¡Descarga articulo Fujiwara y más Apuntes en PDF de Matemáticas solo en Docsity!
ON THE MORPHISMS AND TRANSFORMATIONS OF
TSUYOSHI FUJIWARA (AS A CONCRETION OF A BIDIMENSIONAL
MANY-SORTED GENERAL ALGEBRA AND ITS APPLICATION TO THE EQUIVALENCE BETWEEN CLONES AND ALGEBRAIC THEORIES).
J. CLIMENT VIDAL AND J. SOLIVERES TUR
Abstract. For, not necessarily similar, single-sorted algebras Fujiwara de- fined, through the concept of family of basic mapping-formulas between single- sorted signatures, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras. Subsequently he also defined an equiv- alence relation, the conjugation, on the families of basic mapping-formulas, which corresponds to the relation of inner isomorphism for algebras. In this paper we extend the theory of Fujiwara about morphisms to the, not neces- sarily similar, many-sorted algebras, by defining the concept of fujivor among many-sorted signatures under which the standard signature morphisms, the basic mapping-formulas of Fujiwara, and the derivors of Goguen-Thatcher- Wagner are subsumed. After this, by means of the homomorphisms between B´enabou algebras, we define the composition of fujivors from which we get the corresponding category, and prove that it is isomorphic to the category of Kleisli for a monad on the standard category of many-sorted signatures. Next, by defining the notion of transformation between fujivors, which generalizes the relation of conjugation of Fujiwara, we endow the category of many-sorted sig- natures and fujivors with a structure of 2-category. From this we get a derived 2-category of many-sorted specifications in which we prove the equivalence of the many-sorted specifications of Hall and B´enabou, and, from a suitable pseudo-functor on it to the 2-category of categories, we deduce the equivalence of the categories of Hall and B´enabou algebras. Besides, by defining for each many-sorted signature its corresponding category of generalized many-sorted terms, we prove that the realization of these terms in the many-sorted algebras is invariant under fujivors and compatible with the transformations between fujivors, and from this we get an example, among others, of the new concept of 2-institution.
- Introduction. The closed sets of operations, or clones, on a set A were initially defined and investigated by P. Hall, as pointed out by Cohn in [5], pp. 127 and 132 (who attended the lectures by Professor P. Hall from 1944 to 1951), to show that the crucial mathematical properties of a Σ-algebra A = (A, (Fσ )σ∈Σ) do not depend on the family of primitive operations (Fσ )σ∈Σ on A, but on the system of all operations on A obtainable from (Fσ )σ∈Σ by means of the operations of composition. The concept of an ordinary clone, axiomatized by P. Hall as a single-sorted par- tial algebra subject to satisfy some laws (see [5], p. 132) and, independently but subsequently, by M. Lazard as a compositor (see [22], p. 327), was generalized up to that of a many-sorted clone by Goguen and Meseguer in [13], and axiomatically
Date: February 10, 2005. 2000 Mathematics Subject Classification. Primary: 03C05, 03F25, 08A68, 08A40, 18A23, 18C10, 18D05; Secondary: 03C95, 68Q65. Key words and phrases. Clone, many-sorted algebraic theory, many-sorted set, many-sorted algebra, construction of Ehresmann-Grothendieck, Kleisli category for a monad, many-sorted term, Hall algebra, B´enabou algebra, fujivor, transformation of fujivors, 2-institution. 1
2 J. CLIMENT AND J. SOLIVERES
defined by them (in [13], pp. 318–319) as any many-sorted algebra (of the appro- priate signature) that satisfies a definite system of many-sorted equational laws, concretely, the so-called Projection Axiom, Identity Axiom, Associativity Axiom, and Invariance of Constant Functions Axiom. Given its origin in P. Hall, we agree to refer to the many-sorted algebras that are models of the just named axioms as Hall algebras. Hall algebras, as reflected by the defining axioms, are a species of algebraic construct in which the essential properties of the fundamental procedures of sub- stitution, for the many-sorted terms in the free many-sorted algebras, and of com- position, for the many-sorted-operations on sorted sets are embodied. And this is precisely one of the reasons why Hall algebras are a powerful and fundamental instrument to investigate many-sorted algebras. To this we add that Hall algebras are not only worth of study because of its source in the above mentioned proce- dures. Besides that, Hall algebras are interesting in themselves since they furnish important examples of equationally defined many-sorted algebras, and also because they have been used by Goguen and Meseguer in [13] to prove the Completeness Theorem of finitary many-sorted equational logic (that generalizes the classical Completeness Theorem of finitary equational logic of Birkhoff), providing in this way, a full algebraization of many-sorted equational deduction. Another approximation to the study of many-sorted algebras has been proposed by B´enabou in [2], by making use of the finitary many-sorted algebraic theories, that are the generalization to the many-sorted case of the finitary single-sorted algebraic theories of Lawvere in [21]. The equational presentation of the finitary many-sorted algebraic theories of B´enabou gives rise to what we have called B´enabou algebras. And the B´enabou algebras, even having a many-sorted specification different from that of the Hall algebras, are also models of the essential properties of the clones for the many-sorted operations. For an arbitrary, but fixed, set of sorts S, the many-sorted specifications HS , for Hall algebras, and BS , for B´enabou algebras, are not isomorphic in the category Spf , of many-sorted specifications and many-sorted specification morphisms, be- cause between the corresponding categories of models: Alg(HS ), of Hall algebras, and Alg(BS ), of B´enabou algebras, there is not any isomorphism. However, the many-sorted specifications HS and BS can be considered, in some definite way, as being equivalent, as a consequence of the proof, in the fourth section about Hall and B´enabou algebras, of the categorical equivalence between the categories Alg(HS ) and Alg(BS ). But, the semantical equivalence of the many-sorted specifications HS and BS , or, for that matter, of any two many-sorted specifications, understood, by con- vention, as meaning the categorical equivalence of the canonically associated cat- egories of models, can not be properly reflected at the purely syntactical level of the many-sorted specifications and many-sorted specification morphisms, i.e., can not be mathematically defined in the category Spf. And this is so, essentially, as a consequence of the fact of not having actually endowed Spf with a (non trivial) structure of 2-category. Thus, if one remains anchored in the tradition of view- ing Spf as being, simply, a category, then the only reasonable way of classifying many-sorted specifications from within the category Spf is through the categorical concept of isomorphism, and not, by structural impossibility, by means of some other notion of equivalence between many-sorted specifications, itself being strictly weaker than that of isomorphism (as it would be the case if instead of having a category, we had a 2-category). Therefore, what is really needed to settle the problem of the equivalence be- tween many-sorted specifications (i.e., the problem of determining whether or not
4 J. CLIMENT AND J. SOLIVERES
Every set we consider, unless otherwise stated, will be a U -small set or a U- large set, i.e., an element or a subset, respectively, of a Grothendieck universe U (as defined, e.g., in [23], p. 22), fixed once and for all. Besides, we agree that Set denotes the category which has as set of objects U and as set of morphisms the subset of U of all mappings between U-small sets, and, depending on the context, that Cat denotes either, the category of the U-categories (i.e., categories C such that the set of objects of C is a subset of the Grothendieck universe U , and the hom- sets of C elements of U), and functors between U -categories, or the 2-category of the U-categories, functors between U-categories, and natural transformations between functors. In all that follows we use standard concepts and constructions from category theory, see e.g., [1], [6], [17], [20], and [23]; classical universal algebra, see e.g., [5], [15], and [25]; categorical universal algebra, see e.g., [2] and [21]; many-sorted algebra, see e.g., [2], [3], [13], [18], and [24]; and set theory, see e.g., [7] and [26]. Nevertheless, we have generically adopted the following notational and conceptual conventions:
(1) We denote by N the set of all natural numbers. For two sets A, B we denote by BA^ the set of all functions from A to B and by Hom(A, B) the set of all mappings f : A //B from A to B, i.e., the set of all ordered triples f = (A, F, B) where F is a function from A to B. For a mapping f : A //B, a subset X of A, and a subset Y of B, we denote by f −^1 [Y ] the inverse image of Y under f , and by f [X] the direct image of X under f. However, if Y = {y} is a final set, we will write f −^1 [y] instead of the more accurate f −^1 [{y}]. For sets B, C, a family of sets (Ai)i∈I , a family of mappings (fi)i∈I in
i∈I Hom(B, Ai), and a family of mappings (gi)i∈I in
i∈I Hom(Ai, C), we denote by^ 〈fi〉i∈I , resp., by [gi]i∈I , the unique mapping from B to
i∈I Ai, resp., from^
i∈I Ai^ to^ C, such that, for every i ∈ I, fi = pri ◦ 〈fi〉i∈I , resp., gi = [gi]i∈I ◦ ini, where pri is the canonical projection from
∐^ i∈I^ Ai^ to^ Ai^ and ini^ the canonical injection from^ Ai^ to i∈I Ai. (2) For a set S we agree upon denoting by T?(S) = (S?, f, λ) the free monoid on S, where S?, the underlying set of T?(S), is
n∈N S n, the set of all words on S, f the concatenation of words on S, and λ the empty word on S. For a word w on S, |w| is the length of w. Moreover, T? = (T?, G, f) is the standard monad in Set for the monoid specification, where, T? is the composition of the free monoid functor T? : Set //Mon and the forgetful functor GMon : Mon //Set, for every set S, GS : S //S?^ the inclusion of S into S?, and fS : S??^ //S?^ the merging of strings of words to words. To simplify the notation, we will write (s) instead of GS (s). Furthermore, if ϕ : S //T and ψ : S //T?^ are mappings, then ϕ?^ is the unique homomorphism from T?(S) to T?(T ) such that ϕ?◦ GS =GT ◦ϕ, ψ]^ : S?^ //T?^ the underlying mapping of the canonical extension of ψ to the free monoid T?(S) on S and ψ?^ the unique monoid homomorphism from T?(S) to T?(T ?) such that ψ?◦ GS =GT?^ ◦ψ.
More specific notational conventions will be included and explained in the successive sections.
- The many-sorted term institution. Our main aim in this section is to show that the concept of “derived operation of an algebra”, also known as “term operation of an algebra”, elemental as it is, but fundamental for universal algebra, can be naturally subsumed under the notion of institution (see for this notion, e.g., [12]), provided that an institution is meant
MORPHISMS AND TRANSFORMATIONS OF FUJIWARA 5
not to be an extranatural transformation (as in [12]) but a pseudo-extranatural transformation (as defined at the end of this section). To attain the aim just mentioned we begin by a careful examination of the different types of things that are involved around it, namely many-sorted sets, signatures, algebras, terms, and generalized institutions. More specifically, in this section we define the category MSet of many-sorted sets, in which the many- sorted sets will be labelled with the sets of sorts, by applying the construction of Ehresmann-Grothendieck (see [6], pp. 89–91 and [17], pp. (sub.) 175–177) to a contravariant functor MSet from Set to Cat. Following this we define the categories Sig, of many-sorted signatures, and Alg, of many-sorted algebras, by applying also the construction of Ehresmann-Grothendieck to suitable contravariant functors Sig from Set to Cat, and Alg from Sig to Cat, respectively. Besides we prove the existence of a left adjoint T to a “forgetful”functor G from Alg to MSet ×Set Sig, and from this left adjoint T we define a pseudo-functor Ter from Sig to Cat which formalizes the procedure of translation for many-sorted terms. Finally, to account exactly for the invariant character of the realization of many- sorted terms in many-sorted algebras under change of many-sorted signature, we prove the existence of a pseudo-extranatural transformation from a pseudo-functor on Sigop^ × Sig to Cat, induced by Alg and Ter, to the functor, between the same categories, constantly Set. Then, after providing a generalization of the ordinary concept of institution, we prove that the pseudo-extranatural transformation is, in fact, part of an institution on Set, the so-called many-sorted term institution. Before stating the first proposition of this section, we agree upon calling, from now on, for a set (of sorts) S ∈ U, the objects of the category SetS^ (i.e., the elements A = (As)s∈S of US^ ) S-sorted sets; and the morphisms of the category SetS^ from an S-sorted set A into a like one B (i.e., the ordered triples (A, f, B), abbreviated to f : A //B, where f is an element of
s∈S Hom(As, Bs))^ S-sorted mappings from A to B. Furthermore, we also agree that a pseudo-functor F from a category C to a 2-category D consists of the following data:
(1) An object mapping F : Ob(C) //Ob(D). (2) For every x, y ∈ C, an hom-mapping F : HomC(x, y) //HomD(F (x), F (y)).
(3) For every morphisms f : x //y and g : y //z in C, an isomorphic 2-cell γf,g^ from F (g) ◦ F (f ) to F (g ◦ f ). (4) For every x ∈ C, an isomorphic 2-cell νx^ from idF (x) to F (idx).
These data must satisfy the following coherence axioms:
(1) For morphisms f : x //y, g : y //z, and h : z //t in C,
γg◦f,h^ ◦ (idF (h) ∗ γf,g^ ) = γf,h◦g^ ◦ (γg,h^ ∗ idF (f )). (2) For a morphism f : x //y in C,
idF (f ) = γidx,f^ ◦ (idF (f ) ∗ νx) and idF (f ) = γf,idy^ ◦ (νy^ ∗ idF (f )). In the following proposition, that is basic for a great deal of what follows, for a mapping ϕ from S to T , we prove the existence of an adjunction
ϕ a^ ∆ϕ^ from the category of S-sorted sets to the category of T -sorted sets, as well as the existence of a contravariant functor MSet and of a pseudo-functor MSetq^ (related, respectively, to the right and left components of the adjunction) from Set to Cat.
Proposition 2.1. Let ϕ : S //T be a mapping. Then the functors ∆ϕ from SetT
to SetS^ and
ϕ from^ Set
S (^) to SetT (^) defined, respectively, as follows
MORPHISMS AND TRANSFORMATIONS OF FUJIWARA 7
indicate the covariant situation, and of the superscript to indicate the contravariant one). From this it follows that the functor πMSet is also a split opfibration. Therefore we can assert that MSet is a split bifibration on Set. To prove the bicompleteness of the category MSet and of some other categories defined a bit further on, in this same section, the following definition and proposi- tions, stated in [29] and (partially) in [16], are particularly useful, since they give sufficient conditions that are, mostly, easily verifiable for the cases we are concerned with.
Definition 2.3. (Cf., [29], p. 249) We say that a functor F : Cop^ //Cat is locally reversible if, for every morphism h : c //d in C, the functor F (h) from F (d) to F (c) has a left adjoint.
Proposition 2.4. Let F : Cop^ //Cat be a functor. If C is complete, for every object c ∈ C, the category F (c) is complete and, for every morphism h : c //d in C, the functor F (h) from F (d) to F (c) is continuous (i.e., preserves projective
limits), then
∫ C
F is complete.
Proof. See [29], pp. 247–248. §
Proposition 2.5. Let F : Cop^ //Cat be a functor. If C is cocomplete, for every
object c ∈ C, the category F (c) is cocomplete, and F is locally reversible, then
∫ C
F
is cocomplete.
Proof. See [29], pp. 250–251. §
Corollary 2.6. The category MSet is bicomplete.
Proof. The category MSet is complete because Set is complete, for every set S, MSet(S) = SetS^ is complete, and, for every mapping ϕ : S //T , the functor MSet(ϕ) = ∆ϕ from SetT^ to SetS^ is continuous, since it has
ϕ as a left adjoint. The category MSet is cocomplete because Set is cocomplete, for every set S, MSet(S) = SetS^ is cocomplete, and the contravariant functor MSet is locally re- versible. §
Our next goal is to define the category Sig, of standard many-sorted signatures and many-sorted signature morphisms, by applying the construction of Ehresmann- Grothendieck to a contravariant functor Sig from Set to Cat. The category Sig will be shown to be fundamental to get, by means of the same construction, but applied to a contravariant functor from Sig to Cat, the category of many-sorted algebras, and also to build on it (in the fifth section), through the construction of Kleisli, another category which will be shown to be adequate to prove (in the last section) the categorical equivalence between the specifications for Hall and B´enabou algebras. Before we prove the existence of the contravariant functor Sig in the following proposition, we recall that, for a set of sorts S, the category of S-sorted signatures, denoted by Sig(S), is SetS
?×S
. Therefore an S-sorted signature is a function Σ from S?^ × S to U which sends a pair (w, s) ∈ S?^ × S to the set Σw,s of the formal operations of arity w, sort (or coarity) s, and rank (or biarity) (w, s); and an S- sorted signature morphism from Σ to Σ′^ an ordered triple (Σ, d, Σ′), abbreviated to d : Σ //Σ′, where d is an element of
(w,s)∈S?×S Hom(Σw,s,^ Σ
′ w,s). Thus, for (w, s) ∈ S?^ ×S, dw,s is a mapping from Σw,s to Σ′ w,s which sends a formal operation σ in Σw,s to the formal operation dw,s(σ), abbreviated to d(σ), in Σ′ w,s. Sometimes we will write σ : w //s to indicate that the formal operation σ belongs to Σw,s.
Proposition 2.7. There exists a contravariant functor Sig from Set to Cat defined as follows
8 J. CLIMENT AND J. SOLIVERES
(1) Sig sends a set (of sorts) S to Sig(S) = Sig(S), the category of S-sorted signatures. (2) Sig sends a mapping ϕ from S to T to the functor Sig(ϕ) = ∆ϕ?×ϕ from Sig(T ) to Sig(S) which relabels T -sorted signatures into S-sorted signa- tures, i.e., Sig(ϕ) assigns to a T -sorted signature Λ the S-sorted signature Sig(ϕ)(Λ) = Λϕ?×ϕ, and assigns to a morphism of T -sorted signatures d from Λ to Λ′^ the morphism of S-sorted signatures Sig(ϕ)(d) = dϕ?×ϕ from Λϕ?×ϕ to Λ′ ϕ?×ϕ.
Definition 2.8. The category Sig of many-sorted signatures and many-sorted sig- nature morphisms, obtained by applying the construction of Ehresmann-Grothendieck to the contravariant functor Sig on Set to Cat, is Sig =
∫ (^) Set Sig.
Therefore the category Sig has as objects the pairs (S, Σ), where S is a set of sorts and Σ an S-sorted signature and as many-sorted signature morphisms from (S, Σ) to (T, Λ) the pairs (ϕ, d), where ϕ : S //T is a morphism in Set while d : Σ //Λϕ?×ϕ is a morphism in Sig(S). The composition of
(ϕ, d) : (S, Σ) //(T, Λ) and (ψ, e) : (T, Λ) //(U, Ω),
denoted by (ψ, e) ◦ (ϕ, d), is (ψ ◦ ϕ, eϕ?×ϕ ◦ d), where
eϕ?×ϕ : Λϕ?×ϕ //(Ωψ?×ψ )ϕ?×ϕ = Ω(ψ◦ϕ)?×(ψ◦ϕ).
From now on, unless otherwise stated, we will write Σ, Λ, Ω, and Ξ instead of (S, Σ), (T, Λ), (U, Ω), and (X, Ξ), respectively, and d, e, and h, instead of (ϕ, d), (ψ, e), and (γ, h), respectively. Furthermore, to shorten terminology, we will drop the qualifying adjective “many-sorted”and thus we will say signature and signature morphism instead of many-sorted signature and many-sorted signature morphism, respectively.
Remark. In [18], P.J. Higgins allows the variation of S but holds Σ fixed, while, in [2], J. B´enabou follows precisely the inverse criterium.
The category Sig, as was the case for MSet, is also a split bifibration on Set through the projection functor πSig for Sig. Since the category Sig can be identified to a subcategory of the category Sigf, defined in the fifth section, we refer to that section for examples of signature mor- phisms.
Proposition 2.9. The category Sig is bicomplete.
Proof. The proof of the bicompleteness of the category Sig is the same as that corresponding to the category MSet, but replacing everywhere ϕ by ϕ?^ × ϕ. §
We proceed next to define the category Alg of many-sorted algebras by applying the construction of Ehresmann-Grothendieck to a suitable contravariant functor Alg defined on Sig and taking values in Cat. Besides, we prove that Alg is a concrete and univocally transportable category for a “forgetful”functor from it to an adequate category, that the forgetful functor at issue has a left adjoint, and that Alg is a bicomplete category. Before we realize what has been announced we recall that, for a signature Σ and an S-sorted set A, the S?^ × S-sorted set of the finitary operations on A, HOpS (A) (thus denoted because, as we will show in the fourth section, it is an example of a Hall algebra), is (Hom(Aw, As))(w,s)∈S?×S , where Aw =
i∈|w| Awi^ ; and that a structure of Σ-algebra on A is a morphism F = (Fw,s)(w,s)∈S?×S in Sig(S) from Σ to HOpS (A). For a pair (w, s) ∈ S?^ ×S and a formal operation σ ∈ Σw,s, in order to
10 J. CLIMENT AND J. SOLIVERES
(S, Σ, A) to (T, Λ, B) triples (ϕ, d, f ), such that (ϕ, d) is a signature mor- phism from (S, Σ) to (T, Λ) and (ϕ, f ) a mapping from (S, A) to (T, B), while (2) P 0 is the functor from MSet ×Set Sig to MSet which sends a morphism (ϕ, d, f ) from (S, Σ, A) to (T, Λ, B) to the ms-mapping (ϕ, f ) from (S, A) to (T, B), and P 1 is the functor from MSet ×Set Sig to Sig which sends a morphism (ϕ, d, f ) from (S, Σ, A) to (T, Λ, B) to the signature morphism (ϕ, d) from (S, Σ) to (T, Λ). Then we have that the structural functors P 0 and P 1 are fibrations, and that the unique functor G from Alg to MSet ×Set Sig such that P 0 ◦ G = GMSet and P 1 ◦ G = πAlg makes the category Alg a concrete and univocally transportable category on the category MSet ×Set Sig. §
Before we prove the existence of a left adjoint T to G : Alg //MSet ×Set Sig, we agree on the following notation and terminology.
(1) For a signature Σ in Sig, the functor TΣ from SetS^ to Alg(Σ) is the left adjoint to the forgetful functor GΣ from Alg(Σ) to SetS^ ; (2) For a signature Σ and an S-sorted set of variables X, TΣ(X) is the free (also called the term or word ) Σ-algebra on X, and ηX is the insertion (of the generators) X into TΣ(X), the underlying S-sorted set of TΣ(X); (3) For a Σ-algebra A and a valuation f of the S-sorted set of variables X in A, f ]^ denotes the canonical extension of f up to TΣ(X), i.e., the unique Σ-homomorphism from TΣ(X) to A such that f ]^ ◦ ηX = f ; and (4) For an S-sorted mapping f from X to Y , f @^ denotes the unique Σ-homo- morphism from TΣ(X) to TΣ(Y ) such that f @^ ◦ηX = ηY ◦f , i.e., the value of the functor TΣ in f. Therefore f @^ is also (ηY ◦ f )].
Moreover, transposing to the many-sorted case the terminology coined for the single-sorted case, we call, for s ∈ S, the elements of TΣ(X)s, many-sorted terms for Σ of type (X, s), from now on abbreviated to terms for Σ of type (X, s), or, simply, to terms of type (X, s).
Proposition 2.13. There exists a functor T : MSet ×Set Sig //Alg left adjoint to the functor G : Alg //MSet ×Set Sig.
Proof. The functor T from MSet ×Set Sig to Alg given on objects (S, Σ, X) by T(S, Σ, X) = (Σ, TΣ(X)) and on arrows (ϕ, d, f ) : (S, Σ, X) //(T, Λ, Y ) as
T(ϕ, d, f ) = (d, f d) : (Σ, TΣ(X)) //(Λ, TΛ(Y )),
where f d^ = ((ηY )ϕ ◦f )]^ is the canonical extension of the S-sorted mapping (ηY )ϕ ◦f from X to TΛ(Y )ϕ up to the free Σ-algebra on X, is left adjoint to the functor G. §
For a morphism (ϕ, d, f ) : (S, Σ, X) //(T, Λ, Y ) in MSet ×Set Sig, the functor T : MSet ×Set Sig //Alg acting on (ϕ, d, f ) allows us to get the Σ-homomor- phism f d^ from TΣ(X) to TΛ(Y )ϕ, hence, for s ∈ S, it translates terms for Σ of type (X, s), i.e., elements P of TΣ(X)s, into terms for Λ of type (Y, ϕ(s)), i.e., elements f (^) sd (P ) of TΛ(Y )ϕ(s). In particular, the unit ηϕ^ of the adjunction
ϕ a^ ∆ϕ^ provides, for every^ S- sorted set X, the S-sorted mapping ηϕX : X //(
ϕ X)ϕ^ and if^ d^ :^ Σ^ //Λ is a
morphism of signatures, then (ϕ, d, ηϕX ) : (S, Σ, X) //(T, Λ,
ϕX) is a morphism in MSet×SetSig, hence the functor T acting on (ϕ, d, ηϕX ) determines the morphism
(d, ηd X ) : (Σ, TΣ(X)) //(Λ, TΛ(
ϕ X)),
MORPHISMS AND TRANSFORMATIONS OF FUJIWARA 11
where ηd X = ((η∐ ϕ X )ϕ ◦ ηXϕ )]^ is the Σ-homomorphism from TΣ(X) to TΛ(
ϕ X)ϕ that extends the S-sorted mapping (η∐ ϕ X )ϕ ◦ ηϕX from X to TΛ(
ϕ X)ϕ. There- fore, for s ∈ S, ηd X,s, the s-th component of η Xd , translates terms for Σ of type (X, s)
into terms for Λ of type (
ϕ X, ϕ(s)). The^ Σ-homomorphisms^ η
d X , as stated in the following proposition, are in fact the components of a natural transformation, and this contributes to explain their relevance as translators.
Proposition 2.14. Let d be a morphism of signatures from Σ to Λ. Then the family ηd^ = (ηd X )X∈U , which to an S-sorted set X assigns the Σ-homomorphism η Xd from TΣ(X) to TΛ(
ϕ X)ϕ, is a natural transformation from^ TΣ^ to^ d
∗ ◦TΛ ◦∐
ϕ, and so, for the forgetful functor GΣ from Alg(Σ) to SetS^ , the family GΣ ∗ ηd, i.e., the horizontal composition of the natural transformations ηd^ and idGΣ , also denoted by ηd, is a natural transformation from TΣ = GΣ ◦ TΣ to ∆ϕ ◦ TΛ ◦
ϕ, taking into account that GΣ ◦ d∗^ = ∆ϕ ◦ GΛ and TΛ = GΛ ◦ TΛ.
The category Alg of algebras, as was the case for the categories MSet and Sig, is also bicomplete. These results are already known, although, in particular, we are not aware of any suitably explicit and direct proof, as that provided by us below, of the cocompleteness of Alg.
Proposition 2.15. The category Alg is complete.
Proof. Let d : Σ //Λ be a signature morphism. Since the forgetful functors GΣ and GΛ create projective limits and the functors GΣ ◦d∗^ and ∆ϕ ◦GΛ from Alg(Λ) to SetS^ are identical, the functor d∗^ preserves projective limits, i.e., is continuous. But the category Sig is complete, and, for every signature Σ, Alg(Σ) is complete. Therefore, by Proposition 2.4, the category Alg is complete. §
To prove that the category Alg is cocomplete we begin by proving that, for every signature morphism d : Σ //Λ, the functor d∗^ from Alg(Λ) to Alg(Σ) has a left adjoint d∗.
Proposition 2.16. Let d : Σ //Λ be a signature morphism. Then there exists a functor d∗ : Alg(Σ) //Alg(Λ) that is left adjoint to the functor d∗.
Proof. We begin by defining the action of d∗ on the objects. Let A be a Σ-algebra. Then d∗(A) is the Λ-algebra defined as TΛ(
ϕA)/R
A, where RA (^) is the congruence
on TΛ(
ϕ A) generated by the^ T^ -sorted relation^ R
A, defined, for every t ∈ T , as
RA t =
(F (^) σA (ai | i ∈ |w|), s), d(σ)((ai, wi) | i ∈ |w|)
s ∈ ϕ−^1 [t], w ∈ S?, σ ∈ Σw,s, a ∈ Aw
Following this we define the action of d∗ on the morphisms. Let f be a Σ-homo- morphism from A to A′. Then RA^ ⊆ Ker(prRA′^ ◦ (
ϕ f^ )
@) because, for t ∈ T and
((Fσ (ai | i ∈ |w|), s), d(σ)((ai, wi) | i ∈ |w|)) ∈ RA t , we have that
[(
ϕf^ )
@(F A
σ (ai^ |^ i^ ∈ |w|), s)] = [(fs(F^ A σ (ai^ |^ i^ ∈ |w|)), s)] = [d(σ)(fwi (ai, wi) | i ∈ |w|)] = [(
ϕf^ )
@(d(σ)((a i, wi)^ |^ i^ ∈ |w|))].
From this it follows that there exists a unique Λ-homomorphism d∗(f ) from d∗(A) to d∗(A′) such that d∗(f ) ◦ prRA = prRA′^ ◦ (
ϕ f^ )
After this we prove that the functor d∗, which sends a Σ-algebra A to the Λ-algebra d∗(A) = TΛ(
ϕ A)/R
A (^) and a Σ-homomorphism f from A to A′ (^) to the
Λ-homomorphism d∗(f ) from d∗(A) to d∗(A′), is left adjoint to d∗.
MORPHISMS AND TRANSFORMATIONS OF FUJIWARA 13
It is obvious that f ^ is the unique Λ-homomorphism from d∗(A) into B such that the above diagram commutes, hence d∗ a d∗. §
Proposition 2.17. The category Alg is cocomplete.
Proof. The category Sig is cocomplete. For every signature Σ, the category Alg(Σ) is cocomplete. The functor Alg is locally reversible. Therefore, by Proposition 2.5, the category Alg is cocomplete. §
From Propositions 2.15 and 2.17 we obtain immediately the following
Corollary 2.18. The category Alg is bicomplete.
The contravariant functor Alg from Sig to Cat is not only useful to construct the category Alg. Actually, as we will show from here up to the end of this section, Alg, together with a pseudo-functor Ter from Sig to Cat, and a pseudo-extranatural transformation (Tr, θ) (from a pseudo-functor on Sigop^ × Sig to Cat, induced by Alg and Ter, to the functor, between the same categories, constantly Set), will enable us to construct a new institution on Set, the many-sorted term institution, denoted by Tm = (Sig, Alg, Ter, (Tr, θ)), but for a concept of institution that is strictly more general than that of generalized V-institution in [12]. For the institution Tm on Set, as we will prove, it happens that the existence of the pseudo-functor Ter follows from the fact that, for every signature Σ, the terms for Σ, understood in a generalized sense to be explained below, have a categorical interpretation as the morphisms of a category Ter(Σ). Furthermore, the component Tr of the pseudo-extranatural transformation (Tr, θ) depends for its existence on the fact that the generalized terms have canonically associated generalized term operations on the algebras. Therefore, to proceed properly, we should begin by defining, for a Σ-algebra A and an S-sorted set X, the concept of many-sorted X-ary operation on A, that of many-sorted X-ary term operation on A, and the procedure of realization of terms P of type (X, s) as term operations P A^ on A.
Definition 2.19. Let X be an S-sorted set, A a Σ-algebra, s ∈ S and P ∈ TΣ(X)s a term for Σ of type (X, s). Then
(1) The Σ-algebra of the many-sorted X-ary operations on A, OpX (A), is AAX^ , i.e., the direct AX -power of A, where AX is Hom(X, A), the (ordi- nary) set of the S-sorted mappings from X to A. From now on, to shorten terminology, we will speak of X-ary operations on A instead of many-sorted X-ary operations on A. (2) The Σ-algebra of the many-sorted X-ary term operations on A, TerX (A), is the subalgebra of OpX (A) generated by the subfamily
PXA = (PX,sA )s∈S = ({ prAX,s,x | x ∈ Xs })s∈S
of OpX (A) = AAX^ , where, for every s ∈ S and x ∈ Xs, prAX,s,x is the mapping from AX to As which sends a ∈ AX to as(x). From now on, to shorten terminology, we will speak of X-ary term operations on A instead of many-sorted X-ary term operations on A. (3) We denote by TrX,A^ the unique Σ-homomorphism from TΣ(X) to OpX (A) such that prAX = TrX,A^ ◦ ηX , where prAX is the S-sorted mapping (prAX,s)s∈S from X to OpX (A), with prAX,s = (prAX,s,x)x∈Xs , for every s ∈ S. Further- more, P A^ denotes the image of P under TrX, s A, and we call the mapping P A^ from AX to As, the term operation on A determined by P , or the term realization of P on A.
14 J. CLIMENT AND J. SOLIVERES
From now on, to simplify the notation, we will also denote by TrX,A^ the co- restriction of the Σ-homomorphism TrX,A^ : TΣ(X) //OpX (A) to the subalgebra TerX (A) of its codomain OpX (A). What we want to prove now is the compatibility between the translation of terms and their realization as term operations on the algebras. But for this it will be shown to be useful to take into account the following auxiliary functors and natural transformation.
Definition 2.20. For a mapping ϕ : S //T , an S-sorted set X, a T -sorted set Y , and an S-sorted mapping f : X //Yϕ, we have the following functors and natural transformation
(1) H(Y, ·) is the covariant hom-functor from SetT^ to Set which, we recall, sends a T -sorted set A to the set H(Y, ·)(A) = AY , and a T -sorted mapping u from A to B to the mapping H(Y, ·)(u) from AY to BY which assigns to a T -sorted mapping t from Y to A the mapping u ◦ t from Y to B. (2) H(X, ·) ◦ ∆ϕ is the functor from SetT^ to Set which sends a T -sorted set A to the set (Aϕ)X , and a T -sorted mapping u from A to B to the mapping H(X, ·)(uϕ) from (Aϕ)X to (Bϕ)X which assigns to an S-sorted mapping from X to Aϕ the mapping uϕ ◦ from X to Bϕ. (3) ϑϕ,f^ is the natural transformation from H(Y, ·) to H(X, ·) ◦ ∆ϕ which sends a T -sorted set A to the mapping ϑϕ,fA from AY to (Aϕ)X which assigns to a morphism t : Y //A in AY the morphism tϕ ◦ f in (Aϕ)X.
From this definition, for a T -sorted set A, we get the S-sorted mapping Υϕ,fA
from OpX (Aϕ) = A( ϕA ϕ)Xto OpY (A)ϕ = (AAY^ )ϕ = AA ϕ Y which sends, for s ∈ S, a
mapping a : (Aϕ)X //Aϕ(s) to the mapping a ◦ ϑϕ,fA : AY //Aϕ(s), that we will use in the proof of the following proposition and corollary.
Proposition 2.21. Let (ϕ, d, f ) : (S, Σ, X) //(T, Λ, Y ) be a morphism in the cat- egory MSet×Set Sig. Then, for every Λ-algebra A and term P ∈ TΣ(X)s for Σ of type (X, s), the mappings P d
∗(A) ◦ϑϕ,fA and f (^) sd (P )A^ from AY to Aϕ(s) are identical.
Proof. Let a ∈ AY be a T -sorted mapping from Y to A. Then the following diagram commutes
X
ηX (^) //
f ≤ ≤
TΣ(X)
f d ≤ ≤
ED
BC
(aϕ ◦ f )]
o o
Yϕ
(ηY )ϕ (^) //
aϕ ' '
OOO
OOO
OOO
OOO
OOO
OOO
O TΛ(Y^ )ϕ
(a])ϕ ≤ ≤ Aϕ
hence, for every P ∈ TΣ(X)s, we have that
f (^) sd (P )A(a) = (a])ϕ(s) ◦ f (^) sd (P ) = (aϕ ◦ f )]s(P ) = P d
∗(A) (aϕ ◦ f ) = P d
∗(A) ◦ ϑϕ,fA (a).
Therefore f (^) sd (P )A^ = P d
∗(A) ◦ ϑϕ,fA , as asserted. §
16 J. CLIMENT AND J. SOLIVERES
S-sorted set to the free Σ-algebra on another S-sorted set, i.e., morphisms in a category Ter(Σ). This categorical perspective about terms, in its turn, will allow us to get a functor TrΣ, of realization of terms as term operations, from Alg(Σ) × Ter(Σ) to Set, and therefore to define (in the next section) the validation of equations, understood as ordered pairs of coterminal terms in the corresponding generalized sense, in an algebra. Since it will be fundamental in all that follows, we provide, for a signature Σ, the full definition of the category Ter(Σ) and also the explicit definition of the procedure of realization of the terms for Σ as term operations on a given Σ-algebra. Observe that we depart, in the definition of the category Ter(Σ), but only for this type of category, from the (non-Ehresmannian) tradition, in calling a category by the name of its morphisms.
Definition 2.23. Let Σ be a signature and A a Σ-algebra. Then
(1) The category of terms for Σ, Ter(Σ), is the dual of Kl(TΣ). Therefore Ter(Σ) has (a) As objects the elements of US^ , i.e., the elements of the set of objects of SetS^ , (b) As morphisms from X to Y , that we call terms for Σ of type (X, Y ), or, simply, terms of type (X, Y ), the S-sorted mappings from Y to TΣ(X), (c) As composition, denoted in Ter(Σ) and Kl(TΣ) by ¶, the operation which sends terms P : X //Y and Q : Y //Z in Ter(Σ) to the term Q ¶ P : X //Z in Ter(Σ) defined as Q ¶ P = μX ◦ P @^ ◦ Q, where μX is the value at X of the multiplication μ of the monad TΣ = (TΣ, η, μ) and P @^ the value of the functor TΣ at the S-sorted mapping P : Y //TΣ(X), and (d) As identities the values of η, the unit of the monad TΣ, in the S-sorted sets. (2) If P : X //Y is a term for Σ of type (X, Y ), then P A, the term oper- ation on A determined by P , or the term realization of P on A, is the mapping from AX to AY which assigns to a valuation f of the variables X in A the valuation f ]^ ◦ P of the variables Y in A, i.e., the composition of P : Y //TΣ(X) and the underlying mapping of f ]^ : TΣ(X) //A, the canonical extension of the valuation f : X //A. After associating to every signature Σ the corresponding category Ter(Σ) of terms, we proceed to assign to every signature morphism d : Σ //Λ a corre- sponding functor d¶ from Ter(Σ) to Ter(Λ).
Proposition 2.24. Let d : Σ //Λ be a signature morphism. Then there exists a functor d¶ from Ter(Σ) to Ter(Λ) defined as follows
(1) d¶ sends an S-sorted set X to the T -sorted set d¶(X) =
ϕ X. (2) d¶ sends a morphism P from X to Y in Ter(Σ) to the morphism d¶(P ) = (θϕ)−^1 (ηd X ◦ P ) from
ϕ X^ to^
ϕ Y^ in^ Ter(Λ), where^ η d X is the^ Σ-homo- morphism from TΣ(X) to TΛ(
ϕ X)ϕ^ that extends the^ S-sorted mapping (η∐ ϕ X )ϕ ◦ ηXϕ from X to TΛ(
ϕ X)ϕ, i.e., for^ η
d X we have that ηd X = ((η∐ ϕ X )ϕ ◦ ηϕX )], θϕ^ the natural isomorphism of the adjunction
ϕ a^ ∆ϕ, and^ η
ϕ (^) the unit of the same adjunction.
MORPHISMS AND TRANSFORMATIONS OF FUJIWARA 17
Proof. We restrict ourselves to prove that d¶ preserves compositions since the preservation of identities is easy to do. Let P : X //Y and Q : Y //Z be mor- phisms in Ter(Σ). Then we have the following equations:
d¶(Q ¶ P ) = (θϕ)−^1 (ηd X ◦ P ]^ ◦ Q) = (θϕ)−^1 (ηd X ) ◦
ϕP^
] ◦ ∐
ϕQ, d¶(Q) ¶ d¶(P ) = d¶(P )]^ ◦ d¶(Q) = d¶(P )]^ ◦ (θϕ)−^1 (η Yd ) ◦
ϕQ,
therefore, to prove that d¶(Q ¶ P ) = d¶(Q) ¶ d¶(P ) it is enough to verify the following equation
(θϕ)−^1 (ηd X ) ◦
ϕP^ ] (^) = d¶(P )] (^) ◦ (θϕ)− (^1) (ηd Y ).
But for this, because of the commutativity of the following diagram
ϕ TΣ(Y^ )
GF ED
(θϕ)−^1 (η Yd )
ϕ η
d Y (^) //
ϕ P^
]
≤ ≤
ϕ TΛ(
ϕ Y^ )ϕ
εϕ TΛ(∐ ϕ Y ) / /
∐ ϕ d¶(P^ )
] ϕ ≤ ≤
TΛ(
ϕ Y^ )
d¶(P )] ∐^ ≤^ ≤ ϕ TΛ(X)^ ∐ ϕ η d X
@A BC
(θϕ)−^1 (ηd X )
O O
ϕ TΛ(
ϕ X)ϕ εϕ TΛ(∐ ϕ X)
/ / TΛ(
ϕ X)
it is enough to verify the following equation
(1) ηd X ◦ P ]^ = d¶(P )]ϕ ◦ η Yd.
But equation (1) is valid because the restriction of both terms to the generating ms-set Y coincide:
ηd X ◦ P ]^ ◦ ηY = ηd X ◦ P,
d¶(P )]ϕ ◦ ηd Y ◦ ηY = d¶(P )]ϕ ◦ (η∐ ϕ Y )ϕ ◦ ηϕY = d¶(P )ϕ ◦ ηϕY = (θϕ)−^1 (ηd X ◦ P )ϕ ◦ ηYϕ = (η Xd ◦ P )]^ ◦ ηY = ηd X ◦ P. §
We state now for the generalized terms the homologous of the right-hand diagram in the first part of Corollary 2.22, i.e., the invariant character under signature change of the realization of terms as term operations in arbitrary, but fixed, algebras. We remark that from this fact we will get, in the third section, the invariance of the relation of satisfaction under signature change.
Proposition 2.25. Let d : Σ //Λ be a signature morphism. Then, for every Λ-algebra A and term P : X //Y for Σ of type (X, Y ), the mappings
P d
∗(A) ◦ θϕX,A, θϕY,A ◦ d¶(P )A^ : A∐ ϕX //(Aϕ)Y
are identical.
MORPHISMS AND TRANSFORMATIONS OF FUJIWARA 19
Proof. We restrict ourselves to prove the first part of the lemma. Since (Q ¶ P )A is the mapping from AX to AZ which sends an S-sorted mapping u : X //A to the S-sorted mapping
u]^ ◦ (Q ¶ P ) = u]^ ◦ μX ◦ P @^ ◦ Q : Z //A,
where, we recall, μX is the value in X of the multiplication μ of the monad TΣ = (TΣ, η, μ) and P @^ the value in the S-sorted mapping P : Y //TΣ(X) of the functor TΣ; and QA^ ◦ P A^ the mapping from AX to AZ which sends an S-sorted mapping u : X //A to the S-sorted mapping
(u]^ ◦ P )]^ ◦ Q : Z //A,
to show that (Q ¶ P )A^ = QA^ ◦ P A^ it is enough to prove that the Σ-homomorphisms u]^ ◦ μX ◦ P @^ and (u]^ ◦ P )]^ from TΣ(Y ) to A are identical. But this follows from the equation u]^ ◦ μX ◦ P @^ ◦ ηY = (u]^ ◦ P )]^ ◦ ηY ,
that, in its turn, is a consequence of the laws for the monad TΣ and of the equation
P @^ ◦ ηY = ηTΣ(X) ◦ P,
where ηY is the canonical embedding of Y into TΣ(Y ) and ηTΣ(X) the canonical embedding of TΣ(X) into TΣ(TΣ(X)). §
This lemma has as an immediate consequence the following
Corollary 2.28. Let Σ be a signature and A a Σ-algebra. Then there exists a functor TrΣ,A^ from Ter(Σ) to Set which sends an S-sorted set X to the set TrΣ,A(X) = AX and a term P : X //Y to TrΣ,A(P ) = P A^ : AX //AY , the term operation on A determined by P.
Therefore, from the definition of the object and morphism mappings of the func- tors of the type TrΣ,A, we see that they encapsulate the procedure of realization of terms. And, from the fact that they preserve identities and compositions in Ter(Σ), we conclude that they formally represent the two basic intuitions about the behaviour of the just named procedure, i.e., that the realization of an identity term is an identity term operation, and that the realization of a composite of two terms is the composite of their respective realizations (in the same order).
Remark. By identifying the Σ-algebras with the TΣ-algebras, the just stated corollary can be interpreted as meaning that every Σ-algebra is a functor from Ter(Σ) = Kl(TΣ)op^ to Set.
Before stating the following lemma we recall that, for an S-sorted mapping f from an S-sorted set A into a like one B and an S-sorted set X, fX is the value at X of the natural transformation H(·, f ) from the contravariant functor H(·, A) to the contravariant functor H(·, B), both from (SetS^ )op^ to Set.
Lemma 2.29. Let f be a Σ-homomorphism from A to B and P a term of type (X, Y ) in Ter(Σ). Then the mappings P B^ ◦ fX and fY ◦ P A^ from AX to BY are identical, and we agree to denote it by fP.
Proof. Given an S-sorted mapping u : X //A, we have that (f ◦ u)]^ = f ◦ u], by the universal property of the free Σ-algebra on X and taking into account that f is a Σ-homomorphism from A to B. Therefore, since P B^ ◦ fX (u) = (f ◦ u)]^ ◦ P , and fY ◦ P A(u) = f ◦ (u]^ ◦ P ), we have that P B^ ◦ fX (u) = fY ◦ P A(u). Thus P B^ ◦ fX = fY ◦ P A. §
This lemma has as an immediate consequence the following
20 J. CLIMENT AND J. SOLIVERES
Corollary 2.30. Let Σ be a signature and f a Σ-homomorphism from A to B. Then there exists a natural transformation TrΣ,f^ from the functor TrΣ,A^ to the functor TrΣ,B^ which sends an S-sorted set X to the mapping TrΣX,f = fX from AX to BX. Besides, we have that
(1) For idA, the identity Σ-homomorphism at A, it is the case that TrΣ,idA^ = idTrΣ,A. (2) If g : B //C is another Σ-homomorphism, then TrΣ,g◦f^ = TrΣ,g^ ◦ TrΣ,f^. Therefore, the naturalness of the procedure of realization of terms as term oper- ations on the different algebras is embodied in the natural transformations of the type TrΣ,f^.
Remark. By identifying the Σ-homomorphisms with the TΣ-homomorphisms, the just stated corollary can be interpreted as meaning that every Σ-homomorphism f from A to B is a natural transformation from the functor TrΣ,A^ to the functor TrΣ,B, both from Ter(Σ) = Kl(TΣ)op^ to Set. Actually, each homomorphism (d, f ) from an algebra (Σ, A) into a like one (Λ, B) is identifiable to a morphism (in the category (Cat)//Set, see [17], p. (sub) 186) from the object (Ter(Σ), TrΣ,A) over
Set to the object (Ter(Λ), TrΛ,B) over Set, concretely, to the morphism given by the pair (d¶, (θϕ ·,B )−^1 ◦H(·, f )), where H(·, f ) is the natural transformation from the contravariant hom-functor H(·, A) to the contravariant hom-functor H(·, Bϕ), and (θϕ ·,B )−^1 the natural isomorphism from H(·, Bϕ) to H(
ϕ(·), B). Observe that the naturalness of (θ ·ϕ,B )−^1 ◦ H(·, f ) means that, for every term P for Σ of type (X, Y ),
the mappings (θϕY,B )−^1 ◦ H(Y, f ) ◦ P A^ and d¶(P )B^ ◦ (θϕX,B )−^1 ◦ H(X, f ) from AX to B∐ ϕ Y are identical. From the identification of the homomorphisms between algebras in the category Alg to some convenient morphisms between the associated objects over Set, we can conclude, e.g., that the concept of homomorphism as defined by B´enabou in [2] (that does not allow the variation of the signature and therefore it works between algebras of the same signature (see [2], p. (sub) 16, last paragraph)), corresponds itself, for a signature Σ and a Σ-homomorphism f from A to B, to the (very special) case in which (d¶, (θϕ ·,B )−^1 ◦ H(·, f )) is precisely
(d¶, (θ ·ϕ,B )−^1 ◦ H(·, f )) = (IdTer(Σ), H(·, f )),
i.e., definitely, it corresponds to the natural transformation TrΣ,f^ from the functor TrΣ,A^ to the functor TrΣ,B.
For an arbitrary, but fixed, signature Σ the family of functors (TrΣ,A)A∈Alg(Σ) together with the family of natural transformations (TrΣ,f^ )f ∈Mor(Alg(Σ)) actually
constitute the object and morphism mappings, respectively, of a functor TrΣ,(·)^ from the category Alg(Σ) to the exponential category SetTer(Σ). And it is precisely the functor TrΣ,(·)^ that will allow us to prove, in the following proposition, the existence of a functor TrΣ^ from Alg(Σ) × Ter(Σ) to Set that formalizes the realization of terms as term operations on algebras, but taking into account the variation of the algebras through the homomorphisms between them.
Proposition 2.31. Let Σ be a signature. Then there exists a functor TrΣ^ from Alg(Σ) × Ter(Σ) to Set defined as follows
(1) TrΣ^ sends a pair (A, X), formed by a Σ-algebra A and an S-sorted set X, to the set TrΣ(A, X) = TrΣ,A(X) = AX of the S-sorted mappings from X to the underlying S-sorted set A of A.
