. There is a orgetul 2-unctor PolGam U PolCat which has a let 2-adjoint PolCat U Gam PolGam which constructs the ree polarized game category generated by a polarized category. Proo. We shall sketch the construction o Gam(X). Gam(X) o, Gam(X) p and Gam(X) are deined inductively (this is essentially just the construction o the ree basic game types and terms generated by X): Ob(Gam(X) o ) = Ob(X o ) { I P i P i Gam(X) p, i I, I a inite set} Ob(Gam(X) p ) = Ob(X p ) { I Q i Q i Gam(X) o, i I, I a inite set} Ar(Gam(X) o ) = Ar(X o ) {( i ) I : Q I P i i : Q P i Gam(X), i I, I a inite set} Ar(Gam(X) p ) = Ar(X p ) { i I : I Q i P i : Q i P Gam(X), i I, I a inite set} Gam(X) = X {b k (): Q I Q i : Q Q k Ar(Gam(X) o ), k I, I a inite set} {p k (): I P i P : P k P Ar(Gam(X) p ), k I, I a inite set} where we take the arrows mod the equivalence relation generated by the eight conversions o the basic game type theory. From this description o Gam, the unit η o the adjunction is clear and canonical (it is the evident inclusion). Given any polarized unctor F : X U(X ) we construct F : Gam(X) X, a polarized unctor that preserves polarized sums and products, deined inductively by sending the constructed or in Gam(X) to the selected polarized sum or product in X. Likewise, given a polarized natural transormation α: F F, we may construct a polarized natural transormation α : F F in the same way. It is straightorward to show that this is indeed a 2- adjunction. It is interesting to note that one eect o the game construction is to produce a module which is generated by an adjoint. Indeed i the module has no cross maps then one sideeect o the construction is thereore to produce an adjunction between the two categories. In act, i we restrict the construction to unary polarized products and coproducts the eect is to produce a walking adjunction [SS86]. And so this gives a game theoretic view o an old construction Sotness. In [J95] Joyal describes a property sotness which characterizes the structure o limits and colimits in ree bicompletions o categories. A simpliied version o this was presented in [CS00], dealing with the ree inite product and sum completion. A simple variant o this property also applies to the polarized context Deinition. A polarized game category X is sot i we have the ollowing coproduct (in Sets). X( I X i, J Y j) = X p (X i, J Y j) + X o ( I X i, Y j ) j J POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 23 It is inormative to compare this deinition with that in [CS00], keeping in mind that certain conigurations are ruled out by the typing; it will then be noticed that a pushout in the cartesian case must be replaced by a coproduct in the polarized case Deinition. Given a polarized game category X, an opponent object A X o is atomic i X(A, J X j) = j J X o (A, X j ) and X o ( K X k, A) = and dually a player object B X p is atomic i X( I X i, B) = X p (X i, A) and X p (B, K X k) = Deinition. A polarized category X is Whitman i every object o X is isomorphic to a game (i.e. to a or a ) o atomic objects, and i X is sot. Then one may characterize the image o the (aithul, though not ull) 2-unctor Gam in PolGam as ollows Theorem. A polarized game category X is isomorphic to Gam(Y) or a polarized category Y i and only i X is Whitman. Proo. The unit o the adjunction maps a polarized category into the game category constructed rom it. It is easy to see rom the construction that the atoms o the game category are exactly the objects o the polarized category and that this game category is sot (by construction). Conversely, given a game category which is Whitman we claim it is equivalent to the game category on its atoms. The proo o this ollows the steps in [CS00], which is to say it is a structural induction on the types using the sotness to show that the maps rom polarized products and to polarized coproducts are the same as those rom the ree types.

11 24 J.R.B. COCKETT AND R.A.G. SEELY Part II Multiple channels 4. The logic o polarized cut and its semantics The simple game logic presented so ar does not permit a process (a morphism or proo) to communicate along multiple input and output channels. Without this ability this game logic will be rather inexpressive. In this section we discuss how to add channels to the basic game logic. In order to do this the basic logic has to support dierent kinds o contexts within which a process can listen and send. We shall introduce a (possibly noncommutative) extension to the basic type theory, and we shall show that it is modeled by the AJ combinatorial games. It is worth noting that the exchange rule or the tensors and pars that we shall introduce may be added. Although it is usual or game models to be viewed as commutative, we regard this is an unnecessary restriction. That this is a non-trivial logic ollows immediately rom the act that it is modeled by MALL (multiplicative linear logic with additives). However, the point o the logic is that it aords more separation than MALL so that the categorical coherence problems are much simpler. This is the result o making polarity an explicit part o the system, as we have already seen with ΣΠ-logic and the basic game logic. In particular coherence or the proo theory (that is the underlying ree categories) or game types is a good deal simpler than or the additives in linear logic precisely because all the commuting conversions due to the additives have been removed by the type constraints. We shall discuss this ater completing the description o the logic The logic o polarized cuts. In our extended game logic there are, as beore, three types o sequent; however, this time the sequents have contexts, the orms o which need some preliminary explanation. Player sequents: These take the orm: Γ / X \ Γ p where Γ, Γ are O-phrases, that is lists (or possibly bags) o opponent propositions, X is a player proposition, and is a P-phrase, that is a list (or possibly a bag) o player propositions. Opponent sequents: These are dual to the player sequents, they take the orm: Γ o / Y \ where Γ is an O-phrase, Y is a opponent proposition, and, are P-phrases. Cross sequents: These are sel-dual and have the orm: Γ o where Γ is an O-phrase and is a P-phrase. POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 25 The point to notice here is that in the P and O sequents, we allow a context o the opposite type, and since we wish our logic to allow or non-commutative operators (tensor and par), we allow that context to surround the active ormula. I we were to assume commutative tensor and par, that would not be necessary, and one-sided contexts would suice. This would also reduce the multiplicity o rules below, since we would no longer have to distinguish so many let-right cases. In all sequents, the let side is (primarily) O material, the right is (primarily) P material, with the proviso that pure O sequents allow an additional O proposition on the right, pure P sequents allow an additional P proposition on the let. These may be regarded as in ocus, or active. A mixed or cross sequent has no active proposition. In the ollowing subsections, we describe the inerence rules or this game logic. We shall use upper case letters at the beginning o the alphabet to denote atomic propositions, upper case at end o the alphabet to denote arbitrary propositions, and Greek upper case to denote lists o propositions. We proceed in several steps: the irst indicates the basic context rules, including the cut rules, ollowed by the categorical (or rather polycategorical) semantics or cut. Then we give the rules or the basic ( polarized additive ) game constructors. The remaining constructors, irst the multiplicatives, then negation, and inally the exponentials, will ollow in the next section on representing structure. The logic has twenty our cut rules although this may seem a lot, there is a simple underlying principle, which is we permit all possible well-typed, planar variants o the cut rule. As a single scheme, this would look something like this: Γ x X ΦXΦ y Ψ ΦΓΦ z Ψ where one o, Φ must be empty, and one o, Φ must be empty (this is the planarity condition). There are only six choices o the types o entailment x, y, z that are permitted by the typing, and with our alternatives or each, we end up with the twenty our variants. These are illustrated in Table 5, where we leave to the reader the task o implementing the planarity condition. The opp-cuts are given in two versions, each with two variants, because the cut is being made into one side o the context or the other a similar division is made or the dual opo-cuts. Notice that all these rules preserve the basic duality o this logic obtained by swapping the direction o the sequents, exchanging player or opponent and products or sums. Furthermore, we have represented this duality in the let column right column symmetry in the table. Recall also that the exchange rule can be assumed, in which case the phrases are to be regarded as bags o propositions Polarized polycategories. Corresponding to the logic o polarized cut above is its categorical proo theory which is the notion o a polarized polycategory. In addition polarized polycategories, like polycategories, have a term logic which consists o polarized circuits. The purpose o this section is to introduce these ideas. A polarized polycategory X consists o polycategories X o and X p as well as a polymodule X. Each polyarrow in X o is o the orm Γ o / Y \

12 26 J.R.B. COCKETT AND R.A.G. SEELY Γ 1 / P \ Γ 2 p, X, Φ / X \ Φ p Ψ Φ, Γ 1 / P \ Γ 2, Φ p, Ψ, p-cut Γ o / X \ Φ, X, Φ 1 / P \ Φ 2 p Ψ lopp-cut Φ, Γ, Φ 1 / P \ Φ 2 p, Ψ Γ o / X \ Φ 1 / P \ Φ 2, X, Φ p Ψ Φ 1 / P \ Φ 2, Γ, Φ p Ψ, Table 5: General cut rules ropp-cut Γ o p, X, Φ / X \ Φ p Ψ Φ, Γ, Φ o p, Ψ, cpc-cut Γ o / X \ Φ, X, Φ o Ψ 1 / Q \ Ψ 2 Φ, Γ, Φ o Ψ 1 / Q \ Ψ, o-cut Γ o 1 / Q \ 2, X, / X \ Φ p Ψ Γ, Φ o 1 / Q \ 2, Ψ, ropo-cut Γ o, X, 1 / Q \ 2 Φ / X \ p Ψ Φ, Γ o, Ψ, 1 / Q \ 2 lopo-cut Γ o / X \ Φ, X, Φ o p Ψ Φ, Γ, Φ o p, Ψ, occ-cut where in each rule where they appear, one o, Φ is empty, and one o, Φ is empty. having a sequence o objects Γ rom X o as its domain, and a sequence o objects all but one o which are rom X p as its codomain, with in addition one identiied ( active, or in ocus ) object rom X o :, rom X p, Y rom X o. This collection o arrows must contain an identity arrow Y o / Y \ or each object Y o X o. Dually, each polyarrow in X p is o the orm Γ / X \ Γ p having a sequence o objects all but one o which are rom X o as its domain, with in addition one identiied ( active, or in ocus ) object rom X p in the domain, and a sequence o objects rom X p as its codomain: Γ, Γ rom X o, X, rom X p. This collection o arrows must contain an identity arrow / X \ p X or each object X o X p. We shall usually omit the subscripts on arrows in X o, X p, when the context makes them unnecessary. Each polyarrow in the polymodule has the orm Γ having a sequence Γ o objects rom X o in the domain and a sequence o objects rom X p in the codomain. These arrows may be composed in twenty-our ways, essentially as given by the twentyour cut rules o game logic above. This may seem rather intimidating, but in essence the idea is quite simple: each o X o and X p allow composition much as ordinary polycategories do, but given the non-commutative nature o these sequents, there are minor variants caused by the placement o the active object in the sequents. In addition, X o acts on X on the let, and X p acts on X on the right, in evident ways. This amounts to all the well-typed planar variants o the ollowing generic composition, allowing or the various types o arrows. POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS 27 Γ X ΦXΦ g ΦΓΦ ; g Ψ where one o, Φ is empty, and one o, Φ is empty. (This condition is reerred to as the planarity condition.) In terms o circuits, this is even simpler; it is the usual circuit cut (just join two wires which bear the same label), with the understanding now that the joined wires are o the same type (player or opponent, solid or dotted ). There are standard unit and associativity conditions, analogous to those or ordinary polycategories. For simplicity, we illustrate these rules with generic versions. In these, we suppress the notation or which type o arrows are involved, where the composition (or cut) takes place, and so which type o composition or cut is involved; the reader is supposed to imagine all possible well-typed versions o these rules. Recall that our compositions or cuts are supposed to be planar; we represent that by the convention that in these rules, an expression Γ is to be understood as the trivial concatenation o a sequence and an empty sequence, the assumption being that one o, Γ is empty. (1)[idL] Γ 1 (2)[idR] Γ 1, A, Γ 2 (3)[assoc] = (4)[inter1] = Γ 1 Γ 1 1 Γ 1 Γ 2, A, Γ 3 = Γ 1 Ψ Γ 2, A, Γ 3 A ia A Γ 1 ; ia Γ 2, A, Γ 3 Γ 3 = A ia A Γ 1, A, Γ 2 Γ 1, A, Γ 2 Γ 2, A, Γ 3 1, A, 2 g 1, Γ 1, 2 g ; g ia ; Γ 3 3, B, 4 Γ 3 Γ 2, 3, B, 4, Γ 3 Φ 1, B, Φ 2 h Φ 3 Φ 1, 1, Γ 1, 2, Φ 2 ( ; g) ; h Γ2, 3, Φ 3, 4, Γ 3 1, A, 2 g 3, B, 4 Φ 1, B, Φ 2 h Φ 3 Γ 2, A, Γ 3 Φ 1, 1, A, 2, Φ 2 g ; h 3, Φ 3, 4 Φ 1, 1, Γ 1, 2, Φ 2 ; (g ; h) Γ 1 Γ 2, 3, Φ 3, 4, Γ 3 Γ 2, A, Γ 3 Φ 1, A, Φ 2, B, Φ 3 h Φ 4 2, B, 3 Φ 1, Γ 1, Φ 2, B, Φ 3 ; h Γ2, Φ 4, Γ 3 Φ 1, Γ 1, Φ 2, 1, Φ 3 g ; ( ; h) 1 g 2 Γ 2, Φ 4, 3 Γ 3 2, B, 3 Φ 1, A, Φ 2, B, Φ 3 h Φ 4 g ; h Γ 2, A, Γ 3 Φ 1, A, Φ 2, 1, Φ 3 2, Φ 4, 3 ; (g ; h) Φ 1, Γ 1, Φ 2, 1, Φ 3 2 Γ 2, Φ 4, 3 Γ 3