Cartesian effect categories are Freyd-categories

Cartesian effect categories are F reyd-categories Jean-Guillaume Dumas ∗ Dominique Duv al † Jean-Claude Reynaud ‡ June 12., 2009 Abstract Most often, in a categorical semantics for a programming language, th e substitution of terms is expressed by composition and finite pro duct s. How ever this do es n ot deal with th e order of eva luation of argumen ts, which ma y h ave ma jor consequences when there are side-effects. In this pap er Cartesian effect categories are introdu ced for solving this issue, and they are compared with strong monads, F reyd- categories and Haskell’s Arrows. It is prov ed that a Cartesian effect category is a F reyd- category where the premonoidal structure is provided by a kind of binary p rod u ct, called the sequential pro duct. The universal prop erty of the sequential pro du ct pro vides Cartesi an effect categories with a p ow erful tool for constructions and pro ofs. T o our kn o wledge, b oth effect categories and sequential pro ducts are new notions. Keywords. Categorica l logic, computational effects, monads, F re yd-categories, premonoidal cate- gories, Arrow s, sequ entia l pro duct, effect categories, Cartesian effect categories. 1 In tro duction A categor ical s e mantics for a prog ramming language usually ass o ciates an ob ject to each type, a morphism to e a ch term, and uses co mp os ition a nd finite pro ducts for dealing with the substitution of ter ms. This framework behaves v ery well in a simple equational setting, but it has to b e adapted as so on as there is so me kind of computational effects, for instance non-termination or state updating in an imp erative langua ge. Then ther e are t wo kinds o f ter ms: the g e neral ter ms may cause effects while the pur e terms are e ffect-free. F ollo wing (Moggi, 19 91), a general term ma y be seen as a pr o gr am that returns a value which is pur e. In this pap er we fo cus on the following se quentiality issue: the catego rical pro ducts do not deal with the order of ev aluation of the ar guments, although this or der may have ma jor consequences when there are s ide - effects. F or solving this sequentialit y issue, we introduce Cartesian effe ct c ate gories as a n alter native for Cartesia n categorie s. Other approa ches include str ong monads (Moggi, 1989), F reyd-categ ories (P ow er and Robinson, 1997) and Arrows (H ughes, 200 0). T hes e frameworks are quite similar from several p oints o f view (Heunen and Jacobs, 2006; Atk ey, 2008), while our framework is more pr ecise. A first dra ft for Ca rtesian effect ca tegories can be found in (Dumas et al., 2007), and a s imila r a pproach in (Duv al and Reynaud, 200 5). A category is called Ca rtesian if it has finite pro ducts, and a sub categor y C of a categ ory K is ca lled wide if it ha s the s ame ob jects as K . A F r eyd-c ate gory is a gener alization of a Cartesian catego r y that consists essentially in a category K with a wide subcatego r y C , suc h that C is Cartesian (hence C is symmetric monoidal) and K is symmetric pr emonoidal. A Cartesian effe ct c ate gory , a s defined in this paper, is more precise a nd more homogeneous than a F reyd-catego ry: like the symmetric mono idal structure o n C deriv es ∗ Laboratoire Jean Kuntzmann, Uni v ersit´ e de Grenoble, 38041 Grenoble, F rance – Jean-Guillaume.Dumas@imag.fr – h ttp://ljk.imag.fr/membres/Jean-Guillaume.Dumas † Laboratoire Jean Kun tzmann, Unive rsit´ e de Grenoble, 38041 Grenoble, F rance – Domi nique.Duv al@im ag.fr – h ttp://ljk.imag.fr/membres/Dominique.Duv al ‡ Malhivert, 38640 Claix, F r ance – Jean-Claude.Reynau d@imag.fr 1 from its pro duct, in a Cartesian effect catego ry the symmetric premono idal structur e on K derives from some kind of pro duct, called a se quential pr o duct , which extends the pro duct of C and g eneralizes the usual categoric al pro duct. In fact, there are tw o steps in our definitio n. First an effe ct c ate gory is defined, without men tioning any kind of pro duct: it is made of a category K with a wide sub catego ry C and with a r elation ⊳ called c onsist en cy b etw e e n morphisms. Then a Cartesian effe ct c ate gory is defined as an effect category with a binary pro duct o n C e xtended by a se quential pr o duct on K , which itself is defined thanks to a universal prop erty that generalizes the categor ical pro duct prop er ty and inv olves the co nsistency r elation. Like e very universal prop erty , this provides a p owerful to ol for constructio ns and pro ofs in a Ca rtesian e ffect categor y . Let us lo ok a t tw o basic ex a mples of effect ca tegories (tw o mor phisms in a category a re ca lled parallel if they share the sa me do ma in a nd the sa me co domain). The n on- termination e ffect inv olves par tial functions . As usual, tw o partial functions are called c onsist ent when they coincide on the intersection of their domains of definition. Thu s, on the one hand, tw o partial functions f and f ′ are consis ten t if a nd o nly if there is a tota l function v such that v is co nsistent b oth with f and with f ′ . On the o ther hand, let us say that t w o partial functions hav e the same effe ct if they have the sa me domain of definition. Then clear ly , tw o partia l functions have the sa me effect and ar e cons istent if and only if they are equal. In an imper ative progra mming language, there are side-e ffects due to the mo dification o f the state , since the functions in the sense of the pro g ramming language, in addition to hav e arguments and a return v alue, are allow ed to use the state and to mo dify it. A function is called pur e if it ne ither use nor mo dify the state, and the side-effects are due to the no n-pure functions. Let us say that a function f is c onsistent with a pure function v when b oth r eturn the sa me v alue when they are g iven the sa me arg ument s. Then tw o arbitr a ry functions are called c onsistent when they a re consistent with a common pure function, whic h means that bo th return the same v alue when they are g iven the s ame arg umen ts a nd that in addition this v alue does not depend on the state. It sho uld b e noted tha t this c onsistency relatio n is no t r eflexive. Ther efore, if tw o functions hav e the sa me e ffect and are consistent then they are equa l, but the conv erse is false . More generally , an effe ct c ate gory is a catego ry K with a wide subcatego ry C and with a consistency relation ⊳ betw een parallel morphisms, the first one in K and the s econd one in C , sa tis fying a form of compatibility with the compos itio n. The morphisms in C are called pur e and are denoted with . Two morphisms in K are ca lled c onsistent when there is a pure morphism v such that f ⊳ v a nd f ′ ⊳ v ; this is denoted f ⊳ ⊲ f ′ , and the prop erties of consistency are suc h that the relation ⊳⊲ extends ⊳ . Let 1 be a terminal o b ject in C , the effe ct of a morphism f is defined as the mo rphism E ( f ) = h i Y ◦ f wher e h i Y is the unique pure morphism h i Y : Y 1. It is assumed that the following c omplementarity prop erty holds, which mea ns that the consistency r elation is a kind of “up-to-effects” r e la tion: if t w o morphisms ha v e the same effect and a re co nsistent, then they ar e equal. This notion of consistency coincides with the usual one for partia l functions, but to o ur knowledge it is new in the gener al setting of computational effects. F or instance, we will see in section 2.6 that it is fairly different from the no tio n of having the same re sult that is defined in (Mogg i, 199 5) in the framework of ev aluation logic. Let us lo ok mor e closely at the complemen tarity pro p e rty (for s ome fixed domain and co domain). On the one hand, to ha ve the s ame effect is an e q uiv alence relation ≈ with one distinguishe d equiv alence class, the clas s of the morphisms without effect, which contains all the pure mor phisms. On the other hand, to b e consisten t is a symmetric r elation ⊳ ⊲ , with each maxima l clique made of a unique pure morphism and all the morphisms that are consistent with it. The co mplement arity pr o p erty a sserts that there is a t most one morphism in the in tersection of a given equiv alence class for ≈ and a given maxima l clique for ⊳⊲ . A binary pro duct on a category C provides a bifunctor × on C such that for all v 1 : X 1 → Y 1 and v 2 : X 2 → Y 2 , the morphism v 1 × v 2 : X 1 × X 2 → Y 1 × Y 2 is characterized by the following diagram, where the p i ’s a nd q i ’s are the pr o jections. This prop erty is symmetric in v 1 and v 2 . When C is the categor y of sets, this means that ( v 1 × v 2 )( x 1 , x 2 ) = h v 1 ( x 1 ) , v 2 ( x 2 ) i . 2 X 1 v 1 / / = Y 1 X 1 × X 2 v 1 × v 2 / / p 1 O O p 2   Y 1 × Y 2 q 1 O O q 2   X 2 v 2 / / Y 2 = A Cartesian effe ct c ate gory is defined as an effect ca tegory with a binary pro duct on C , extended b y t wo symmetric semi-pur e pr o ducts v ⋉ f and f ⋊ v wher e v is pur e. The left semi-pure pro duct v ⋉ f is characterized b y the following diagra m, which means that q 1 ◦ ( v ⋉ f ) ⊳ v ◦ p 1 and q 2 ◦ ( v ⋉ f ) = f ◦ p 2 (the right semi-pure pro duct is characterized by a s ymmetric diagra m). Y 1 v / / /o /o /o /o /o /o /o /o ⊳ Y 1 Y 1 × X 2 v ⋉ f / / p 1 O O O p 2   O Y 1 × Y 2 q 1 O O O q 2   O X 2 f / / Y 2 = This pr op erty mea ns that the effect of v ⋉ f is the effect of f , and that “up to effects” v ⋉ f looks like an or dinary binary pro duct. Then the le ft se qu en tial pr o duct of t wo a rbitrary morphisms f 1 and f 2 is eas ily obtained by co mpo sing tw o semi- pur e pro ducts: f 1 ⋉ f 2 = (id 1 ⋉ f 2 ) ◦ ( f 1 ⋊ id 2 ) where id 1 and id 2 denote the ident ities of Y 1 and X 2 , resp ectively . T his definition formalizes the notion of s e quentiality : “firs t f 1 , then f 2 ”. The right sequential pro duct is defined in a symmetric wa y . W e will chec k tha t the sequential pro duct extends the semi-pure pro duct, so that there is no ambiguity in using the same symbols ⋉ and ⋊ fo r b oth. This approa ch, to o ur knowledge, is co mpletely new. It can b e s umma r ized as follows: while the universal prop erty o f a binary pro duct consists in t wo equalities , the universal pr op erty o f a semi-pure pro duct consists in one equality a nd one consistency . F or instance, in the category of s ets with partial functions, v ⋉ f is the partial function suc h th at ( v ⋉ f )( x 1 , x 2 ) = h y 1 , y 2 i where y 1 = v ( x 1 ) and y 2 = f ( x 2 ) whenever f ( x 2 ) is defined, otherwise ( v ⋉ f )( x 1 , x 2 ) is not defined. When side-effects ar e due to the up da ting o f the state, v ⋉ f is such that for each state s , ( v ⋉ f )( s, x 1 , x 2 ) = h s 2 , y 1 , y 2 i where h s, y 1 i = v ( s, x 1 ) and h s 2 , y 2 i = f ( s, x 2 ). The prop er ties o f the sequential pro duct imply that a Cartesia n e ffect category is a F rey d-categor y . On the other hand, each stro ng monad defines a F reyd-catego r y (Po wer and Robinson, 1997). W e prov e that a F reyd-c ategory defined from a strong monad is a w eak Cartesian effect catego ry if and only if, ro ughly sp eaking: the strength of the monad is consistent with the identit y . Section 2 is dev oted to effect categories a nd section 3 to Cartesian effect categor ies. Then Car tesian effect categ o ries a re related to F r eyd-catego ries, Arr ows a nd strong monads in section 4. Several exa mples are consider ed in se c tio ns 2.5, 3.8 and 4.4. 2 Effect categories 2.1 Pure morphisms Definition 2.1 . A sub categ ory C of a c ategory K is wide if it ha s the same ob jects as K ; this is denoted C j K . Given C j K , a morphism of K is called pur e if it is in C ; then it is denoted with “ ”. An ob ject 1 is a pur e terminal ob ject in C j K if it is terminal in C , then fo r each o b ject X the unique pure morphism from X to 1 is denoted h i X : X 1. R emark 1 . Pure morphisms i n a Kle i sli category . Let C 0 be a categ ory (called the ba s e category) with a mona d ( M , µ, η ) (or simply M ) and let K M be the Kleisli category of M . Then K M has the same ob jects as C 0 and for all ob jects X and Y ther e is a bijection betw e en C 0 ( X, M Y ) a nd K M ( X, Y ). In this pap er, for each morphism f : X → Y in K M the corr esp onding morphism in C 0 is denoted [ f ] : X → M Y , and we 3 say that f stands for [ f ], and for ea ch morphism ϕ : X → M Y in C 0 the corres po nding morphis m in K M is denoted ] ϕ [: X → Y . So, ]([ f ])[= f f or every f in K M and [(] ϕ [)] = ϕ for every ϕ in C 0 with co doma in M Y for so me Y . Let J : C 0 → K M denote the functor asso ciated with M a nd let C M = J ( C 0 ). Then J is the identit y o n ob jects, so that C M is a wide subcateg ory o f K M . A pure morphism v : X Y in K M is a morphism v = J ( v 0 ) for some v 0 : X → Y in C 0 ; this means that [ v ] = η Y ◦ v 0 : X → M Y in C 0 . Ea ch ident ity id X in K M henceforth stands for [id X ] = η X and the c omp osition g ◦ f of f : X → Y and g : Y → Z stands for [ g ◦ f ] = [ g ] ∗ ◦ [ f ] where [ g ] ∗ = µ Z ◦ M [ g ]. It follo ws that when v : X Y and w : Y Z , then [ g ◦ v ] = [ g ] ◦ v 0 , [ w ◦ f ] = M w 0 ◦ [ f ] a nd [ w ◦ v ] = η Z ◦ w 0 ◦ v 0 . It should be noted that it doe s not make se ns e to say that a mor phism in C 0 is pure or not. Indeed, ea ch morphism ϕ : X → M Y in C 0 gives rise in K M bo th to a pur e morphism v = J ( ϕ ) : X → M Y and to a mor phism f =] ϕ [: X → Y , related by [ v ] = η M Y ◦ [ f ] in C 0 . C 0 X [ f ] / / M Y X [ v ] / / v 0 ( ( Q Q Q Q Q Q Q Q Q Q M Y = Y η Y O O = M 2 Y X ϕ / / [ J ( ϕ )] 6 6 m m m m m m m m M Y η M Y O O K M X f / / Y X v = J ( v 0 ) / / /o /o /o /o Y M Y X ] ϕ [ / / J ( ϕ ) 6 6 6v 6v 6v 6v 6v Y In addition, the functor J : C 0 → K M has a right adjoint, which means that for each ob ject X there is a n ob ject X † called the lifting of X , with a n isomo rphism K M ( X, Y ) ∼ = C 0 ( X, Y † ) na tural in X a nd Y . Le t us assume that the mono r e quir ement is satisfied by the mona d, which means that η X is a mono for every ob ject X , or equiv alen tly tha t the functor J is faithful, so that it defines an isomo r phism from C 0 to C M . 2.2 Effects In this section we define the effect of a mo rphism f as a kind of measure of ho w fa r f is fro m b eing pur e: pure morphisms are effect-fr ee and the effect o f v ◦ f , when v is pure, is the same as the effect of f . Definition 2.2. Let K be a ca tegory with a wide subcatego ry C and with a pure terminal ob ject 1 . The effe ct o f a morphism f : X → Y is the morphism E ( f ) = h i Y ◦ f : X → 1. W e denote f ≈ f ′ when f : X → Y and f ′ : X → Y ′ hav e the same effect: ∀ f : X → Y , ∀ f ′ : X → Y ′ , f ≈ f ′ ⇐ ⇒ h i Y ◦ f = h i Y ′ ◦ f ′ . A morphism f : X → Y is effe ct-fr e e if E ( f ) = E (id X ), which means that E ( f ) = h i X . The fo llowing prop er ties ar e easily derived fro m the definition. Prop ositi o n 1 . The same-effe ct r elatio n ≈ is an e quivalenc e re lation b etwe en morph isms with the same domain that satisfies: • Pur e morphisms ar e effe ct-fr e e. ∀ v : X Y , v ≈ id X . • Substitut ion. ∀ f : X → Y , ∀ g : Y → Z , ∀ g ′ : Y → Z ′ , g ≈ g ′ = ⇒ g ◦ f ≈ g ′ ◦ f . • Pur e wiping. ∀ f : X → Y , ∀ w : Y Z , w ◦ f ≈ f . R emark 2 . E ff ects in a Kleisli category . Within the s ame fra mework as in remark 1, let us a ssume that there is a terminal ob ject 1 in C 0 , or equiv a lently in C M . F or each ob ject X , the pur e morphism h i X : X 1 stands for [ h i X ] = η 1 ◦ h i X : X → M 1 in C 0 , and for each morphism f : X → Y in K M the effect E ( f ) o f f stands for [ h i Y ◦ f ] = M h i Y ◦ [ f ] : X → M 1 in C 0 . Let ≈ 0 denote the relation b etw een morphisms in C 0 defined by [ f ] ≈ 0 [ f ′ ] if and only if f ≈ f ′ . Then in C 0 : ∀ ϕ : X → M Y , ∀ ϕ ′ : X → M Y ′ , ϕ ≈ 0 ϕ ′ ⇐ ⇒ M h i Y ◦ ϕ = M h i Y ′ ◦ ϕ ′ . 4 2.3 Consistency Now we define a cons is tency relation b etw een tw o pa rallel morphisms . Definition 2.3. Let K b e a catego ry w ith a wide subca tegory C . A c onsistency relation ⊳ is a relation betw een par a llel mor phisms, the second one b eing pure, which satisfies : • Pur e r eflexivity . ∀ v : X Y , v ⊳ v . • Comp atibility with c omp osition . ∀ f : X → Y , ∀ g : Y → Z , ∀ u : Y Y ′ , ∀ v : X Y ′ , ∀ w : Y ′ Z , ( u ◦ f ⊳ v ) ∧ ( g ⊳ w ◦ u ) = ⇒ g ◦ f ⊳ w ◦ v . X f - - v ⊳ / / /o /o /o /o Y ′ w ⊳ / / /o /o /o /o Z Y g > > u O O O = ⇒ X g ◦ f 4 4 w ◦ v ⊳ / / /o /o /o /o /o /o /o /o /o /o Z Two par allel morphisms f and f ′ are called c onsistent when f ⊳ v ⊲ f ′ for some pure mo r phism v , this is denoted f ⊳ ⊲ f ′ . The fo llowing prop er ties ar e easily derived fro m the definition. Prop ositi o n 2. L et K b e a c ate gory with a wide sub c ate gory C and with a c onsistency r elation ⊳ . Then: • Pr eservation by c omp osition. ∀ f : X → Y , ∀ v : X Y , ∀ g : Y → Z , ∀ w : Y Z , ( f ⊳ v ) ∧ ( g ⊳ w ) = ⇒ g ◦ f ⊳ w ◦ v . X f ; ; v ⊳ / / /o /o /o /o Y g ; ; w ⊳ / / /o /o /o /o Z = ⇒ X g ◦ f 4 4 w ◦ v ⊳ / / /o /o /o /o /o /o /o /o /o /o Z • Pur e substitu tion. ∀ v : X Y , ∀ g : Y → Z , ∀ w : Y Z , g ⊳ w = ⇒ g ◦ v ⊳ w ◦ v . • Pur e r eplac ement . ∀ f : X → Y , ∀ v : X Y , ∀ w : Y Z , f ⊳ v = ⇒ w ◦ f ⊳ w ◦ v . Definition 2.4. An effe ct c ate gory ( C j K , ⊳ ) is made o f a c ategory K and a wide subcatego ry C of K , with a pure terminal ob ject 1 and the same-effect relation ≈ as in definition 2.2, tog ether with a consistency relation ⊳ which satisfie s : • Complementarity with ≈ . ∀ f , f ′ : X → Y , ( f ≈ f ′ ) ∧ ( f ⊳ ⊲ f ′ ) = ⇒ f = f ′ . In essence, the co mplemen tarity prop erty can b e stated as follows: if t wo morphisms have the same e ffect and are consistent, then they are equal. The fo llowing prop er ties ar e easily derived. Prop ositi o n 3. L et ( C j K, ⊳ ) b e an effe ct c ate gory. Then: • Consistency on effe cts. ∀ f : X → Y , ( ∃ v , f ⊳ v ) = ⇒ E ( f ) ⊳ h i X . • Consistency on pur e morphisms. ∀ v, v ′ : X Y , v ⊳ v ′ ⇐ ⇒ v = v ′ . • Consistency is unambiguous. ∀ f : X → Y , ∀ v : X Y , f ⊳ ⊲ v ⇐ ⇒ f ⊳ v . R emark 3 . It follows that a pure morphism v is consistent with itself and with no other pur e morphism. In general a morphism f may be consistent with no pure mo rphism or with sev eral ones . The relatio n ⊳ ⊲ is symmetric but in gene r al it is not r eflexive. R emark 4 . Let K be a catego ry with a wide sub catego ry C and with a pure terminal ob ject 1 . Then the same-effect relation ≈ is uniquely defined, and ther e is a “trivial” consistency re la tion: the e quality of pure morphisms. But neither the existence nor the unicity of a non-trivial cons istency re lation ⊳ is gua ranteed. 5 2.4 Extended consistency The co nsistency ⊳ is a relatio n b etw een tw o mor phisms, the seco nd o ne b eing pure. It can b e extended to pairs of arbitra ry mo rphisms. Definition 2. 5. In an effect categ o ry ( C j K , ⊳ ), an extende d c onsistency is a relatio n ◭ between parallel morphisms such that: • Extension . ∀ f : X → Y , ∀ v : X Y , f ⊳ v = ⇒ f ◭ v . • Substitut ion . ∀ f : X → Y , ∀ g , g ′ : Y → Z , g ◭ g ′ = ⇒ g ◦ f ◭ g ′ ◦ f . The sy mmetric r elation ◭ ◮ is defined b y f ◭◮ f ′ if and only if there is a morphism f ′′ such that f ◭ f ′′ ◮ f ′ . This rela tio n ◭◮ is weak er than the re lation ⊳⊲ . It follows easily that ◭ is r eflexive and that f ◭ f ′ implies f ◭◮ f ′ . R emark 5 . It is easy to chec k tha t in an effect ca tegory ( C j K, ⊳ ) there is a smalles t ex tended consistency ◭ , which is defined a s follows: ∀ h, h ′ : X → Y , h ◭ h ′ ⇐ ⇒ ∃ f : X → Y , ∃ g : Y → Z , ∃ w : Y Z , ( h = g ◦ f ) ∧ ( h ′ = w ◦ f ) ∧ ( g ⊳ w ) X f / / Y g ; ; w ⊳ / / /o /o /o /o Z ⇐ ⇒ X g ◦ f 4 4 w ◦ f ◭ / / /o /o /o /o /o /o /o /o /o /o Z In addition, this re lation ◭ satisfies pure r eplacement: ∀ f , f ′ : X → Y , ∀ w : Y Z , f ◭ f ′ = ⇒ w ◦ f ◭ w ◦ f ′ . 2.5 Examples of effect categories Several examples are introduced in this sectio n. F or each example, the s ame-effect relatio n ≈ is descr ib ed, then a consistency relation ⊳ is chosen in such a w a y tha t we get an e ffect category , and the smalle s t extended consistency rela tion ◭ is de s crib ed. It will b e check ed in sections 3.8 a nd 4.4 that in each ex ample the c hosen consistency relation gives rise to a Ca rtesian effect categor y . The examples ab out err ors, lists, finite multisets and finite sets are provided directly by a mo nad M , then K M and C M are defined as in rema rk 1. States could be treated with mo nads, at the cost of using an ex tra adjunction, but this would no t b e p os sible for partiality over an arbitrar y base categ o ry . Err ors . Let C 0 be a catego ry with an initial ob ject 0 and w ith a distinguished ob ject E (for “erro rs”), hence with a unique morphism ! E : 0 → E . Let us assume that there are copro ducts of the form X + E that b ehave wel l in the sense of extensivity (Carb o ni et al., 1 993): for every ϕ : X → Y + E , there is a copro duct X = D ϕ + D ϕ with tw o mo rphisms ϕ Y : D ϕ → Y and ϕ E : D ϕ → E such that ϕ = ϕ Y + ϕ E . The err or monad on C 0 has M X = X + E a s endofunctor a nd the copro jection η X : X → X + E as unit. A morphism f : X → Y in the Kleisli ca tegory K M stands fo r a morphism [ f ] : X → Y + E in C 0 , such that [ f ] = [ f ] Y + [ f ] E as e xplained above. A pur e morphism v = J ( v 0 ) : X Y in K M stands for [ v ] = η Y ◦ v 0 : X → Y + E in C 0 , suc h tha t [ v ] = v 0 +! E : X → Y + E in C 0 . Let us a ssume that C 0 has a terminal ob ject 1. F or each morphism f : X → Y in K M , the effect E ( f ) = h i Y ◦ f : X → 1 is suc h that [ E ( f )] = ( h i Y + id E ) ◦ [ f ] = h i D [ f ] + [ f ] E . All this can b e illustr a ted a s fo llows in C 0 , first for a pure 6 morphism v then for a morphism f and finally for the effect E ( f ); the vertical arr ows ar e the copro jections: X v 0 / / id X   Y   X [ v ] / / = = Y + E 0 ! E / / ! X O O E O O D [ f ] [ f ] Y / /   Y   X [ f ] / / = = Y + E D [ f ] [ f ] E / / O O E O O D [ f ] [ f ] Y / /   h i D [ f ] = ' ' Y h i Y / / 1   X [ E ( f )] / / = = 1 + E D [ f ] [ f ] E / / O O [ f ] E = 7 7 E id E / / E O O Let i [ f ] : D [ f ] → X deno te the copro jection and let ≃ → deno te an isomor phism in C 0 . • ∀ f : X → Y , ∀ f ′ : X → Y ′ , f ≈ f ′ ⇐ ⇒ ∃ i : D [ f ] ≃ → D [ f ′ ] , [ f ] E = [ f ′ ] E ◦ i . • ∀ f : X → Y , ∀ v = J ( v 0 ) : X Y , f ⊳ v ⇐ ⇒ [ f ] Y = v 0 ◦ i [ f ] . When C 0 is the category of s ets, we say that D ϕ is the domai n of definition of ϕ a nd that ϕ r aises t he err or e at x whenever ϕ ( x ) = e ∈ E , so that a morphism v is pure if and only if [ v ] do es no t rais e any er ror. Then, f ≈ f ′ means that [ f ] and [ f ′ ] rais e the same errors for the same ar guments, he nc e they hav e the s ame domain o f definition. F urthermor e , f ⊳ v mea ns that [ f ] co incides with [ v ] on D [ f ] , hence f ⊳ ⊲ f ′ means that [ f ] a nd [ f ′ ] coincide on D [ f ] ∩ D [ f ′ ] . Then the smalles t extended consis tency relation is suc h that for all f , f ′ : X → Y , f ◭ f ′ if and only if D [ f ] ⊆ D [ f ′ ] and [ f ] co incides with [ f ′ ] o n D [ f ] and also on D [ f ′ ] . It follows that ◭ is tr a nsitive and that ◭◮ is the same relation as ⊳ ⊲ . Partiality . A c ate gory of p artial morphisms is defined here, as in (Curien and Obtulowitz, 19 89), a s a categor y K with a wide sub ca tegory C suc h that the c a tegory K is enriched with a par tial o rder ≤ and ev ery pure arrow is maximal fo r ≤ . The n the mo r phisms in K are called the p artial functions and the morphisms in C the total functions , as in the fundamen tal situatio n of sets. In addition, let us assume that ther e is a pure terminal ob ject 1, and wher efore the effect of a morphism f : X → Y is the mor phism h i Y ◦ f (in (Curien and Obtulowitz, 1 9 89) this morphism is ca lled the domain of definition of f ). • ∀ f : X → Y , ∀ f ′ : X → Y ′ , f ≈ f ′ ⇐ ⇒ h i Y ◦ f = h i Y ◦ f ′ . • ∀ f : X → Y , ∀ v = J ( v 0 ) : X Y , f ⊳ v ⇐ ⇒ f ≤ v . • ∀ f , f ′ : X → Y , f ◭ f ′ ⇐ ⇒ f ≤ f ′ . W e add, as a new axiom, the co mplemen tarity of ≈ and ⊳ . On sets, with the usual no tion of partial function, the inclusion of C in K has a right a djoint with lifting X † = X + 1, so that the partial functions fr om X to Y c a n be identified to the (total) functions from X to Y + 1 and the partial o rder ≤ corr esp onds to the inclusion of the domains of definition (in their usual sense, as subsets). Then b oth p o ints o f view (partiality a nd er ror) are equiv alen t. State . Let C 0 be a category with a distinguished o b ject S (for “states” ) and with pro ducts of the form S × X . F or eac h set X let σ X : S × X → S and π X : S × X → X denote the pro jections. Le t K be the category with the the sa me ob jects a s C 0 and with a mo rphism f : X → Y for each [ f ] : S × X → S × Y in C 0 ; we say that f in K stands for [ f ] in C 0 . Let C b e the wide s ubca tegory o f K w ith the pur e morphisms v = J ( v 0 ) : X Y standing for [ v ] = id S × v 0 : S × X → S × Y . Let us assume that C 0 has a terminal ob ject 1. W e ma y identify S × 1 with S , so that the morphism h i X : X 1 stands for the pro jection σ X : S × X → S and the effect of a morphism f : X → Y stands for σ Y ◦ [ f ] : S × X → S . • ∀ f : X → Y , ∀ f ′ : X → Y ′ , f ≈ f ′ ⇐ ⇒ σ Y ◦ [ f ] = σ Y ′ ◦ [ f ′ ]. 7 • ∀ f : X → Y , ∀ v = J ( v 0 ) : X Y , f ⊳ v ⇐ ⇒ π Y ◦ [ f ] = v 0 ◦ π X . • ∀ f , f ′ : X → Y , f ◭ f ′ ⇐ ⇒ π Y ◦ [ f ] = π Y ◦ [ f ′ ]. It follows that ◭ is an equiv alence relation, so that ◭◮ is the same as ◭ . On sets, f ≈ f ′ means that [ f ] a nd [ f ′ ] mo dify the s tate in the same w ay , and f ⊳ v means that [ f ] alwa ys returns the same v a lue as v 0 , so that f ⊳ ⊲ f ′ means that [ f ] and [ f ′ ] b oth always r eturn the same v alue, which in a ddition do es not dep end on the state, while f ◭ f ′ (as well as f ◭ ◮ f ′ ) means that [ f ] and [ f ′ ] b oth always r eturn the same v alue, which may dep end o n the state. Lists . Let us consider the list monad with endofunctor L on the categ ory of sets. The unit η maps each x to ( x ) and the multiplication µ flattens each list of lists. Since 1 is a sing leton, a lis t ℓ in L (1) may be identified to its leng th len(() ℓ ) in N , a nd the effect o f a morphism f : X → Y to len( ◦ ) f : X → N . Then, a mo r phism f is effect-free when len( ◦ ) f is the consta nt function 1. F or each x ∈ X a nd k ∈ N , w e denote by ( x ) k the list ( x, . . . , x ) where x is r e pea ted k times. Mo re generally , fo r each list x = ( x 1 , . . . , x n ) ∈ L ( X ) a nd each list of naturals k = ( k 1 , . . . , k n ) with the same length as x , we deno te by x k the list ( x 1 , . . . , x 1 , . . . , x n , . . . , x n ) where each x i is rep eated k i times. • ∀ f : X → Y , ∀ f ′ : X → Y ′ , f ≈ f ′ ⇐ ⇒ ∀ x ∈ X , len(() f ( x )) = len(() f ′ ( x )). • ∀ f : X → Y , ∀ v = J ( v 0 ) : X Y , f ⊳ v ⇐ ⇒ ∀ x ∈ X , ∃ k ∈ N , [ f ]( x ) = ( v 0 ( x )) k . • ∀ f , f ′ : X → Y , f ◭ f ′ ⇐ ⇒ ∀ x ∈ X , ∃ k ∈ L ( N ) , [ f ]( x ) = [ f ′ ]( x ) k . It follows that f ⊳ ⊲ f ′ if and only if for ea ch x ∈ X there is so me y ∈ Y that is the unique element (if a ny) in the lists [ f ]( x ) and [ f ′ ]( x ), and that f ◭◮ f ′ as so on as f and f ′ are parallel. Finite (multi)sets . The example of lists can easily b e adapted to the finite multiset monad and to the fi nite set monad on the categor y o f sets. F o r the finite m ultiset monad, M fin (1) can b e iden tified to N a nd the effect of a mor phism to the cardinal of its ima ge. • ∀ f : X → Y , ∀ f ′ : X → Y ′ , f ≈ f ′ ⇐ ⇒ ∀ x ∈ X , ca rd(() f ( x )) = card(() f ′ ( x )). • ∀ f : X → Y , ∀ v = J ( v 0 ) : X Y , f ⊳ v ⇐ ⇒ ∀ x ∈ X , [ f ]( x ) ⊆ { v 0 ( x ) } . • ∀ f , f ′ : X → Y , f ◭ f ′ ⇐ ⇒ ∀ x ∈ X , [ f ]( x ) ⊆ [ f ′ ]( x ) . F or the finite set monad, the definitions o f ⊳ and ◭ are similar, but ≈ is different. Since P fin (1) has only t wo elements ∅ and 1 , w e get f ≈ f ′ if a nd only if for all x ∈ X either bo th f ( x ) and f ′ ( x ) ar e empty or bo th are non- e mpt y . 2.6 Results in ev aluation logic In (Moggi, 19 9 5), within the framework of evaluatio n lo gic and with respect to a stro ng monad satisfying some extra prop erties, Mo g gi defines the r elation c ⇓ a , whic h means that the v alue a is a r esu lt of the computation c . With the sa me notations as in r emark 1, c : 1 → M X and a : 1 → X are morphisms in C 0 , o r equiv alently c = [ f ] for a morphism f : 1 → X in K M and a = v 0 : X → Y yields a pur e morphism v = J ( v 0 ) : 1 X . Then it may happ en that f is c onsisten t with v in the s ense of this pap er. The following table compar es b oth notions for several mona ds on sets. Monad Results (Moggi, 199 5) Consistency (this pap er ) M Y c ⇓ a f ⊳ v Y + E c = a (thus, c is to tal) c ∈ Y = ⇒ c = a ( Y × S ) S ∃ s ∈ S , ∃ s ′ ∈ S , c ( s ) = ( a, s ′ ) ∀ s ∈ S , ∃ s ′ ∈ S , c ( s ) = ( a, s ′ ) L ( Y ) a ∈ c ∃ k ∈ N , c = ( a ) k P fin ( Y ) a ∈ c c = { a } or c = ∅ 8 F rom this table we see that in genera l f ⊳ v 6⇒ c ⇓ a and c ⇓ a 6⇒ f ⊳ v . It ca n easily be seen from the example of the state monad that having the same results is not a consistency relation in genera l, since t wo different morphisms may hav e the same effect and the s ame results. T he r efore, the no tio n of result in ev aluation lo g ic do es not eas ily fit with our notion o f cons istency . 3 Cartesian effect categories 3.1 Cartesian categories In this pap er a Cartesia n c ate gory is a catego ry with chosen finite pro ducts. W e deno te b y 1 the terminal ob ject, × for the pro ducts and p, q , r, s, t, . . . (with indices) for the pr o jections. The binary pro duct defines a functor × : C 2 → C such that for all v 1 : X 1 → Y 1 and v 2 : X 2 → Y 2 , the mor phism v 1 × v 2 : X 1 × X 2 → Y 1 × Y 2 is the unique mor phism tha t satisfies the binary pr o duct pr op erty : q 1 ◦ ( v 1 × v 2 ) = v 1 ◦ p 1 q 2 ◦ ( v 1 × v 2 ) = v 2 ◦ p 2 X 1 v 1 / / = Y 1 X 1 × X 2 v 1 × v 2 / / p 1 O O p 2   Y 1 × Y 2 q 1 O O q 2   X 2 v 2 / / Y 2 = In a Cartes ia n category C , the swap natural transformation c , with compo nents c X 1 ,X 2 : X 1 × X 2 → X 2 × X 1 , is defined fr o m the pro jections p i : X 1 × X 2 → X i and p ′ i : X 2 × X 1 → X i by p ′ i ◦ c X 1 ,X 2 = p i for i = 1 , 2. It fo llows that c X 2 ,X 1 = c − 1 X 1 ,X 2 . Now, Cartesian pro ducts in a category are generalized, first as s emi-pure pro ducts, then as s equential pro ducts, in an effect categor y . 3.2 Semi-pure pro ducts Let us consider an effect category ( C j K, ⊳ ) where C is a Cartesia n category . W e define the semi-pur e pr o ducts as tw o gra ph homomorphisms ⋉ : C × K → K and ⋊ : K × C → K that e x tend × and that satisfy some generalizatio n of the bina ry pro duct prop erty inv olving the co nsistency relation ⊳ . while the univ ersal prop erty o f a binary pro duct consists in t wo equalities , the universal pr op erty o f a semi-pure pro duct consists in one equality a nd one consistency . Definition 3.1 . Let ( C j K, ⊳ ) b e an effect categor y with a binar y pr o duct × o n C . A graph ho momor- phism ⋉ : C × K → K is the left semi-pur e pr o duct o n ( C j K, ⊳ , × ) if it extends × a nd satisfies the left semi-pur e pr o duct pr op erty : for all v 1 : X 1 Y 1 and f 2 : X 2 → Y 2 , the morphism v 1 ⋉ f 2 : X 1 × X 2 → Y 1 × Y 2 is the unique mor phism s uch that: q 1 ◦ ( v 1 ⋉ f 2 ) ⊳ v 1 ◦ p 1 q 2 ◦ ( v 1 ⋉ f 2 ) = f 2 ◦ p 2 X 1 v 1 / / /o /o /o /o /o /o /o /o ⊳ Y 1 X 1 × X 2 v 1 ⋉ f 2 / / p 1 O O O O p 2   O O Y 1 × Y 2 q 1 O O O O q 2   O O X 2 f 2 / / Y 2 = Symmetrically , a graph ho momorphism ⋊ : K × C → K is the r ight semi-pur e pr o duct on ( C j K , ⊳ , × ) if it extends × and satisfies the right semi-pu re pr o duct pr op erty : for all f 1 : X 1 → Y 1 and v 2 : X 2 Y 2 , the 9 morphism f 1 ⋊ v 2 : X 1 × X 2 → Y 1 × Y 2 is the unique mor phism s uch that: q 1 ◦ ( f 1 ⋊ v 2 ) = f 1 ◦ p 1 q 2 ◦ ( f 1 ⋊ v 2 ) ⊳ v 2 ◦ p 2 X 1 = f 1 / / Y 1 X 1 × X 2 f 1 ⋊ v 2 / / p 2   O O p 1 O O O O Y 1 × Y 2 q 1 O O O O q 2   O O X 2 v 2 / / /o /o /o /o /o /o /o /o Y 2 ⊳ A Ca rtesian effe ct c ate gory is an effect c a tegory ( C j K , ⊳ ) with a binary pro duct × on C and with semi-pure pro ducts ⋉ and ⋊ (for short, it may be denoted C j K o r simply K ). A s traightforw ard co ns equence o f definition 3 .1 is that the right semi-pure pro duct can be determined from the left o ne, as follows. Consequen tly , fro m now on, we gener ally o mit the right se mi- pure pro ducts. Prop ositi o n 4. In a Cartesian effe ct c ate gory. for al l f 1 : X 1 → Y 1 and v 2 : X 2 Y 2 : ( f 1 ⋊ v 2 ) = c Y 2 ,Y 1 ◦ ( v 2 ⋉ f 1 ) ◦ c X 1 ,X 2 . In a binary pr o duct v 1 × v 2 , obviously the firs t pro jection q 1 ◦ ( v 1 × v 2 ) does no t dep end on v 2 , and symmetrically the s e cond pro jection q 2 ◦ ( v 1 × v 2 ) does not depend on v 1 . F or a left semi-pure pr o duct v 1 ⋉ f 2 , this r e mains true for the second pro jection but no t fo r the first one. How e ver, a conseq uence of the complementarit y o f ⊳ with ≈ is that q 1 ◦ ( v 1 ⋉ f 2 ) depends o n f 2 precisely through its effect E ( f 2 ), as stated in the next pr op osition. Prop ositi o n 5. In a C artesian effe ct c ate gory, for al l v 1 : X 1 Y 1 , f 2 : X 2 → Y 2 and f ′ 2 : X 2 → Y 2 , E ( q 1 ◦ ( v 1 ⋉ f 2 )) = E ( v 1 ⋉ f 2 ) = E ( f 2 ◦ p 2 ) and: E ( f 2 ) = E ( f ′ 2 ) = ⇒ q 1 ◦ ( v 1 ⋉ f 2 ) = q 1 ◦ ( v 1 ⋉ f ′ 2 ) . Pr o of. The first result derives from the pur e w iping pr op erty of the effect. F or the sec o nd r esult, let h = v 1 ⋉ f 2 and h ′ = v 1 ⋉ f ′ 2 . The left semi-pure pro duct pr op erty implies that q 1 ◦ h ⊳ ⊲ q 1 ◦ h ′ and q 2 ◦ h = q 2 ◦ h ′ . The latter implies that q 2 ◦ h ≈ q 2 ◦ h ′ , and thus by pure wiping w e hav e also q 1 ◦ h ≈ q 1 ◦ h ′ . The r e sult now follows from the complementarit y of ⊳ with ≈ . The next propos ition fo llows from the fact that the restrictio n o f ⋉ to C 2 coincides with the binary pro duct functor × on C . Prop ositi o n 6. In a Cartesian effe ct c ate gory, for al l obje cts X 1 and X 2 : id X 1 ⋉ id X 2 = id X 1 × id X 2 = id X 1 × X 2 . R emark 6 . Let us as sume that the following unicity c ondition holds: ∀ h, h ′ : X → Y 1 × Y 2 , ( q 1 ◦ h ⊳ ⊲ q 1 ◦ h ′ ) ∧ ( q 2 ◦ h = q 2 ◦ h ′ ) = ⇒ h = h ′ . In this case, if there is a graph ho momorphism ⋉ : C × K → K extending × a nd satisfying the left semi-pure pro duct pro p er ty , then ⋉ is the left semi-pure pro duct. 3.3 Sequen tial products In accorda nce with the intended meaning of “ sequential”, we define sequen tial pr o ducts as comp osed from t wo consecutive semi-pure pr o ducts. 10 Definition 3.2. In a Cartesian effect category , the pa ir of se quen tial pr o ducts comp osed from the semi- pro ducts ⋉ , ⋊ is made of the graph homomorphisms ⋉ seq , ⋊ seq : K 2 → K (the left and right seq ue ntial pro ducts, resp ectively) defined as follows: • for all f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 : f 1 ⋉ seq f 2 = (id Y 1 ⋉ f 2 ) ◦ ( f 1 ⋊ id X 2 ) • for all f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 : f 1 ⋊ seq f 2 = ( f 1 ⋊ id Y 2 ) ◦ (id X 1 ⋉ f 2 ) X 1 = f 1 / / Y 1 id / / /o /o /o /o /o /o /o ⊳ Y 1 X 1 × X 2 f 1 ⋊ id / / p 2   O O O p 1 O O O O O Y 1 × X 2 r 1 O O O O O r 2   O O O id ⋉ f 2 / / Y 1 × Y 2 q 1 O O O O O q 2   O O O X 2 id / / /o /o /o /o /o /o /o X 2 ⊳ f 2 / / Y 2 = X 1 id / / /o /o /o /o /o /o /o ⊳ X 1 = f 1 / / Y 1 X 1 × X 2 id ⋉ f 2 / / p 1 O O O O O p 2   O O O X 1 × Y 2 s 1 O O O O O s 2   O O O f 1 ⋊ id / / Y 1 × Y 2 q 1 O O O O O q 2   O O O X 2 f 2 / / Y 2 = id / / /o /o /o /o /o /o /o Y 2 ⊳ It follows easily fro m prop o sition 4 that the right s equential pro duct can be determined from the left one, as follows. Consequen tly , from now on, we g enerally omit the right seq uent ial pro ducts. Prop ositi o n 7. In a Cartesian effe ct c ate gory, for al l f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 : ( f 1 ⋊ seq f 2 ) = c Y 2 ,Y 1 ◦ ( f 2 ⋉ seq f 1 ) ◦ c X 1 ,X 2 . Prop ositi o n 8. In a Cartesian effe ct c ate gory, the left se quential pr o duct ⋉ seq extends the left semi-pur e pr o duct ⋉ . Pr o of. Let v : X 1 Y 1 and f : X 2 → Y 2 . Since v ⋉ seq f = (id Y 1 ⋉ f ) ◦ ( v ⋊ id X 2 ) and since ⋊ e xtends the binary pro duct × on C 2 : v ⋉ seq f = (id Y 1 ⋉ f ) ◦ ( v × id X 2 ) . The left semi-pur e pr o duct pr op erty yields: q 1 ◦ (id Y 1 ⋉ f ) ⊳ r 1 and q 2 ◦ (id Y 1 ⋉ f ) = f ◦ r 2 so that by pure substitution: q 1 ◦ ( v ⋉ seq f ) ⊳ r 1 ◦ ( v × id X 2 ) and q 2 ◦ ( v ⋉ seq f ) = f ◦ r 2 ◦ ( v × id X 2 ) hence from the binar y pro duct prop erty we g et: q 1 ◦ ( v ⋉ seq f ) ⊳ v ◦ p 1 and q 2 ◦ ( v ⋉ seq f ) = f ◦ p 2 which is the left se mi-pure pro duct pro p erty . R emark 7 . It follows from prop osition 8 that we may drop the s ubs cript “seq”. Definition 3.3. In a Cartesian effect category , for all f 1 : X → Y 1 and f 2 : X → Y 2 the left p airing of f 1 and f 2 is h f 1 , f 2 i l = ( f 1 ⋉ f 2 ) ◦ h id X , id X i : X → Y 1 × Y 2 and the righ t p airing of f 1 and f 2 is h f 1 , f 2 i r = ( f 1 ⋊ f 2 ) ◦ h id X , id X i : X → Y 1 × Y 2 . R emark 8 . Another point of view on sequen tial pr o ducts, as “direct” generaliza tions of binar y pro ducts (independently fro m a ny a priori semi-pure pr o ducts) is given in sectio n 3.7. 11 3.4 Pure morphisms are cen tral The next definition is similar to the definition o f central morphisms in a binoidal categ ory , see section 4.1. Definition 3.4. In a Cartesian effect catego ry , a morphism k 1 is c entr al if fo r each morphism f 2 : k 1 ⋉ f 2 = k 1 ⋊ f 2 . Then it follows fro m pro p osition 7 that f 2 ⋉ k 1 = f 2 ⋊ k 1 . The c enter C K of K is made of the ob jects of K together with the c e ntral mor phisms, we will prove in theorem 12 that C K is a sub categ o ry o f K . R emark 9 . According to definitio n 3 .2, in a Cartesia n effect ca teg ory a mor phism k 1 : X 1 → Y 1 is central if and only if for each morphism f 2 : X 2 → Y 2 : ( k 1 ⋊ id Y 2 ) ◦ (id X 1 ⋉ f 2 ) = (id Y 1 ⋉ f 2 ) ◦ ( k 1 ⋊ id X 2 ) . R emark 10 . It follows from definition 3.2 a nd prop osition 6 that the ident ities are central. Theorem 9 now prov es that this is v alid for all pure morphisms. Theorem 9 . In a Cartesian effe ct c ate gory, every pur e morphism is c entr al. Pr o of. Giv en v : X 1 Y 1 and f : X 2 → Y 2 , let us prov e tha t the left semi-pure pro duct v ⋉ f is equa l to the right sequential pro duct v ⋊ f . Let: h = v ⋊ f = ( v ⋊ id Y 2 ) ◦ (id X 1 ⋉ f ) = ( v × id Y 2 ) ◦ (id X 1 ⋉ f ) . Using the binary pro duct prop erty: q 1 ◦ h = v ◦ s 1 ◦ (id X 1 ⋉ f ) and q 2 ◦ h = s 2 ◦ (id X 1 ⋉ f ) then the left semi-pur e pro duct prop er ty: s 1 ◦ (id X 1 ⋉ f ) ⊳ p 1 and s 2 ◦ (id X 1 ⋉ f ) = f ◦ p 2 we get by pure r e pla cement: q 1 ◦ h ⊳ v ◦ p 1 and q 2 ◦ h = f ◦ p 2 which means that the left semi-pure pro duct pro p e rty is satisfied: h = v ⋉ f , as requir ed. R emark 11 . In view of theorem 9 ther e would b e no ambiguit y in denoting × for the semi-pure pr o ducts ⋉ and ⋊ , howev er we will not us e this o pp ortunity , in order to keep in mind tha t the semi-pure pr o ducts are not real pro ducts. 3.5 F un ctoriality properties As reminded in section 3.1, the binary pro duct in a Cartesian categor y is a functor. In this section it is prov ed that similar ly the semi-pure pro ducts in a Cartesia n effect ca tegory are functors. Lemma 10. In a Cartesian effe ct c ate gory, for all X 1 , f 2 : X 2 → Y 2 and g 2 : Y 2 → Z 2 : (id X 1 ⋉ g 2 ) ◦ (id X 1 ⋉ f 2 ) = id X 1 ⋉ ( g 2 ◦ f 2 ) . Pr o of. The pro of is easily obta ined by chasing the following diagr a m a nd using the compatibility o f consis- tency with comp os ition. X 1 id / / /o /o /o /o /o /o ⊳ X 1 id / / /o /o /o /o /o /o ⊳ X 1 X 1 × X 2 p 1 O O O O p 2   O O id ⋉ f 2 / / X 1 × Y 2 s 1 O O O O s 2   O O id ⋉ g 2 / / X 1 × Z 2 s ′ 1 O O O O s ′ 2   O O X 2 f 2 / / Y 2 = g 2 / / Z 2 = 12 Lemma 11. In a C artesian effe ct c ate gory, for al l f 1 : X 1 → Y 1 , k 1 : Y 1 → Z 1 , f 2 : X 2 → Y 2 and g 2 : Y 2 → Z 2 with k 1 c entra l: ( k 1 ⋉ g 2 ) ◦ ( f 1 ⋉ f 2 ) = ( k 1 ◦ f 1 ) ⋉ ( g 2 ◦ f 2 ) Pr o of. According to definition 3.2: ( k 1 ⋉ g 2 ) ◦ ( f 1 ⋉ f 2 ) = (id Z 1 ⋉ g 2 ) ◦ ( k 1 ⋊ id Y 2 ) ◦ (id Y 1 ⋉ f 2 ) ◦ ( f 1 ⋊ id X 2 ) . Since k 1 is central, this is equa l to (id Z 1 ⋉ g 2 ) ◦ (id Z 1 ⋉ f 2 ) ◦ ( k 1 ⋊ id X 2 ) ◦ ( f 1 ⋊ id X 2 ). The result now follows from lemma 10 and definition 3.2 aga in. Theorem 12. In a Cartesian effe ct c ate gory C j K , the c enter C K is a wide sub c ate gory of K that c ontains C , and the r estrictions of the se quential pr o ducts ar e functors ⋉ : C K × K → K and ⋊ : K × C K → K . Pr o of. The central mor phisms form a sub categor y of K : this comes from remar k 10 for identities and from lemma 11 and its symmetric version for comp os ition. The center C K is wide by definition, and it contains C b ecaus e of theorem 9. The res tr ictions of the left sequential pr o duct is a functor : by pr op osition 6 for ident ities and lemma 1 1 for co mpo sition. Symmetrically , the restr ictions of the right s equential pro duct is a functor. 3.6 Naturalit y prop erties As reminded in section 3.1, a Cartes ian ca tegory C with × : C 2 → C and 1 forms a symmetric monoidal category , which means that the pro jections ca n b e com bined in order to get natura l isomor phisms a, r, l , c with comp onents: • a X = a X 1 ,X 2 ,X 3 : ( X 1 × X 2 ) × X 3 → X 1 × ( X 2 × X 3 ), • r X : 1 × X → X , l X : X × 1 → X , • c X = c X 1 ,X 2 : X 1 × X 2 → X 2 × X 1 , which s atisfy the symmetric mo no idal coherence conditions (Mac La ne, 19 97). In this s ection we prov e that in a Cartesian effect ca tegory C j K , the natural isomorphis ms a, r, l, c that are defined from C satisfy more general natura lity co nditions, inv olving the sequen tial pr o ducts ⋉ , ⋊ . The verification o f the next r esult is straightforward fr om the definitions. Lemma 13. In a Cartesian effe ct c ate gory, for all f 1 , f 2 , f 3 and pur e v 1 , v 2 , v 3 :        a Y ◦ ( f 1 ⋊ ( v 2 ⋊ v 3 )) = (( f 1 ⋊ v 2 ) ⋊ v 3 ) ◦ a X a Y ◦ ( v 1 ⋉ ( f 2 ⋊ v 3 )) = (( v 1 ⋉ f 2 ) ⋊ v 3 ) ◦ a X a Y ◦ ( v 1 ⋉ ( v 2 ⋉ f 3 )) = (( v 1 ⋉ v 2 ) ⋉ f 3 ) ◦ a X Theorem 14. In a C artesian effe ct c ate gory, for al l f : X → Y , f 1 : X 1 → Y 1 , f 2 : X 2 → Y 2 and f 3 : X 3 → Y 3 :                  r Y ◦ (id 1 ⋉ f ) = f ◦ r X l Y ◦ ( f ⋊ id 1 ) = f ◦ l X c Y ◦ ( f 1 ⋊ f 2 ) = ( f 2 ⋉ f 1 ) ◦ c X a Y ◦ ( f 1 ⋉ ( f 2 ⋉ f 3 )) = (( f 1 ⋉ f 2 ) ⋉ f 3 ) ◦ a X a Y ◦ ( f 1 ⋊ ( f 2 ⋊ f 3 )) = (( f 1 ⋊ f 2 ) ⋊ f 3 ) ◦ a X 13 Pr o of. Since r X and l X are the pro jections, the first tw o lines comes from the definition of semi-pur e pro ducts. Since c X is the swap morphism from se c tion 3.1, the third line is pro p osition 7. As for the fourth line, let us use the definition of sequential pro ducts: f 1 ⋉ ( f 2 ⋉ f 3 ) = (id ⋉ ( f 2 ⋉ f 3 )) ◦ ( f 1 ⋊ id) a nd f 2 ⋉ f 3 = (id ⋉ f 3 ) ◦ ( f 2 ⋊ id) hence by lemma 10: id ⋉ ( f 2 ⋉ f 3 ) = (id ⋉ (id ⋉ f 3 )) ◦ (id ⋉ ( f 2 ⋊ id)) and finally: f 1 ⋉ ( f 2 ⋉ f 3 ) = (id ⋉ (id ⋉ f 3 )) ◦ (id ⋉ ( f 2 ⋊ id)) ◦ ( f 1 ⋊ id) . In a symmetric wa y: ( f 1 ⋉ f 2 ) ⋉ f 3 = (id ⋉ f 3 ) ◦ ((id ⋉ f 2 ) ⋊ id) ◦ (( f 1 ⋊ id) ⋊ id) . Hence the res ult follows from the three lines of lemma 13, together with pro po sition 6 for dealing with ident ities. 3.7 The sequen t ial pro duct prop erties Sequential pro ducts also sa tisfy the left and ri ght se quential pr o duct pr op erties , a s defined b elow, whic h generalize the binary pr o duct prop erty . W e use an extended consistency ◭ , as defined in section 2.4. Definition 3.5. Let ( C j K, ⊳ ) be an effect ca tegory with an extended c o nsistency relation ◭ and with a pair of gr aph homomorphisms ⋉ ′ , ⋊ ′ : K 2 → K extending × . Then the left se quential pr o duct pr op erty states that for all f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 , the morphis m f 1 ⋉ ′ f 2 : X 1 × X 2 → Y 1 × Y 2 satisfies: q 1 ◦ ( f 1 ⋉ ′ f 2 ) ◭ f 1 ◦ p 1 q 2 ◦ ( f 1 ⋉ ′ f 2 ) = f 2 ◦ r 2 ◦ ( f 1 ⋊ ′ id X 2 ) X 1 f 1 / / ◭ Y 1 X 1 × X 2 f 1 ⋉ ′ f 2 / / p 1 O O O O O O f 1 ⋊ ′ id / / Y 1 × Y 2 q 1 O O O O O O q 2   O O O O O Y 1 × X 2 r 2   O X 2 f 2 / / Y 2 = Symmetrically , the right s e quential pr o duct pr op erty says that for a ll f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 , the morphism f 1 ⋊ ′ f 2 : X 1 × X 2 → Y 1 × Y 2 satisfies: q 1 ◦ ( f 1 ⋊ ′ f 2 ) = f 1 ◦ s 1 ◦ (id X 1 ⋉ ′ f 2 ) q 2 ◦ ( f 1 ⋊ ′ f 2 ) ◭ f 2 ◦ p 2 X 1 = f 1 / / Y 1 X 1 × Y 2 s 1 O O O X 1 × X 2 f 1 ⋊ ′ f 2 / / p 2   O O O O id ⋉ ′ f 2 / / Y 1 × Y 2 q 1 O O O O O O O q 2   O O O O X 2 f 2 / / Y 2 ◭ Prop ositi o n 15. In a Cartesia n effe ct c ate gory, the se quential pr o duct s ⋉ , ⋊ satisfy the se quential pr o duct pr op erties. 14 Pr o of. The left se q uent ial pro duct is defined a s f 1 ⋉ f 2 = (id Y 1 ⋉ f 2 ) ◦ ( f 1 ⋊ id X 2 ). Since ◭ extends ⊳ , the left semi-pure pro duct prop erty yields: q 1 ◦ (id Y 1 ⋉ f 2 ) ◭ r 1 and q 2 ◦ (id Y 1 ⋉ f 2 ) = f 2 ◦ r 2 so that by the subs titution prop erty o f ◭ : q 1 ◦ ( f 1 ⋉ f 2 ) ◭ r 1 ◦ ( f 1 ⋊ id X 2 ) a nd q 2 ◦ ( f 1 ⋉ f 2 ) = f 2 ◦ r 2 ◦ ( f 1 ⋊ id X 2 ) . The right semi- pur e pro duct prop erty implies that r 1 ◦ ( f 1 ⋊ id X 2 ) = f 1 ◦ p 1 , hence: q 1 ◦ ( f 1 ⋉ f 2 ) ◭ f 1 ◦ p 1 and q 2 ◦ ( f 1 ⋉ f 2 ) = f 2 ◦ r 2 ◦ ( f 1 ⋊ id X 2 ) which is the left se quential pro duct pr o p erty . R emark 12 . The following condition is calle d the extende d un icity condition: ∀ h, h ′ : X → Y 1 × Y 2 , ( q 1 ◦ h ◭◮ q 1 ◦ h ′ ) ∧ ( q 2 ◦ h = q 2 ◦ h ′ ) = ⇒ h = h ′ Since ◭◮ is w eaker than ⊳⊲ , the extended unicit y condition implies the unicity c o ndition of remark 6. Whenever the extended unicity condition holds, the sequential pro duct prop er ties can b e used as a definition of the s equential pro ducts, instead of definition 3.2. In addition, although this lo o ks like a m utually recur sive definition of the left and right sequential pro ducts, this recurs ivity has o nly tw o steps. Indeed, let ⋉ , ⋊ b e the seque ntial pro ducts and let f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 . First let h = f 1 ⋊ id X 2 . The right semi-pure pro duct pro p e rty states that q 1 ◦ h = f 1 ◦ p 1 and q 2 ◦ h ⊳ v 2 ◦ p 2 , thanks to the unicity condition this is a character ization of h . Now let k = f 1 ⋉ f 2 , from prop ositio n 1 5 w e get q 1 ◦ k ◭ f 1 ◦ p 1 and q 2 ◦ k = f 2 ◦ r 2 ◦ h , and thanks to the extended unicity condition this is a characterizatio n of k . 3.8 Some examples of Cartesian effect categor ies In this section and in se c tio n 4.4 we chec k that the effect categories from section 2.5 can b e seen a s Cartesian effect categ o ries. In ea ch example, fo r any pure morphism v and mor phis m f we build a morphism v ⋉ f , and it is left as an exercis e to chec k that v ⋉ f actually is the left semi-pur e product o f v and f . In addition, it happ ens tha t the extended unicity condition is satisfied, so that the sequential pro ducts ar e characterized by the sequential pro duct prop erties . Err ors . According to (Car b o ni et al., 1993), an extensive catego ry with pr o ducts is distributive. So, in the category C 0 , for all X , Y , Z the cano nical map from X × Y + X × Z to X × ( Y + Z ) is a n isomorphis m. Let v = J ( v 0 ) : X 1 Y 1 and f : X 2 → Y 2 in K , so that b y distributivity X 1 × X 2 is isomo r phic to ( X 1 × D [ f ] ) + ( X 1 × D [ f ] ). W e define v ⋉ f : X 1 × X 2 → Y 1 × Y 2 by D [ v ⋉ f ] = X 1 × D [ f ] , D [ v ⋉ f ] = X 1 × D [ f ] , [ v ⋉ f ] Y = v 0 × [ f ] Y and [ v ⋉ f ] E = [ f ] E ◦ π , where π : X 1 × D [ f ] → D [ f ] is the pro jection. D [ f ] [ f ] Y / /   Y 2   X 2 [ f ] / / = = Y 2 + E D [ f ] [ f ] E / / O O E O O X 1 × D [ f ] v 0 × [ f ] Y / /   Y 1 × Y 2   X 1 × X 2 [ v ⋉ f ] / / = = Y 1 × Y 2 + E X 1 × D [ f ] [ f ] E ◦ π / / O O E O O On sets, as exp ected, this provides the left sequential pro duct: ∀ x 1 ∈ X 1 , ∀ x 2 ∈ X 2 , ( f 1 ⋉ f 2 )( x 1 , x 2 ) =        h [ f 1 ]( x 1 ) , [ f 2 ]( x 2 ) i if [ f 1 ]( x 1 ) ∈ Y 1 and [ f 2 ]( x 2 ) ∈ Y 2 [ f 2 ]( x 2 ) if [ f 1 ]( x 1 ) ∈ Y 1 and [ f 2 ]( x 2 ) ∈ E [ f 1 ]( x 1 ) if [ f 1 ]( x 1 ) ∈ E 15 When E has one element a ll morphisms a re central, but as so on as E has more than one element there are non-central morphisms. Partiality . Given a categor y of partial morphisms , if we impose the existence of seque ntial pro ducts and the fact that a ll mor phisms are central, then w e get a notion that is rather similar to the notion o f p artial Cartesian catego ry o f pa rtial morphisms in (Cur ien a nd Obtulowitz, 19 89). On sets, up to adjunction, the left se quential pro duct is the same as for the monad X + 1: D ( f 1 ⋉ f 2 ) = D f 1 ⋉ D f 2 and ∀ x 1 ∈ D f 1 , ∀ x 2 ∈ D f 2 , ( f 1 ⋉ f 2 )( x 1 , x 2 ) = h [ f 1 ]( x 1 ) , [ f 2 ]( x 2 ) i . State . Let v = J ( v 0 ) : X 1 Y 1 and f : X 2 → Y 2 in K . Let us define v ⋉ f : X 1 × X 2 → Y 1 × Y 2 , up to the relev ant co mm utations, by [ v ⋉ f ] = v 0 × [ f ] : S × X 1 × Y 1 → S × Y 1 × Y 2 . X 1 v 0 / / Y 1 S × X 1 × X 2 [ v ⋉ f ] / / O O   = = S × Y 1 × Y 2 O O   S × X 2 [ f ] / / S × Y 2 On sets, as exp ected, this provides the left sequential pro duct: ∀ x 1 ∈ X 1 , ∀ x 2 ∈ X 2 , ∀ s ∈ S , [ f 1 ⋉ f 2 ]( s, x 1 , x 2 ) = h s 2 , y 1 , y 2 i where [ f 1 ]( s, x 1 ) = h s 1 , y 1 i and [ f 2 ]( s 1 , x 2 ) = h s 2 , y 2 i . The left seq uent ial pro duct f 1 ⋉ f 2 is usually distinct from the right seq ue ntial pro duct f 1 ⋊ f 2 . 4 Comparisons The use of stro ng monads for dealing with computational effects has bee n introduced b y Mog g i for r easoning ab out progr ams (Mog gi, 1989, 1991; W adler, 1 992). This has b een g eneralized by Pow er and Robinson, who defined F r eyd-c ate gories and proved that a strong monad is eq uiv alent to a F reyd-ca tegory with an adjunction (Po w er and Robinso n, 1997; Pow er and Thieleck e, 19 99). Independently , Arrows have b een introduced b y Hughes for genera lizing strong monads in Haskell (Hughes, 2 000; P aterson, 200 1); it was b elieved that Arrows are “ essentially” equiv alent to F reyd-categ ories, un til A tk ey pr ov ed that Arrows are in fact mor e general than F re y d catego ries (A tk ey, 2008). In this section we directly compar e each of these three framew orks to Ca r tesian effect categories: F reyd-categ ories in section 4.1, Ar rows in section 4 .2 and strong mo nads in section 4.3. Examples ar e conside r ed in section 4.4. 4.1 F reyd- categories In this section, it is prov ed that Car tes ian effect categories are F reyd-ca tegories (Pow er and Robinson, 1997; Po w er and Thielecke , 1 9 99; Selinger, 20 01). Let | K | deno te the s mallest wide sub categ ory of K , made of the ob jects and identities o f K . Definition 4.1. A binoidal c ate gory is a category K together with tw o functors ⊗ : | K | × K → K and ⊗ : K × | K | → K whic h coincide on | K | 2 (so that the notation ⊗ is not ambiguous). The functors ⊗ can be extended as tw o g raph homomorphisms ⋉ F r , ⋊ F r : K 2 → K , as follo ws. F or all f 1 : X 1 → Y 1 and f 2 : X 2 → Y 2 in K , le t: ( f 1 ⋉ F r f 2 = (id Y 1 ⊗ f 2 ) ◦ ( f 1 ⊗ id X 2 ) : X 1 ⊗ X 2 → Y 1 ⊗ Y 2 f 1 ⋊ F r f 2 = ( f 1 ⊗ id Y 2 ) ◦ (id X 1 ⊗ f 2 ) : X 1 ⊗ X 2 → Y 1 ⊗ Y 2 16 A morphism k 1 : X 1 → Y 1 is c entra l if for all f 2 : X 2 → Y 2 , k 1 ⋉ F r f 2 = k 1 ⋊ F r f 2 and symmetrically f 2 ⋉ F r k 1 = f 2 ⋊ F r k 1 . Let t : Φ ⇒ Ψ b e a natura l transforma tio n b etw een tw o functor s Φ , Ψ : K ′ → K , then t is c en t r al if every co mpo nent of t is central. In theor em 1 6 the gr aph homo mo rphisms ⋉ F r , ⋊ F r will b e r elated to the sequential pr o ducts ⋉ , ⋊ fro m section 3. In the next definition, “natural” means natura l in each comp onent sepa r ately . Definition 4.2. A symmetric pr emonoidal c ate gory is a binoidal categ ory K together with a n ob ject I o f K and central na tural is o morphisms with co mpo nents a X,Y ,Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ), l X : X ⊗ I → X , r X : I ⊗ X → X and c X,Y : X ⊗ Y → X ⊗ Y , sub ject to the usual coherence equations for symmetric monoidal categorie s (Mac La ne , 1997). Note that every symmetric monoidal ca tegory , hence every categor y with finite pro ducts, is s ymmetric premonoida l. A symmetric pr emonoidal functor betw een t wo symmetric pr emonoidal categorie s is a functor that preserves the partial functor ⊗ , the ob ject I a nd the natural isomo rphisms a, l , r, c . It is strict if in a ddition it maps central morphisms to central morphisms. A F re yd-c ate gory is an identit y- on-ob jects functor J : C → K where the catego ry C has finite pro ducts, the catego ry K is sy mmetr ic premonoidal and the functor J is strict s ymmetric premonoidal. The following result s tates that every Cartesia n effect catego ry is a F reyd-categ o ry . It is an eas y conse- quence of the r e s ults in section 3. Theorem 16. L et C j K b e a Cartesian effe ct c ate gory. L et a, l , r, c b e the natur al isomorphisms on C define d as in se ction 3.6. L et J : C → K b e the inclusion, let ⊗ : | K | × K → K and ⊗ : K × | K | → K b e the r est rictions of ⋉ and ⋊ , r esp e ctively, and let I = 1 . This f orms a F r eyd-c ate gory, whe r e ⋉ F r and ⋊ F r c oincide with ⋉ and ⋊ , r esp e ctively. Pr o of. The graph homomor phis ms ⊗ : | K | × K → K and ⊗ : K × | K | → K coincide on | K | 2 , a nd they are functors by theorem 12, hence K with ⊗ is a binoidal catego r y . Then, definitions 3.2 a nd 4.1 state that the graph homomorphisms ⋉ F r , ⋊ F r are the sequential pr o ducts ⋉ , ⋊ . It follows that both notions of central morphism (definitions 3.4 and 4.1) coincide. The fact tha t the transformations a, l , r, c are natur al, in the sense o f symmetric pr emonoidal ca tegories, is an immediate consequence of theo r em 1 4 (in fact fo r a it is lemma 13). Since all the comp onents of a, l, r, c are defined fro m the symmetric monoida l ca teg ory C , we know that they are isomorphisms a nd that they satisfy the coherence equa tions. In addition, since all pure morphisms a re c e n tral by theore m 9, it follows that a, l , r, c are central. Hence K with ⊗ , I a nd a, l , r, c is a symmetric premono idal category . Clearly the inclusion functor J : C → K is symmetric premono idal, and it is stric t b eca use of theorem 9. 4.2 Arrows In view of the similarities b etw een F rey d-categor ies and Arrows, it can b e gues sed that every Cartesian effect category gives rise to a n Arrow (Hughes, 2 000; Paterson, 2001); this is stated in this sectio n. Definition 4.3. An Arr ow typ e is a binary type co nstructor A of the for m: class Arrow A where arr :: ( X → Y ) → A X Y ( > > > ) :: A X Y → A Y Z → A X Z first :: A X Y → A ( X , Z ) ( Y , Z ) satisfying the following equa tions: (1) arr id > > > f = f (2) f > > > arr id = f (3) ( f > > > g ) > > > h = f > > > ( g > > > h ) (4) arr ( w .v ) = arr v > > > ar r w (5) first ( arr v ) = arr ( v × id) (6) first ( f > > > g ) = first f > > > fir st g (7) first f > > > arr (id × v ) = arr (id × v ) > > > fir st f (8) first f > > > arr fst = arr fst > > > f (9) first ( firs t f ) > > > ar r asso c = arr ass oc > > > f irst f 17 where the functions ( × ), fst and ass oc are defined as: ( × ) :: ( X → X ′ ) → ( Y → Y ′ ) → ( X , Y ) → ( X ′ , Y ′ ) such that ( f × g )( x, y ) = ( f x, g y ) fst :: ( X, Y ) → X such that fst ( x, y ) = x assoc :: (( X , Y ) , Z ) → ( X, ( Y , Z )) suc h that assoc (( x, y ) , z ) = ( x, ( y , z )) Let C H denote the category of Haskell t ypes and or dinary functions, so that the Haskell notation ( X → Y ) represents C H ( X, Y ), made o f the Hask ell ordinary functions from X to Y . An arrow A constructs a t y p e A X Y for all t yp es X and Y . W e slightly mo dify the definition of Arr ows b y allowing ( X → Y ) to represent C ( X , Y ) for any Ca rtesian category C and by requir ing tha t A X Y is a s e t r ather than a type: more on this issue can be found in (At key, 2 008). In addition, we use ca tegorical notations instead of Haskell sy ntax. F or this reason, from no w o n, for an y Cartesian category C , an Arr ow A on C asso cia tes to each o b jects X , Y of C a set A ( X , Y ), tog ether with thr e e o pe r ations: arr : C ( X , Y ) → A ( X, Y ) , > > > : A ( X, Y ) → A ( Y , Z ) → A ( X , Z ) , first : A ( X , Y ) → A ( X × Z, Y × Z ) , t hat satisfy the eq uations (1 )– (9). Basically , the corres p o ndence b etw een a Ca rtesian effect categor y C j K and an Arrow A o n C identifies K ( X , Y ) with A ( X, Y ) for a ll types X and Y . This is stated more precis e ly in prop osition 17. Prop ositi o n 17. Every Cartesian effe ct c ate gory C j K gives rise t o an Arr ow A on C , ac c or ding to the fol lowing table: Cartesian effe ct c ate gories Arr ows K ( X, Y ) A ( X, Y ) C ( X , Y ) ⊆ K ( X , Y ) arr : C ( X , Y ) → A ( X, Y ) f 7→ ( g 7→ g ◦ f ) > > > : A ( X , Y ) → A ( Y , Z ) → A ( X , Z ) f 7→ f × id first : A ( X , Y ) → A ( X × Z , Y × Z ) Pr o of. The fir st and second line in the table sa y that A ( X, Y ) is made o f the morphisms fro m X to Y in K and that arr is the conversion from pure morphisms to a r bitrary mo rphisms. The third and fourth lines say that > > > is the (re verse) comp osition of morphisms and that fi rst is the right semi-pure pro duct with the identit y . Now we prove that A is an Arrow b y transla ting ea ch pr op erty (1)–(9) in terms of Ca rtesian effect categor ie s and giving the argument for its pro o f. Note that fst is the common name for pro jections like p 1 , q 1 , . . . (in section 3) and that asso c is the na tural isomor phism a a s in section 3.6. (1) f ◦ id = f ident ity in K (2) id ◦ f = f identit y in K (3) h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f asso ciativity in K (4) w ◦ v in C = w ◦ v in K C ⊆ K is a functor (5) v × id in C = v × id in K × in K e x tends × in C (6) ( g ◦ f ) × id = ( g × id) ◦ ( f × id) lemma 1 0 (7) (id × v ) ◦ ( f × id) = ( f × id) ◦ (id × v ) theor em 9 (8) q 1 ◦ ( f × id) = f ◦ p 1 definition 3.1 (9) a ◦ (( f × id) × id) = ( f × id) ◦ a lemma 13 The Ar r ow combinators sec ond , ( ∗ ∗ ∗ ) and (& & &) can b e derived from arr , ( > > > ) and f irst , see e.g (Hughes, 20 00; Paterson, 2001). The corr esp ondence in prop os itio n 17 is easily e x tended to these functions . The left pair ing h f 1 , f 2 i l and the na tur al isomorphism c (cor r esp onding to swa p ) a re defined in se c tion 3.3 and 3.6, resp ectively . Cartesian effect ca tegories Arrows ( id × f ) = c ◦ ( f × id ) ◦ c second f = arr swap > > > fi rst f > > > a rr swap f 1 ⋉ f 2 = (id × f 2 ) ◦ ( f 1 × id) f 1 ∗ ∗ ∗ f 2 = fi rst f 1 > > > s econd f 2 h f 1 , f 2 i l = ( f 1 ⋉ f 2 ) ◦ h id , id i f 1 & & & f 2 = ar r ( λx → ( x, x )) > > > ( f 1 ∗ ∗ ∗ f 2 ) 18 F or instance in (Hughes , 2000, § 4.1) it is s tated that & & & is no t a categorical pr o duct since in gener al f 1 is different from ( f 1 & & & f 2 ) > > > arr fst : “ ther e is no r e ason to exp e ct Haskel l’s p air typ e, & & & , to b e a c ate goric al pr o duct in the c ate gory of arr ows, or inde e d to exp e ct any c ate goric al pr o duct to ex ist ”. W e can state this more precisely in a Cartesian effect catego ry , where ( f 1 & & & f 2 ) > > > a rr f st corresp onds to q 1 ◦ h f 1 , f 2 i l . Indeed, b oth morphisms are c onsistent: it follows fr o m prop ositio n 15 and pur e subs titution that q 1 ◦ h f 1 , f 2 i l ◭ f 1 . 4.3 Strong monads Strong monads corres po nd to F reyd-ca teg ories J : C → K with a right a djoint for J (Pow er and Robinson, 1997), while Cartesian effect categories cor resp ond to F reyd-categor ies with a sequential pro duct (theo- rem 16). In this section, w e giv e a condition which c haracterizes the strong mo na ds such that the corre- sp onding F r eyd-catego ry is a we ak Cartesian effect categor y , which mea ns that there are tw o g raph homo- morphisms ⋉ : C × K → K and ⋊ : K × C → K which sa tisfy the left and right semi- pur e pro duct prop erty resp ectively , but which may not b e uniq ue . W e use the same nota tions as in remar k 1. It has b een seen in remark 2 that the effect of a morphism f : X → Y of K stands for [ h i Y ◦ f ] = M h i Y ◦ [ f ] : X → M 1 in C 0 , s o that in C 0 : ∀ ϕ : X → M Y , ∀ ϕ ′ : X → M Y ′ , ϕ ≈ 0 ϕ ′ ⇐ ⇒ M h i Y ◦ ϕ = M h i Y ′ ◦ ϕ ′ . Let ⊳ be a consis tency r e lation on C j K , then the r elation ⊳ 0 in C 0 is defined by [ f ] ⊳ 0 [ v ] ⇐ ⇒ f ⊳ v , or equiv a lent ly: ∀ ϕ, ϕ ′ : X → M Y in C 0 , ϕ ⊳ 0 ϕ ′ ⇐ ⇒ ∃ v 0 : X → Y in C 0 , ( ϕ ′ = η Y ◦ v 0 ) ∧ (] ϕ [ ⊳ J ( v 0 )) . The pure substitution pro p erty of ⊳ (prop o s ition 3) corresp onds to the following substitution pr op erty of ⊳ 0 : ∀ v 0 : X → Y , ∀ w 0 : Y → Z , ∀ ψ : Y → M Z , ψ ⊳ 0 η Z ◦ w 0 = ⇒ ψ ◦ v 0 ⊳ 0 η Z ◦ w 0 ◦ v 0 . Now in addition let us assume that C 0 , hence C , is Ca rtesian. In (Mog gi, 19 89), it is explained why the monad ( M , µ, η ) and the product × are not sufficient for dealing with several v ariables: there is a typ e mismatch fro m Y 1 × M Y 2 to M ( Y 1 × Y 2 ). This issue is solved by adding a str ength , i.e., a natural transformatio n t with co mpo nent s t Y 1 ,Y 2 : Y 1 × M Y 2 → M ( Y 1 × Y 2 ) sa tisfying four axio ms (Mogg i, 19 89). One of these axioms is that for all X , r M X = M r X ◦ t 1 ,X : 1 × M X → M X , where the natural iso morphism r is made o f the pro jections r X : 1 × X → X as in section 3.6. Let us a ssume that we ar e giv en a strength t for our monad. In K , let v : X 1 Y 1 and f : X 2 → Y 2 ; in order to form a kind of product of v and f , the usual metho d cons ists in co mp o sing in C 0 the pro duct v 0 × [ f ] : X 1 × X 2 → Y 1 × M Y 2 with the streng th t Y 1 ,Y 2 : Y 1 × M Y 2 → M ( Y 1 × Y 2 ); we call this construction the left Kleisli pr o duct . The right Kleisli pro duct is defined symmetrically . Definition 4.4 . F or all v = J ( v 0 ) : X 1 Y 1 and f : X 2 → Y 2 in K , the left Kleisli pr o duct of v and f in K is defined by: [ v ⋉ Kl f ] = t Y 1 ,Y 2 ◦ ( v 0 × [ f ]) : X 1 × X 2 → M ( Y 1 × Y 2 ) in C 0 . Lemma 18. The stre ngth c an b e expr esse d as a left Kleisli pr o duct: ] t Y 1 ,Y 2 [= id Y 1 ⋉ Kl ]id M Y 2 [ in K . F or al l Y 1 , Y 2 , with pr oje ctions q 2 : Y 1 × Y 2 Y 2 and q ′ 2 : Y 1 × M Y 2 M Y 2 : q 2 ◦ ] t Y 1 ,Y 2 [= q ′ 2 in K . Pr o of. In K , let v = id Y 1 : Y 1 Y 1 and f =]id M Y 2 [: M Y 2 → Y 2 , so that v 0 = id Y 1 and [ f ] = id M Y 2 in C 0 . Then v 0 × [ f ] = id Y 1 × M Y 2 so that [ v ⋉ Kl f ] = t Y 1 ,Y 2 , this is the first prop er t y . Now, for reada bilit y , we omit the subscript 0 for naming the pr o jections in C 0 . The result is equiv alen t to M q 2 ◦ t Y 1 ,Y 2 = q ′ 2 in C 0 . The 19 pro jection q 2 can b e dec o mp o sed as q 2 = r 2 ◦ ( h i Y 1 × Y 2 ), w he r e r 2 = r Y 2 : 1 × Y 2 → Y 2 is the pr o jection. Hence on the one hand M q 2 = M r 2 ◦ M ( h i Y 1 × Y 2 ), and on the other hand q ′ 2 = r ′ 2 ◦ ( h i Y 1 × M Y 2 ) where r ′ 2 = r M Y 2 : 1 × M Y 2 → M Y 2 is the pro jection. Y 1 × M Y 2 q ′ 2 = ) ) h i× M id   t Y 1 ,Y 2 / / = M ( Y 1 × Y 2 ) M ( h i× id)   M q 2 = u u 1 × M Y 2 r ′ 2   t 1 ,Y 2 / / = M (1 × Y 2 ) M r 2   M Y 2 id / / M Y 2 In the previous diagra m, the square on the top is commutativ e since t is natural, and the square on the bo ttom is commutativ e thanks to the pro p er ty of the streng th with resp ect to r . Hence the lar ge squa re is commutativ e , a nd the result follows. Theorem 1 9 . L et C 0 b e a Cartesian c ate gory with a str ong monad ( M , µ, η , t ) and with a c onsistency r elation ⊳ on C j K . Then C 0 with the left and right Kleisli pr o ducts is a we ak Cartesian effe ct c ate gory if and only if for al l Y 1 , Y 2 (with the pr oje ctions q 1 : Y 1 × Y 2 → Y 1 and q ′ 1 : Y 1 × M Y 2 → Y 1 ): q 1 ◦ ] t Y 1 ,Y 2 [ ⊳ q ′ 1 in K , or e quivalently M q 1 ◦ t Y 1 ,Y 2 ⊳ 0 η Y 1 ◦ q ′ 1 in C 0 . If in addition ∀ ϕ, ϕ ′ : X → M ( Y 1 × Y 2 ) in C 0 , ( M q 1 ◦ ϕ ⊳ ⊲ 0 M q 1 ◦ ϕ ′ ) ∧ ( M q 2 ◦ ϕ = M q 2 ◦ ϕ ′ ) = ⇒ ϕ = ϕ ′ in C 0 , then C 0 with the left and right Kleisli pr o ducts is a Cartesian effe ct c ate gory. Roughly sp eaking (i.e., forg etting the pro jections), this means that C 0 with the Kle is li pro ducts is a weak Cartesian effect catego ry if and only if: the streng th o f the monad is cons is tent with the identit y . Pr o of. Let us consider t he morphism ] t Y 1 ,Y 2 [. By the first part of lemma 18 ] t Y 1 ,Y 2 [= id Y 1 ⋉ Kl ]id M Y 2 [. Therefore, if the left Kle isli pro duct do es satisfy the left semi-pure pro duct prop er t y , then q 1 ◦ ] t Y 1 ,Y 2 [ ⊳ q ′ 1 . Now, let us a ssume that q 1 ◦ ] t Y 1 ,Y 2 [ ⊳ q ′ 1 ; this is illustra ted b elow, together with q 2 ◦ ] t Y 1 ,Y 2 [= q ′ 2 (second part of lemma 18), first in K then in C 0 : Y 1 id / / /o /o /o /o /o /o /o /o /o /o Y 1 Y 1 × M Y 2 q ′ 1 O O O O q ′ 2   O O ] t [ / / = ⊳ Y 1 × Y 2 q 1 O O O O q 2   O O M Y 2 ]id[ / / Y 2 Y 1 η / / M Y 1 Y 1 × M Y 2 q ′ 1 O O q ′ 2   t / / = ⊳ 0 M ( Y 1 × Y 2 ) M q 1 O O M q 2   M Y 2 id / / M Y 2 F or a ny v : X 1 Y 1 and f : X 2 → Y 2 , the morphism v ⋉ Kl f in K is defined by [ v ⋉ Kl f ] = t Y 1 ,Y 2 ◦ ( v 0 × [ f ]) in C 0 . In the diag ram b elow, in C 0 , the left-hand side illustra tes the bina ry pro duct prop erty of v 0 × [ f ] and 20 the right-hand side is a s ab ov e. X 1 v 0 / / [ v ] = * * Y 1 η / / M Y 1 X 1 × X 2 p 1 O O p 2   v 0 × [ f ] / / = = Y 1 × M Y 2 q ′ 1 O O q ′ 2   t / / = ⊳ 0 M ( Y 1 × Y 2 ) M q 1 O O M q 2   X 2 [ f ] / / [ f ] = 4 4 M Y 2 id / / M Y 2 It follows immediately from the b otto m part o f this diag ram that M q 2 ◦ [ v ⋉ Kl f ] = [ f ] ◦ p 2 , which means that q 2 ◦ ( v ⋉ Kl f ) = f ◦ p 2 in K . Moreov er, it follows from the top part, using the s ubstitution prop erty of ⊳ 0 , that M q 1 ◦ [ v ⋉ Kl f ] ⊳ 0 [ v ] ◦ p 1 , which means that q 1 ◦ ( v ⋉ Kl f ) ⊳ v ◦ p 1 in K . The left semi-pur e pro duct prop erty is hence satisfied by ⋉ Kl . Then the last par t o f the theorem follows immediately from remark 6. 4.4 More examples of Cartesian effect categories In this s e ction we consider the effect categ ories in section 2 .5 whic h ar e defined from a s tr ong monad. In each example the strength is describ ed, then it is easy to ch eck that the conditions of theorem 19 are satisfied, so tha t the Kleisli categor y g ives rise to a cartesian effect categ ory with the Kleisli pro ducts as semi-pure pro ducts. Ho wev e r , for the monads of lists and of finite (multi) sets, the extended consistency relatio n is so weak that the sequential pr o duct prope r ties (definition 3.5) are no t sufficien t for c haracterizing the sequen tial pro ducts. Err ors . The str ength t X 1 ,X 2 is obtained by composing the isomorphism X 1 × ( X 2 + E ) ∼ = ( X 1 × X 2 ) + ( X 1 × E ) with id X 1 × X 2 + σ X 1 : ( X 1 × X 2 ) + ( X 1 × E ) → ( X 1 × X 2 ) + E , where σ X 1 is the pro jection. The Kleisli pro ducts are semi-pur e pro ducts from section 3.8. Lists . The strength is such that for all x 1 ∈ X 1 and x 2 = ( x 2 , 1 , . . . , x 2 ,k ) ∈ L ( X 2 ), t X 1 ,X 2 ( x 1 , x 2 ) = ( h x 1 , x 2 , 1 i , . . . , h x 1 , x 2 ,k i ). It follows tha t M p 1 ◦ t X 1 ,X 2 ( x 1 , x 2 ) = ( x 1 ) k while η X 1 ◦ p ′ 1 ( x 1 , x 2 ) = ( x 1 ). So, the left sequential pro duct is: ∀ x 1 ∈ X 1 , ∀ x 2 ∈ X 2 , ( f 1 ⋉ f 2 )( x 1 , x 2 ) = ( h y 1 , z 1 i , . . . , h y 1 , z p i , . . . , h y n , z 1 i , . . . , h y n , z p i ) , where f 1 ( x 1 ) = ( y 1 , . . . , y n ) and f 2 ( x 2 ) = ( z 1 , . . . , z p ), so that there are non-central morphisms. Finite (mu lti)sets . Finite multisets and finite s ets ha ve similar prop erties. F or sets, the strength is such that for all x 1 ∈ X 1 and x 2 ∈ P fin ( X 2 ), t X 1 ,X 2 ( x 1 , x 2 ) = {h x 1 , x ′ i | x ′ ∈ x 2 } , and both the left and the rig ht sequential pro duct ar e : ∀ x 1 ∈ X 1 , ∀ x 2 ∈ X 2 , ( f 1 ⋉ f 2 )( x 1 , x 2 ) = ( f 1 ⋊ f 2 )( x 1 , x 2 ) = {h y , z i | y ∈ f 1 ( x 1 ) ∧ z ∈ f 2 ( x 2 ) } . 5 Conclusion This pa pe r deals with the ma jor issue of formalizing computational effects, especia lly while using multiv ariate functions. F or this purp ose, we hav e intro duced sev eral new features: first a c onsistency relation and the asso ciated notio n of effe ct c ate gory , then the semi-pur e and se quential pro ducts for getting a Cartesian effe ct c ate gory . Thank s to the universal prop erty of the semi-pure pro ducts, each Cartesia n effect catego ry is 21 endow e d with a powerful too l for definitions and pro ofs. This has been used for proving that every Cartesian effect categor y is a F reyd-ca tegory and for giving conditions which ensur e that a stro ng monad gives rise to a Cartesian effect categor y . W e hav e s tudied several ex amples of effects, in each cas e we get a Cartes ian effect categor y . Since the notio ns of e ffect c a tegory and Ca rtesian effect category are new, there is still a large amount of work to do in order to study their applications and their limitations. F or instance, in or der to define some kind of clo sure, o ne could try to gener alize the r esults of (Curien and Obtulowitz, 19 8 9) on par tiality to other effects. F urther investigations include: enhancing the comparison with (Moggi, 1995) in orde r to clarify the rela tio ns b etw een Car tesian effect categor ies and ev aluation logic; fitting mor e ex amples in our framework (e.g. contin uations). In a ddition, the is sue of co mbining effects, as in (Hyland et al., 2006), mig h t be revisited from the p oint of view of effect categor ies. Ac knowledgmen ts The a uthors would like to thank Eugenio Mogg i for p ointing o ut the pa p e r s (Curien and Obtulowitz, 1 989) and (Moggi, 199 5). References A tkey , R., 200 8 . What is a categorical mo del of arrows? In: Mathematically Structured F unctional Pro- gramming (MSFP’08). Carb oni, A., Lack, S., W alter s , R., 19 93. Intro duction to extensiv e and distributive categories. 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