Diagrammatic logic applied to a parameterization process

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1 Diarammatic loic applied to a parameterization process César Domínuez Dominique Duval Auust 25., 2009 Abstract. This paper provides an abstract deinition o some kinds o loics, called diarammatic loics, toether with a deinition o morphisms and o 2-morphisms between diarammatic loics. The deinition o the 2-cateory o diarammatic loics rely on cateory theory, mainly on adjunction, cateories o ractions and limit sketches. This ramework is applied to the ormalization o a parameterization process. This process, which consists in addin a ormal parameter to some operations in a iven speciication, is presented as a morphism o loics. Then the parameter passin process, or recoverin a model o the iven speciication rom a model o the parameterized speciication and an actual parameter, is seen as a 2-morphism o loics. 1 Introduction This paper provides an introduction to the ramework o diarammatic loics with an application to the ormalization o a parameterization process. The ramework o diarammatic loics is presented in section 2. It stems rom [Duval 2003, Duval 2007], where the aim was to et an abstract deinition o loics, with relevant notions o models and proos, toether with a ood notion o morphism between loics: we were lookin or kinds o loics or dealin with computational eects and or morphisms or expressin the meanin o the eects into more usual loics. This work is based on adjunction [Kan 1958] and cateories o ractions [Gabriel and Zisman 1967] with an additional level o abstraction provided by limit sketches [Ehresmann 1968], which leads to a notion o entailment apparented to [Makkai 1997]. Our point o view is more abstract than the institutions [Gouen and Burstall 1984], see [Duval 2003] or a comparison. This new paper does not depend on [Duval 2003, Duval 2007]. On the other hand, the EAT and Kenzo sotware systems have been developed by F. Sereraert or symbolic computation in alebraic topoloy [Rubio et al. 2007, Dousson et al. 1999]. The data types used in EAT and Kenzo have been speciied throuh a parameterization process in [Domínuez et al. 2006, Domínuez et al. 2007], which is described in [Lambán et al. 2003] in terms o object-oriented technoloies like hidden alebras [Gouen and Malcolm 2000] or coalebras [Rutten 2000]. The parameterization process consists in addin a ormal parameter to some operations in a iven speciication. It is ollowed by the parameter passin process, which recovers a model o the iven speciication rom any model o the parameterized speciication and any actual parameter. A irst attempt to use diarammatic loics in order to ormalize this parameterization process is iven in [Domínuez et al. 2005]. In section 3 we present a simple ormalization o the parameterization and parameter passin processes as a morphism and a 2-morphism o diarammatic loics, respectively. The ocus in this application is on the models, but in [Dumas et al. 2009] another kind o application is studied, where proos in a diarammatic loic play an important role. Most cateorical notions used in this paper can be ound in [Mac Lane 1998] or [Barr and Wells 1999]. For simplicity, we omit most size issues and we do not always distinuish between equivalent cateories. The Departamento de Matemáticas y Computación, Universidad de La Rioja, Ediicio Vives, Luis de Ulloa s/n, E Loroño, La Rioja, Spain, cesar.dominuez@unirioja.es. Laboratoire Jean Kuntzmann, Université de Grenoble, 51 rue des mathématiques, BP 53, F Grenoble Cédex 9, France, Dominique.Duval@ima.r. 1

2 class o morphisms rom to Y in a cateory C is denoted C[, Y ]. A raph means a directed multiraph, and in order to distinuish between various kinds o structures with an underlyin raph we speak about the objects and morphisms o a cateory, the types and terms o a theory or a speciication and the points and arrows o a limit sketch. The diarammatic loics which are considered in this paper are the equational loic and several apparented loics. However diarammatic loics can be much richer, or instance irst-order loic as well as simple lambda calculus and loics with induction or coinduction can be seen as diarammatic loics. 2 Diarammatic loics The 2-cateory o diarammatic loics and its related notions are deined in sections 2.1, 2.2 and 2.3, then the diarammatic equational loic is described in section Limit sketches There are several deinitions o limit sketches (also called projective sketches), all o them are such that a limit sketch enerates a cateory with limits [Coppey and Lair 1984, Barr and Wells 1999]. While a cateory with limits is a raph with identities, compositions, limit cones and tuples, satisyin a bunch o axioms, we deine a limit sketch E as a raph with potential identities, compositions, limit cones and tuples, which become real eatures in the enerated cateory with limits C(E). For instance a point in E may have a potential identity, this is an arrow id : in E which becomes the identity morphism at the object in C(E). As another instance, a diaram in E may have a potential limit cone, which becomes a limit cone in C(E). Potential eatures are not required to satisy any axiom in E. In addition, or the simplicity o notations, we assume that each potential eature is unique: a point has at most one potential identity, a diaram has at most one potential limit cone, and so on. A morphism o limit sketches e : E 1 E 2 is a raph morphism which maps the potential eatures o E 1 to potential eatures o E 2. This orms the cateory o limit sketches. A realization (or loose model) o a limit sketch E with values in a cateory C is a raph morphism which maps the potential eatures o E to real eatures o C. A morphism o realizations is (an obvious eneralization o) a natural transormation. This ives rise to the cateory Real(E,C) o realizations o E with values in C, denoted simply Real(E) when C is the cateory o sets. The cateory Real(E) has colimits and we will use the act that let adjoint unctors preserve colimits. The Yoneda contravariant realization Y E o a limit sketch E takes its values in Real(E). It is deined as Y E (E) P(E)[E, ] where P(E) is the prototype o E, which means, the cateory enerated by E such that every potential eature o E becomes a real eature o P(E). Thanks to Y E, up to contravariance the limit sketch E can be identiied to a part o Real(E) which will be called the elementary part o Real(E) (with respect to E) and denoted Real el (E). It is a raph with distinuished eatures, deined as the identities, compositions, colimits and cotuples which are the imaes o the potential eatures o E. A undamental property is that the elementary part o Real(E) is dense in Real(E): every realization or morphism o realizations o E can be obtained by colimits and cotuples rom Real el (E). Moeover, a undamental theorem due to Ehresmann states that every morphism o limit sketches e:e 1 E 2 ives rise to an adjunction F e G e where the riht adjoint G e is the precomposition with e [Ehresmann 1968]: Real(E 1 ) F e G e Real(E 2 ) Then the unctor F e contravariantly extends e via the Yoneda contravariant realizations, in the sense that there is a natural isomorphism: F e Y E1 YE2 e. A locally presentable cateory [Gabriel and Ulmer 1971] is a cateory C which is equivalent to the cateory o set-valued realizations o a limit sketch E, then E is called a limit sketch or the cateory C. In addition, 2

3 we deine a locally presentable unctor as a unctor F:C 1 C 2 which is the let adjoint to the precomposition with some morphism o limit sketches e, so that C 1 and C 2 are locally presentable cateories. Then e is called a morphism o limit sketches or the unctor F. 2.2 Diarammatic loic: models and proos The ramework o diarammatic loics stems rom [Duval 2003, Duval 2007]. Deinition 2.1 A diarammatic loic is a locally presentable unctor L such that its riht adjoint R is ull and aithul. The act that R is ull and aithul is equivalent to the act that the counit natural transormation ε: L R Id is an isomorphism. Accordin to [Gabriel and Zisman 1967], it is also equivalent to the act that L is a localization, up to an equivalence o cateories: it consists o addin inverse morphisms or some morphisms, constrainin them to become isomorphisms. Let us consider a diarammatic loic L: S L R T Deinition 2.1 also means that R deines an isomorphim rom T to its imae, which is a relective subcateory o S. Deinition 2.2 The cateories S and T are the cateory o speciications and the cateory o theories, respectively, o the diarammatic loic L. A speciication Σ presents a theory Θ i Θ is isomorphic to L(Σ). Two speciications are equivalent i they present the same theory. The act that R is ull and aithul means that every theory Θ, when seen as a speciication R(Θ), presents itsel. With the next deinition, we claim that every model o a speciication takes its values in some theory. Deinition 2.3 A (strict) model M o a speciication Σ in a theory Θ is a morphism o theories M: LΣ Θ or equivalently (thanks to the adjunction) a morphism o speciications M: Σ RΘ. It ollows that equivalent speciications have the same models. A model M o Σ in Θ is sometimes called an oblique morphism, it is denoted M: Σ Θ. Whenever in addition S and T are 2-cateories with a natural isomorphism between T[LΣ, Θ] and S[Σ, RΘ], then T[LΣ, Θ] is the cateory o models o Σ in Θ, denoted L[Σ, Θ]. Otherwise, L[Σ, Θ] is simply the discrete cateory with the models o Σ in Θ as objects. Deinition 2.4 An entailment is a morphism τ in S such that Lτ is invertible in T. A similar notion can be ound in [Makkai 1997]. Two speciications which are related by entailments are equivalent. Deinition 2.5 An instance ρ o a speciication Σ in a speciication Σ 1 is a cospan in S made o a morphism σ : Σ Σ 1 and an entailment τ : Σ 1 Σ 1. It is also called a raction with numerator σ and denominator τ, and it is denoted ρ τ\σ : Σ Σ 1. Let us illustrate an instance ρ τ\σ o Σ in Σ 1 as: Σ σ Σ 1 Σ 1 τ this provides easily a diaram in the cateory S, by omittin the dotted arrow, and a diaram in the cateory T, by makin the dotted arrow a solid one, inverse to Lτ: in S: Σ σ Σ 1 Σ 1 τ in T: LΣ Lσ LΣ 1 (Lτ) 1 LΣ 1 Lτ 3

4 Since the cateory S has colimits and since the composition o entailments is an entailment, the instances can be composed in the usual way as cospans, thanks to pushouts. This orms the bicateory o instances o the loic, denoted S 2. Let ρ τ\σ: Σ Σ 1 in S 2, then we deine Lρ (Lτ) 1 Lσ: LΣ LΣ 1 in T. The instances are better suited than the morphisms o speciications or presentin the morphisms o theories, because or every morphism o theories θ : LΣ LΣ 1 there is an instance ρ such that Lρ θ. Since L is a localization, the quotient cateory o the bicateory S 2 is equivalent to T. Deinition 2.6 An inerence system or a diarammatic loic L is a morphism o limit sketches e:e S E T or the locally presentable unctor L. Thanks to the Yoneda contravariant realization, the morphism e has properties similar to the unctor L. In particular, e can be chosen so as to consist o addin inverse arrows or some collection o arrows in E S ; see [Duval 2003, theorem 3.13] or a systematic construction o e. The next deinitions depend on the choice o an inerence system e:e S E T or L; more details are iven in [Duval 2007]. Deinition 2.7 An inerence rule r with hypothesis H and conclusion C is a span in E S, made o two morphisms t : H H and s : H C such that e(t) is invertible in E T. It is also called a raction with numerator s and denominator t, and it is denoted r s/t : H C. With this deinition we claim that an inerence rule with hypothesis H and conclusion C can be seen, via the Yoneda contravariant realization, as an instance o Y(C) in Y(H). So, we can deine an inerence step simply as a composition o ractions, which means, as a pushout in the cateory S. Deinition 2.8 Given an inerence rule r s/t : H C and an instance κ: Y(H) Σ o the hypothesis Y(H) in a speciication Σ, the correspondin inerence step provides the instance κ Y(r): Y(C) Σ o the conclusion Y(C) in Σ. Deinition 2.9 A proo (or derivation, or derived rule) is the description o a raction in S 2 in terms o inerence rules (thanks to composition and cotuples). Typically, by derivin ρ τ\id σ or a iven morphism τ : Σ 1 Σ, we et the property that τ is an entailment. For instance, in equational loic, let τ be the inclusion o a iven speciication Σ 1 into the speciication Σ made o Σ 1 toether with an equation made o two terms, in Σ 1 ; then τ is an entailment i and only i the equation holds in the theory presented by Σ The 2-cateory o diarammatic loics Deinition 2.10 A morphism o loics F: L 1 L 2 is a pair o locally presentable unctors (F S, F T ) toether with a natural isomorphism F T L 1 L2 F S. This means that there are inerence systems e 1 and e 2 or L 1 and L 2 respectively, and morphisms o limit sketches e S and e T or F S and F T respectively, which orm a commutative square o limit sketches: F L 1 F S S 1 L 1 T 1 F T E 1,S e 1 E 1,T e S e T L 2 S 2 L 2 T2 E 2,S e 2 E2,T Usin the Yoneda contravariant realization, a morphism o loics F: L 1 L 2 can be determined by any raph morphism on S 1,el (the elementary part o S 1 with respect to E 1 ) with values in S 2 preservin the distinuished eatures o S 1,el and the entailments o L 1. Some morphisms o loics are easier to describe at the sketch level (as the undecoration morphism in section 3.1) while others are easier to describe at the loic level (as the parameterization morphism in section 3.2). The next result is a straihtorward application o adjunction. 4

5 Proposition 2.11 Given a morphism o loics F: L 1 L 2 and the correspondin adjunctions F T G T between theories and F S G S between speciications, or each speciication Σ 1 o L 1 and each theory Θ 2 o L 2 the adjunctions provide an isomorphism, natural in Σ 1 and Θ 2, between the cateories o models: L 1 [Σ 1, G T (Θ 2 )] L 2 [F S (Σ 1 ), Θ 2 ]. Deinition 2.12 A 2-morphism o loics l: F F : L 1 L 2 is a pair o natural transormations (l S, l T ) where l S : F S F S :S 1 S 2 and l T : F T F T :T 1 T 2 are such that l T L 1 L 2 l S. Given a morphism o loics F (F S, F T ) or a 2-morphism o loics l (l S, l T ), we will usually omit the subscripts S and T. The diarammatic loics toether with their morphisms and 2-morphisms orm a 2-cateory. By ocusin on theories we et a unctor rom the 2-cateory o diarammatic loics to the 2-cateory o cateories. The other parts o the loic (the cateory o speciications, the adjunction, and the inerence system) provide a way to answer some issues about theories, typically whether some morphisms o theories are invertible. 2.4 The diarammatic equational loic The equational loic provides a undamental example o a diarammatic loic. As usual in cateorical loic (see [Pitts 2000]), the equational theories are deined as the cateories with chosen inite products; with the unctors which preserve the chosen inite products they orm a cateory T eq. Similarly (see [Lellahi 1989, Barr and Wells 1999, Wells 1993]), the equational speciications are deined as the inite product sketches, which means, the limit sketches (as in section 2.1) such that their potential limits are only potential products; with the morphisms o inite product sketches they orm a cateory S eq. Since all inite products may be recovered rom binary products and a terminal type, we restrict the arity o products to either 2 or 0. We will oten omit the word equational. Every theory Θ can be seen as a speciication R eq Θ and every speciication Σ enerates, or presents, a theory L eq Σ. This corresponds to an adjunction: S eq L eq R eq The cateory o sets with the cartesian products as chosen products orms an equational theory denoted Set. By deault the models o an equational speciication Σ are the models o Σ in Set, called the set-valued models o Σ. It is a classical exercise to build limit sketches or T eq and S eq, then it is easy to check that L eq is a diarammatic loic. A simpliied description is iven now, see [Domínuez and Duval 2009] or a detailed construction. The startin point is the limit sketch or raphs E r, where the points Type and stand or the sets o vertices (or types) and edes (or terms) and the arrows dom and codom or the unctions source (or domain) and taret (or codomain): T eq Type dom codom Fiure 1 presents the main part o the raph underlyin E eq,s, in addition there are potential limits, includin the speciication o potential monomorphisms, and equalities o arrows. We have represented this raph in such a way that the bottom line, which is made o E r with potential limits and tuples, is equivalent to E r. The point Type has been duplicated or readability, and the point Unit is a potential terminal type, interpreted as a sinleton. The point Comp stands or the set o pairs o composable terms, the arrow i or the inclusion into the set o pairs o consecutive terms and comp or (, ). The point Selid stands or the set o types with a potential identity, the arrow i0 or the inclusion and selid or id. 5

6 The point 2-Prod stands or the set o pairs o types with a potential binary product, the arrow j or the inclusion into the set o pairs o types and 2-prod or (Y 1, Y 2 ) (pr i : Y 1 Y 2 Y i ) i1,2. The point 2-Tuple stands or the set o binary cones with a potential binary tuple, the arrow k or the inclusion into the set o binary cones, 2-base or recoverin the base ( i : Y i ) i1,2 (Y 1, Y 2 ), and 2-tuple stands or the construction o the potential binary tuple ( i : Y i ) i1,2 1, 2 : Y 1 Y 2. The point0-prod stands or the set o potential terminal types, the arrowj0 or the injection (ensurin that there is at most one terminal type) and 0-prod or the selection o the potential terminal type (i any). The point 0-Tuple stands or the set o types with a potential collapsin term (or nullary tuple), the arrow k0 or the inclusion into the set o types, 0-base or recoverin the potential terminal type and 0-tuple stands or the construction o the potential collapsin term : 1. 0-tuple 2-tuple 0-Prod j0 Unit 0-base 0-prod 0-base 0-Tuple k0 Type id Selid i0 Type selid dom codom comp st snd Comp i Cons Fiure 1: The raph underlyin E eq,s c1 c2 2-Tuple 2-base k 2-Cone b1 b2 2-prod 2-base 2-Prod j Type 2 A limit sketch E eq,t or equational theories is obtained rom E eq,s by choosin the entailments and mappin them to equalities, the correspondin morphism is the diarammatic equational loic L eq. Fiure 2 provides the correspondence between the usual rules o equational loic and the diarammatic inerence rules, as ractions. Since only a part o E eq,s is considered, some rules are missin, it is an exercise to enlare E eq,s so as to et them. It should be noted that in this deinition o the equational theories and speciications, the equations are identities o terms; a more subtle point o view, where the equations in a theory orm a conruence, can be ound in [Domínuez and Duval 2009]. 3 A parameterization process Several variants o the diarammatic equational loic, related by morphisms, are deined in section 3.1. The parameterization process and the parameter passin process are ormalized in sections 3.2 and 3.3, respectively. 3.1 Some diarammatic loics The theories o the parameterized equational loic L A are the equational theories toether with a distinuished type, called the type o parameters and usually denoted A. The speciications are the equational speciications with maybe a distinuished type A. The inclusion o limit sketches determines a morphism o loics F A : L eq L A. 6

7 name rule raction composition : Y :Y Z : Z Cons Comp i comp identity id : Type Selid i0 selid binary product Y 1 Y 2 pr i :Y 1 Y 2 Y i i1,2 Type 2 2-Prod j 2-prod 2-Cone binary tuple 1: Y 1 2: Y 2 2-tuple 1, 2 : Y 1 Y 2 2-Cone 2-Tuple k terminal type 1 Unit 0-Prod j0 0-prod Type collapsin 0-tuple : 1 Type 0-Tuple k0 Fiure 2: Rules or the equational loic The theories o the equational loic with a parameter L a are the parameterized equational theories toether with a distinuished constant o type A, called the parameter and usually denoted a: 1 A. The speciications are the parameterized equational speciications with maybe a distinuished term a: 1 A. The inclusion o limit sketches determines a morphism o loics F a : L A L a. The theories o the decorated equational loic L dec are the equational theories toether with a wide subtheory called pure (wide means with the same types). The speciications are the equational speciications toether with a wide subspeciication. Here is a way to build E dec,t rom E eq,t which relects the meanin o the word decoration, a smaller choice or E dec,t can be ound in [Domínuez and Duval 2009]. The decorations in this context are simply made o two keywords p or pure and or eneral ; some terms are pure, all terms are eneral, and there are rules or dealin with the decorations: identities and projections are always pure, and the compositions or tuples o pure terms are pure. This inormation can be encoded as a realization o E eq,t with values in the cateory o equational theories, as ollows. First let us describe the set-valued realization 0 o E eq,t underlyin. The set 0 (Type) is made o one type D and the set 0 () o two terms p and, so that 0 (Cons) {(p, p), (p, ), (, p), (, )}, 0 (2-Cone) {(p, p), (p, ), (, p), (, )} and 0 (Type 2 ) {(D, D)}, and we denote 0 (Unit) { }. Then 0 (selid) maps D to p, 0 (comp) maps (p, p) to p and everythin else to, 0 (2-prod) maps (D, D) to (p, p), 0 (2-tuple) maps (p, p) to p and everythin else to, 0 (0-prod) maps to p and 0 (0-tuple) maps D to p. The structure o equational theory on each set 0 (E) is induced by a monomorphism p in (). Then E dec,t is the sketch o elements (similar to the more usual cateory o elements) o the realization o E eq,t : the points o E dec,t include one point Type.D over the point Type o E eq,t, two points.p and. over the point o E eq,t, our points over Cons, and so on, and the arrows o E dec,t include an arrow c:.p. over id which is a potential monomorphism, or the conversion o pure terms to eneral terms. Clearly by orottin the decorations we et a morphism o diarammatic loics F und : L dec L eq, called the undecoration morphism. And by mappin every eature o E eq,t to the correspondin pure eature o E dec,t we et a morphism o diarammatic loics F p : L eq L dec such that F und F p id Leq. 3.2 The parameterization process is a morphism o loics In this section we deine a morphism o loics F par : L dec L A. We deine F par on speciications, its deinition on theories ollows easily. We will use the act, which ollows rom the deinition o a morphism o loics, 7

8 that a speciication may be replaced by an equivalent one whenever needed. The parameterization process starts rom a decorated speciication and returns a parameterized speciication. Rouhly speakin, it replaces every eneral eature in a decorated speciication by a parameterized one, in such a way that a pure eature does not really depend on the parameter. More precisely, types and pure terms are unchaned, while every eneral term : Y is replaced by : A Y where A is the type o parameter. Fiure 3 deines the imae o the elementary decorated speciications (pure terms are denoted with and the projections pr : A A and ε : A are oten omitted): or each point E.x in E dec,s, the parameterization process replaces the elementary decorated speciication Y(E.x) by the parameterized speciication F par (Y(E.x)). The morphisms between elementary decorated speciications are transormed in a straihtorward way. For instance, the imae o the morphism Y(c), where c:.p. is the conversion arrow, maps : A in F par (Y(.)) to ε : A Y in F par (Y(.p)), or more precisely in a parameterized speciication equivalent to F par (Y(.p)). This provides a raph morphism F par :Real el (E dec,s ) Real(E A,S ). point E.x Y(E.x) F par (Y(E.x)) type Type.p pure term.p term. pure composition Comp.p composition Comp. Y Y selection o identity Selid.p id binary product 2-Prod.p Y 1 p 1 Y 1 Y 2 pure pairin 2-Tuple.p Y 1 Y 2, Y2 Y Y A Y Z Z A pr, Y A Y Y pr, id p 1 Y 1 Y 2 Y 1 p 1 Y 1 Y 2 Y 2 Y 1, Y2 p 1 Z Y 1 Y 2 Z pairin 2-Tuple. Y 1, Y2 p 1 Y 1 Y 2 A terminal type 0-Prod.p 1 1 Y 1, Y2 p 1 Y 1 Y 2 pure collapsin 0-Tuple.p 1 1 Fiure 3: The parameterization morphism on elementary decorated speciications 8

9 Theorem 3.1 The raph morphism F par deines a morphism o diarammatic loics: F par : L dec L A which is the inclusion on the pure part o L dec, in the sense that F par F p F A. It is called the parameterization morphism. Proo. It can be checked that this raph morphism preserves the distinuished eatures o Real el (E dec,s ) and the entailments o the decorated loic, so that it provides a morphism o diarammatic loics. The equality F par F p F A is easily checked on elementary speciications. The morphisms o loics F und, F par and F A orm a (non-commutative) trianle, which becomes commutative when restricted to the pure part o L dec : id F und F p L eq L dec F A F par L eq F A L A The parameterization morphism F par ormalizes the parameterization process. The span made o F und and F par ormalizes the process o startin rom an equational speciication Σ eq, choosin a pure subspeciication Σ 0 o Σ eq so as to et a decorated speciication Σ dec such that Σ eq F und (Σ dec ), then ormin the parameterized speciication Σ A F par (Σ dec ). 3.3 The parameter passin process is a 2-morphism o loics The diaram o loics in section 3.2 composed with the inclusion F a : L A L a, which adds the parameter a: 1 A, provides another diaram with in addition a 2-morphism l as described below: L eq id F und F p L eq L dec l F a F A F a F A F a F par Each decorated speciication Σ dec, with Σ eq F und (Σ dec ), ives rise to two speciications with parameter: on the one hand Σ eq,a F a (F A (Σ eq )), which is simply Σ eq seen as a speciication with a parameter, and on the other hand Σ a F a (F par (Σ dec )). Let us deine the morphism l Σdec : Σ eq,a Σ a. When Σ dec is some Y(E.p) (where p means pure ) it is easy to check that Σ eq,a Σ a ; then l Σdec is the identity. When Σ dec Y dec (.) (where means eneral ), then l Σdec is deined by l Σdec () a,id : Y (where 1 is identiied with ). The deinitions when Σ dec Y dec (Comp.) and when Σ dec Y dec (2-Tuple.) are similar. L a Theorem 3.2 The morphisms l Σdec : Σ eq,a Σ a deine a 2-morphism o diarammatic loics: l: F a F A F und F a F par : L dec L a which is the identity on the pure part o L dec. It is called the parameter passin 2-morphism. Proo. The deinition o l Σdec on the elementary decorated speciications is extended to all speciications by colimits, and the result ollows. Theorem 3.2 has the expected consequence on models, stated as proposition 3.3: iven a set-valued model M A o the paramererized speciication Σ A, each α M A (A), called an actual parameter or an arument, 9

10 ives rise to a model M(α) o the equational speciication Σ eq. Let us introduce some notations. For each set A, let Set A denote the object o T A made o the equational theory o sets with A as the interpretation o A, so that R A (Set A ) Set. For each set A and element α A, let Set A,α denote the object o T a made o the equational theory o sets with A and α as the interpretations o A and a respectively, so that R a (Set A,α ) Set A. For each decorated speciication Σ dec (Σ eq, Σ 0 ), made o an equational speciication Σ eq and a wide subspeciication Σ 0, and or each set-valued equational model M 0 o Σ 0, let L eq [Σ eq,set] M0 denote the set o models o Σ eq extendin M 0. Let Σ A F par (Σ dec ), the deinition o F par is such that Σ 0 is also a subspeciication o Σ A and or each : Y in Σ eq there is a : A Y in Σ A, with ε when is pure. Proposition 3.3 Let Σ dec (Σ eq, Σ 0 ) be a decorated speciication and let Σ A F par (Σ dec ). For each set A and each set-valued model M A : Σ A Set A in L A, let M 0 : Σ eq Set denote the restriction o M A to Σ 0. Then there is a unction: M: A L eq [Σ eq,set] M0 which maps each α A to the model M(α) o Σ eq extendin M 0 and such that M(α)() M A ( )(α, ) or each : Y in Σ eq. Proo. Let Σ eq,a F a (F A (Σ eq )) and Σ a F a (F par (Σdec)). The precomposition with the morphism l Σdec : Σ eq,a Σ a ives rise to a unctor L a [Σ a,set A,α ] L a [Σ eq,a,set A,α ]. Proposition 2.11 provides the isomorphisms L a [Σ a,set A,α ] L A [Σ A,Set A ] and L a [Σ eq,a,set A,α ] L eq [Σ eq,set]. So, or each α A we et a unctor L A [Σ A,Set A ] L eq [Σ eq,set]. Let M A,α denote the imae o M A, because o the deinition o l Σdec it extends M 0 and satisies M A,α () M A ( )(α, ) or each : Y in Σ eq. Now, when M A is ixed, the result ollows by deinin M(α) M A,α. The unction M is not a bijection in eneral. However this may happen, under the conditions o proposition 3.4: this is the exact parameterization property rom [Lambán et al. 2003], which is also proved in [Domínuez and Duval 2009]. Proposition 3.4 With the speciications Σ eq, Σ 0 and Σ A as in proposition 3.3, let M 0 be a model o Σ 0 and M A a terminal model o Σ A extendin M 0. Then the unction M rom proposition 3.3 is a bijection: M A (A) L eq [Σ eq,set] M0. It ollows rom [Rutten 2000] and [Hensel and Reichel 1995] that there is a terminal model o Σ A over M 0. Proposition 3.4 corresponds to the way alebraic structures are implemented in the systems Kenzo/EAT. In these systems the parameter set is encoded by means o a record o Common Lisp unctions, which has a ield or each operation in the alebraic structure to be implemented. The pure terms correspond to unctions which can be obtained rom the ixed data and do not require an explicit storae. Then, each particular instance o the record ives rise to an alebraic structure. Reerences [Barr and Wells 1999] Barr, M. and Wells, C. (1999) Cateory Theory or Computin Science. Centre de Recherches Mathématiques (CRM) Publications, 3rd Edition. [Coppey and Lair 1984] Coppey, L. and Lair, C. (1984) Leçons de Théorie des Esquisses. Diarammes 12. [Domínuez and Duval 2009] Domínuez, C. and Duval, D. (2009) A parameterization process as a cateorical construction. [Domínuez et al. 2005] Domínuez, C., Duval, D., Lambán, L. and Rubio, J. (2005) Towards diarammatic speciications o symbolic computation systems. In: Mathematics, Alorithms, Proos. Coquand, T., Lombardi, H. and Roy, M. (Eds.) Dastuhl Seminar php?semnr

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