On the category of algebraizable logics

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On the category of algebraizable logics

March 30, 2006

P.Arndt¹, R.A.F reire¹, O.O.Luciano², H.L.M ariano²

(1) CLE-UNICAMP: peter.arndt@gmail.com, freire@cle.unicamp.br (2) IME-USP: odilon.luciano@gmail.com, hugomar@ime.usp.br

This research has been supported by FAPESP, under the Thematic Project “ConsRel” (number 2004/1407-2). March 2006

Abstract

We explore the possibility and some potential payoffs of using the theory of accessible categories in the study of categories of logics: the category of finitary structural logics and its subcategory of algebraizable logics.

1 Introduction

This work responds to an increasing tendency to consider logics by their relations to other logics. Accordingly, the (potential) use of categories in logic we are considering here is not to give semantics for formal languages or perform proof-theoretical considerations. Rather we are considering the use of categories for what was their original purpose, to study the “sociology of mathematical objects”, and thus we are considering categories of logics, i.e. categories whose objects are logical systems and whose morphisms are translations.

This is a relatively recent point of view which has largely come into consideration through the topic of combination of logics: The goal of combining two logicsL1 andL2 has been described as to obtain “the smallest logic system for the combined language which is a conservative extension of bothL1 andL2”. In [14] it was proposed that this rather informal statement could be given precise meaning by considering the combination of logics as a colimit construction in an appropriate category of logics. Following this idea the notions of modulated fibring, metafibring and combination of institutions and π-institutions have been presented as colimit constructions in different categories. It has also been observed that a presentation of a logic as a colimit of others can be seen as a splitting of this logic into other, simpler, logics, possibly helping to understand the more complex logic. Possible translation semantics and remote algebraization (see e.g. [6]) are concrete developments of this point of view. We observe that the situation of a logicLthat is completely determined by the translation of other logicsLi intoL can be seen as a “covering” ofLby the Li. In [1] we turn this intuition into a mathematical statement choosing a category of logics and giving a rigorous definition of covering.

Thus it seems reasonable to adopt a global perspective on logic and consider not only constructions with particular logics but the whole category of (some kind of) logics. Defining such a category is not a completely straightforward enterprise. It means giving a partial answer to the identity problem, i.e. the question of when two given presentations of logic systems can be considered to describe the same logic: Two such presentations should be expected to describe isomorphic objects in their ambient category. This gives only a partial answer, since the identity problem also includes the task of comparing logic systems which are given in different styles of presentation, e.g. a Hilbert calculus and a sequent calculus. It seems difficult to unite two such differently presented systems in one category. However, the identity problem is not our main concern here and we shall only briefly readdress it towards the end of the article.

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Here we focus a category whose objects are the Blok-Pigozzi algebraizable logics, taking first natural choices of categories of signatures and of logics. As previous research about algebraizable logics, in categorial perspective, we have the work of Victor Fern´andez and Marcelo E. Coniglio, about coproducts ([13]) and of Juliana Bueno- Soler, Walter Carnielli and Marcelo E. Coniglio ([6], see also [7] about “bounded” products, where they adopt an alternative definition of signature morphism. Also in choosing such categories it should ultimately be of advantage to take into account global categorial properties and not only the ad hoc requirements of the constructions one wishes to perform. We believe that the theory of accessible categories has to offer theorems and intuitions that can be of use in tackling technical as well as conceptual questions arising in Universal Logic. Technically, a categoryCis (finitely) accessible whenChas filtered colimits and asetAof objects that arefinitely presentable (the Hom-functor C(a, .) preserves filtered colimits, a ∈ A) and all object in the category is a filtered colimit of a diagram of finitely presentable objects in that setA. Intuitively to take a colimit of a filtered diagram of objects in a category is “glue compatible partial information”; the finitely presented objects are the “fundamental bricks” of the accessible category: any object is a gluing of these basic pieces. Categories with “algebraic soul”

are natural examples of accessible categories. The main point of these case studies, in each of which we prove a certain category of logics to be accessible, is twofold: first, to show that and how the notion of accessibilitycan apply to categories of logics in a natural way (this is not obvious, see the last section) and, second that some benefits can be gained thereof. We are, however, aware that the categories we present are not a good ambient for proper logical studies since they give unsatisfactory answers to the identity problem. It remains a project for the future to give a category of logics with good global propertiesand an appropriate notion of isomorphism of logics; we outline a possible solution for this task in the section 5.

2 Locally presentable and accessible categories

With the advent of category theory came the task of characterizing categories “of an algebraic character” by the means of categorial language. One answer that has been given is through the theory of monads and its variations;

see [4] for a survey. Another one came from sketch theory; the varieties from Universal Algebra are exactly the categories ofSet-models of finite product sketches — this is close in spirit to the usual characterization of varieties as categories of Set-models of equational theories.

A more general class of “algebraic” categories are the locally finitely presentable categories. The key observation leading to the definition of these is that in the familiar algebraic categories every object is a directed colimit of finitely presentable objects, i.e. objects specifiable by a finite number of generators and relations (a directed colimit is the colimit of a directed poset, i.e. one in which for every pair of elements there is one greater than each of the two). In these familiar cases the property of an objectAbeing finitely presentable has an equivalent categorial description: Ais finitely presentable iff the functorHom(A,−) preserves directed colimits. A category is called locally finitely presentable if it is cocomplete and has aset of finitely presentable objects such that every object is a directed (or equivalently, filtered, see [2]) colimit of objects from this set. Examples of such categories are all varieties of finitary many-sorted algebras as well as the categories of sets and posets and categories of Set-valued functors on a small category.

A further generalization is the notion of locally λ-presentable category, where λ is a regular cardinal: A poset is called λ-directed if every set of elements of cardinality strictly lesser than λ has an upper bound, an objectAis calledλ-presentable ifHom(A,−) preserves colimits of (diagrams over) such posets and a category is λ-presentable iff it is cocomplete and there is a set ofλ-presentable objects such that each object is aλ-directed colimit of these objects. Finally, a category is called locally presentable if it is locally λ-presentable for some regular cardinalλ. Model-theoretically the locally presentable categories have been described as

1. Categories of Set-models of limit sketches;

2. Categories of Set-models of essentially algebraic theories, i.e. equational theories of partial operations in which the domain of each operation is defined by equations in the preceding operations;

3. Categories of Set-models of so-called limit theories, which are certain infinitary first order theories. A locally λ-presentable category is the category of models of an Lλλtheory. In particular, for λ = ω, this means that locally finitely presentable theories are categories of models of finitary first order theories.

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Examples of locally presentable (but not finitely presentable) categories include the category of categories, that of convergence spaces and any complete lattice considered as a category.

The last notion we will use here is that ofλ-accessible category, which are defined like the locallyλ-presentable categories except for requiring onlyλ-directed colimits to exist instead of arbitrary ones. An accessible category is, again, a category which isλ-accessible for someλ. These categories are exactly the categories ofSet-models of arbitrary sketches or, alternatively, the categories of Set-models of so-called basic sentences of the infinitary first order logicL∞∞(which allows disjunctions and quantification over arbitrary (small) sets of formulas/variables).

Examples not included in the previous classes are the categories of fields and Hilbert spaces.

For an introduction to these notions and their theories see [2] and [20]. There are also other related notions like locally multipresentable categories, weakly locally presentable categories andD-accessible categories (where D is some class of small categories) to some of which the related types of sketches and first order theories have also been identified.

The theory of accessible and locally presentable categories provides some powerful tools and also some conceptual clarifications. As an example of the first, we mention the theorem of [16] which says that an accessible category is locally presentable iff it is complete iff it is cocomplete. In Section 4 we prove a certain category of logics to be accessible and, since we know it not to have an initial object (hence not to be cocomplete), we know that it is not complete either. By investigating whether this category is of any of the types of categories mentioned in the previous paragraph we could further try to discover which type of limit is missing. For the second recall that the above types of categories were meant to capture the essential properties of categories of algebraic objects. Now there is a general duality phenomenon in mathematics between algebraic and geomet- rical/topological objects as witnessed by Stone duality, Gelfand duality, the duality between algebraic sets and k-algebras and many others. In [16] this informal observation is turned into a real mathematical statement by a theorem saying that the dual of a locally presentable category can not also be locally presentable except when it is a (category coming from a) poset. This seems to indicate that the notion of local presentability has succeeded in capturing some of the features of algebraic categories. In view of the ever returning question of “how algebraic”

is logic, it thus seems to be interesting to investigate whether categories of logics are locally presentable.

Having seen these attempts to characterize categories of algebras, one is naturally led to think about whether the parallel between universal logic and universal algebra could be continued here. It has been argued, as mentioned in [8], that logical structures should be seen as one of the fundamental species of mathematical structures in the sense of Bourbaki. Assuming this to be the case one can wonder if it is possible to recognize the logical character of a category by categorial properties. If so, it should be in some different vein than the limit closure and generator properties defining locally presentable categories; after all we show below that a certain category of logics is locally presentable. The theory of fibred categories seems to fit very well for considering categories of logics as is convincingly shown in [9]. Indeed it seems reasonable that a category of logics should be fibred over a base category of signatures and finding further decisive properties (maybe like the base category being a category of free algebras) of the involved categories could give a sharp categorial picture.

3 The categories of signatures and logics

3.1 The category S

The categorySis the category of signatures and morphisms of signatures. In what follows, letX ={x0, x1, . . . , xn, . . .}

be an enumerable set (written in a fixed order) as in [12].

3.1.1 What isS?

The objects of S are signatures. A signature Σ is a sequence of sets Σ = (Σn)_n∈ω such that Σi∩Σj =∅for all i < j < ω . We write|Σ|=F

n∈ωΣn = S

n∈ωΣn× {n} and we denote by F(Σ) the set of all (propositional) formulas built with signature Σ over the variables in X.The notion of complexity l(ϕ) of the formula ϕ is the usual:

• l(ϕ) = 1 ifϕ∈ X∪Σ0;

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• l(ϕ) = 1 +l(ψ0) +. . . l(ψn−1) ifϕ=c(ψ0, . . . , ψn−1), wherec∈Σn andn >0.

If Σ,Σ⁰ are signatures then a morphismf : Σ−→Σ⁰ is a sequence of functionsf = (fn)_n∈ω, wherefn: Σn−→Σ⁰_n. For each morphismf : Σ−→Σ⁰there is only one functionfb:F(Σ)−→F(Σ⁰), called theextension off, such that:

• fb(x) =xifx∈X;

• fb(c) =f0(c) ifc∈Σ0;

• fb(c(ψ0, . . . , ψ_n−1) =fn(c)(fb(ψ0), . . . ,f(ψb _n−1)) if c∈Σn, n >0.

Composition in S is componentwise. The extension of the formula algebra of a composition is the extensions’

composition. Identities inSare the sequences of identities on each leveln, forn∈ω. The extension of an identity is the identity function on the formula algebra.

3.1.2 Some facts aboutS

Remark 1 About stratification: For each signature Σ and for each n ∈ ω we consider the set of Σ-formulas : F(Σ)[n] = {ϕ ∈ F(Σ) : the set of variables that occur in ϕ is precisely {x₀, . . . , x_n−1}}. Let f : Σ−→Σ⁰ be a signature morphism and fb: F(Σ)−→F(Σ⁰) be the induced formula algebra function; we can see directly by induction on the complexity of Σ-formulas thatfb“preserves stratification”: ifϕ∈F(Σ)[n] thenfb(ϕ)∈F(Σ⁰)[n].

Fact 2 About substitution:

• For any substitution functionσ:X−→F(Σ), there is only one extension eσ:F(Σ)−→F(Σ)such that eσis an “homomorphism”: σ(x) =e σ(x), for allx∈X andeσ(cn(ψ0, . . . , ψn−1) =cn(eσ(ψ0)), . . . ,eσ(ψn−1)), for all c_n∈Σ_n,n∈ω; it follows that for anyθ(x₀, . . . , x_n−1)∈F(Σ)σ(θ(xe ₀, . . . , x_n−1)) =θ(σ(x₀), . . . , σ(x_n−1)).

The identity substitution induces the identity homomorphism on the formula algebra; the composition substitutionof the substitutionsσ⁰, σ:X−→F(Σ)is the substitutionσ⁰⁰:X−→F(Σ),σ⁰⁰=σ⁰? σ .

=σe⁰◦σ andσf⁰⁰=σ^⁰? σ .

=σe⁰◦eσ.

• Let f : Σ−→Σ⁰ be a S-morphism. Then for any substitutionσ:X−→F(Σ) there is another substitution σ⁰ :X−→F(Σ⁰)such thatσe⁰◦fb=fb◦σ.e

F(Σ) F(Σ⁰)

F(Σ⁰) F(Σ)

fb

eσ σe⁰

?

-

Proposition 3S is a complete and cocomplete category.

Proof. Observe thatS is equivalent to the functor categorySetN, whereNis the discrete category with object classN, thenS has all small limits and colimits and they are componentwise.

Here we write the constructions but omit the (standard) verifications:

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Limits LetI be a small category andD :I−→S, (Σⁱ ^f

h

−→Σ^j)_(h:i→j)∈I a diagram. Then (Σ,(πⁱ)_i∈Obj(I)) is the limit of this diagram if we take:

• Σn={c= (ci)_i∈Obj(I)∈Q

i∈objIΣⁱ_n: for allI-arrow (i−→^h j),f_n^h(ci) =cj};

• πⁱ_n: Σn−→Σⁱ_n such that ifc= (ci)_i∈Obj(I)∈Σn thenπⁱ(c) =ci,n∈ωandi∈I.

In fact:

(a) By construction, Σ∈Obj(S) andπⁱ∈ S(Σ,Σⁱ),i∈Obj(I);

(b) For all (i−→^h j)∈M or(I)π^j=f^h◦πⁱ. This is because for all n∈Nand c= (ci)_i∈Obj(I)∈Σn, we have π^j(c) =f^h(πⁱ(c));

(c) (Σ,(πⁱ)_i∈Obj(I⁾) is the universal cone over the diagramD: if (Σ⁰,(αⁱ)_i∈Obj(I⁾) is a commutative cone over the diagram D then, for all n ∈ ω, define αn : Σ⁰_n−→Σn , if c⁰ ∈ Σn take αn(c⁰) = (αⁱ_n(c⁰))_i∈Obj(I). It follows thatα_n is a well defined function,αis a signature morphism such thatαⁱ=πⁱ◦α,i∈I andαis the only signature morphism satisfying that property.

Now we describe the most important kind of colimits in this work.

Filtered colimits Let (I,≤) be a directed ordered set and D : (I,≤)−→S, (Σⁱ ^f

ij

−→ Σ^j)_(i≤j)∈I a diagram.

Then (Σ,(γⁱ)_i∈I) is the colimit of this diagram if we take:

• Σn = (F

i∈I(Σⁱ)n)/ ∼n where, if ci ∈ Σⁱ_n, cj ∈ Σ^j_n (ci, i) ∼n (cj, j) iff there is a k ≥ i, j such that (f^ik)n(ci) = (f^jk)n(cj),n∈ω: it follows from the directness assumption that∼nis an equivalence relation onF

i∈I(Σⁱ)n;

• γ_nⁱ : Σⁱ_n−→Σn such that ifci∈Σⁱ_n thenγ_nⁱ(ci) = [(ci, i)],n∈ω andi∈I.

In fact:

(a) By construction, Σ∈Obj(S) andγⁱ∈ S(Σⁱ,Σ),i∈I;

(b) For all (i≤k)∈I,γⁱ=γ^k◦f^ik. This is because, for alln∈Nandc_i∈Σⁱ_n, we have (c_i, i)∼_n(f_n^ik(c_i), k);

(c) (Σ,(γⁱ)_i∈I) is the universal cocone over the diagram D: if (Σ⁰,(αⁱ)_i∈I) is a commutative cocone over the diagram D then, for alln∈ω, defineαn : Σn−→Σ⁰_n, if c∈Σn,choose ani∈I such thatc= [(ci, i)] and take αn(c) = αⁱ_n(ci). It follows that αn is a well defined function, αis a signature morphism such that αⁱ=α◦γⁱ,i∈Iand αis the only signature morphism satisfying that property.

Colimits LetIbe a small category andD:I−→S, (Σⁱ ^f

h

−→Σ^j)_(h:i→j)∈I a diagram. Then (Σ,(γⁱ)_i∈Obj(I)) is the colimit of this diagram if we take:

• Σn = (F

i∈Obj(I)(Σⁱ)n)/∼n, n∈ω where ∼n is the smallest equivalence relation on F

i∈I(Σⁱ)n such that for allI-arrow (i−→^h j) ifc_i∈Σⁱ_n then (c_i, i)∼_n (f_n^h(c_i), j);

• γ_nⁱ : Σⁱ_n−→Σn such that ifci∈Σⁱ_n thenγ_nⁱ(ci) = [(ci, i)],n∈ω andi∈Obj(I).

In fact:

(a) By construction, Σ∈Obj(S) andγⁱ∈ S(Σⁱ,Σ),i∈Obj(I);

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(b) For all I-arrow (i −→^h j), γⁱ =γ^j◦f^h. This is because, for all n ∈ N and ci ∈ Σⁱ_n, we have (ci, i) ∼n

(f_n^h(ci), j);

(c) (Σ,(γⁱ)_i∈Obj(I)) is the universal cocone over the diagramD: if (Σ⁰,(αⁱ)_i∈Obj(I)) is a commutative cocone over the diagram D then, for alln ∈ω, define α_n : Σ_n−→Σ⁰_n, if c∈Σ_n, choose ani ∈Obj(I) such that c= [(ci, i)] and takeαn(c) =αⁱ_n(ci). It follows thatαnis a well defined function,αis a signature morphism such thatαⁱ=α◦γⁱ,i∈Obj(I) andαis the only signature morphism satisfying that property.

Fact 4 About monomorphisms and epimorphisms inS: let f : Σ−→Σ⁰ be a signature morphism. Then

(i) f is aS-monomorphism iff, for alln∈ω,fn: Σn−→Σ⁰_n is injective;f is aS-epimorphism iff, for alln∈ω, fn: Σn−→Σ⁰_n is surjective;

(ii) iff is aS-monomorphism thenfb:F(Σ)−→F(Σ⁰)is injective; iff is aS-epimorphism thenfb:F(Σ)−→F(Σ⁰) is surjective.

3.1.3 S is a locally presentable category

We have just seen that the category S is complete and cocomplete. Furthermore, it has other nice categorial property: it is a finitely accessible category¹. Therefore S is a finitely locally presentable category (a complete and cocomplete finitely accessible category).

Fact 5 Aditional facts on filtered colimits in S: Let D : (I,≤)−→S, (Σ^{i f}

ij

−→Σ^j)_(i≤j)∈I be a directed diagram and let(Σ⁰,(αⁱ)_i∈I)be a commutative cocone over the diagramD:

(i) (Σ⁰,(αⁱ)i∈I)is “the” universal colimit cocone of diagramD iff:

• Σ⁰_n=S

i∈Iαⁱ_n[Σⁱ_n],n∈ω;

• ifci∈Σⁱ_n, cj ∈Σ^j_n are such that αⁱ_n(ci) =α^j_n(cj), then there is ak≥i, j such thatf_n^ik(ci) =f_n^jk(cj), n∈ω.

(ii) If (Σ⁰,(αⁱ)_i∈I) is “the” universal colimit cocone of diagram D, as noticed above, for all n ∈ ω, Σ⁰_n = S

i∈Iαⁱ_n[Σⁱ_n]. It follows easily from the directness condition, by induction on complexity, that any formula in the colimit signature can be “obtained at given defined time”, that is, F(Σ⁰) = S

i∈Iαbⁱ[F(Σⁱ)] and, analogously, any finite set of formulas in the colimit signature can be “obtained at a given defined time”;

(iii) if, for all(i≤j)∈Iand alln∈ω,f_n^ij : Σⁱ_n−→Σ^j_n is injective, then if(Σ⁰,(αⁱ)_i∈I)is “the” universal colimit cocone of diagramD, then for alln∈ω,αⁱ_n: Σⁱ_n−→Σ⁰_n is injective.

Proof. Here we only prove item (iii). Letj∈Iandn∈ω. Letcⁱ, dⁱ∈Σⁱ_nsuch thatαⁱ_n(cⁱ) =αⁱ_n(dⁱ). Therefore, by item (i) above, there isk≥isuch thatf_nîk(cⁱ) =f_nîk(dⁱ) and, asf_nîk: Σⁱ_n−→Σ^k_n is injective, we havecⁱ=dⁱ.

Proposition 6Any signature is a directed colimit of finite type signatures.

Proof. ConsiderI as the set of all Σ⁰ such that, for alln∈ω, Σ⁰_n⊆Σn and|Σ⁰|⊆_{f in}|Σ|. Take inI the pointwise order relation Σ⁰ ≤ Σ⁰⁰iff for alln∈ω, Σ⁰_n⊆Σn. Then:

• (I,≤) is a directed ordered set;

1A categoryC isfinitely accessible if it has filtered colimits and a set Aof objects that are finitely presentable(A⊆C_{f p}) and every object in the category is a filtered colimit of a diagram of finitely presentable objects in that setA; an objectcis called finitely presentable if the hom functorHomC(c,?) preserves all filtered colimits. (See [2, 11, 21])

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• The obvious diagram D : (I,≤)−→S : (Σ⁰ ≤ Σ⁰⁰) 7→ (Σ⁰ ,→ Σ⁰⁰) is such that (Σ,(Σ⁰ ⁱ,→^Σ⁰ Σ)_Σ⁰_∈I) is a commutative D-cocone;

• By the characterization in Fact 5.(i), (Σ,(Σ⁰ ,ⁱ→^Σ⁰ Σ)_Σ0∈I) is a colimit cocone over D, so Σ is a directed colimit of finite type (sub)signatures.

Proposition 7A signature is finitely presentable if and only if it is of finite type.

Proof.

(⇐) Let Σ⁰ be a signature of finite type, that is, |Σ⁰| is finite, and consider D: (I,≤)−→S, (Σ^{i f}

ij

−→ Σ^j)_(i≤j)∈I a directed diagram of signatures. Then the canonical arrow k:colim_i∈IS(Σ⁰,Σi)−→S(Σ⁰, colim_i∈IΣi) is an isomorphism.

[(Σ⁰ ^h

i

−→ Σi), i)] 7→ (Σ⁰ ^h

i

−→Σi γi

−→ colim_i∈IΣi).

In order to prove thatkis surjective, we have to prove that, for each signature morphismh: Σ⁰−→colim_i∈IΣ_i, there is an i ∈ I and a signature morphism hⁱ : Σ⁰−→Σ_i such that (Σ⁰ −→^h colim_i∈IΣ_i) = (Σ⁰ −→^hⁱ Σ_i −→^γⁱ colim_i∈IΣi). As|Σ⁰| is a finite set, there is only a finite set{n0, . . . , n_t−1}⊆N such that, for allr < t, Σ⁰_n_r is a finite non-empty set. Since that for each nr, Σn_r is finite and (I,≤) is a directed ordered set, there is anir∈I such that hn_r[Σ⁰n_r]⊆γ_nⁱ^r_r[Σⁱ_n^r_r]. As there is only a finite set {n0, . . . , n_t−1}⊆Nsuch that for all r < t, Σ⁰_n_r 6=∅ and (I,≤) is a directed ordered set, take an i ≥i0, . . . , i_t−1. As γⁱ^r =γⁱ◦fⁱ^rⁱ, it follows that, for all n ∈ω, hn[Σ⁰n]⊆γⁱ_n[Σⁱ_n]. Just take, for eachn∈ω,hⁱ_n: Σ⁰_n−→Σⁱ_n such that, for eachc⁰_n∈Σ⁰_n, hⁱ_n(c⁰_n)∈Σⁱ_n is such that hn(c⁰_n) = [(hⁱ_n(c⁰_n), i)] and soh=γⁱ◦hⁱ.

In order to prove that kis injective, we have to prove that, for each signature morphismh: Σ⁰−→colim_i∈IΣi

such that there are i0, i1 ∈ I and signature morphisms hⁱ⁰ : Σ⁰−→Σi₀, hⁱ¹ : Σ⁰−→Σi₁ such that γⁱ⁰ ◦hⁱ⁰ = h = γⁱ¹ ◦hⁱ¹, then there is a j ≥ i0, i1 and a h^j : Σ⁰−→Σj such that fⁱ⁰^j ◦hⁱ⁰ = h^j = fⁱ¹^j ◦hⁱ¹. Since that for each i ∈ {i0, i1} and n ∈ ω, hⁱ_n : Σ⁰_n−→Σⁱ_n is such that for each c⁰_n ∈ Σ⁰_n, hⁱ_n(c⁰_n) ∈ Σⁱ_n is such that [(hⁱ_n⁰(c⁰_n), i0)] = hn(c⁰_n) = [(hⁱ_n¹(c⁰_n), i1)], because h = γⁱ◦ hⁱ, then there is a j(c⁰_n) ≥ i0, i1 such that fⁱ⁰^j(c

0 n)

n (hⁱ_n⁰(c⁰_n)) =fⁱ¹^j(c

0 n)

n (hⁱ_n¹(c⁰_n))∈Σ^j(c

0 n)

n . As|Σ⁰|is a finite set then {j(c⁰_n) :c⁰_n ∈Σ⁰_n} is a finite set and, as (I,≤) is a directed ordered set, there is a jn ≥j(c⁰_n) for each c⁰_n ∈Σ⁰_n. Again, as |Σ⁰| is a finite set, there is only a finite set {n0, . . . , nt−1}⊆Nsuch that, for allr < t, Σ⁰_n_r 6=∅. As (I,≤) is a directed ordered set, take a j≥jn₀, . . . , jn_t. Then, asD is a diagram, for eachn∈ω,f_nⁱ⁰^j◦hⁱ_n⁰ =f_nⁱ¹^j◦hⁱ_n¹ takeh^j=f^ij◦hⁱ,i∈ {i0, i1}.

(⇒) Let Σ be a finitely presentable logic. Then, by Proposition 6, there is a directed diagram of finite type logics D : (I,≤)−→S, (Σⁱ −→^f^ij Σ^j)_(i≤j)∈I such that there is an isomorphism h : l −→^∼⁼ colim_i∈IΣⁱ. Then, as the canonical morphism is invertible k : colimi∈IS(Σ,Σⁱ) −→ S(Σ, colim^∼⁼ i∈IΣⁱ), there is a factorization of h: (Σ−→^h colim_i∈IΣⁱ) = (Σ ^h

i

−→Σⁱ ^γ

i

−→colim_i∈IΣⁱ). Then, ashis an isomorphism,hⁱ: Σ−→Σⁱis anS-section.

In particular there is a sequence of injections (hⁱ_n: ΣnΣⁱ_n)_n∈ω so, as|Σⁱ|is finite, then|Σ|is finite.

Theorem 8 The category S is a finitely locally presentable category, i.e., S is an accessible category that is cocomplete and complete.

Proof. Direct consequence of Propositions 6, 7 and 3.

Corollary 9

(i) The Yoneda functorY :Sf p−→Set⁽Sf p)^op has an extension to a functorY⁰ :S−→Set⁽Sf p)^op,Σ7→Y⁰(Σ) = L(ι(.),Σ)that is full and faithful;

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(ii) Let F lat(S_{f p}, Set) be the full subcategory of Set⁽Sf p)^op whose objects are the functors that are filtered colimits of representable functors. ThenF lat(S_{f p}, Set)is the “essential image” ofY⁰and so his restriction functor E:S−→F lat(S_{f p}, Set)is an equivalence of categories;

(iii) F lat(Sf p, Set)coincides with the category ofSet-valued functors that preserve finite limits;

(iv) Y⁰ has a left adjoint.

Proof. For (i) and (ii) see [11] Theorem 5.3.5 (pag 265) or [2] Theorem 2.26 (pag 83) or [21] Observation 1.6 (pag 46). For (iii) and (iv) see [2] Theorem 1.46 (pag 38).

3.2 The category L

The category Lis the category of propositional logics and translations as morphisms. This is a category “built above” the categoryS, that is, there is an obvious forgetful functorU :L−→S.

3.2.1 What isL?

The objects of L are logics. A logic is an ordered pair l = (Σ,`) where Σ is an object of S and ` codifies the “consequence operator” on F(Σ) — ` is a binary relation, a subset of P arts(F(Σ))×F(Σ), such that Cons(Γ) ={ϕ∈F(Σ) : Γ`ϕ}, for all Γ⊆F(Σ), gives a structural finitary closure operator on F(Σ):

• inflationary: Γ⊆Cons(Γ);

• increasing: Γ₀⊆Γ₁⇒Cons(Γ₀)⊆Cons(Γ₁);

• idempotent: Cons(Cons(Γ))⊆Cons(Γ);

• finitary: Cons(Γ) =S

{Cons(Γ⁰) : Γ⁰⊆_{f in}Γ};

• structural: σ(Cons(Γ))⊆Cons(e σ(Γ)), for each substitutione σ:X→F(Σ).

In [17], Lo´s and Suszko give a characterization of usual provability notion in Hilbert calculi by the consequence operators with the above properties.

We say that a logicl= (Σ,`) is offinite typewhen|Σ|is a finite set and`is determined, in the sense of [17]², by a finite set of axioms and inference rules.

Ifl= (Σ,`), l⁰= (Σ⁰,`⁰) are logics then atranslation morphismf :l−→l⁰is a signature morphismf : Σ−→Σ⁰ that “preserves the consequence relation”, that is, for all Γ∪ {ψ}⊆F(Σ), if Γ` ψ then fb[Γ]`⁰ fb(ψ). We say that a morphismf :l−→l⁰ is aconservative translation morphism if for all Γ∪ {ψ}⊆F(Σ), Γ`ψif and only if fb[Γ]`⁰fb(ψ). Composition and identities are similar toS.

3.2.2 Some facts aboutL

Definition 10There is a natural definition of order between consequence relations on each signatureΣ: for each pair `,`⁰ of consequence relations over Σwe have the equivalence between the items below:

• For each Γ∪ {ψ}⊆F(Σ),Γ`ψ⇒Γ`⁰ ψ;

• The identity signature morphism over Σ,id_Σ: Σ−→Σ, is a translation morphismid_Σ: (Σ,`)−→(Σ,`⁰).

We write` ≤ `⁰ when the conditions above are satisfied.

2A formula is demonstrable from a given set of hypothesis iff there is a finite sequence of formulas such that the last one is the thesis and each formula is an hypothesis or an instance of axiom or is obtained from the previous formulas in the sequence by an instantiation of a (finitary) inference rule.

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Fact 11 The set of consequence relations on a signature Σ, denoted by LΣ, is a complete lattice. It is in fact an algebraic lattice where the compact elements are the “finitely generated logics”, the logics overΣgiven by a finite set of axioms and a finite set of (finitary) inference rules.

Proof. Here we just give a sketch of proof. In the following we have other similar (but more general) propositions where we supply full proofs.

Let Σ be a signature.

Infs Consider I a set and D = {lⁱ = (Σ,`_i)}_i∈I a family of logics over the signature Σ. Now, for each Γ∪ {ψ} ⊆ F(Σ), define that Γ ` ψ⇔ there is Γ⁰⊆_{f in}Γ such that (∀i ∈ I)(Γ⁰ `i ϕ), then (Σ,`) is a logic and l= (Σ,`) is the infimum of the familyDin L. In fact, this follows from them items below:

• l∈Obj(L);

• id_Σ∈ L(l, l^j) for all j∈I;

• ifl⁰= (Σ,`⁰) is a logic over the signature Σ such thatidΣ∈ L(l⁰, l^j) for all j∈I, thenidΣ∈ L(l⁰, l).

Directed sups Consider I a set and D = {lⁱ = (Σ,`_i)}_i∈I an upward directed family of logics over the signature Σ, that is, for each i, j ∈I there is a k∈I such thatid_Σ∈ L(lⁱ, l^k) ,id_Σ ∈ L(l^j, l^k). Now, for each Γ∪ {ψ}⊆F(Σ), define that Γ `ψ⇔ there is Γ⁰⊆_{f in}Γ and there is an i ∈I such that Γ⁰ `_i ϕ, then (Σ,`) is a logic andl= (Σ,`) is the supremum of the familyD inL. In fact, this follows from the items below:

• l∈Obj(L);

• id_Σ∈ L(l^j, l) for all j∈I;

• ifl⁰= (Σ,`⁰) is a logic over the signature Σ such thatidΣ∈ L(l^j, l⁰) for all j∈I, thenidΣ∈ L(l, l⁰).

Sups As usual, the supremum of a family of logics can be obtained taking the infimum of the set of upper bounds of that family of logics. A more objective characterization of suprema can be given but we postpone that because this can be easily described by more general results below (see Proposition 16).

Compact consequence relations A consequence relation `⁰ over Σ is compact if for each setI, each D = {lⁱ = (Σ,`_i)}_i∈I a upward directed family of logics over the signature Σ, if `⁰≤W

i∈I `_i then there is ani ∈I such that `⁰≤`_i. It follows easily that this condition is equivalent to the “stronger” condition: for each set J, each D = {l^j = (Σ,`_j)}_j∈J a family of logics over the signature Σ, if `⁰≤ W

j∈J `_j then there is a finite subset J⁰⊆J such that `⁰ ≤ W

j∈J⁰ `j.³ A consequence relation on Σ is compact if and only if it is a finite generated consequence relation on Σ. Any consequence relation on Σ is the directed supremum of its compact (sub)consequence relations on Σ.

Remark 12As the set of consequence operators (or consequence relations) on a signature Σ is a complete lattice, there exists the logic generated by any function W :P(F(Σ))−→P(F(Σ)): it is enough to take the infimum of the family of all consequence relations on Σ that are upper bounds of the “proto-consequence relation” associated with the “proto-consequence operator”W.

Definition 13 Direct image and inverse image: letf : Σ−→Σ⁰ be aS-morphism:

• Inverse image: if l⁰= (Σ⁰,`⁰)∈Obj(L)then for all Γ∪ {ψ}⊆F(Σ) defineΓ`_f?(`⁰)ψ ifffb[Γ]`⁰ fb(ψ);

3Just observe that any sup of a family coincides with a sup of a directed family: for each setJ takeI=Pf in(J) then, for each J⁰⊆_{f in}J, define`_J0=W

j∈J⁰`j. . . .

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• Direct image: if l= (Σ,`)∈Obj(L)then for all Γ⁰∪ {ψ⁰}⊆F(Σ⁰)define Γ⁰`f_?(`)ψ⁰ iff there is a finite sequence of Σ⁰-formulas(φ⁰₀, . . . , φ⁰_t) such that:

– φ⁰_t=ψ⁰;

– for all p≤t at least one of the alternatives below occurs:

∗ “φ⁰_p is a hypothesis”: φ⁰_p∈Γ⁰;

∗ “φ⁰_p is an instance of anl-axiom”: there is a θ_p∈F(Σ) such that`θ_p and there is a substitution σ⁰ :X−→F(Σ⁰)such that eσ⁰(fb(θ_p)) =φ⁰_p;

∗ “φ⁰_p is a direct consequence of an instance of l-inference rule applied over previous members in the sequence”: there is a ∆p∪ {θp}⊆_{f in}F(Σ) such that ∆p ` θp and there is a substitution σ⁰ :X−→F(Σ⁰)such that eσ⁰(fb(θp)) =φ⁰_j andeσ⁰[fb[∆p]]⊆{φ⁰₀, . . . , φ⁰_j−1}.

Fact 14About direct image and inverse image: letf : Σ−→Σ⁰ be aS-morphism and letl= (Σ,`), l⁰ = (Σ⁰,`⁰) be logicsl, l⁰ ∈Obj(L). Then(i)^?and(i)_? hold, and(ii)^?,(ii)_tm and(ii)_?are equivalent:

(i)^? ifl⁰= (Σ⁰,`⁰)∈Obj(L)thenf^?(l⁰) = (Σ,`f^?(`⁰))∈Obj(L);

(i)_? ifl= (Σ,`)∈Obj(L)thenf_?(l) = (Σ⁰,`f_?(`))∈Obj(L).

(ii)^? ` ≤ f^?(`⁰);

(ii)_tm f : (Σ,`)−→(Σ⁰,`⁰)is a translation morphism;

(ii)? f?(`) ≤⁰ `⁰. Proof. (Sketch)

(i) The proof of (i)^? is omitted; the proof of (i)_? is analogous to the item Colimits.(a) in Proposition 16;

(ii) The equivalence (ii)^?⇔(ii)tm follows directly from the definitions; the implication (ii)_?⇒(ii)tm is analogous to the item Colimits.(b) in Proposition 16; the implication (ii)_tm⇒(ii)_? is analogous to the item Colimits.(c) in Proposition 16.

Remark 15 It follows easily from the facts above that the forgetful functor U :L−→S : ((Σ,`)−→^f (Σ⁰,`⁰))7→(Σ−→^f Σ⁰) has left and right adjoint functors: the left adjoint T : S−→L and the right adjoint V :S−→Ltake a signature Σ to, respectively, T(Σ) = (Σ,`_min) (the first element of L_Σ) andV(Σ) = (Σ,`max) (the last element ofLΣ).

Proposition 16The categoryLis complete and cocomplete and the forgetful functorU :L−→S creates all small limits and colimits.

Proof. This proof has three sections: limits⁴, filtered colimits and colimits.

Limits Let I be a small category and D : I−→L, ((Σⁱ,`i) ^f

h

−→ (Σ^j,`j))_(h:i→j)∈I a diagram, and take (Σ,(πⁱ)_i∈Obj(I)) the limit of the underlying diagram (I −→ S^D −→ L). For all Γ^U ∪ {ψ}⊆F(Σ), define that Γ ` ψ⇔ there is Γ⁻⊆f inΓ such that for all i ∈ Obj(I) πbⁱ[Γ⁻] `i bπⁱ(ψ)⁵, then l = (Σ,`) is a logic and (l,(πⁱ)_i∈Obj(I⁾) is the limit ofD inL. In fact, this follows from (a), (b) and (c) below:

4In [12] there is a similar proof for products, but with another notion of signature morphism.

5This definition also works for the terminal logicl= (Σ,`) where Σ is the terminal signature (card(Σn) = 1,∀n∈ω), and for all Γ∪ {ψ}⊆F(Σ), Γ`ψ.

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(a) l∈Obj(L);

(b) π^j ∈ L(l, l^j), for allj∈I;

(c) if (l⁰,(αⁱ)_i∈Obj(I)) is a commutative cone over the diagram D then the unique signature morphism α : Σ⁰−→Σ such thatαⁱ=πⁱ◦α,i∈Obj(I) preserves the consequence relation.

Now we prove (a), (b) and (c).

(a)it follows directly from the definition of`that it gives afinitaryandincreasing consequence operator. It is alsoinflationarybecause ifψ∈Γ, take any Γ⁻⊆_{f in}Γ such thatψ∈Γ⁻ then, for alli∈Obj(I),πbⁱ(ψ)∈πbⁱ[Γ⁻].

Since`i is inflationary, thenπbⁱ[Γ⁻]`iπbⁱ(ψ). We have Γ`ψ.

Idempotent: Let ψ ∈ F(Σ) such that Γ ` ψ, where Γ = {θ ∈ F(Σ) : Γ ` θ}. Let us prove that Γ ` ψ.

Since` is finitary, let ∆⊆_{f in}Γ be such that ∆`ψ. Then, for each θ∈∆, let Γ^θ⊆_{f in}Γ be such that Γ^θ `θ. It follows that Γ⁻ =S

θ∈∆Γ^θ satisfies Γ⁻⊆_{f in}Γ and, as ìs increasing, for each θ∈∆, Γ⁻ `θ. Then, asì gives an inflationary operator, for each i∈Obj(I), it follows from the definition of ∆`ψ that for each i∈Obj(I), bπⁱ[∆]ìπbⁱ(ψ). Analogously, as Γ⁻ `θfor eachθ∈∆, then for eachi∈Obj(I),πbⁱ[Γ⁻]ìπbⁱ(θ), for eachθ∈∆.

Now, as`i gives an idempotent operator, thenπbⁱ[Γ⁻]`iπbⁱ(ψ), for eachi∈Obj(I). Therefore, as Γ⁻⊆_{f in}Γ, we have Γ`ψ.

Structural: Let Γ∪ {ψ}⊆F(Σ) be such that Γ`ψ. We have to prove that for any substitutionσ:X−→F(Σ) we have σ[Γ]e ` eσ(ψ). Let Γ⁻⊆_{f in}Γ be such that, for all i ∈ Obj(I), bπⁱ[Γ⁻] ì bπⁱ(ψ). Since eσ[Γ⁻]⊆_{f in}σ[Γ],e we have σ[Γ]e ` eσ(ψ) if we prove that bπⁱ[σ[Γe ⁻]] ì bπⁱ(eσ(ψ)), for each i ∈ Obj(I). Now, from Fact 2.(ii), for all i ∈Obj(I) there is a substitutionσⁱ : X−→F(Σⁱ) such that bπⁱ◦σe =σeⁱ◦bπⁱ. Then, for eachi ∈ Obj(I), since πbⁱ[Γ⁻] ì πbⁱ(ψ) and ì gives a structural operator, we have that eσⁱ[bπⁱ[Γ⁻]] ì eσⁱ(bπⁱ(ψ)). So we have bπⁱ[σ[Γe ⁻]]ì bπⁱ(σ(ψ)), for eache i∈Obj(I). Thereforeeσ[Γ]èσ(ψ).

(b)Let Γ∪ {ψ}⊆F(Σ) be such that Γ`ψ. Then select a Γ⁻⊆_{f in}Γ such that for alli∈Obj(I),bπⁱ[Γ⁻]`iπbⁱ(ψ).

Sinceπbⁱ[Γ⁻]⊆bπⁱ[Γ] andìis inflationary, for eachi∈Obj(I), we havebπⁱ[Γ]ìbπⁱ(ψ). So the signature morphism πⁱ: Σ−→Σⁱ gives also atranslation morphism πⁱ: (Σ,`)−→(Σⁱ,ì), for eachi∈Obj(I).

(c)Let Γ⁰∪ {ψ⁰}⊆F(Σ⁰) be such that Γ⁰ `⁰ ψ⁰. We have to prove thatα[Γb ⁰]`α(ψb ⁰). Since `⁰ is finitary, select Γ⁰⁻⊆_{f in}Γ⁰ such that Γ⁰⁻ `⁰ ψ⁰. Then, as αⁱ : l⁰−→lⁱ is a translation morphism, αbⁱ[Γ⁰⁻] `i αbⁱ(ψ⁰), for each i∈Obj(I). So, asαⁱ=πⁱ◦α, we havebπⁱ[α[Γb ⁰⁻]]`_ibπⁱ(α(ψb ⁰)), for eachi∈Obj(I). Now, sinceα[Γb ⁰⁻]⊆_{f in}α[Γb ⁰], we have from the definition that α[Γb ⁰]`α(ψb ⁰). So the signature morphism α: Σ⁰−→Σ gives also a translation morphismα: (Σ⁰,`⁰)−→(Σ,`).

Filtered colimits Let (I,≤) be a directed ordered set andD: (I,≤)−→L, ((Σⁱ,`i) ^f

ij

−→ (Σ^j,`j))_(i≤j)∈I be a diagram. Take (Σ,(γⁱ)i∈I) the colimit of the underlying diagram (I −→^D S −→ L). Now, for all Γ^U ∪ {ψ} ⊆F(Σ), define that Γ`ψ⇔ there is Γ⁻⊆f inΓ and there is an i∈I such that Γ⁻∪ {ψ}⊆bγⁱ[F(Σⁱ)] and there is Γ⁻ⁱ∪ {ψⁱ}⊆_{f in}F(Σⁱ) such thatγbⁱ[Γ⁻ⁱ] = Γ⁻ , γbⁱ(ψⁱ) = ψ and Γ⁻ⁱ `i ψⁱ. Thenl= (Σ,`) is a logic and (l,(γⁱ)_i∈I) is the colimit ofDinL. In fact, this follows from (a), (b) and (c) below:

(a) l∈Obj(L);

(b) γ^j∈ L(l^j, l), for allj∈I;

(c) if (l⁰,(αⁱ)_i∈I) is a commutative cocone over the diagramD then the unique signature morphismα:l−→l⁰ such thatαⁱ=α◦γⁱ,i∈I preserves the consequence relations.

Now we prove (a), (b) and (c).

(a)It follows directly from the definition of`that it gives afinitaryandincreasingconsequence operator. It is alsoinflationarybecause ifψ∈Γ, take Γ⁻={ψ}and anyi∈I such thatψ∈bγⁱ[Σⁱ], takeψⁱ∈F(Σⁱ) such that bγⁱ(ψⁱ) =ψ and Γ⁻ⁱ={ψⁱ}then, as`i gives a inflationary operator, Γ⁻ⁱ`iψⁱ.

Idempotent: Letψ ∈ F(Σ) be such that Γ ` ψ where Γ = {θ ∈ F(Σ) : Γ ` θ}. Let us prove that Γ` ψ.

Since ` is finitary, let ∆⊆_{f in}Γ be such that ∆ ` ψ. Then, for each θ ∈ ∆, let Γ^θ⊆_{f in}Γ be such that Γ^θ ` θ.