A Global Glance on Categories in Logic

Logica universalis 1 (2007), 3–391661-8297/010003-37, DOI 10.1007/s11787-006-0002-7c© 2007 Birkhauser Verlag Basel/Switzerland Logica Universalis

A Global Glance on Categories in Logic

Peter Arndt, Rodrigo de Alvarenga Freire, Odilon OtavioLuciano and Hugo Luiz Mariano

Abstract. We explore the possibility and some potential payoffs of using thetheory of accessible categories in the study of categories of logics. We illustratethis by two case studies focusing on the category of finitary structural logicsand its subcategory of algebraizable logics.

Mathematics Subject Classification (2000). Primary 03B22; Secondary 18C35.

Keywords. Logics, signatures, categories, accessible categories, locally present-able categories.

1. Introduction

1.1. Categories of logics

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

This is a relatively recent point of view which has largely come into con-sideration through the topic of combination of logics: The goal of combining twologics L1 and L2 has been described as to obtain “the smallest logic system forthe combined language which is a conservative extension of both L1 and L2”.In [24] it was proposed that this rather informal statement could be given pre-cise meaning by considering the combination of logics as a colimit constructionin an appropriate category of logics. Following this idea the notions of modulatedfibring, metafibring and combination of institutions and π-institutions have beenpresented as colimit constructions in different categories.

Research supported by FAPESP, under the Thematic Project “ConsRel” (2004/1407-2).

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4 P. Arndt et al. Logica universalis

It has also been observed that a presentation of a logic as a colimit of otherscan be seen as a splitting of this logic into other, simpler, logics, possibly helpingto understand the more complex logic. Possible translation semantics and remotealgebraization (see, e.g., [11]) are concrete developments of this point of view.We observe that the situation of a logic L that is completely determined by thetranslation of other logics Li into L can be seen as a “covering” of L by the Li.In Section 4 we turn this intuition into a mathematical statement by choosing acategory of logics and giving a rigorous definition of covering.

Thus it seems reasonable to adopt a global perspective on logic and considernot only constructions with particular logics but the whole category of (some kindof) logics. Defining such a category is not a completely straightforward task. Itmeans giving a partial answer to the identity problem, amply discussed in the bookLogica Universalis: Towards a general theory of logic (see [7]), i.e., the questionof when two given presentations of logic systems can be considered to describethe same logic: Two such presentations should be expected to describe isomorphicobjects in their ambient category. This gives only a partial answer, since the iden-tity problem also includes the task of comparing logic systems which are given indifferent styles of presentation, e.g. a Hilbert calculus and a sequent calculus. Itseems difficult to unite two such differently presented systems in one category.

However, the identity problem is not our main concern here and we shall onlybriefly readdress it towards the end of the article. What is our main concern, andthis gives the second intended meaning to the title, is to investigate global prop-erties of categories of logics, like those of being complete, connected, accessibleor locally presentable (the latter two notions will be explained in a moment). Webelieve that in choosing a category of logics it should ultimately be of advantageto take into account such global categorial properties and not only the ad hocrequirements of the constructions one wishes to perform. In particular we believethat the theory of accessible categories has to offer theorems and intuitions thatcan be of use in tackling technical as well as conceptual questions arising in Uni-versal Logic. To confirm and illustrate this we will in the following present two casestudies taken from our previous publications [4,5], of which this article is an amal-gamation and expansion. The main point of these case studies, in each of whichwe prove a certain category of logics to be accessible, is twofold: first, to show thatand how the notion of accessibility can apply to categories of logics in a naturalway (this is not obvious, see the last section) and, second that some benefits canbe gained thereof. We are, however, aware that the categories we present are nota good ambient for proper logical studies since they give unsatisfactory answersto the identity problem. It remains a project for the future to give a category oflogics with good global properties and an appropriate notion of isomorphism oflogics; we outline a possible solution for this task in Section 5.

1.2. 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

Vol. 1 (2007) A Global Glance on Categories in Logic 5

has been given is through the theory of monads and its variations; see [14] for asurvey. Another one came from sketch theory; the varieties from Universal Algebraare exactly the categories of Set-models of finite product sketches — this is closein spirit to the usual characterization of varieties as categories of Set-models ofequational theories.

A more general class of “algebraic” categories are the locally finitely pre-sentable categories. The key observation leading to the definition of these is thatin the familiar algebraic categories every object is a directed colimit of finitelypresentable objects, i.e., objects specifiable by a finite number of generators andrelations (a directed colimit is the colimit of a directed poset, i.e., a non emptyposet such that for every pair of elements there is one greater than each of thetwo). In these familiar cases the property of an object A being finitely presentablehas an equivalent categorial description: A is finitely presentable iff the functorHom(A,−) preserves directed colimits (or equivalently, filtered colimits)1. A cat-egory is called locally finitely presentable if it is cocomplete and has a set offinitely presentable objects such that every object is a directed (or filtered) colimitof objects from this set. Examples of such categories are all varieties of finitarymany-sorted algebras as well as the categories of sets and posets and categories ofSet-valued functors on a small category.

A further generalization is the notion of locally λ-presentable category, whereλ is a regular cardinal: A poset is called λ-directed if every set of elements of cardi-nality strictly lesser than λ has an upper bound, an object A is called λ-presentableif Hom(A,−) preserves colimits of (diagrams over) such posets and a category isλ-presentable iff it is cocomplete and there is a set of λ-presentable objects suchthat each object is a λ-directed colimit of these objects. Finally, a category iscalled locally presentable if it is locally λ-presentable for some regular cardinal λ.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 the-

ories of partial operations in which the domain of each operation is definedby equations in the preceding operations;

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

Examples of locally presentable (but not finitely presentable) categories in-clude the category of Banach spaces and linear contractions as morphisms, thatof convergence spaces and any complete lattice considered as a category.

1The technical definition of filtered category is not needed in this article. The curious reader

who wants know this definition and the sense of the “equivalence” between directed colimit andfiltered colimit must see the results in the pages 13–16 of [1].


The last notion we will use here is that of λ-accessible category, which are de-fined like the locally λ-presentable categories except for requiring only λ-directedcolimits to exist instead of arbitrary ones. An accessible category is, again, a cat-egory which is λ-accessible for some λ. These categories are exactly the categoriesof Set-models of arbitrary sketches or, alternatively, the categories of Set-modelsof so-called basic sentences of the infinitary first order logic L∞∞ (which allowsdisjunctions and quantification over arbitrary (small) sets of formulas/variables).Examples not included in the previous classes are the categories of fields andHilbert spaces.

For an introduction to these notions and their theories see [1] and [21]. Thereare also other related notions like locally multipresentable categories, weakly lo-cally presentable categories and D-accessible categories (where D is some class ofsmall categories) to some of which the related types of sketches and first ordertheories have also been identified.

The theory of accessible and locally presentable categories provides somepowerful tools and also some conceptual clarifications. As an example of the first,we mention the theorem of [18] which says that an accessible category is locallypresentable iff it is complete iff it is cocomplete. In Section 3 we prove a certaincategory of logics to be accessible and, since we know it not to have an initialobject (hence not to be cocomplete), we know that it is not complete either. Byinvestigating whether this category is of any of the types of categories mentionedin the previous paragraph we could further try to discover which type of limit ismissing. For the second recall that the above types of categories were meant tocapture the essential properties of categories of algebraic objects. Now there isa general duality phenomenon in mathematics between algebraic and geometri-cal/topological objects as witnessed by Stone duality, Gelfand duality, the dualitybetween algebraic sets and k-algebras and many others. In [18] this informal ob-servation is turned into a real mathematical statement by a theorem saying thatthe dual of a locally presentable category can not also be locally presentable ex-cept when it is a (category coming from a) poset. This seems to indicate that thenotion of local presentability has succeeded in capturing some of the features ofalgebraic categories. In view of the ever returning question of “how algebraic” islogic, it thus seems to be interesting to investigate whether categories of logics arelocally presentable.

Having seen these attempts to characterize categories of algebras, one is nat-urally led to think about whether the parallel between universal logic and universalalgebra could be continued here. It has been argued, as mentioned in [7], that log-ical structures should be seen as one of the fundamental species of mathematicalstructures in the sense of Bourbaki. Assuming this to be the case one can wonderif it is possible to recognize the logical character of a category by categorial prop-erties. If so, it should be in some different vein than the limit closure and generatorproperties defining locally presentable categories; after all we show below that acertain category of logics is locally presentable. The theory of fibred categoriesseems to fit very well for considering categories of logics as is convincingly shown

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in [19]. Indeed it seems reasonable that a category of logics should be fibred over abase category of signatures and finding further decisive properties (maybe like thebase category being a category of free algebras) of the involved categories couldgive a sharp categorial picture.

2. The categories of signatures and logics

In the following subsection we will define a category of signatures which is equiv-alent to SetN and the main results will be that this category is locally finitelypresentable (a special case of [1], ex. 1.12, p. 18) and that the finitely presentableobjects are precisely those signatures which have finitely many connectives (a spe-cial case of [1], ex. 1.2(2), p. 9). The reader who is content with these explanationscan skip to section 2.2 without many problems, for the others we will give a detailedexposition.

2.1. The category SThe category S is the category of signatures and morphisms of signatures. Inwhat follows, let X = {x0, x1, . . . , xn, . . . } be an enumerable set (written in afixed order) as in [10].

2.1.1. What is S? The objects of S are signatures. A signature Σ is a sequenceof sets Σ = (Σn)n∈ω such that Σi ∩ Σj = ∅ for all i < j < ω . We write |Σ| =⊔

n∈ω Σn =⋃

n∈ω Σn ×{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 complexityl(ϕ) of the formula ϕ is the usual:

• l(ϕ) = 1 if ϕ ∈ X ∪ Σ0;• l(ϕ) = 1+l(ψ0)+ · · · l(ψn−1) if ϕ = c(ψ0, . . . , ψn−1), where c ∈ Σn and n > 0.

If Σ, Σ′ are signatures then a morphism f : Σ −→ Σ′ is a sequence of functionsf = (fn)n∈ω, where fn : Σn −→ Σ′

n. For each morphism f : Σ −→ Σ′ there is onlyone function f : F (Σ) −→ F (Σ′), called the extension of f , such that:

• f(x) = x if x ∈ X ;• f(c) = f0(c) if c ∈ Σ0;• f(c(ψ0, . . . , ψn−1) = fn(c)(f(ψ0), . . . , f(ψn−1)) if c ∈ Σn, n > 0.

Then f(ϕ(ψ0, . . . , ψn−1) = f(ϕ)(f (ψ0), . . . , f(ψn−1)), by induction on l(ϕ).Composition in S is componentwise. The extension of the formula algebra of

a composition is the extensions’ composition. Identities in S are the sequences ofidentities on each level n, for n ∈ ω. The extension of an identity is the identityfunction on the formula algebra.

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2.1.2. Some facts about S.

Remark 2.1. About stratification: For each signature Σ and for each n ∈ ω we con-sider the set of Σ-formulas: F (Σ)[n] = {ϕ ∈ F (Σ) : the set of variables that occurin ϕ is precisely {x0, . . . , xn−1}}. Let f : Σ −→ Σ′ be a signature morphism andf : F (Σ) −→ F (Σ′) be the induced formula algebra function; we can see directlyby induction on the complexity of Σ-formulas that f “preserves stratification”: ifϕ ∈ F (Σ)[n] then f(ϕ) ∈ F (Σ′)[n].

Fact 1. About substitution:• For any substitution function σ : X −→ F (Σ), there is only one extension

σ : F (Σ) −→ F (Σ) such that σ is an “homomorphism”: σ(x) = σ(x), for allx ∈ X and σ(cn(ψ0, . . . , ψn−1) = cn(σ(ψ0)), . . . , σ(ψn−1)), for all cn ∈ Σn,n ∈ ω; it follows that for any θ(x0, . . . , xn−1) ∈ F (Σ)σ(θ(x0, . . . , xn−1)) =θ(σ(x0), . . . , σ(xn−1)). The identity substitution induces the identity homo-morphism on the formula algebra; the composition substitution of the sub-stitutions σ′, σ : X −→ F (Σ) is the substitution σ′′ : X −→ F (Σ) , σ′′ =σ′ � σ

.= σ′ ◦ σ and σ′′ = ˜σ′ � σ.= σ′ ◦ σ.

• Let f : Σ −→ Σ′ be a S-morphism. Then for any substitution σ : X −→ F (Σ)there is another substitution σ′ : X −→ F (Σ′) such that σ′ ◦ f = f ◦ σ.

F (Σ) F (Σ′)

F (Σ′)F (Σ)

f

f

σ σ′

��

�

�

�

Proposition 2.2. S is a complete and cocomplete category.

Proof. Observe that S is equivalent to the functor category SetN, where N is thediscrete category with object class N, then S has all small limits and colimits andthey are componentwise.

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

Limits. Let I be a small category and D : I −→ S, (Σi fh

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

• Σn = {c = (ci)i∈Obj(I) ∈∏

i∈objI Σin: for all I-arrow (i h−→ j), fh

n (ci) = cj};• πi

n : Σn −→ Σin such that if c = (ci)i∈Obj(I) ∈ Σn then πi(c) = ci, n ∈ ω and

i ∈ I.


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 fij

−→ Σj)(i≤j)∈I a diagram. Then (Σ, (γi)i∈I) is the colimit of this diagram ifwe take:

• Σn = (⊔

i∈I(Σi)n)/ ∼n where, if ci ∈ Σi

n, cj ∈ Σjn(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 thedirectness assumption that ∼n is an equivalence relation on

⊔i∈I(Σ

i)n;• γi

n : Σin −→ Σn such that if ci ∈ Σi

n then γin(ci) = [(ci, i)], n ∈ ω and i ∈ I.

Colimits. Let I be a small category and D : I −→ S, (Σi fh

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

• Σn = (⊔

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

relation on⊔

i∈I(Σi)n such that for all I-arrow (i h−→ j) if ci ∈ Σi

n then(ci, i) ∼n (fh

n (ci), j);• γi

n : Σin −→ Σn such that if ci ∈ Σi

n then γin(ci) = [(ci, i)], n ∈ ω and

i ∈ Obj(I). �

Fact 2. About monomorphisms and epimorphisms in S: Let f : Σ −→ Σ′ be asignature morphism. Then

(i) f is a S-monomorphism iff, for all n ∈ ω, fn : Σn −→ Σ′n is injective; f is

a S-epimorphism iff, for all n ∈ ω, fn : Σn −→ Σ′n is surjective;

(ii) if f is a S-monomorphism then f : F (Σ) −→ F (Σ′) is injective; if f isa S-epimorphism then f : F (Σ) −→ F (Σ′) is surjective.

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

Fact 3. Aditional facts on filtered colimits in S: Let D : (I,≤) −→ S,

(Σi fij

−→ Σj)(i≤j)∈I be a directed diagram and let (Σ′, (αi)i∈I) be a commutativecocone over the diagram D:

(i) (Σ′, (αi)i∈I) is “the” universal colimit cocone of diagram D iff:• Σ′

n =⋃

i∈I αin[Σi

n], n ∈ ω;• if ci ∈ Σi

n, cj ∈ Σjn are such that αi

n(ci) = αjn(cj), then there is a k ≥ i, j

such that f ikn (ci) = f jk

n (cj), n ∈ ω.(ii) If (Σ′, (αi)i∈I) is “the” universal colimit cocone of diagram D, as noticed

above, for all n ∈ ω, Σ′n =

⋃i∈Iα

in[Σi

n]. It follows easily from the directnesscondition, by induction on complexity, that any formula in the colimit signa-ture can be “obtained at given defined time”, that is, F (Σ′) =

⋃i∈I α

i[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) ∈ I and all n ∈ ω, f ijn : Σi

n −→ Σjn is injective, then

if (Σ′, (αi)i∈I) is “the” universal colimit cocone of diagram D, then for alln ∈ ω, αi

n : Σin −→ Σ′

n is injective.

Proof. Here we only prove item (iii). Let j ∈ I and n ∈ ω. Let ci, di ∈ Σin such

that αin(ci) = αi

n(di). Therefore, by item (i) above, there is k ≥ i such thatf ik

n (ci) = f ikn (di) and, as f ik

n : Σin −→ Σk

n is injective, we have ci = di. �

Proposition 2.3. Any signature is a directed colimit of finite type signatures.

Proof. Consider I as the set of all Σ′ such that, for all n ∈ ω, Σ′n ⊆ Σn and

|Σ′| ⊆fin |Σ|. Take in I the pointwise order relation Σ′ ≤ Σ′′ iff for all n ∈ ω,Σ′

n ⊆ Σn. Then:• (I,≤) is a directed ordered set;• The obvious diagram D : (I,≤) −→ S : (Σ′ ≤ Σ′′) �→ (Σ′ ↪→ Σ′′) is such

that (Σ, (Σ′ iΣ′↪→ Σ)Σ′∈I) is a commutative D-cocone;

• By the characterization in Fact 3.(i), (Σ, (Σ′ iΣ′↪→ Σ)Σ′∈I) is a colimit cocone

over D, so Σ is a directed colimit of finite type (sub)signatures. �

Proposition 2.4. A signature is finitely presentable if and only if it is of finite type.

Proof.(⇐). Let Σ′ be a signature of finite type, that is, |Σ′| is finite, and consider

D : (I,≤) −→ S, (Σi fij

−→ Σj)(i≤j)∈I a directed diagram of signatures. Then thecanonical arrow k : colimi∈IS(Σ′, Σi) −→ S(Σ′, colimi∈IΣi) is an isomorphism.

[(

Σ′ hi

−→ Σi

)

, i)]

�→(

Σ′ hi

−→ Σiγi−→ colimi∈IΣi

)

.

In order to prove that k is surjective, we have to prove that, for each signaturemorphism h : Σ′ −→ colimi∈IΣi, there is an i ∈ I and a signature morphismhi : Σ′ −→ Σi such that (Σ′ h−→ colimi∈IΣi) = (Σ′ hi−→ Σi

γi−→ colimi∈IΣi). As|Σ′| is a finite set, there is only a finite set {n0, . . . , nt−1} ⊆ N such that, for allr < t, Σ′

nris a finite non-empty set. Since that for each nr, Σnr is finite and (I,≤)

is a directed ordered set, there is an ir ∈ I such that hnr [Σ′nr ] ⊆ γir

nr[Σir

nr]. As

there is only a finite set {n0, . . . , nt−1} ⊆ N such that for all r < t, Σ′nr

= ∅ and(I,≤) is a directed ordered set, take an i ≥ i0, . . . , it−1. As γir = γi◦f iri, it followsthat, for all n ∈ ω, hn[Σ′

n] ⊆ γin[Σi

n]. Just take, for each n ∈ ω, hin : Σ′

n −→ Σin

such that, for each c′n ∈ Σ′n, hi

n(c′n) ∈ Σin is such that hn(c′n) = [(hi

n(c′n), i)] andso h = γi ◦ hi.

In order to prove that k is injective, we have to prove that, for each signaturemorphism h : Σ′ −→ colimi∈IΣi such that there are i0, i1 ∈ I and signaturemorphisms hi0 : Σ′ −→ Σi0 , hi1 : Σ′ −→ Σi1 such that γi0 ◦ hi0 = h = γi1 ◦ hi1 ,then there is a j ≥ i0, i1 and a hj : Σ′ −→ Σj such that f i0j ◦hi0 = hj = f i1j ◦hi1 .Since that for each i ∈ {i0, i1} and n ∈ ω, hi

n : Σ′n −→ Σi

n is such that foreach c′n ∈ Σ′

n, hin(c′n) ∈ Σi

n is such that [(hi0n (c′n), i0)] = hn(c′n) = [(hi1

n (c′n), i1)],


because h = γi ◦ hi, then there is a j(c′n) ≥ i0, i1 such that fi0j(c′n)n (hi0

n (c′n)) =f

i1j(c′n)n (hi1

n (c′n)) ∈ Σj(c′n)n . As |Σ′| is a finite set then {j(c′n) : c′n ∈ Σ′

n} is a finiteset and, as (I,≤) is a directed ordered set, there is a jn ≥ j(c′n) for each c′n ∈ Σ′

n.Again, as |Σ′| is a finite set, there is only a finite set {n0, . . . , nt−1} ⊆ N such that,for all r < t, Σ′

nr = ∅. As (I,≤) is a directed ordered set, take a j ≥ jn0 , . . . , jnt .

Then, as D is a diagram, for each n ∈ ω, f i0jn ◦ hi0

n = f i1jn ◦ hi1

n take hj = f ij ◦ hi,i ∈ {i0, i1}.(⇒). Let Σ be a finitely presentable logic. Then, by Proposition 2.3, there is a di-

rected diagram of finite type logics D : (I,≤) −→ S, (Σi fij

−→ Σj)(i≤j)∈I such that

there is an isomorphism h : l∼=−→ colimi∈IΣi. Then, as the canonical morphism

is invertible k : colimi∈IS(Σ, Σi)∼=−→ S(Σ, colimi∈IΣi), there is a factorization of

h : (Σ h−→ colimi∈IΣi) = (Σ hi

−→ Σi γi

−→ colimi∈IΣi). Then, as h is an isomor-phism, hi : Σ −→ Σi is an S-section. In particular there is a sequence of injections(hi

n : Σn � Σin)n∈ω so, as |Σi| is finite, then |Σ| is finite. �

Theorem 2.5. The category S is a finitely locally presentable category, i.e., S isan accessible category that is cocomplete and complete.

Proof. Direct consequence of Propositions 2.3, 2.4 and 2.2. �Corollary 2.6. (i) The Yoneda functor Y : Sfp −→ Set(Sfp)op

has an extensionto a functor Y ′ : S −→ Set(Sfp)op

, Σ �→ Y ′(Σ) = L(ι(.), Σ) that is full andfaithful;

(ii) Let Flat(Sfp, Set) be the full subcategory of Set(Sfp)op

whose objects are thefunctors that are filtered colimits of representable functors. ThenFlat(Sfp, Set) is the “essential image” of Y ′ and so his restriction functorE : S −→ Flat(Sfp, Set) is an equivalence of categories;

(iii) Flat(Sfp, Set) coincides with the category of Set-valued functors that preservefinite limits;

(iv) Y ′ has a left adjoint.

Proof. For (i) and (ii) see [9], Theorem 5.3.5 (p. 265) or [1], Theorem 2.26 (p. 83)or [22], Observation 1.6 (p. 46). For (iii) and (iv) see [1], Theorem 1.46 (p. 38). �2.2. The category LThe category L is the category of propositional logics and translations as mor-phisms. This is a category “built above” the category S, that is, there is an obviousforgetful functor U : L −→ S.

2.2.1. What is L? 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 Parts(F (Σ)) × F (Σ), such that Cons(Γ) = {ϕ ∈F (Σ) : Γ � ϕ}, for all Γ ⊆ F (Σ), gives a structural finitary closure operator onF (Σ):

• inflationary: Γ ⊆ Cons(Γ);


• increasing: Γ0 ⊆ Γ1 ⇒ Cons(Γ0) ⊆ Cons(Γ1);• idempotent: Cons(Cons(Γ)) ⊆ Cons(Γ);• finitary: Cons(Γ) =

⋃{Cons(Γ′) : Γ′ ⊆fin Γ};

• structural: σ(Cons(Γ)) ⊆ Cons(σ(Γ)), for each substitution σ : X → F (Σ).In [20], �Los and Suszko give a characterization of usual provability notion in Hilbertcalculi by the consequence operators with the above properties.

We say that a logic l = (Σ,�) is of finite type when |Σ| is a finite set and �is determined, in the sense of [20]2, by a finite set of axioms and inference rules.

If l = (Σ,�), l′ = (Σ′,�′) are logics then a translation morphism f : l −→ l′

is a signature morphism f : Σ −→ Σ′ that “preserves the consequence relation”,that is, for all Γ∪{ψ} ⊆ F (Σ), if Γ � ψ then f [Γ] �′ f(ψ). We say that a morphismf : l −→ l′ is a conservative translation morphism if for all Γ ∪{ψ} ⊆ F (Σ), Γ � ψ

if and only if f [Γ] �′ f(ψ). Composition and identities are similar to S.

2.2.2. Some facts about L.

Definition 2.7. There is a natural definition of order between consequence relationson each signature Σ: for each pair �,�′ of consequence relations over Σ we havethe equivalence between the items below:

• For each Γ ∪ {ψ} ⊆ F (Σ), Γ � ψ ⇒ Γ �′ ψ;• The identity signature morphism over Σ, idΣ : Σ −→ Σ, is a translation

morphism idΣ : (Σ,�) −→ (Σ,�′).We write � ≤ �′ when the conditions above are satisfied.

Fact 4. The set of consequence relations on a signature Σ, denoted by LΣ, is acomplete lattice. It is in fact an algebraic lattice where the compact elements arethe “finitely generated logics”, the logics over Σ given by a finite set of axioms anda 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 thesignature Σ. Now, for each Γ∪{ψ} ⊆ F (Σ), define that Γ � ψ ⇔ there is Γ′ ⊆fin Γsuch that (∀i ∈ I)(Γ′ �i ϕ), then (Σ,�) is a logic and l = (Σ,�) is the infimum ofthe family D in L. In fact, this follows from them items below:

• l ∈ Obj(L);• idΣ ∈ L(l, lj) for all j ∈ I;• if l′ = (Σ,�′) is a logic over the signature Σ such that idΣ ∈ L(l′, lj) for all

j ∈ I, then idΣ ∈ L(l′, l).

2A formula is demonstrable from a given set of hypothesis iff there is a finite sequence of formulassuch 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) inferencerule.


Directed sups. Consider I a set and D = {li = (Σ,�i)}i∈I an upward directedfamily of logics over the signature Σ, that is, for each i, j ∈ I there is a k ∈ I suchthat idΣ ∈ L(li, lk) , idΣ ∈ L(lj , lk). Now, for each Γ ∪ {ψ} ⊆ F (Σ), define thatΓ � ψ ⇔ there is Γ′ ⊆fin Γ and there is an i ∈ I such that Γ′ �i ϕ, then (Σ,�) isa logic and l = (Σ,�) is the supremum of the family D in L. In fact, this followsfrom the items below:

• l ∈ Obj(L);• idΣ ∈ L(lj , l) for all j ∈ I;• if l′ = (Σ,�′) is a logic over the signature Σ such that idΣ ∈ L(lj , l′) for all

j ∈ I, then idΣ ∈ L(l, l′).Sups. As usual, the supremum of a family of logics can be obtained taking theinfimum of the set of upper bounds of that family of logics. A more objectivecharacterization of suprema can be given but we postpone that because this canbe easily described by more general results below (see Proposition 2.11).Compact consequence relations. A consequence relation �′ over Σ is compact if foreach set I, each D = {li = (Σ,�i)}i∈I a upward directed family of logics over thesignature Σ, if �′ ≤

∨i∈I �i then there is an i ∈ I such that �′ ≤�i. It follows easily

that this condition is equivalent to the “stronger” condition: for each set J , eachD = {lj = (Σ,�j)}j∈J a family of logics over the signature Σ, if �′ ≤

∨j∈J �j

then there is a finite subset J ′ ⊆ J such that �′ ≤∨

j∈J′ �j .3 A consequencerelation on Σ is compact if and only if it is a finitely generated consequence relationon Σ. Any consequence relation on Σ is the directed supremum of its compact(sub)consequence relations on Σ. �

Remark 2.8. As the set of consequence operators (or consequence relations) on asignature Σ is a complete lattice, there exists the logic generated by any functionW : P (F (Σ)) −→ P (F (Σ)): it is enough to take the infimum of the family ofall consequence relations on Σ that are upper bounds of the “proto-consequencerelation” associated with the “proto-consequence operator” W .

Definition 2.9. Direct image and inverse image: let f : Σ −→ Σ′ be a S-morphism:• Inverse image: if l′ = (Σ′,�′) ∈ Obj(L) then for all Γ ∪ {ψ} ⊆ F (Σ) define

Γ �f�(�′) ψ iff f [Γ] �′ f(ψ);• Direct image: if l = (Σ,�) ∈ Obj(L) then for all Γ′ ∪ {ψ′} ⊆ F (Σ′) define

Γ′ �f�(�) ψ′ iff there is a finite sequence of Σ′-formulas (φ′0, . . . , φ

′t) such that:

– φ′t = ψ′;

– for all p ≤ t at least one of the alternatives below occurs:∗ “φ′

p is a hypothesis”: φ′p ∈ Γ′;

∗ “φ′p is an instance of an l-axiom”: there is a θp ∈ F (Σ) such that

� θp and there is a substitution σ′ : X −→ F (Σ′) such thatσ′(f(θp)) = φ′

p;

3Just observe that any sup of a family coincides with a sup of a directed family: for each set Jtake I = Pfin(J) then, for each J ′ ⊆fin J , define �J′ =

∨j∈J′ �j . . . .


∗ “φ′p is a direct consequence of an instance of l-inference rule applied

over previous members in the sequence”: there is a ∆p ∪{θp} ⊆fin

F (Σ) such that ∆p � θp and there is a substitution σ′ : X −→F (Σ′) such that σ′(f(θp)) = φ′

j and σ′[f [∆p]] ⊆ {φ′0, . . . , φ

′j−1}.

Fact 5. About direct image and inverse image: Let f : Σ −→ Σ′ be a S-morphismand let l = (Σ,�), l′ = (Σ′,�′) be logics l, l′ ∈ Obj(L). Then (i)� and (i)� hold,and (ii)�, (ii)tm and (ii)� are equivalent:

(i)� if l′ = (Σ′,�′) ∈ Obj(L) then f�(l′) = (Σ,�f�(�′)) ∈ Obj(L);(i)� if l = (Σ,�) ∈ Obj(L) then f�(l) = (Σ′,�f�(�)) ∈ Obj(L).(ii)� � ≤ f�(�′);

(ii)tm f : (Σ,�) −→ (Σ′,�′) is a translation morphism;(ii)� f�(�) ≤′ �′.

Proof. (Sketch)(i) The proof of (i)� is omitted; the proof of (i)� is analogous to the item Colim-

its.(a) in Proposition 2.11;(ii) The equivalence (ii)� ⇔ (ii)tm follows directly from the definitions; the impli-

cation (ii)� ⇒ (ii)tm is analogous to the item Colimits.(b) in Proposition 2.11;the implication (ii)tm ⇒ (ii)� is analogous to the item Colimits.(c) in Propo-sition 2.11. �

Remark 2.10. It follows easily from the facts above that the forgetful functorU : L −→ S : ((Σ,�)

f−→ (Σ′,�′)) �→ (Σf−→ Σ′) has left and right adjoint

functors: the left adjoint T : S −→ L and the right adjoint V : S −→ L takea signature Σ to, respectively, T (Σ) = (Σ,�min) (the first element of LΣ) andV (Σ) = (Σ,�max) (the last element of LΣ).

Proposition 2.11. The category L is complete and cocomplete and the forgetfulfunctor U : L −→ S creates all small limits and colimits.

Proof. Before begining the proof (that is long and tedious!) let us claim that thisresult is not surprising and has an “easy proof” in terms of the concepts of in-fima/suprema and inverse/direct image: the limit cones in L are the underlyinglimit cones in S when we take the “limit consequence relation” on the limit sig-nature as the infimum of the set of the inverse images of consequence relationsthrough each S-morphism in the S-limit cone; analogously the colimit cocones in Lare the underlying colimit cocones in S when we take the “colimit consequencerelation” on the colimit signature as the supremum of the set of the direct imagesof consequence relations through each S-morphism in the S-colimit cocone. How-ever we choose present the full proofs by two reasons: first we have not present anyexplicit calculation for the concepts of infima/suprema and inverse/direct image(see, respectively Fact 4 and Fact 5), second, and most importantly, we do needthe explicit construction of the direct colimit logic in the proofs of Section 3.

We split the proof in three sections: limits4, filtered colimits and colimits.

4In [10] there is a similar proof for products, but with another notion of signature morphism.


Limits. Let I be a small category and D : I −→ L, ((Σi,�i)fh

−→ (Σj ,�j))(h:i→j)∈I

a diagram, and take (Σ, (πi)i∈Obj(I)) the limit of the underlying diagram (I D−→S U−→ L). For all Γ ∪ {ψ} ⊆ F (Σ), define that Γ � ψ ⇔ there is Γ− ⊆fin Γsuch that for all i ∈ Obj(I) πi[Γ−] �i πi(ψ)5, then l = (Σ,�) is a logic and(l, (πi)i∈Obj(I)) is the limit of D in L. In fact, this follows from (a), (b) and (c)below:(a) l ∈ Obj(L);(b) πj ∈ L(l, lj), for all j ∈ I;(c) if (l′, (αi)i∈Obj(I)) is a commutative cone over the diagram D then the unique

signature morphism α : Σ′ −→ Σ such that αi = πi ◦α, i ∈ Obj(I) preservesthe consequence relation.Now we prove (a), (b) and (c).

(a) it follows directly from the definition of � that it gives a finitary and increasingconsequence operator. It is also inflationary because if ψ ∈ Γ, take any Γ− ⊆fin Γsuch that ψ ∈ Γ− then, for all i ∈ Obj(I), πi(ψ) ∈ πi[Γ−]. Since �i is inflationary,then πi[Γ−] �i πi(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 ∆ ⊆fin Γ be such that ∆ � ψ. Then,for each θ ∈ ∆, let Γθ ⊆fin Γ be such that Γθ � θ. It follows that Γ− =

⋃θ∈∆Γθ

satisfies Γ− ⊆fin Γ and, as � is increasing, for each θ ∈ ∆, Γ− � θ. Then, as �i

gives an inflationary operator, for each i ∈ Obj(I), it follows from the definitionof ∆ � ψ that for each i ∈ Obj(I), πi[∆] �i πi(ψ). Analogously, as Γ− � θ for eachθ ∈ ∆, then for each i ∈ Obj(I), πi[Γ−] �i πi(θ), for each θ ∈ ∆. Now, as �i givesan idempotent operator, then πi[Γ−] �i πi(ψ), for each i ∈ Obj(I). Therefore, asΓ− ⊆fin Γ, we have Γ � ψ.

Structural: Let Γ ∪ {ψ} ⊆ F (Σ) be such that Γ � ψ. We have to prove thatfor any substitution σ : X −→ F (Σ) we have σ[Γ] � σ(ψ). Let Γ− ⊆fin Γ besuch that, for all i ∈ Obj(I), πi[Γ−] �i πi(ψ). Since σ[Γ−] ⊆fin σ[Γ], we haveσ[Γ] � σ(ψ) if we prove that πi[σ[Γ−]] �i πi(σ(ψ)), for each i ∈ Obj(I). Now,from Fact 1.(ii), for all i ∈ Obj(I) there is a substitution σi : X −→ F (Σi)such that πi ◦ σ = σi ◦ πi. Then, for each i ∈ Obj(I), since πi[Γ−] �i πi(ψ) and�i gives a structural operator, we have that σi[πi[Γ−]] �i σi(πi(ψ)). So we haveπi[σ[Γ−]] �i πi(σ(ψ)), for each i ∈ Obj(I). Therefore σ[Γ] � σ(ψ).(b) Let Γ ∪ {ψ} ⊆ F (Σ) be such that Γ � ψ. Then select a Γ− ⊆fin Γ such thatfor all i ∈ Obj(I), πi[Γ−] �i πi(ψ). Since πi[Γ−] ⊆ πi[Γ] and �i is inflationary, foreach i ∈ Obj(I), we have πi[Γ] �i πi(ψ). So the signature morphism πi : Σ −→ Σi

gives also a translation morphism πi : (Σ,�) −→ (Σi,�i), for each i ∈ Obj(I).(c) Let Γ′ ∪ {ψ′} ⊆ F (Σ′) be such that Γ′ �′ ψ′. We have to prove that α[Γ′] �α(ψ′). Since �′ is finitary, select Γ′− ⊆fin Γ′ such that Γ′− �′ ψ′. Then, as αi :l′ −→ li is a translation morphism, αi[Γ′−] �i αi(ψ′), for each i ∈ Obj(I). So,

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


as αi = πi ◦ α, we have πi[α[Γ′−]] �i πi(α(ψ′)), for each i ∈ Obj(I). Now, sinceα[Γ′−] ⊆fin α[Γ′], we have from the definition that α[Γ′] � α(ψ′). So the signaturemorphism α : Σ′ −→ Σ gives also a translation morphism α : (Σ′,�′) −→ (Σ,�).Filtered colimits. Let (I,≤) be a directed ordered set and D : (I,≤) −→ L,

((Σi,�i)fij

−→ (Σj ,�j))(i≤j)∈I be a diagram. Take (Σ, (γi)i∈I) the colimit of the

underlying diagram (I D−→ S U−→ L). Now, for all Γ ∪ {ψ} ⊆ F (Σ), define thatΓ � ψ ⇔ there is Γ− ⊆fin Γ and there is an i ∈ I such that Γ− ∪ {ψ} ⊆ γi[F (Σi)]and there is Γ−i ∪ {ψi} ⊆fin F (Σi) such that γi[Γ−i] = Γ− , γi(ψi) = ψ andΓ−i �i ψi. Then l = (Σ,�) is a logic and (l, (γi)i∈I) is the colimit of D in L. Infact, this follows from (a), (b) and (c) below:(a) l ∈ Obj(L);(b) γj ∈ L(lj , l), for all j ∈ I;(c) if (l′, (αi)i∈I) is a commutative cocone over the diagram D then the unique

signature morphism α : l −→ l′ such that αi = α ◦ γi, i ∈ I preserves theconsequence relations.Now we prove (a), (b) and (c).

(a) It follows directly from the definition of � that it gives a finitary and increasingconsequence operator. It is also inflationary because if ψ ∈ Γ, take Γ− = {ψ} andany i ∈ I such that ψ ∈ γi[Σi], take ψi ∈ F (Σi) such that γi(ψi) = ψ andΓ−i = {ψi} then, as �i gives a inflationary operator, Γ−i �i ψi.

Idempotent: Let ψ ∈ F (Σ) be such that Γ � ψ where Γ = {θ ∈ F (Σ) : Γ � θ}.Let us prove that Γ � ψ. Since � is finitary, let ∆ ⊆fin Γ be such that ∆ � ψ.Then, for each θ ∈ ∆, let Γθ ⊆fin Γ be such that Γθ � θ. Then Γ− =

⋃θ∈∆Γθ

is such that Γ− ⊆fin Γ and, since � is increasing, for each θ ∈ ∆, Γ− � θ. Wecan choose i ∈ I and ∆i ∪ {ψi} ⊆fin F (Σi) with γi[∆i] = ∆ , γi(ψi) = ψ and∆i �i ψi because, by definition, ∆ � ψ iff there is a subset ∆− ⊆fin ∆ and aj ∈ I ∆−j ∪{ψj} ⊆fin F (Σj) with γj [∆−j ] = ∆, γj(ψj) = ψ and ∆−j �j ψj and,since ∆ is a finite set and (I,≤) is a directed ordered set, there is an i ≥ j and a∆i ⊆fin F (Σi) such that f ji[∆−j ] ⊆ ∆i and γi[∆i] = ∆. Now we have f ji[∆−j ] �i

f ji(ψj) and since (Σ, (γi)i∈I) is a commutative cocone over the diagram D, takingψi = f ji(ψj) we have, since �i is increasing, ∆i∪{ψi} ⊆fin F (Σi) with γi[∆i] = ∆,γi(ψi) = ψ and ∆i �i ψi. Analogously we can choose, for each θ ∈ ∆, an iθ ∈ I

such that there is Γ−iθ ∪ {θiθ} ⊆fin F (Σiθ ) with γiθ [Γ−iθ ] = Γ−, γiθ (θiθ ) = θ

and Γ−iθ �iθθiθ . Then, since (I,≤) is a directed ordered set, take j ≥ i, iθ for

all θ ∈ ∆. Now, since (Σ, (γi)i∈I) is a commutative cocone over the diagram D :(I,≤) −→ L, then, with Γ−j =

⋃θ∈∆f iθj [Γ−iθ ] ⊆fin F (Σj) and θj = f iθj(θiθ ),

we have, since �j is increasing, Γ−j �j θj for all θ ∈ ∆. We can also suppose jsuch that ∆j = {θj : θ ∈ ∆} satisfies ∆j ∪ {ψj} ⊆fin F (Σj) with γj[∆j ] = ∆,γj(ψj) = ψ and ∆j �j ψj . Finally, since �j is idempotent, we have Γ−j �j ψj andwe have γj[Γ−j ] =

⋃θ∈∆γj[f iθj [Γ−iθ ]] =

⋃θ∈∆γiθ [Γ−iθ ] =

⋃θ∈∆Γ− = Γ− ⊆fin Γ

and γj(ψj) = γj(f ij(ψi)) = γi(ψi) = ψ.


Structural: Let Γ ∪ {ψ} ⊆ F (Σ) be such that Γ � ψ. We have to prove thatfor any substitution σ : X −→ F (Σ) we have σ[Γ] � σ(ψ).

Let Γ− ⊆fin Γ, an i ∈ I such that Γ− ∪ {ψ} ⊆ γi[F (Σi)] and let Γ−i ∪{ψi} ⊆fin F (Σi) be such that γi[Γ−i] = Γ−, γi(ψi) = ψ and Γ−i �i ψi. Now,since Γ−∪{ψ} is a finite set of Σ-formulas then also Γ−∪{ψ}∪σ[Γ−]∪{σ(ψ)} ⊆fin

F (Σ). So there is an n ∈ ω such that the X-variables that occur in Γ− ∪ {ψ} ∪σ[Γ−]∪{σ(ψ)} ⊆fin F (Σ) are in the finite set {x0, . . . , xn−1} ⊆fin X . Then, since(I,≤) is a directed ordered set, there is a k ≥ i such that σ[{x0, . . . , xn−1}] ⊆γk[Σk]. Now take a substitution σk : X −→ F (Σk) such that for any m < n,σk(xm) ∈ (γk)−1[σ(xm)]. Now, with Γ−k = f ik[Γ−i] and ψk = f ik(ψi), we haveΓ−k ∪ {ψk} ⊆fin F (Σk) such that γk[Γ−k] = Γ−, γk(ψk) = ψ and Γ−k �k ψk.So, since �k is structural, it follows that σk[Γ−k] �k σk(ψk). Since σk[Γ−k] ∪{σk(ψk)} ⊆fin F (Σk) it is enough to show that for all Σ-formula θ in the finiteset Γ− ∪ {ψ} ⊆fin F (Σ) the Σk-formula θk ∈ Γ−k ∪ {ψk} ⊆fin F (Σk) such thatγk(θk) = θ also satisfies σ(θ) = γk(σk(θk)), in order to prove that σ[Γ] � σ(ψ) . Aswe have seen above, θ = θ(x0, . . . , xn−1). This equation together with Fact 1.(i)give σ(θ(x0, . . . , xn−1)) = θ(σ(x0), . . . , σ(xn−1)) = γk(θk)(σ(x0), . . . , σ(xn−1)) =γk(θk)(γk(σk(x0)), . . . , γk(σk(xn−1))) = γk(θk(σk(x0), . . . , σk(xn−1))) =γk(σk(θk(x0, . . . , xn−1))).(b) Let j ∈ I and Γj ∪ {ψj} ⊆ F (Σj) be such that Γj �j ψj . Now since �j isfinitary, select Γ−j ⊆fin Γj such that Γ−j �j ψj . For any Γ∪{ψ} ⊆ F (Σ) such thatΓ = γj [Γj ], ψ = γj(ψj). We have that Γ � ψ. In fact, there is a Γ− ⊆fin Γ (takeΓ− = γj[Γ−j ]) and there is an i ∈ I (take i = j) such that Γ− ∪ {ψ} ⊆ γi[F (Σi)]and there is Γ−i ∪ {ψi} ⊆fin F (Σi) such that γi[Γ−i] = Γ−, γi(ψi) = ψ andΓ−i �i ψi.(c) Let Γ∪{ψ} ⊆ F (Σ) be such that Γ � ψ. Then, by definition, there is Γ− ⊆fin Γand there is an i ∈ I such that Γ− ∪{ψ} ⊆ γi[F (Σi)] and there is Γ−i ∪{ψi} ⊆fin

F (Σi) such that γi[Γ−i] = Γ−, γi(ψi) = ψ and Γ−i �i ψi. Since αi : li −→ l′

is a translation morphism, we have αi[Γ−i] �′ αi(ψi) and since αi = α ◦ γi,α[γi[Γ−i]] �′ α(γi(ψi)) we have α[Γ−] �′ α(ψ) and α[Γ] �′ α(ψ) (�′ is increasing).

Colimits. Let I be a small category and D : I −→ L, ((Σi,�i)fh

−→ (Σj ,�j))(h:i→j)∈I be a diagram, and take (Σ, (γi)i∈Obj(I)) the colimit of the under-

lying diagram (I D−→ S U−→ L). Now, for all Γ∪{ψ} ⊆ F (Σ), define that Γ � ψ ⇔there is a finite sequence of Σ-formulas (φ0, . . . , φt), where φt = ψ and for all p ≤ tone of these alternative occurs:

• “φp is an hypothesis”: φp ∈ Γ;• “φp is an axiom”: there are i ∈ Obj(I), θi ∈ F (Σi), σ : X −→ F (Σ) such

that �i θi and φp = σ(γi(θi));


• “φp is a consequence of a inference rule”: there are i ∈ Obj(I), ∆i∪{θi} ⊆fin

F (Σi), σ : X −→ F (Σ) such that ∆i �i θi and σ[γi[∆i]] ⊆ {φ0, . . . , φp−1},φp = σ(γi(θi));

Then l = (Σ,�) is a logic and (l, (γi)i∈Obj(I)) is the colimit of D in L. In fact, thisfollows from (a), (b) and (c) below:

(a) l ∈ Obj(L);(b) γj ∈ L(lj , l), for all j ∈ Obj(I);(c) if (l′, (αi)i∈I) is a commutative cocone over the diagram D then the unique

signature morphism α : l −→ l′ such that αi = α ◦ γi, i ∈ Obj(I) preservesthe consequence relation.

Now we prove (a), (b) and (c).(a) It follows directly from the definition of � that it gives a finitary, increasingand inflationary consequence operator.

Idempotent: Let ψ ∈ F (Σ) be such that Γ � ψ, where Γ = {θ ∈ F (Σ) : Γ � θ}.Let us prove that Γ � ψ. Choose a “proof of ψ from hypothesis in Γ”, that is, afinite sequence of Σ-formulas (φ0, . . . , φt) where φt = ψ and for all p ≤ t, “φp is aΓ-hypothesis” or “φp is an instance of axiom” or “φp is a consequence of previousformulas in the sequence by an instance of a inference rule”. Now, for each p ≤ tsuch that “φp is an Γ-hypothesis”, that is, Γ � φp, select a “proof of φp fromhypothesis in Γ”: a finite sequence of Σ-formulas (ϕp

0, . . . , ϕptp

) where ϕptp

= φp

and for all m ≤ tp, “ϕpm is a Γ-hypothesis” or “ϕp

m is an instance of axiom” or“ϕp

m is a consequence of previous formulas in the sequence by an instance of ainference rule”. Now merge the original sequence (φ0, . . . , φt) with the sequences(ϕp

0, . . . , ϕptp

), for each p ≤ t such that φp is a Γ-hypothesis in the obvious way:replace that φp by the sequence (ϕp

0, . . . , ϕptp

). Then since ϕptp

= φp, we get afinite sequence of Σ-formulas such that the last one is ψ and each formula is “aΓ-hypothesis” or is “an instance of axiom” or “a consequence of previous formulasin the sequence by an instance of a inference rule”. So this resulting finite sequenceof Σ-formulas is a “proof of ψ from hypothesis in Γ”.

Structural: Let Γ ∪ {ψ} ⊆ F (Σ) be such that Γ � ψ. We have to prove thatfor any substitution σ′ : X −→ F (Σ) we have σ′[Γ] � σ′(ψ). Choose a “proof of ψfrom hypothesis in Γ”, that is, a finite sequence of Σ-formulas (φ0, . . . , φt) whereφt = ψ and for all p ≤ t, “φp is a Γ-hypothesis” or “φp is an instance of axiom”or “φp is a consequence of previous formulas in the sequence by an instance ofa inference rule”. We will see that (σ′(φ0), . . . , σ′(φt)) is a “proof of σ′(ψ) fromhypothesis in σ′[Γ]”: Since σ′(φt) = σ′(ψ), then we have to show that for all p ≤ t,“σ′(φp) is an σ′[Γ]-hypothesis” or “σ′(φp) is an instance of axiom” or “σ′(φp) isa consequence of previous formulas in the sequence by an instance of a inferencerule”:

• If “φp is a Γ-hypothesis” then φp ∈ Γ so σ′(φp) ∈ σ′[Γ], that is, “σ′(φp) is aσ′[Γ]-hypothesis”;


• If “φp is an instance of axiom” then select i ∈ Obj(I), θi ∈ F (Σi), σ :X −→ F (Σ) such that �i θi and φp = σ(γi(θi)), then take the substitution

σ′′ : X −→ F (Σ) such that (X σ′′−→ F (Σ)) = (X σ−→ F (Σ) σ′

−→ F (Σ)). Then,as σ′′ = σ′ ◦ σ, we have, by uniqueness of extensions, σ′′ = σ′ ◦ σ. It followsthat we have i ∈ Obj(I), θi ∈ F (Σi), σ′′ : X −→ F (Σ) such that �i θi andσ′(φp) = σ′(σ(γi(θi))) = σ′′(γi(θi)). So “σ′(φp) is an instance of axiom”;

• If “φp is a consequence of previous formulas in the sequence by an instance of ainference rule” then select i ∈ Obj(I), ∆i ∪{θi} ⊆fin F (Σi), σ : X −→ F (Σ)such that ∆i �i θi and σ[γi[∆i]] ⊆ {φ0, . . . , φp−1}, φp = σ(γi(θi)). Then, asabove, take the substitution σ′′ : X −→ F (Σ) such that σ′′ = σ′ ◦ σ, then wehave i ∈ Obj(I), ∆i ∪ {θi} ⊆fin F (Σi), σ′′ : X −→ F (Σ) such that ∆i �i θi

and σ′′[γi[∆i]] = σ′[σ[γi[∆i]]] ⊆ σ′[{φ0, . . . , φp−1}] = {σ′(φ0), . . . , σ′(φp−1)}and (σ′′(γi(θi)) = σ′(σ(γi(θi))) = σ′(φp). So “σ′(φp) is a consequence ofprevious formulas in the sequence by an instance of a inference rule”.

(b) Let j ∈ Obj(I) and Γj ∪ {ψj} ⊆ F (Σj) be such that Γj �j ψj . Since �j

is finitary, select Γ−j ⊆fin Γj such that Γ−j �j ψj . Now take {φ0, . . . , φt−1} aenumeration of the finite set γj[Γ−j ] and consider the finite sequence of Σ-formulas(φ0, . . . , φt−1, φt) such that φt = γj(ψj). Then γj[Γj ] � γj(ψj). In fact, the finitesequence of Σ-formulas (φ0, . . . , φt) is a “proof of γj(ψj) from hypothesis in γj[Γj ]”:for each p < t, φp is a “γj[Γj ]-hypothesis”, because φp ∈ γj [Γ−j ] ⊆ γj [Γj ] and,for p = t, φp = γj(ψj) is a “consequence of previous formulas in the sequence”,because we have a j ∈ Obj(I) and ∆j ∪ {θj} ⊆fin F (Σj) such that ∆j �j θj

(take ∆j = Γ−j and θj = ψj) and we have a substitution σ : X −→ F (Σ)(take σ(xn) = xn, for all n ∈ ω. Clearly σ = id : F (Σ) −→ F (Σ)) such thatσ(γj(θj)) = φt = γj(ψj) and σ[γj [∆j ]] = γj [Γ−j ] ⊆ {φ0, . . . , φt−1}.(c) Let Γ ∪ {ψ} ⊆ F (Σ) be such that Γ � ψ. We show that α[Γ] �′ α(ψ), by“induction on the rank of a formula demonstrable from hypothesis in Γ”: if Λ � ϕ,rk(ϕ) is the least t ∈ ω such there is a sequence of Σ-formulas (φ0, . . . , φt) that isa “proof of ϕ from hypothesis in Λ”.

Let t = rk(ψ):• If t = 0 then ψ = φt is a hypothesis or an axiom:

– If ψ is a hypothesis then ψ ∈ Γ, so α(ψ) ∈ α[Γ] and, as �′ is inflationary,α[Γ] �′ α(ψ);

– If ψ is an axiom then select i ∈ Obj(I), θi ∈ F (Σi), σ : X −→ F (Σ)such that �i θi and ψ = σ(γi(θi)). As αi is a translation morphism, wehave �′ αi(θi) and as αi = α◦γi, �′ α(γi(θi)). Now, by Fact 1.(ii), take asubstitution σ′ : X −→ F (Σ′) such that σ′◦α = α◦σ. As �′ is structuralwe get �′ σ′(α(γi(θi))) and then, as σ′(α(γi(θi))) = α(σ(γi(θi))) =α(ψ), we have �′ α(ψ) so, as �′ is increasing, α[Γ] �′ α(ψ).

• If t > 0 then ψ = φt is a consequence of a inference rule: there are i ∈ Obj(I),∆i ∪ {θi} ⊆fin F (Σi), σ : X −→ F (Σ) such that ∆i �i θi, ψ = φt =σ(γi(θi)) and σ[γi[∆i]] ⊆ {φ0, . . . , φt−1}. As for each m < t the subsequence


(φ0, . . . , φm) is a “proof of φm from hypothesis in Γ” then, for each m < t,the formula φm is such that rk(φm) ≤ m < t so, by the induction hypothesis,α[Γ] �′ α(φm), for each m < t. As αi is a translation morphism, we haveαi[∆i] �′ αi(θi) and as αi = α◦γi, α[γi[∆i] �′ α(γi(θi)). Now, by Fact 1.(ii),take a substitution σ′ : X −→ F (Σ′) such that σ′ ◦ α = α ◦ σ. As �′ is struc-tural we get σ′[α[γi[∆i]]] �′ σ′(α(γi(θi))). As σ′(α(γi(θi))) = α(σ(γi(θi))) =α(ψ) and σ′[α[γi[∆i]]] = α[σ[γi[∆i]]] we have α[σ[γi[∆i]]] �′ α(ψ) so, as �′

is inflationary, α[{φ0, . . . , φt−1}] �′ α(ψ). Finally, as α[Γ] �′ α(φm) for eachm < t, and �′ is idempotent, we get α[Γ] �′ α(ψ). �

Proposition 2.12. Monomorphisms and epimorphisms in L: let U : L −→ S bethe forgetful functor. Then a morphism f in L is monic (epic) iff U(f) is monic(epic) in S.

Proof. The right to left implication is easy. Let f : l −→ l′ be a morphism in Lsuch that U(f) : U(l) −→ U(l′) is not S-monic. Then there exists Σ′′ ∈ Obj(S)and g, h : Σ′′ −→ U(l) such that g = h and U(f) ◦ g = U(f) ◦ h. These g, h areL-morphisms from the least logic over Σ (i.e., the logic whose closure operator isthe identity) to l which satisfy g = h and f ◦ g = f ◦ h, showing that f is notL-monic. For the “epic” part proceed similarly, taking two “counterexample” ar-rows in S, g, h : U(l′) −→ Σ′′ such that g = h and g ◦U(f) = h ◦U(f). These g, hbecome L-morphisms equipping their codomain with the greatest consequencerelation there (where “everything can be deduced from anything and/or noth-ing”) or with the “logic generated” by the direct image logic of this S-morphism(�′′= g�(�) ∨ h�(�)). �

2.2.3. L is a locally presentable category.

Fact 6. Additional facts on filtered colimits in L: Let D : (I,≤) −→ L, (lifij

−→lj)(i≤j)∈I be a directed diagram and let (l′, (αi)i∈I) be a commutative cocone overthe diagram D:

(i) (l′, (αi)i∈I) is “the” universal colimit cocone of diagram D iff:• Σ′

n =⋃

i∈I αin[Σi

n], n ∈ ω;• If ci ∈ Σi

n, cj ∈ Σjn are such that αi

n(ci) = αjn(cj) then there is k ≥ i, j

such that (f ik)n(ci) = (f jk)n(cj), n ∈ ω;• For all Γ′ ∪ {ψ′} ⊆ F (Σ′) such that Γ′ �′ ψ′ ⇔, there is Γ′− ⊆fin

Γ′ and there is i ∈ I such that Γ′− ∪ {ψ′} ⊆ αi[F (Σi)] and there isΓ−i ∪ {ψi} ⊆fin F (Σi) such that αi[Γ−i] = Γ′−, αi(ψi) = ψ′ andΓ−i �i ψi.

(ii) If, for all (i ≤ j) ∈ I, f ij : li −→ lj is a conservative monomorphism, then if(l′, (αi)i∈I) is “the” universal colimit cocone of diagram D then αi : li −→ l′

is a conservative monomorphism.

Proof. We prove only the part of item (ii) concerning conservativeness: Let j ∈ Iand Γj ∪ {ψj} ⊆ F (Σj) such that αj [Γj ] �′ αj(ψj). Now take Γ′ ∪ {ψ′} ⊆ F (Σ′)


such that Γ′ = αj [Γj] , ψ′ = αj(ψj). As Γ′ �′ ψ′ then, by item (i) above, thereis Γ′− ⊆fin Γ′ and there is i ∈ I such that Γ′− ∪ {ψ′} ⊆ αi[F (Σi)] and there isΓ−i ∪ {ψi} ⊆fin F (Σi) such that αi[Γ−i] = Γ′−, αi(ψi) = ψ′ and Γ−i �i ψi.As (I,≤) is a directed ordered set, there is k ≥ i, j, then as αi = αk ◦ f ik,Γ′−∪{ψ′} ⊆ αk[F (Σk)] and there is Γ−k∪{ψk} ⊆fin F (Σk) (take Γ−k = f ik[Γ−i]and ψk = f ik(ψi)) such that αk[Γ−k] = Γ′−, αk(ψk) = ψ′ and Γ−k �k ψk

(this because f ik : li −→ lk is a translation morphism). As αj , αk, f jk are S-monomorphisms (by Proposition 2.12 and Fact 3.(iii)) and αj = αk ◦ f jk wehave, by Fact 2.(ii), that the formula algebra functions αj , αk, f jk are injectivesand αj = αk ◦ f jk. From this we can conclude that ψk = f ik(ψj) and there isΓ−j ⊆fin Γj such that Γ−k = f jk[Γ−j ] (take Γ−j = (αj)−1[Γ′−]). As Γ−k �k ψk

and f jk : lj −→ lk is a conservative translation morphism, we have Γ−j �j ψj .Finally, as Γ−j ⊆ Γj and �j is increasing, we have Γj �j ψj . �

Proposition 2.13. Any logic is a directed colimit of finite type logics.

Proof. Let l = (Σ,�) ∈ Obj(L) and take the set I of all l′ = (Σ′,�′) ∈ Obj(L) suchthat |Σ′| ⊆fin |Σ|; �′ is given by a finite set of axioms and a finite set of finitaryinference rules; the signature morphism of inclusion Σ′ ↪→ Σ is also a translationmorphism l′ ↪→ l. Then:

(a) This diagram is “directed by inclusions” (clear);(b) (l, (l′ ↪→ l)l′∈I) is the colimit of this diagram.

This follows from the characterization of � in filtered colimits.It is clear that the first two conditions in Fact 6.(i) are satisfied. Now consider

Γ ∪ {ψ} ⊆ F (Σ) such that Γ � ψ: take Γ− ⊆fin Γ such that Γ− � ψ. Take allsymbols ∈ |Σ| that occur in formulas in Γ− ∪ {ψ}: this is a finite set S ⊆fin |Σ|,and take the unique subsignature Σ′ ↪→ Σ such that |Σ′| = S. Take l′ = (Σ′,�′)the unique logic that is generated (by substitutions) by the unique basic axiom (ifΓ− = ∅) or inference rule (if Γ− = ∅) “from hypothesis Γ− conclude ψ”, then theinclusion l′ ↪→ l is in fact a translation morphism and Γ � ψ iff there is Γ− ⊆fin Γsuch that if i′ : l′ ↪→ l then Γ− ∪ {ψ} ⊆ i′[F (Σ′)] and Γ− �′ ψ (in l′). �

Remark 2.14. Through an analogous argument we can prove that any logic is afiltered colimit of “conservative sublogics” with finite type underlying signature.However, the next proposition say that we have chosen the “correct definition” offinite type logic.

Proposition 2.15. A logic is finitely presentable if and only if it is of finite type.

Proof.(⇐). Let l′ = (Σ′,�′) be a logic of finite type and consider D : (I,≤) −→ L,

(lifij

−→ lj)(i≤j)∈I a directed diagram of logics. Then the canonical arrow k :


colimi∈IL(l′, li) −→ L(l′, colimi∈I li) is an isomorphism.

[(

l′hi

−→ li)

, i)]

�→(

l′hi

−→ liγi

−→ colimi∈I li

)

.

Let us prove that k is surjective. Let h : l′ −→ colimi∈I li be a L-morphism.

Because |Σ′| is a finite set, as in Proposition 2.4, there exists i ∈ I such that, foreach n ∈ ω, hn[Σ′

n] ⊆ γin[Σi

n]. As �′ is given by a finite set of axioms and a finiteset of (finitary) inference rules and (I,≤) is a directed ordered set, there existsj ∈ I such that j ≥ i and the finite image set of formulas in this chosen axioms andrules by h : F (Σ′) −→ F (colimi∈IΣi) are contained in the set γj[F (Σj)]. Since h isa translation morphism whose codomain is a filtered colimit, then we can assumealso, by the definition of � in filtered colimits, that j is such that the images ofthese axioms and inference rules under h are in fact �j-derivable in lj. So if wetake for each n ∈ ω and each c′n ∈ Σ′

n, hjn : Σ′

n −→ Σjn such that hj

n(c′n) ∈ Σjn with

hn(c′n) = [(hjn(c′n), j)] then, hj : l′ −→ lj is a translation morphism (by Fact 5.(ii),

because �′ ≤ (hj)�(�j)) and h = γj ◦ hj .Now we prove that k is injective. This is analogous to the correspondent part

of Proposition 2.4; in fact, here we need only the information that |Σ′| is a finiteset.(⇒). Let l = (Σ,�) be a finitely presentable logic. Then, by the proof of Propo-sition 2.13, the logic l is the colimit of the directed diagram of its finite typesublogics. Then, as l is a finitely presentable logic, there is l′, a finite type sublogicof l, such that the identity translation morphism idl : l −→ l must factor throughthe (colimit) canonical inclusion l′ ↪→ l, because the canonical morphism k :coliml′∈IL(l, l′) −→ L(l, coliml′∈I l

′) is surjective, that is, there is a translation

morphism h′ : l −→ l′ such that (l idl−→ l) = (l h′−→ l′ ↪→ l). Then the L-inclusion

l′ ↪→ l must be a S-isomorphism. Then l′ ↪→ l and h′ : l −→ l′ have as subjacentthe identity S-morphism. So, by Fact 5.(ii), as l′ ↪→ l is a translation morphism,then �′ ≤ � and as h′ : l −→ l′ is a translation morphism, then � ≤ �′. Then wehave l = l′ and l is a finite type logic. �

Theorem 2.16. The category L is a finitely locally presentable category, that is,L is an accessible category that is cocomplete and complete.

Proof. Direct consequence of the Propositions 2.13, 2.15 and 2.11. �

Corollary 2.17. (i) The Yoneda functor Y : Lfp −→ Set(Lfp)op

has an extensionto a functor Y ′ : L −→ Set(Lfp)op

, l �→ Y ′(l) = L(ι(.), l) that is full andfaithful;

(ii) Let Flat(Lfp, Set) be the full subcategory of Set(Lfp)op

whose objects are thefunctors that are filtered colimits of representable functors. ThenFlat(Lfp, Set) is the “essential image” of Y ′, so its restriction functor E :L −→ Flat(Lfp, Set) is an equivalence of categories;


(iii) Flat(Lfp, Set) coincides with the category of Set-valued functors that pre-serve finite limits;

(iv) Y ′ has a left adjoint.

Proof. For (i) and (ii) see [9], Theorem 5.3.5 (p. 265) or [1], Theorem 2.26 (p. 83)or [22], Observation 1.6 (p. 46). For (iii) and (iv) see [1], Theorem 1.46 (p. 38). �

Remark 2.18. L is an “algebraic category” and is not a “topological category”:

• L is an “algebraic category”: all the usual categories in algebra are accessible;• L has a “topological appeal”: because its objects have a particular kind of

closure operator and its morphisms are continuous functions relative to theclosure operators and the set of consequence relations over any given signa-ture, it is a complete lattice;

• L is not a “topological category”: the category of commutative C∗-algebrasis ω1-locally presentable and has as dual the category of compact Hausdorfftopological spaces. There is a general non-duality principle for categorieslocally presentables: If a category and its opposite are both locally presentablethen they are equivalent to a complete lattice6: see Theorem 1.64 in [1] (p. 51).

3. The category of algebraizable logics A3.1. What is A?

The category A is the category of algebraizable logics and translation morphismsthat preserves algebraizing pair. An algebraizable logic is a logic l = (Σ,�) ∈Obj(L) that is algebraizable in a Blok–Pigozzi sense: There is an ordered pair((δ ≡ ε), ∆), called an algebraizing pair, and a class K of Σ-structures (i.e. Σ-algebras), called an equivalent algebraic semantic, such that:

• (δ ≡ ε) is a finite set of ordered pair of Σ-formulas (δ ≡ ε) = {(δr, εr) : r < s},called the set of defining equations, such that {δr, εr′ : r ≤ r′ < s} ⊆F (Σ)[1];

• ∆ is a finite set of Σ-formulas ∆ = {∆u : u < v}, called the set of equivalenceformulas, such that, {∆u : u < v} ⊆ F (Σ)[2];

and ((δ ≡ ε), ∆) satisfies conditions (i) and (ii) (and/or conditions (i)′ and (ii)′)below, with Γ ∪ Θ ∪ {ψ, ϕ, ζ, η, ϑ} ⊆ F (Σ):

(i) Γ � ϕ ⇔ {(δ(ψ) ≡ ε(ψ)) : ψ ∈ Γ} �K (δ(ϕ) ≡ ε(ϕ));(ii) (ϕ ≡ ψ) �K� (δ(ϕ∆ψ) ≡ ε(ϕ∆ψ));7

(i)′ Θ �K (ϕ ≡ ψ) ⇔ {ζ∆η : (ζ∆η) ∈ Θ} � ϕ∆ψ;(ii)′ ϑ �� δ(ϑ)∆ε(ϑ).

6This result connects, in some sense, the three fundamental species of structures of Bourbaki...7This is an abbreviation for (ϕ ≡ ψ) �K {δr(ϕ∆uψ) ≡ εr(ϕ∆uψ) : r < s, u < v} and{δr(ϕ∆uψ) ≡ εr(ϕ∆uψ) : r < s, u < v} �K (ϕ ≡ ψ).


If l = (Σ,�), Σ′ = (Σ′,�′) are algebraizable logics, then a morphism f : Σ −→Σ′ is a translation morphism that also “preserves the algebraizing pairs”: if ((δ ≡ε), ∆) is an algebraizing pair for l, then ((f [δ] ≡ f [ε]), f [∆]) is an algebraizing pairfor l′, where (f [δ] ≡ f [ε]) = {(f(δr), f(εr)) : r < s} and f [∆] = {f(∆u) : u < v}.By Remark 2.1, we have (f [δ] ≡ f [ε]) ⊆ F (Σ′)[1] and f [∆] ⊆ F (Σ′)[2].

Composition and identities are similar to L.

Proposition 3.1. For each l ∈ A, let ((δi ≡ εi), ∆i), an algebraizing pair, and Ki anequivalent algebraic semantic, for each i ∈ {0, 1}. For any class K ′ of Σ-algebraslet us denote (K ′)Q the Σ-quasivariety generated by K ′. Then some uniquenessconditions holds: on quasivariety semantics: (K0)Q = (K1)Q; on equivalence for-mulas: ∆0 �l� ∆1; on defining equations: (δ0 ≡ ε0) �K� (δ1 ≡ ε1) (whereK

.= (K0)Q = (K1)Q).

Proof. Theorem 2.15 in [8]. �Remark 3.2. By Proposition 3.1, a L-morphism f : (Σ,�) −→ (Σ′,�′) betweenalgebraizable logics l, l′ is a A-morphism if and only if there exists ((δ ≡ ε), ∆),an algebraizing pair for l, such that ((f [δ] ≡ f [ε]), f [∆]) is an algebraizing pair forl′. So we believe that the terminology “category of algebraizable logics” fits betterthan “category of algebrized logics”, as the notion of morphism between A-objetsdoes not depend of a preservation of any particular choice of algebraizing pairs ofthe logics source and target.

The theorem below (Theorem 4.7 in [8]) gives a useful characterization ofalgebraizable logics through an algebraizing pair.

Theorem 3.3. Let l = (Σ,�) a logic and ∆ ⊆fin F (Σ)[2], (δ ≡ ε) ⊆fin (F (Σ)[1]×F (Σ)[1]) such that the conditions below are satisfied(a) � ϕ∆ϕ8, for all ϕ ∈ F (Σ);(b) ϕ∆ψ � ψ∆ϕ, for all ϕ, ψ ∈ F (Σ);(c) ϕ∆ψ, ψ∆ϑ � ϕ∆ϑ, for all ϕ, ψ, ϑ ∈ F (Σ);(d) ϕ0∆ψ0, . . . , ϕn−1∆ψn−1 � c(ϕ0, . . . , ϕn−1)∆c(ψ0, . . . , ψn−1), for all c ∈ Σn

and all ϕ0, ψ0, . . . , ϕn−1, ψn−1 ∈ F (Σ);(e) ϑ � � δ(ϑ)∆ε(ϑ), for all ϑ ∈ F (Σ).

Then l is an algebraizable logic with ∆ as equivalence formulas and (δ ≡ ε) asdefining equations.

It follows easily that:

Corollary 3.4. If f : l � l′ is an L-epimorphism where l is an algebraizable logicthen l′ is an algebraizable logic and f is a A-epimorphism.

We might ask if there are algebraizable logics with a given number of equiva-lence formulas and/or of defining equations. The example below gives an affirma-tive to the first question.

8That is, � ϕ∆uϕ, for all u < v.


Example. For each natural number k > 1 we consider the following algebraizablelogic lk = (Σk,�k):

Σk : as exactly two connectives, c1 a unary connective and ck a k-ary con-nective.

�k : is the consequence operator determined by the following axioms andinference rules:

First let us write:∆k(x0, x1)

.= {ck(x0, x1, . . . , x1), ck(x0, x0, x1, . . . , x1), . . . , ck(x0, . . . , x0, x1)} a setwith k − 1-formulas in F (Σ)[2];(ε(x0) ≡ δ(x0))

.= {(x0, c1(x0))} a unitary set with a pair of formulas in F (Σ)[1]

[axiom:]• �k ck(x0, . . . , x0)

[inference rules:]• ∆k(x0, x1) �k ∆k(x1, x0) 9

• ∆k(x0, x1), ∆k(x1, x2) �k ∆k(x0, x2)• ∆k(x0, x1) �k ∆k

(c1(x0), c1(x1)

)

• ∆k(x0, x1), ∆k(x2, x3), . . . , ∆k(x2k−2, x2k−1) �k ∆k(ck(x0, x2, . . . , x2k−2),ck(x1, x3, . . . , x2k−1))

• x0 �k ∆k (ε(x0), δ(x0))• ∆k (ε(x0), δ(x0)) �k x0

It follows from Theorem 3.3 that lk is an algebraizable logic with algebraizingpair ((ε ≡ δ), ∆k). Besides, if ∆′(x0, x1) is a set of formulas in F (σ)[2] such thatx0∆′x1 �k x0∆kx1 then ∆k ⊆ ∆′ so, by Proposition 3.1, k−1 is the least numberof equivalence formulas for the logic lk.

The example above suggests the:

Definition 3.5. (i) Let l = (Σ,�) be an algebraizable logic. We write neqv =min{k ∈ N : there is an algebraizable pair ((δ, ε), ∆) of l such that card(∆) =k}; neqt = min{k ∈ N : there is an algebraizable pair ((δ ≡ ε), ∆) of l suchthat such that card((δ ≡ ε)) = n}.

(ii) Let I be a small category and D : I −→ A a diagram; we write D(i) = li =(Σi,�i), for each i ∈ I. We say that diagram D of algebraizable logics isbounded if both the sets of natural numbers {ni

eqv : i ∈ I} and {nieqt : i ∈ I}

have an upper bound in N.

Remark 3.6. (i) For each signature Σ, l = (Σ,�top), the greatest logic over Σ,is algebraizable and neqv = neqt = 0.

(ii) Let l, l′ algebraizable logics. Then n′eqv ≤ neqv and n′

eqt ≤ neqt are necessaryconditions for the existence of a A-morphism f : l −→ l′.

(iii) A diagram D : I −→ A of algebraizable logics is bounded if and only if{D(i) : i ∈ obj(I)}, the discrete diagram underlying to D, is bounded.

9That are k − 1 distinct inference rules!


3.2. Filtered Colimits in A3.2.1. Obtaining the filtered colimits.

Theorem 3.7. The category A has all filtered colimits and the (obvious) underlyingfunctor U : A −→ L creates all such colimits.

Proof. Let (I,≤) be a directed ordered set and D : (I,≤) −→ A, (lifij

−→ lj)(i≤j)∈I

a diagram. Consider (l, (γi)i∈I) the colimit in L of the underlying diagram((I,≤) D−→ A U−→ L and choose an i ∈ I and an algebraizing pair ((δi ≡ εi), ∆i) =({(δi

r ≡ εir) : r < s}, {∆i

u : u < v}) of li and take ((δ ≡ ε), ∆) = (γi[δi] ≡γi[εi], γi[∆i]). Then l is an algebraizable logic with algebraizing pair ((δ ≡ ε), ∆)and (l, (γi)i∈I) is the colimit of D in A. In fact, this follows from (a), (b) and (c)below:(a) l = (Σ,�) ∈ Obj(A).10 To check this, we invoke the characterization of

algebraizable logics through an algebraizing pair in Theorem 3.3:(a1) � ϕ∆ϕ11, for all ϕ ∈ F (Σ): By definition of Σ and by Fact 3.(ii), for

each u < v there is an ju ∈ I and a ϕju ∈ F (Σju) such that γju(ϕju ) =ϕ. Because (I,≤) is a directed ordered set and D : (I,≤) −→ A isa diagram, there exists k ≥ i, j0, . . . , jv−1 such that γk(ϕk) = ϕ. Asf ik : li −→ lk is an A-morphism, then ((f ik[δi] ≡ f ik[εi]), f ik[∆i]) is analgebraizing pair for lk. By Theorem 3.3 , �k ϕkf ik(∆i

u)ϕk, for all u < v.As γk : lk −→ l is an L-morphism, then � γk(ϕk)γk(f ik(∆i

u))γk(ϕk),for all u < v. As (γj : lj −→ l)j∈I is a commutative cocone overthe diagram ((I,≤) −→ A −→ L), then � γk(ϕk)γi(∆i

u)γk(ϕk), forall u < v. Therefore, for all u < v, � ϕ∆uϕ, because we have taken∆ = γi[∆i].

(a2) ϕ∆ψ � ψ∆ϕ, for all ϕ, ψ ∈ F (Σ): is analogous to (a1)(a3) ϕ∆ψ, ψ∆ϑ � ϕ∆ϑ, for all ϕ, ψ, ϑ ∈ F (Σ): is analogous to (a1)(a4) ϕ0∆ψ0, . . . , ϕn−1∆ψn−1 � c(ϕ0, . . . , ϕn−1)∆c(ψ0, . . . , ψn−1), for all c ∈

Σn and all ϕ0, ψ0, . . . , ϕn−1, ψn−1 ∈ F (Σ): We can find k ≥ i andck ∈ (Σk)n, ϕk

0 , ψk0 , . . . ϕk

n−1, ψkn−1 ∈ F (Σk) such that γk(ck) = c and,

for all m < n, γk(ϕkm) = ϕm, γk(ψk

m) = ψm. As f ik : li −→ lk is anA-morphism, then ((f ik[δi] ≡ f ik[εi]), f ik[∆i]) is an algebraizing pairfor lk. Hence by Theorem 3.3 ϕk

0 f ik(∆iu)ψk

0 , . . . , ϕkn−1f

ik(∆iu)ψk

n−1 �k

ck(ϕk0 , . . . , ϕ0

n−1)fik(∆i

u)ck(ψk0 , . . . , ψ0

n−1), for all u < v. As γk : lk −→ l

is an L-morphism, then γk(ϕk0)γk(f ik(∆i

u))γk(ψk0 ), . . . , γk(ϕk

n−1)γk(f ik(∆i

u))γk(ψkn−1) � γk(ck(ϕk

0 , . . . , ϕ0n−1))γ

k(f ik(∆iu))γk(ck(ψk

0 , . . . ,

ψ0n−1)), for all u < v. As (γj : lj −→ l)j∈I is a commutative cocone

over the diagram ((I,≤) −→ A −→ L), then γk(ϕk0)γi(∆i

u)γk(ψk0 ), . . . ,

γk(ϕkn−1)γ

i(∆iu)γk(ψk

n−1) � γk(ck(ϕk0 , . . . , ϕ0

n−1))γi(∆i

u)γk(ck(ψk0 , . . . ,

10This is independent of the chosen i ∈ I and the algebraizing pair ((δi ≡ εi), ∆i) of li.11That is, � ϕ∆uϕ, for all u < v.


ψ0n−1)), for all u < v. Therefore, for all u < v, ϕ0∆uψ0, . . . ,

ϕn−1∆uψn−1 � c(ϕ0, . . . , ϕn−1)∆uc(ψ0, . . . , ψn−1) because we havetaken ∆ = γi[∆i].

(a5) ϑ � � δ(ϑ)∆ε(ϑ), for all ϑ ∈ F (Σ): We can find k ≥ i such thatγk(ϑk) = ϑ. As f ik : li −→ lk is an A-morphism, then ((f ik[δi] ≡f ik[εi]), f ik[∆i]) is an algebraizing pair for lk. Hence by Theorem 3.3ϑk �k� (f ik(δi

r))(ϑk) (f ik(∆i

u)) (f ik(εir))(ϑ

k), for all u < v andr < s. As γk : lk −→ l is an L-morphism, then for each u < v andr < s γk(ϑk) � � γk((f ik(δi

r))(ϑk)) γk(f ik(∆i

u)) γk((f ik(εir))(ϑ

k)).As (γj : lj −→ l)j∈I is a commutative cocone over the diagram((I,≤) −→ A −→ L), then for each u < v and r < s, γk(ϑk) � �γi(δi

r)γk(ϑk)γi(∆iu) γi(εi

r)γk(ϑk). Therefore, for all u < v and r < s,ϑ � � δr(ϑ)∆uεr(ϑ), because we have taken (δ ≡ ε) = (γi[δi] ≡ γi[εi])and ∆ = γi[∆i].

(b) γj ∈ A(lj , l), for all j ∈ I: As (γj : lj −→ l)j∈I is a commutative coconeover the diagram ((I,≤) −→ A −→ L), to show that γj : lj −→ l is anA-morphism it is enough to find k ≥ j such that γk : lk −→ l is anA-morphism because, in this case, γj is a composition of A-morphisms:γj = γk ◦ f jk. So, as (I,≤) is a directed ordered set, it is enough to showthat γk : lk −→ l is a A-morphism, for all k ≥ i. Take a k ≥ i: becausef ik : li −→ lk is an A-morphism, we have that ((f ik[δi] ≡ f ik[εi]), f ik[∆i])is an algebraizing pair for lk; because γi = γk ◦ f ik and, by definition,((δ ≡ ε), ∆) = ((γi[δi] ≡ γi[εi]), γi[∆i]), we have that the L-morphism γk

takes an algebraizing pair of lk onto an algebraizing pair of l (by item (a)just above), hence by Remark 3.2, γk : lk −→ l is an A-morphism.

(c) If (l′, (αj)j∈I) is a commutative cocone over the diagram D then the uniquetranslation morphism α : l −→ l′ such that αj = α ◦ γj , j ∈ I is anA-morphism: By definition of l and item (a) above, ((δ ≡ ε), ∆) = ((γi[δi] ≡γi[εi]), γi[∆i]) is an algebraizing pair of l, so by Remark 3.2, to prove thatα : l −→ l′ is an A-morphism it is enough to show that ((α[δ] ≡ α[ε]), α[∆])is an algebraizing pair of l′. As αi : li −→ l′ is an A-morphism, then((αi[δi] ≡ αi[εi]), αi[∆i]) is an algebraizing pair of l′. Finally, as αi = α ◦ γi,we have that the L-morphism α takes an algebraizing pair of l onto an alge-braizing pair of l′, so by Remark 3.2, α : l −→ l′ is an A-morphism. �

3.2.2. On the algebraic semantics for filtered colimits.

Fact 7. About structures and morphisms:

(i) For each signature morphism Σf−→ Σ′, there is a functor Σ−Str

f−→ Σ′−Strbetween categories of structures over the signatures such that:

• A′ = (A′, (c′nA′

: A′n → A′)c′n∈Σ′n,n∈ω) �→ f(A′) = (A′, (cf(A′)

n : A′n →A′)cn∈Σn,n∈ω), where (cf(A′)

n : A′n → A′) = ((fn(cn))A′

: A′n → A′),cn ∈ Σn, n ∈ ω;


• (A′ h−→ B′) �→ (f(A′) h−→ f(B′)).(ii) Connectives (c′n ∈ Σ′

n, n ∈ ω) and (propositional) Σ′-formulas(ϕ′(x0, . . . , xn−1)) give functions A′n −→ A′. The first order atomic formulasare equations between Σ′-terms (= propositional Σ′-formulas);

(iii) Let A′ be a Σ′-structure. Then, for every first order valuation in A′:• For each Σ-equation ϕ(x0, . . . , xn−1) ≡ ψ(x0, . . . , xn−1): A′ �Σ′−Str

(f(ϕ)[a′0, . . . , a

′n−1]) ≡ (f(ψ)[a′

0, . . . , a′n−1]) ⇔ f(A′) �Σ−Str

(ϕ)[a′0, . . . , a

′n−1]) ≡ (ψ)[a′

0, . . . , a′n−1]);

• For each first order Σ-formula P(x0, . . . , xn−1): A′ �Σ′−Str

(f(P))[a′0, . . . , a

′n−1] ⇔ f(A′) �Σ−Str (P)[a′

0, . . . , a′n−1].

Proposition 3.8. On the quasivariety semantics for filtered colimits: Let (I,≤) be

a directed ordered set and D : (I,≤) −→ A, (lifij

−→ lj)(i≤j)∈I a diagram. Consider(l, (γi)i∈I) the colimit in A. For each i ∈ I, take Ki the unique quasivariety-semantics for li and let K be the unique quasivariety-semantics for l. Then, forall A ∈ Σ − Str:

A ∈ K ⇔ γi(A) ∈ Ki, for all i ∈ I.

Proof. This follows from Theorem 2.17 in [8], by the construction in Theorem 3.7and the Fact 7 above. �

3.3. More on colimits and limits in A3.3.1. Colimits.

Proposition 3.9. The category A has colimits of all non-empty diagrams and the(obvious) underlying functor U : A −→ S creates all such colimits.12

Proof. (Sketch) Let I = ∅ be a nonempty category and D : I −→ A a diagram.We write D(i) = li = (Σi,�i) for each i ∈ Obj(I). Consider (Σ, (γi)i∈Obj(I)) the

colimit in S of the underlying diagram (I D−→ A U−→ S). As LΣ is a completelattice, take � as the least consequence relation over the signature Σ such that:

• For each i ∈ Obj(I) and Θi ∪ {ψi} ⊆ F (Σi), if Θi �i ψi then γi[Θi] � γi(ψi);• For each i, j ∈ Obj(I) and each algebraizing pairs ((δi ≡ εi), ∆i) of li and

((δj ≡ εj), ∆j) of lj, then: γi[∆i] �� γj[∆j ].

Now choose any i ∈ Obj(I) and any algebraizing pair ((δi ≡ εi), ∆i) of li andtake ((δ ≡ ε), ∆) = ((γi[δi] ≡ γi[εi]), γi[∆i]). Then l is an algebraizable logic withalgebraizing pair ((δ ≡ ε), ∆) and (l, (γi)i∈Obj(I)) is the colimit of D in A. �

Remark 3.10. The restriction on the proposition above is essential, i.e., the cate-gory A does not have initial object. In fact, A does not have weak initial objects: asthere is a unbounded family of algebraizable logics (see, for instance, Example 3.1)

12The construction in this result is similar to the one in [16], but they work with another notionof morphism.


and if l− is any algebraizable logic then, by Remark 3.6.(b), the subset of an un-bounded family of the algebraizable logics that are target of some A-morphismwith source l− is a proper subset.

3.3.2. Products. As we saw in the Remark 3.10 above there is no (projective) coneover an unbounded diagrams. But there are universal commutative cones in thecase of discrete bounded diagrams:

Proposition 3.11. The category A has products of all bounded families and the(obvious) inclusion functor J : A −→ L creates all such limits.13

Proof. (Sketch) Let I be a set and D : I −→ A a bounded discrete diagram. Wewrite D(i) = li = (Σi,�i), for each i ∈ I. Consider (Σ, (πi)i∈I) the product in Lof the underlying diagram (I D−→ A J−→ L):

• If I = ∅: observe that the empty family can not be unbounded and theterminal logic in L is algebraizable, so it is the terminal algebraizable logic(in A);

• If I = ∅: then, with the notation just above, choose n, k ∈ N and anyalgebraizing pair ((δi ≡ εi), ∆i) of li such that card((δi ≡ εi)) ≤ n andcard(∆i) ≤ k, for each i ∈ I. We take ((δ ≡ ε), ∆), with card((δ ≡ ε)) = nand card(∆) = k, as the finite sets of formulas in the product signature thatare obtained as I-indexed sequences of the corresponding finite set of formu-las in the chosen algebraizing pair ((δi ≡ εi), ∆i) (with some repetition, whennecessary). Then l is an algebraizable logic with algebraizing pair ((δ ≡ ε), ∆)and (l, (πi)i∈I) is the product of D in A. �

3.4. A is an accessible category

Theorem 3.12. The category A is a finitely accessible category.

Proof. (Sketch) Through analogous arguments to others in the category L, weobtain that:

• Each algebraizable logic is the colimit of a directed diagram of finite typelogics that are also algebraizable;

• An algebraizable logic is finitely presentable in A if and only if it is a fi-nite type logic that are also algebraizable if and only if it is a logic finitelypresentable in L that is also an algebraizable logic.

It follows that any algebraizable logic is the direct colimit of finitely presentablealgebraizable logics, that is, A is a finitely accessible category. �

Remark 3.13. Note that A is a finitely accessible category but, on the contrary to Sand L, A is not a finitely locally presentable category because it is not cocomplete,as it has no initial object, and is not complete, as it has no commutative cone overunbounded diagrams. These results are related with a general fact about accessible

13The construction in this result is similar to the one in [11], but they work with another notionof morphism.


categories: an accessible category has all non-empty colimits if and only if it haslimits of diagrams that are base of some commutative cone14 (see [2, 3]).

3.5. Application: Remote algebrization revisited

As another consequence of the theory of accessible categories to our context, weobtain a general (for any logic) weak form of the concept of remote algebrizationfirstly introduced in [12]: since E : A −→ L is an accessible functor, that is, L and Aare accessible categories and E preserves filtered colimits then, by Proposition 6.1.2in [21], for each logic l there is a set I and a family of L-arrows ηi : l −→ E(li),where the li are A-objects such that, for each L-arrow of l to an algebraizablelogic l′, f : l −→ E(l′), there exists i ∈ I and an A-morphism f i : li −→ l′ suchthat f = f i ◦ ηi.

It follows that for any logic l there exists a L-arrow η : l −→∏

i∈I li (takeη = (ηi)i∈I) such that, for any L-morphism f : l −→ l′ with l′ ∈ Obj(A), thereis an L-morphism f :

∏i∈I li −→ l′ such that f = f ◦ η. Consequently, the

logic l is remotely algebraizable if an only if the L-morphism η is a conservativetranslation. It seems to be an interesting problem to characterize the class of logicsthat have such a “canonical” family (in case that the existential conditions aboveare substituted by “there is only one”). In this situation it follows easily that theset I and the L-arrows ηi : l −→ li are unique up to isomorphism. This will providea kind of “algebraizable spectre” of that logic.

4. Fibrings, coverings and sheaves

4.1. Fibrings and coverings

Now that we have described the objects of L as colimits of the essentially smallcategory Lfp and thus gained an embedding of L into Set(Lfp)op

(which is thecocompletion of Lfp), we are in the position to pursue the intuition, mentionedin the introduction, that a logic l, whose consequence relation is generated by theimages of translations of other logics li into l, can be considered to be “covered”by the li. To this end we introduce Grothendieck topologies on Lfp and investi-gate how the related sheaf theoretic notions apply to the logics in L. The overallpicture is: We are relating the categories displayed in the following diagram, whereFlat((Lfp)op, Set) denotes the category of functors in Set(Lfp)op

that are filteredcolimits of representables, where k, i denote the inclusion functors from sheavesinto separated presheaves and from the latter to presheaves (we will recall thesenotions below) and where a and s are the associated sheaf and separated presheaffunctors. a and s are left adjoint to k and i respectively. j denotes the inclusionfunctor of flat presheaves into general presheaves and, by the theory of locallypresentable categories, there is a left adjoint to j which we call e. Further we de-fine L := e ◦ i ◦ k and S := a ◦ s ◦ j, L and S are adjoint to each other being acomposition of adjoints; given any Grothendieck topology on Lfp we thus get an

14We thank the referee who pointed to us this result and references.


https://www.researchgate.net/publication/242983388_Sur_les_categories_accessibles_multicompletes?el=1_x_8&enrichId=rgreq-9cd6bcf71de15ed5f0458914078007c0-XXX&enrichSource=Y292ZXJQYWdlOzIyNjY5OTcxMDtBUzoxNTA0ODY3MDM0MTUyOTZAMTQxMjg5MDIzNDA2OA==

https://www.researchgate.net/publication/242620841_Possible-translations_algebraization_for_paraconsistent_logics?el=1_x_8&enrichId=rgreq-9cd6bcf71de15ed5f0458914078007c0-XXX&enrichSource=Y292ZXJQYWdlOzIyNjY5OTcxMDtBUzoxNTA0ODY3MDM0MTUyOTZAMTQxMjg5MDIzNDA2OA==


adjunction between Flat((Lfp)op, Set) and the category of sheaves. Further, bythe theory of accessible categories, L and Flat((Lfp)op, Set) are equivalent (by anequivalence denoted E in the diagram). In the following we shall make free use ofthese functors and their properties.

L Sh(Lfp, F ib)Sep(Lfp, F ib)Set(Lfp)op

Flat((Lfp)op, Set)

�Y ′ �s�i

�a�k

�

j

�

e

��

E �

��

S

��

��

��

��

��

L��

First we will take a look at some possible notions of covering.

Definition 4.1. Let l = (Σ,�) be a finitely presentable logic and H a set of trans-lation morphisms with codomain l and with domain a finitely presentable logic:

(i) H is a covering(i) of l iff for all Γ ∪ {ψ} ⊆ F (Σ)Γ � ψ ⇔ there is Γ− ⊆fin Γand there exists h ∈ H such that Γ− ∪ {ψ} ⊆ h[F (Σdom(h))] and thereis Γ−h ∪ {ψh} ⊆fin F (Σdom(h)) such that h[Γ−h] = Γ− , h(ψh) = ψ andΓ−h �dom(h) ψh.

Since we have coproducts in L, we can also express properties of familiesof morphisms with common codomain by means of the induced arrow from thecoproduct of the occurring domains:

(ii) H is a covering(ii) of l iff the canonical translation morphism cH :∐

h∈H dom(h) −→ l, the unique arrow such that for all h ∈ H , (dom(h) h→−→ l) = (dom(h) ih−→

∐h∈H dom(h) cH−→ l), is such that � = cH�(�H).

(iii) H is a covering(iii) of l iff the canonical translation morphism cH :∐

h∈H dom(h) −→ l, the unique arrow such that for all h ∈ H , (dom(h) h→−→ l) = (dom(h) ih−→

∐h∈H dom(h) cH−→ l), is such that cH is a conservative

translation morphism that is also an L-epimorphism.(iv) H is a covering(iv) of l iff the morphism cH of (ii) is an isomorphism.

The latter is equivalent to saying that H covers l iff l is a fibring in the senseof [24] of the domains of the morphisms in H and thus suggests the more generaldefinition:

(v) Given any notion of fibring, define H to be a covering of l iff l is the resultof fibring (in the given sense) the domains of the arrows in H .



Remark 4.2. Adopting either of the above covering notions15 (or any other notionwhich gives conditions only involving the consequence relations), we have somespace for “fine tuning” according to the intended meaning of “covering”:

(i) We can require covering families to be epimorphic families (or, equivalently,to be cH in (ii) to be epic). On the logical side, by Proposition 2.12, thismeans that every generating connective of the underlying language of thecovered logic will occur in the image of some covering morphism and so rulesout coverings by proper linguistic fragments as given by the one-elementcovering family ({→}, modus ponens) −→ ({¬,→}, modus ponens). On thesheaf-theoretical side it implies that every representable functor will be aseparated presheaf (see next subsection).

(ii) We can require the members of a covering family to be monos, thus (again byProposition 2.12) allowing only coverings by proper linguistic fragments butnot, for example, ({∧1,∧2}, 〈A ∧1 B � A, A ∧2 B � B〉) −→ ({∧}, 〈A ∧ B �A, A ∧ B � B〉) (where the expressions 〈. . . 〉 denote the consequence rela-tions generated by the given inference rules and/or axioms; see Remark 2.8).This condition amounts to allowing only proper decompositions of logics ascoverings and seems appropriate to treat the splitting of logics.

4.2. Sheaves

A notion of covering (i.e., a mapping Cov associating to each object X a collectionCov(X) of families of morphisms to X) gives rise to a unique Grothendieck topol-ogy on Lfp.16 The covering sieves of an object X in this Grothendieck topologyare the compositional right ideals of pullbacks to X of covering families in Lfp

and will also be denoted by Cov(X).The purpose of this section is to investigate what it means for a logic to be

a separated presheaf or a sheaf.We first recall the relevant definitions.

Definition 4.3. Given a presheaf F ∈ |Set(Lfp)op |, an object X ∈ |Lfp| and acovering sieve S = {fi : Xi → X | i ∈ I} ∈ Cov(X), a compatible family is afamily {si ∈ F (Xi) | i ∈ I} such that F (f : Xi → Xj)(sj) = si for all i, j ∈ I.A presheaf F is called a separated presheaf (a sheaf, respectively) if, for each suchX ∈ |Lfp|, S ∈ Cov(X) and compatible family {si ∈ F (Xi) | i ∈ I}, there is atmost one (exactly one, respectively) s ∈ F (X) such that si = F (fi : Xi → X)(s)for all i ∈ I.

Remark 4.4. As we see in the diagram above, there is a pair of functors LL′

�S′

Sh(Lfp, F ib) such that S′ : L −→ Sh(Lfp, F ib) preserves finite limits and filtered

15Some of these covering notions are obviously related: covering(iv) is the strongest notion be-tween (i), (ii), (iii) and (iv); covering(iii) and covering(i) are stronger than covering(ii).16Notation for that kind of site: (Lfp, F ib).


colimits.17 As we know that every logic is a filtered colimit of finitely presentablelogic, we get a “good codification” of L in Sh(Lfp, F ib) in the sense that codifi-cation preserves the glue (the filtered colimits) between fundamental bricks (theFP-logics). We also know that the category L of all logics has nice categorialproperties, but the category has no nice logical properties! This “defect” of L ismitigated by the codification into the category of sheaves, a complete, cocomplete,locally finitely presentable category that does have nice logical properties: as anelementary topos it has exponential objects and a classifying object.

Now we will take a closer look at the notions of sheaf and separated sheafand what they say about the behavior of a logic; we rephrase these definitions intostatements about F in the category Set(Lfp)op

.Under the Yoneda lemma isomorphism, elements si ∈ F (Xi), s ∈ F (X) corre-

spond to unique natural transformations τi : Hom(−, Xi) → F, τ : Hom(−, X) →F respectively and F (f : Xi → Xj)(sj) = si, F (f : Xi → X)(s) = si translate toτi = τj ◦Hom(−, f), τi = τ ◦Hom(−, fi) respectively, so that we get the followingcharacterization: F is a sheaf iff for each X ∈ |Lfp|, S ∈ Cov(X) the Yoneda em-bedded cone Y (S) is universal for F in the sense that for every compatible family{τi | i ∈ I} of morphisms from the domains of Y (S), there exists a unique arrowas in the following diagram:

Hom(−, Xi) Hom(−, Xj)Hom(−, X)

F

�gi

�gj

�

∃!

��

τi

τj

��

��

�

��

So that F is a sheaf roughly says that “for F all (Yoneda embedded) coveringsieves are colimit cones”. From this formulation it is immediate to see the followingequivalences which we note in passing:

Fact 8. (i) The representable functors are separated presheaves iff all coveringfamilies are epimorphic.

(ii) The representable functors are sheaves iff all covering sieves are colimit cones.

17Because Y ′ = j◦E : L −→ Set(Lfp)oppreserves limits and filtered colimits and the “associated

sheaf functor” a ◦ s : Set(Lfp)op−→ Sh(Lfp, F ib) preserves finite limits and colimits.


So separatedness for a logic l means that a translation h from a finitarylogic l′ into l is completely determined by the translations obtained by composingwith the morphisms of a covering family of l′ and the sheaf property says that,given translations from the domains of a covering family of l′, such a translationl′ → l in fact always exists.

Example. A logic which is not a separated presheaf: take as (generating) coveringsall families {fi : li → l | i ∈ I} such that �l ≤ sup{f�

i (�li) |i ∈ I}. Let l1 bethe logic with signature given by one connective c and the minimal consequencerelation, and l2 the logic with the signature consisting of two connectives c1, c2

with the same arity as c and the minimal consequence relation. Then the signaturemorphism mapping c to c1 is clearly a morphism of logics and moreover gives a(one-element) covering family of l, but we have two different endomorphisms of lmaking the diagram below commute, namely the identity and the morphism thatmaps both c1 and c2 to c1:

({c},�min) ({c1, c2},�min)

({c1, c2},�min)

�c �→ c1

��

c �→ c1

�

id

�

t�

This example (and lots of similar ones) suggests that separatedness is a kindof “Occam’s Razor property” in the sense that a separated logic has no redun-dant (e.g., doubly occurring or interchangeable) connectives. Thus the separationfunctor s (the left adjoint to the inclusion of separated presheaves into presheaves)would give a procedure to cut down redundancies in the presentation of a logic —and in fact s is defined by a quotient construction.18 Be careful to note that wedo not know a priori whether the application of s actually yields a presheaf cor-responding to a logic; this has to be investigated separately after fixing a specificnotion of covering. However, at any rate we get an equivalence relation betweenlogics given by l ∼ l′ iff s(l) ∼= s(l′) which, in the spirit of the above, could beunderstood to hold when two (presentations of) logics have the same essence aftercleaning up the redundancies. In the same vein, the sheaf property for a logic lindicates a good behavior with respect to translations from finitary logics into land the sheafification functor yields an equivalence relation between logics.

18For F a presheaf, s(F ) is defined by s(F )(X) := F (X)/ ∼, where a ∼ b iff there is a covering{fi | i ∈ I} such that F (fi)(a) = F (fi)(b) for all i.


5. Operads and the need for better categories of logics

Our chosen framework, the category S of signatures and the category L of logics,excludes many kinds of logics: the propositional logics with infinitary connectives,the non-reflexive logics, the linear logics and the relevant logics. Our category Sof signatures is, for its simplicity and good categorial properties, often chosen as atest bed for categorial constructions like combination of logics, as is the case here.In practice it leads, however, to a much restricted notion of morphism betweenlogics: for example, two presentations of classical propositional logic taking asprimitive connectives (i.e., signatures) {¬,→}, {¬,∨} respectively, do not admitany morphism between each other (since it would have to take → to ∨) while theycould intuitively be expected to be isomorphic. In terms of the introduction, thismeans that our categories give an unsatisfactory (partial) answer to the identityproblem.

To remedy this defect, [10] and others have taken as signature morphisms thesubstitutions from Section 2.1.2, thus allowing to take → to the derived connective(¬ (−) ∨ −). The resulting category S of signatures has, however, bad categorialproperties — for instance, it does not have all pullbacks nor all colimits, whichimplies that a category of logics built above it (and thus coming with a limit cre-ating forgetful functor to S) can not be accessible. The reason for these categorialinsufficiencies lies in the fact that F (Σ) (the language freely generated by Σ) isthe absolutely free algebra over its signature, so S is a category of free algebrasand colimits of free structures are hardly free again. That formal languages arefree algebras over some signature seems to be a crucial feature in logic, so how canwe overcome this difficulty?

It could be helpful to rephrase the situation using the language of operads: Anoperad is a multicategory (a structure coming from proof theory and thus maybeknown to logicians) with only one object. Such a structure can be seen as an ax-iomatization of the behavior of a collection of finitary operations on a set, closedunder the formation of derived operations. Thus it consists of a set of “operations”of finite arity and bears a structure given by substitution of operations into otheroperations. A morphism of operads is a function between the sets of operationspreserving arities and commuting with substitution. An algebra over an operad isan interpretation of the sets of operations as actual operations on some set; moreformally, it is an operad morphism into the operad of all finitary operations onsome set. An introduction to operads would go beyond the scope of this article,for this we refer to [6], which, although treating much more general operads thanwe are needing here, gives an excellent intuition of this notion. It is also possi-ble to describe an operad by giving generating operations and relations betweenthem, and accordingly there are free operads generated by a series of sets of n-aryoperations — those with no relations. The formation of an absolutely free algebraover a signature can now be described in two steps: First form the free operadover the signature, then the free algebra over that operad. The substitutions fromabove are just morphisms between (free) operads which of course induce a function

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between the corresponding free operad-algebras. The above category of signaturesis thus just the category of free operads and arbitrary operad morphisms. Theoperad-algebras only come into play when on wants to define a category of logicsabove the signatures: Since consequence relations are defined on sets (of formulas)we now need to form the free algebra over the operad to get such a set.

To escape from the dilemma that categories of free structures are bad behavedwe propose a simple step: One could take as a category of signatures the categoryof all operads which is known to have good categorial properties. In particular it islocally presentable, and thus possibly allows to reproduce the results of this article.This would lead to a category of logics over languages with possibly interdefinableconnectives (since now the operations can satisfy relations) and would thus includea common practice in logic into the formal treatment. To build a category oflogics above this category of signatures one could proceed as before: Formallyone could define an object in this category as a pair consisting of an operad Oand a consequence relation (possibly satisfying some further conditions) on theunderlying set of the associated free O-algebra. A morphism would be a morphismof operads such that the induced function between the associated free algebraspreserves consequence (again possibly satisfying further conditions).

The step from free operads to arbitrary ones may seem to go away fromthe usual intuitions of logic, where one is used to the sets of formulas havingthe structure of an absolutely free algebra, and at first one may feel to havethus introduced pathological objects into one’s category of logics. But first, theinformal use of interdefinable connectives in logical practice does not cause anyproblems, and if the language of operads is an adequate formalization of thispractice, which it seems, this gives a reason to hope not to encounter difficultieson the formal side as well. Second, on a more abstract level, it is a highly successfulmathematical practice to admit pathological objects in a category in order to make(the global properties of) the category itself less pathological — the passage frommanifolds to C∞-schemes in Differential Geometry illustrates well this point, asdoes the functorial approach to algebraic geometry, where one passes by the Yonedaembedding from schemes into a category of functors where most objects have nogeometric appeal at all.

6. Conclusion

The main concrete purpose of this work was to bring the theory of accessiblecategories into the study of logics. Let us summarize what we have achieved forthis goal and point out some further consequences:

1. The category of finitary structural logics is locally finitely presentable.2. The category of algebraizable logics is accessible but not locally presentable.3. We have identified which are the finitely presentable logics: They are precisely

those whose signature has a finite number of symbols and whose consequencerelation can be generated by a finite number of axioms and deduction rules.


Thus the categorical notion of finite presentability makes sense in the contextof logics — a fact which was not a priori clear and encourages further studiesof the accessibility of categories of logics.

4. There exist direct and inverse image logics.5. It is a question of Universal Logic whether there exist irreducible logics, i.e.,

logics which cannot be obtained by combining proper non-trivial sublogics,and, if so, what characterizes such logics. We know that, in a λ-accessiblecategory of logics, any logic is a colimit of λ-presentable logics, and thereforeit receives morphisms from its colimit cocone. Taking the images of thesemorphisms we obtain sublogics whose combination gives the original logic.This indicates that an irreducible logic would have to be λ-presentable.

The question about irreducible logics is, of course, aimed at finding thefundamental building bricks out which any other logic can be constructed.In a λ-accessible category of logics it would suffice to show that every λ-presentable logic can be built out of irreducible ones, the rest being given bycolimits (i.e. combinations) of these.

6. Another question of Universal Logic is to what extent Logic (as a field ofstudy) has an algebraic character. The notions of accessibility and local pre-sentability bring new aspects into this question. Of course it is still not anentirely settled question how well local presentability really corresponds toalgebraicity.

7. About the metatheory of logics: That the categories studied here are accessi-ble, implies, by the model theoretical characterizations of accessible categoriesmentioned in the introduction, that they are categories of models of first or-der theories. The locally finitely presentable category L is even a categoryof models of a theory in usual classical first order logic, which is somewhatastonishing, given the fact that the definition of a consequence relation in-volves the powerset. It is possible that there is a reasonable model theory forlogics which form a locally finitely presentable category.

For the moment the main task for starting the exploitation of the theory ofaccessible categories is to find “serious” categories of logics, not suffering from thedefects pointed out in the previous section — the possible use of operads suggestedthere could do the job. Independently of that we hope to have convinced the readerthat global properties of categories of logics are not an esoteric subject matter,but make themselves felt in concrete situations and are worthwhile to care about.

Acknowledgement

We are greatly indebted to Augusto Jun Devegili for compensating our lack ofLATEX abilities and substantially improving the readability of this text.


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Peter Arndt and Rodrigo de Alvarenga FreireGroup for Theoretical and Applied LogicIFCH and CLE — UNICAMPP.O. Box 6133BR-13083-970 Campinas, SPBrazile-mail: [email protected]

[email protected]

Odilon Otavio Luciano and Hugo Luiz MarianoIME — USPRua do Matao, 1010BR-05508-090 Sao Paulo, SPBrazile-mail: [email protected]

[email protected]

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A Global Glance on Categories in Logic

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