on reﬂective subcategories of varieties · 686 j. adámek, l. sousa / journal of algebra 276...

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with asolution

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Journal of Algebra 276 (2004) 685–705

www.elsevier.com/locate/jalgebr

On reflective subcategories of varieties

Jirí Adámeka,1,∗ and Lurdes Sousab,2

a Department of Theoretical Computer Science, Technical University of Braunschweig,Postfach 3329, 38023 Braunschweig, Germany

b Department of Mathematics, School of Technology, Polytechnic Institute of Viseu, 3504-510 Viseu, Po

Received 28 May 2003

Available online 16 December 2003

Communicated by Kent R. Fuller

Abstract

Full reflective subcategories of varieties are characterized as the cocomplete categoriesregular generator, or as classes of algebras presented by “preequations.” As a byproduct, ais presented to the problem of describingω-orthogonality classes of locallyfinitely presentablecategories in terms of closure properties. 2004 Elsevier Inc. All rights reserved.

1. Introduction

By the Birkhoff Variety Theorem, equational classes of algebras (varieties) are ethe classes closed under products, subalgebras and quotient algebras. Analogoquasivarieties, i.e., classes presented byquasiequationsor implications of the followingform

∀(xu)u∈U

[∧i∈I

αi(xu) →∧j∈J

βj (xu)

]

* Corresponding author.E-mail address:[email protected] (J. Adámek).

1 Supported by the Czech Grant Agency, Project 201/02/0148.2 Financial support by the Center of Mathematics of the University of Coimbra and the School of Technol

of Viseu.

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2003.09.039

686 J. Adámek, L. Sousa / Journal of Algebra 276 (2004) 685–705

ed

Alg

letectives,

plete

already

be

notit is a

by its

whereαi andβj are equations in the variables{xu}u∈U , are precisely the classes closunder products and subalgebras. That is, the full subcategories of AlgΣ , whereΣ is a(potentially infinitary, many-sorted) signature, which are reflective, and the reflections areregular epimorphisms. In the present paper we study full reflective subcategories ofΣ

in general. We call themprevarieties.Whereas quasivarieties (and varieties) have been characterized as the cocomp

categories with a regular generator formed by regular projectives (or exact projerespectively), see [8,10], and [2], we prove that prevarieties are just the cocomcategories with a regular generator. All these results assume that the signatureΣ is allowedto be large (a proper class of operations); in that case the definition of a prevarietyV has tobe supplemented by the requirement that free algebras exist. Large signatures havebeen used by J.R. Isbell [9] and other authors later.

Prevarieties can be characterized syntactically as classes of algebras which canpresented bypreequations, i.e., formulas of the following form

∀(xu)u∈U

[∧i∈I

αi(xu) → ∃!(yv)v∈V

∧j∈J

βj (xu, yv)

]. (1)

These are precisely the limit sentences in the logicL∞∞ in the sense of [5] and [12].

Example ( posets). The categoryPos of posets and order-preserving functions doeshave a regularly projective regular generator, that is, this is not a quasivariety. Butprevariety, presented by two 2-sorted unary operations (source and target)

s, t : e → v

where the setS = {e, v} of sorts has two members:e for “edges” andv for “vertices.”A natural presentation by preequations specifies that (1) an edge is determineddomain and codomain:

∀(y, z)([

(sy = sz) ∧ (ty = tz)] → (y = z)

),

and that (2) the resulting relation is reflexive:

∀p ∃!z [(sz = p) ∧ (tz = p)

],

antisymmetric:

∀(y, z)([

(sy = tz) ∧ (sz = ty)] → (y = z)

),

and transitive:

∀(y, z)((ty = sz) → ∃!x [

(sx = sy) ∧ (tx = tz)])

.

(Herep is a variable of sortv andx, y, z are variables of sorte.)

J. Adámek, L. Sousa / Journal of Algebra 276 (2004) 685–705 687

l and,

of the,

s (zero-

ll thee

ucedvious

vey.

ty—k 5.5).

aning

Prevarieties naturally generalize the locally presentable categories of GabrieUlmer: if the given regular generator is assumed to consist ofλ-presentable objectsthen the prevariety is locallyλ-presentable. And conversely, every locallyλ-presentablecategory is equivalent to such a prevariety, see [3]. To mention examples outsiderealm of locally presentable categories: the category of compactT2-spaces is a varietythus, every reflective subcategory, e.g., the dual category of that of boolean algebradimensional compactT2-spaces) is a prevariety.

The most interesting special case of prevarieties are thefinitary prevarieties, i.e., classesof finitary algebras presented by preequations of the finitary first-order logic (i.e., aindexing setsI , J , U andV in (1) are finite) as the examplePosabove demonstrates. Wcharacterize finitary prevarieties as the classesA of finitary algebras closed in AlgΣ under

(i) products,(ii) directed colimits, and(iii) A-pure subobjects.

The last notion is a relativization of the concept of a pure subobject which is introdin the present paper in order to solve the more general problem left open in prework [6]: a characterization ofω-orthogonality classes; see Section 5 for a short surHere we just recall that a homomorphismm :B → A in Alg Σ is calledpure providedthat every positive-primitive formula of the first-order logic valid inA is valid in B.Categorically, this means that in every commutative square

Xf

u

Y

v

Bm

A

where X and Y are finitely presentableΣ-algebras the homomorphismu factorizesthrough f . Unfortunately, it is not true in general that every class ofΣ-algebrasclosed under limits, directed colimits and pure subojects is a finitary prevariea counterexample, essentially due to H. Volger [15], is given in 4.5 (see also RemarWe therefore introduce, for every full subcategoryA of Alg Σ , the following concept of anA-pure subobject: it is precisely as above except that we requestf to be anA-epimorphism(i.e., given a parallel pairp1,p2 :Y → Z with Z ∈ A thenp1f = p2f impliesp1 = p2).We prove that the above conditions (i)–(iii) characterize finitary prevarieties; the meof (iii) is, as expected, that for every algebraA ∈ A and everyA-purem :B → A we haveB ∈ A. A surprising corollary is that if a classA of algebras iscogenerating, i.e., if forevery pair of distinct homomorphismsp1,p2 :Y → Z in Alg Σ there existsq :Z → A,A ∈A, with qp1 �= qp2, then

A is a finitary prevariety ⇔ A is a finitary quasivariety.


atticethe

y.ing

les

of alleties are,l andory is

hism.

Thus, for example, every finitary prevariety of lattices containing the two-elements lis a quasivariety. In the above examplePoscannot be cogenerating—in fact, considertwo graph homomorphisms

Y

• ��

��

p1

p2 •

•��

Z

as homomorphisms ofΣ-algebras: we haveqp1 = qp2 for every homomorphismq wherethe codomain is antisymmetric.

2. An abstract characterization

2.1. Definition. A category is called aprevariety if it is equivalent to a full reflectivesubcategory of a category monadic over a power ofSet.

2.2. Examples. (1) Every locally presentable category of Gabriel and Ulmer is a prevarietIn fact, let K be locally λ-presentable and letA be a small subcategory representall λ-presentable objects. Then the canonical functorE :K → SetA

op, given by K →

K(−,K)/Aop, is a full and faithfull right adjoint, see [3]. The presheaf categorySetAop

isof course monadic overSetS , whereS = obj(A), via the forgetful functorF :SetA

op →SetS . AndK is equivalent to the full reflective subcategoryE[K].

(2) Every monadic category onSetS is, of course, a prevariety. This includes exampsuch as compact Hausdorff topologicalspaces and complete semilattices.

(3) The dual of the category of boolean algebras, equivalently, the categoryzero-dimensional compact Hausdorff spaces, is aprevariety: the latter is a full reflectivsubcategory of the category of compact Hausdorff spaces. This shows that prevariein fact, a substantial extension of locally presentable categories (for which GabrieUlmer showed that, with the exception of partially ordered classes, the dual categnever locally presentable).

2.3. Remark. Recall that aregular generatorin a categoryK is a small collectionG ofobjects such that for everyK the canonical morphism

eK :∐G∈G

K(G,K) ◦ G → K

is well-defined (i.e., the coproduct in the domain exists) and is a regular epimorp(HereM ◦ G denotes the copower ofG indexed byM.)

Examples. (1) In anS-sorted quasivariety of algebras the collection{Gs}s∈S , whereGs isa free algebra on one element of sorts, is a regular generator.


tor,

ular

nctor,

s by

r,

s withegular

ss

al

(2) In [1] a cocomplete categoryB is found which (a) does not have a regular generabut (b) has an objectB such that all objects ofB are regular quotients of copowers ofB.

2.4. Theorem. Prevarieties are precisely the cocomplete categories with a reggenerator.

Proof. Sufficiency follows from the well-known fact that, given an adjoint situation

F � U :K →L (L cocomplete),

if the counitε :FU → Id has regular epimorphic components then the comparison fuK :K →LT of the corresponding monadT is full and faithful; and, ifL has coequalizersthenK is a right adjoint. Thus, given a regular generatorG = {Gs}s∈S in K, apply theabove to the adjunctionF � U whereU :K → SetS is the forgetful functor

UK = (K(Gs,K)

)s∈S

andF is its left adjoint

F(Ms)s∈S =∐s∈S

Ms ◦ Gs.

Since ε is formed by the canonical morphisms, which are regular epimorphismassumption onG, we obtain a full and faithful right adjoint

K :K → (SetS

)T

for the monadT = (U,F, ε, η). Consequently,K is equivalent to a full, reflectivesubcategory of the category(SetS)T.

For the necessity, letK be a full reflective subcategory of(SetS)T. Then K iscocomplete because(SetS)T is: the latter follows from the fact thatSetS is cocompleteand has all epimorphisms split, see 7.9 in [11]. Moreover,(SetS)T has a regular generatoe.g.,(FTXs)s∈S whereXs is the object ofSetS with all sorts empty except the sorts witha single element (andFT is the left adjoint induced by the monadT). It is obvious that forevery full reflective subcategoryK of (SetS)T the reflections of the free algebrasFTXs inK form a regular generator ofK. �2.5. Remark. (a) Analogously, quasivarieties are precisely the cocomplete categoriea regularly projective regular generator, see [8] or [2]. Observe that the concept of a rgenerator is equivalent toE-projectiveE-generator for some classE ⊆ RegEpi. Moreprecisely, a collectionG of objects in a categoryK is a regular generator iff there is a claE of regular epimorphisms such thatG is E-projective (i.e., every hom-functorK(G,−),G ∈ G, mapsE-morphisms to epimorphisms) and anE-generator (i.e., the above canonicmorphismseK lie in E for all K). In fact, it is sufficient to denote byE the class of allregular epimorphisms w.r.t. whichG is projective.


ure

l

efy,rem 2.4nden all

ronges notis

ere

matet

ory

r

(b) Clearly a regular generatorG in K is colimit-dense, in the sense that the closunder colimits ofG is the whole categoryK. Given a class of epimorphismsF , let F ′denote the largest pullback stable subclass ofF . In [14] it is shown that, under mildconditions onF , a cocomplete category with pullbacks having anF ′-projective colimit-denseF -generatorG is a prevariety. Under these circumstances,F ′ is just the class of alF -morphisms to whichG is projective.

2.6. Remark. In Lawvere’s classical characterization of finitary varieties [10] the existencof colimits is weakened to that of (i) coproducts of objects fromG and (ii) coequalizers oequivalence relations. In [4] the concept ofpseudoequivalenceswas introduced; essentiallthese are just equivalence relations precomposed with a regular epimorphism. Theoremains valid if cocompleteness is restricted to coproducts of objects of the generator acoequalizers of pseudoequivalences. This follows from the fact, proved in [2], that thcoequalizers exist.

2.7. Example. Recall that for locally presentable categories, we can work with stgenerators rather than regular ones: a category is locallyλ-presentable iff it is cocompletand has a strong generator formed byλ-presentable objects. The analogous result doehold for prevarieties: the categoryB of Example 2.3(2) is not a prevariety, although itcocomplete and has a strong generatorB.

3. A concrete characterization

3.1. Recall that, for every setS of sorts, monadic categories onSetS are precisely thosequivalent toS-sorted varieties. More detailed: consider any (possibly large) signatuΣ

of S-sorted operation symbolsσ of arities

σ : (si )i<n → s

wheren is a (small) cardinal andsi ands are sorts. We can form the quasicategory

Alg Σ

of all S-sortedΣ-algebras and homomorphisms—this is, in general, not a legiticategory since, wheneverΣ is a large signature, the collection of allΣ-algebras on the se{0,1} is as large as expCard (the collection of all subclasses of the proper classCard). Byavarietyof Σ-algebras we mean a classA of Σ-algebras (considered as a full subcategof Alg Σ and equipped with the natural forgetful functorU :A → SetS) such that

(1) A has free algebras, i.e.,U is a right adjoint, and(2) A can be presented by equations.

For every varietyA the forgetful functorU :A → SetS is monadic. Conversely, foevery monadic functorU0 :A0 → SetS there exists a varietyU :A → SetS of S-sorted


ctor

-sorted

ince

f

the

algebrasconcretely equivalentto A0, i.e., such that there exists an equivalence funE :A0 → A for which the following triangle

A0E

U0

∼=

A

U

SetS

commutes up to natural isomorphism. This has been proved in [11, 5.45], in the onecase. A generalization toSetS is straightforward.

3.2. Remark. Let Σ be anS-sorted signature andX anS-sorted set, i.e., an object ofSetS .We can form thetermsoverX in the usual manner, but we do not obtain an algebra (sall terms will typically form a proper class). That is, we define anS-sorted collection

TΣX = (TΣ,sX)s∈S

of terms overX to be the collection of the smallest classes such that

(1) every variable of sorts is a term of sorts: Xs ⊆ TΣ,sX; and(2) given an operation symbolσ ∈ Σ of arity σ : (si)i∈I → s then for every collection o

termsti of sortsi (i ∈ I ) we have a termσ(ti )i∈I of sorts.

For everyΣ-algebraA and everyS-sorted functionf :X → UA we denote by

f :TΣX → A

the computation of terms, i.e., theS-sorted functionf = (fs )s∈S extendingf and such

that for every termσ(ti ) above we havef s (σ (ti)i∈I ) = σA(f

si (ti))i∈I .

3.3. Definition. By a preequationis meant a formula of the form

∀(xu)u∈U

(E → ∃!(yv)v∈V E′), (2)

whereE is a conjunction of equations (between terms of the same sort over theS-sortedsetX = {xu}u∈U of variables) andE′ is a conjunction of equations (between terms ofsame sort overX + Y whereY is theS-sorted setY = {yv}v∈V ).

Remark. A Σ-algebraA is said tosatisfy the preequation(2) provided that for everyS-sorted functionf :X → UA such thatf

s (t (xu)) = fs (t ′(xu)) for every equation

t (xu) = t ′(xu) of sorts in E there exists a uniqueS-sorted functiong :Y → UA such that[f,g]s(u(xu, yv)) = [f,g]s(u′(xu, yv)) for everyu(xu, yv) = u′(xu, yv) of sorts in E′.


Thisnted by

oft

re

f

tion

isms

f

tor

3.4. Examples. (1) In the variety of monoids we have the full subcategory of groups.is, obviously, not a subquasivariety. But it is a subprevariety because it can be presethe following preequation

∀x∃!y (xy = e).

(2) See Section 1 for a preequational presentation of posets.

3.5. In the following definition, by anS-sorted set is simply meant an objectX = (Xs)s∈S

of SetS . If∑

s∈S cardXs = n, we say thatX hasn elements.

Definition. Let A be a class ofΣ-algebras. We say that an algebraA ∈ A is A-gene-rated by an S-sorted subsetX of UA provided thatA has no proper subalgebra inAcontainingX.

A is said to havebounded generationprovided that for every cardinaln there is, up toisomorphism, only a set of objects inA which areA-generated by a set ofn elements.

Remark. Bounded generation ofA, jointly with closedness under intersectionsubalgebras, implies that the forgetful functorU :A → SetS satisfies the solution-secondition. Not conversely: in 3.8 we present an example of a categoryA of algebras on twounary operations which does nothave bounded generation, althoughU is a right adjoint.

3.6. Theorem. For a classA of Σ-algebras with bounded generation the following aequivalent:

(i) A is closed under limits inAlg Σ ;(ii) A is reflective inAlg Σ ;(iii) A can be presented by preequations.

Remark. The main part of the proof below, the implication (ii)→ (iii), is an adaptation othe use of “orthogonality formulas” in [3, 5.6].

Proof. (i) → (ii). This follows from the Adjoint Functor Theorem: bounded generayields the solution-set condition for the embeddingE :A → Alg Σ . In fact, for everyΣ-algebraB onn elements, a solution set is obtained by considering all homomorphh :B → A such thatA ∈ A and the seth[B] (of at mostn elements)A-generatesA. Everyhomomorphismf :B → C with C ∈ A factorizes through one of those: denote byA

the intersection of all subalgebras ofC lying in A and containingf [B]. The codomainrestrictionh :B → A fulfils f = mh for the inclusionm :A → C. There is only a set osuch homomorphismsh :B → A becauseA is generated by at mostn elements.

(ii) → (iii). Bounded generation and the fact that, being reflective in AlgΣ , A isclosed under intersections, provide the solution-set condition of the forgetful funcU :A → SetS , thusU has a left adjointF with unit η : Id → UF . For everyS-sortedsetX of variables let≈X denote the kernel equivalence of

η :TΣX → FX.
X


r every

ry

SinceFX is a set,≈X has a set of representatives, and we choose one such set. Foterm t ∈ TΣX we denote by[t] ∈ TΣX the representative of the class oft . Then everyalgebraA ∈ A fulfils the equationt = [t]: given any interpretationf :X → UA of thevariables, then the unique homomorphismf :FX → A extendingf forms a commutativetriangle

TΣX

f

ηX

FX

f

A

and thus fromηX(t) = η

X([t]) we concludef (t) = f ([t]).

For an arbitraryΣ-algebraB we form a conjunction of equations called theA-graphof B as follows. These equations use the setX = UB of variables. Consider an arbitraoperation symbolσ : (si)i<n → s in Σ and arbitrary elementsxi ∈ Bsi andx ∈ Bs suchthat

σB(xi)i<n = x. (3)

Thenσ(xi)i<n andx are two terms inTΣX, and we can turn to their representatives[σ(xi)]and[x], respectively. We define theA-graph as the following conjunction

grA B =∧([

σ(xi)] = [x])

ranging over allσ , xi andx as in (3) above.TheA-graph ofB has the following property:

Given a Σ-algebraA satisfying [t] = t for all terms t and anS-sorted functionf :X → UA, thenf is a homomorphism fromB to A iff grA B holds inA underthe interpretationf (i.e., iff (3) impliesf

s ([σ(xi)]) = fs([x])).

In fact, if f is a homomorphism, then (3) implies

f s

([σ(xi)

]) = f s

(σ(xi)

) (A fulfils [t] = t

)= σA

(fsi (xi)

) (definition off

)= fs

(σA(xi)

)(f is a homomorphism)

= fs(x)(see (3)

)= f

s

([x]) (A fulfils [t] = t

).


of

s

sn

,l

onal

ofy, withesnts

Conversely, iff s ([σ(xi)]) = f

s ([x]) holds whenever (3) does, then we havef

s (σ (xi)) =

fs (x), due to[t] = t in A, i.e., σA(fsi (xi)) = fs(x)—this proves thatf is a homomor-

phism.We are prepared to define the preequation prB which is satisfied by every algebra

A—we derive, then, that these preequations and the equations[t] = t present the classA.Let r :B → B∗ be the reflection ofB intoA and letY = UB∗. We assume, without los

of generality, thatX andY are disjoint in every sort. Observe that for every variablex ∈ X

we have an equationx = r(x) in the variablesX ∪ Y . Put

prB ≡ (∀�x) [grA B → (∃!�y)(

grA B∗ ∧∧x∈X

(x = r(x)

))]

where�x is a list of all elements ofX and �y is a list of all elements ofY . We claim thatevery algebraA ∈A satisfies prB. In fact, letf :X → UA be an interpretation of variablefrom X under which grA B holds. Equivalently, letf :B → A be a homomorphism. Thethere exists a unique homomorphismf ∗ :B∗ → A with f = f ∗ · r—that is, a uniqueinterpretationf ∗ :Y → UA of the variables inY such that grB∗ is satisfied andx = r(x)

are satisfied (by[f,f ∗] :TΣ(X + Y ) → A), equivalently,f (x) = f ∗(r(x)) holds for allx ∈ X.

The classA is presented by the collection of

(α) all preequations prB, whereB ranges over allΣ-algebras, and(β) all equationst = [t], wheret ranges over all terms.

In fact, every algebra inA satisfies (α) and (β). Conversely, ifB satisfies (α) and (β),we show that the reflectionr :B → B∗ is a split subobject;A, being closed under limitsis closed under split subobjects, thusB ∈ A. SinceB satisfies prB and since the triviainterpretation idX of variables has the property that all equations of grA B hold in B, weconclude that there exists a unique interpretationg :Y → UB of the variables inY suchthat (a) grA B∗ holds inB under the interpretationg and (b)x = g(r(x)) holds for allx ∈ X. Now (a) guarantees thatg :B∗ → B is a homomorphism and (b) yieldsg · r = id,as desired.

(iii) → (i). It is straightforward (see [3, 5.7]).�3.7. Corollary. Prevarieties are precisely the categories equivalent to preequaticlasses of algebras with bounded generation.

Proof. In fact, monadic categoriesA over SetS are precisely the equational classesS-sorted algebras (over large signatures) with bounded generation, or, equivalentlfree algebras; see, e.g., [11]. Every reflective subcategory ofA is preequational, as whave proved above. Conversely, a preequational classA with bounded generation ireflective in AlgΣ . The closure�A of the classA under subalgebras and regular quotie(homomorphic images) has the same free algebras asA, therefore,�A is a variety, i.e.,a category monadic overSetS . AndA is reflective in �A. �


ns

.6, be

our

ring.dinals,

t

:

ahism

3.8. Example. We now present an example of a classA of unary algebras on two operatiowhich

(i) is closed under limits,(ii) has free algebras, and(iii) is not a reflective subcategory of AlgΣ .

This shows that the assumption of bounded generation cannot, in Theorem 3weakened to the existence of free algebras.

We use 2-sorted algebras with sortsS = {e, v} and with two unary operations,s andt ,of sorte → v. Thus, AlgΣ = Gra is the category of graphs and homomorphisms. Forexample we need to assume that

a full embeddingE :Ordop → Gra exists

whereOrdop is the linearly ordered class of all ordinals with the dual of the usual ordeThis assumption is fulfilled whenever our set theory does not have measurable carsee A7 in [3].

Given the embeddingE as above, we denote byA the class of all graphsG such thateither there exists an ordinali such that

(1) hom(Ej ,G) ={∅, if j < i,

a singleton set, if j � i,

or G has no path of length 2, in other words,

(2) hom(P,G) = ∅.

HereP denotes the graph

0 → 1→ 2

with Pv = {0,1,2} and Pe = {(0,1), (1,2)} whose operationss and t are the twoprojections.

The classA clearly has all free algebras: a free algebra on a setA of arrows and a setXof elements is the graph having pairwise disjoint arrows indexed byA and nodes withouarrows indexed byX—it has no path of length 2. The collectionA1 of all graphs satisfying(1) above is obviously closed under limits. It follows thatA is also closed under limitscondition (2) is namely equivalent to

hom(G,Q) �= ∅

whereQ is the single arrow, i.e.,Qe = {q} andQv = {0,1} with s(q) = 0 andt (q) = 1.(In fact, if G has no path of length 2, we have a homomorphismh :G → Q mapping anelementx of G to 0 iff x lies in the image ofs; the converse is also evident.) A limit ofdiagram lying inA1 lies inA1, and for a diagram where some object has a homomorpinto Q the limit also has such a homomorphism.


tion:

testrict

peakn

,f

Assuming thatP has a reflection

r :P → P

in A, we derive a contradiction. SinceP does not satisfy (2) above, it lies inA1, thus, thereexists a homomorphism

h :Ei → P

for some ordinali. Observe thatEi+1 ∈A and conclude that

hom(P,Ei+1) = ∅.

(In fact, every homomorphismP → Ei+1 extends uniquely to a homomorphismP →Ei+1 which, composed withh above, yields a homomorphismEi → Ei+1—a contradic-tion to the fullness ofE.) In other words, we have proved that

hom(Ei+1,Q) �= ∅.

Since certainly

hom(Q,Ei+2) �= ∅

(the graph Ei+2 has at least one arrow), this yields the desired contradichom(Ei+1,Ei+2) �= ∅.

4. λ-ary prevarieties

4.1. Remark. So far we have worked in the logicL∞∞ in which conjunctions over any se(of equations) and quantifications over any set of variables are allowed. We want to rourselves to the finitary logicLωω in which afinitary preequationis a formula

∀(x1, . . . , xn)(E → ∃!(y1, . . . , yt )E′)

whereE andE′ are finite conjunctions of equations. Or, more generally, to the logicLλλ,whereλ is an infinite regular cardinal (i.e., a cardinal equal to its cofinality). Here we saboutλ-ary preequationsof the form 3.3 whereU andV are sets of cardinality less thaλ and alsoE andE′ are conjunctions of less thanλ equations.

4.2. Definition. By aλ-ary prevariety ofΣ-algebras, whereΣ is a (small)λ-ary signatureis meant a full subcategory of AlgΣ which can be presented byλ-ary preequations. Iλ = ω we speak aboutfinitary prevarieties.


o

ehthe

whichbols).

e

by

Examples. (1) The category of posets is a finitary prevariety, see Section 1.(2) Every locally finitely presentable categoryA of Gabriel and Ulmer is equivalent t

a finitary prevariety. In fact,A is equivalent to anω-orthogonalityclass ofSetB for somesmall subcategoryB, see [3, 1.46], i.e., there exists a setM of morphismsm :X → Y inSetB with X andY finitely presentable such that the full subcategoryM⊥ of all objectsZ of SetB orthogonal to eachm (i.e., for every morphismX → Z there exists a uniqufactorization throughm) is equivalent toA. NowSetB is a variety of unary algebras witS = Bobj andΣ = Bmor (and the sorting given by the domain and codomain). Andorthogonality to m can be expressed by a limit sentence in this signature, see [3, 5.6],is another name for finitary preequation (in any signature without relational symUsing the same technique as in [3] we prove, more generally:

4.3. Proposition. For everyλ-ary preequation there exists a homomorphismm :A → �Abetweenλ-presentableΣ-algebras A and �A such that aΣ-algebra K satisfies thepreequation iffK is orthogonal tom.

Proof. We are given a preequation as follows

∀(xi)i∈I

(( ∧u∈U

tu(xi) = t ′u(xi)

)→ ∃!(yj )j∈J

( ∧v∈V

sv(xi, yj ) = s′v(xi, yj )

))(4)

whereI , U , J andV are sets of less thanλ elements. We define a homomorphism

m :A → �A

in Alg Σ with the following property:A and�A areλ-presentable algebras and

satisfaction of (4) ⇔ orthogonality to m.

That is, aΣ-algebraK satisfies (4) iff for every homomorphismf :A → K there exists auniquef : �A → K with f = f · m.

Let F :SetS → Alg Σ and η : Id → UF denote the left adjoint and the unit of thforgetful functorU (i.e.,FX is a freeΣ-algebra onX). We denote by

e :FX → A

the quotient of the free algebra onX = {xi}i∈I modulo the congruence generatedtu(xi) = t ′u(xi) for all u ∈ U . Then an algebraK satisfies

∧u∈U(tu(xi) = t ′u(xi)) under

the interpretationho :X → UK of variables iff there is a homomorphismh :A → K with

ho = U(he)ηX;

andh is uniquely determined byho. We also have a quotient, forY = {yj }j∈J ,

e∗ :F(X + Y ) → A∗


of

les

of the free algebra onX + Y modulo the congruence generated bysv(xi, yj ) = s′v(xi, yj )

for all v ∈ V . Then homomorphisms fromA∗ to K correspond to the interpretationsvariables inX+Y satisfying the latter equations: The coproduct injectionm1 :X → X+Y

yields a homomorphismsFm1 :FX → F(X + Y ). Let us form a pushout

FXFm1

e

F (X + Y )e∗

A∗

e

Am

�A

in Alg Σ . SinceFX, A andA∗ areλ-presentable algebras, so is�A.(I) If an algebra K satisfies (4), then it is orthogonal tom. In fact, given a

homomorphism

h :A → K

then the interpretation of variables

ho = U(he)ηX :X → UK

satisfies all equationstu(xi) = t ′u(xi), thus, there exists a unique interpretation of variabfrom X + Y extendingho and satisfying all the equationssv(xi, yj ) = s′

v(xi, yj )—in otherwords, there exists a unique homomorphism

h∗ :A∗ → K

such that

ho = U(h∗e∗Fm1

)ηX.

We conclude

h∗e∗Fm1 = he :FX → K

since both sides are homomorphisms extendingho. We obtain a unique homomorphismhsuch that the following diagram

FXFm1

e

F (X + Y )e∗

A∗

e

h∗A

m

h

�Ah

K


ess

ion

fulle

sLet

commutes. To prove thath is uniquely determined byhm = h, recall thatK satisfies (4),consequently, a homomorphism fromA∗ to K (which is an interpretation of the variablin X + Y satisfyingsv(xi, yj ) = s′

v(xi, yj ) for all v ∈ V ) is uniquely determined by itvalues onm1 :X → X + Y . That is, given a homomorphismk : �A → K with

km = h,

we prove thatk = h by verifying

ke = he :A∗ → K

which is equivalent to

U(kee∗)ηX+Y m1 = U

(hee∗)ηX+Y m1 :X → UK.

The last equation follows easily:

U(kee∗)ηX+Y m1 = U

(kee∗Fm1

)ηX = U

(hme

)ηX = U

(hee∗)ηX+Y m1.

(II) If an algebraK is orthogonal tom, then for every interpretationho :X → UK ofvariables satisfyingtu(xi) = t ′u(xi) for all u ∈ U we have the homomorphismh :A → K

determined byho = U(he)ηX . Let h : �A → K be the unique homomorphism withh = hm.Thenhe :A∗ → K corresponds to an interpretation of the variables inX+Y which satisfiesall sv(xi, yj ) = s′

v(xi, yj ), and we conclude thathe is uniquely determined byho, since itacts onX asho:

U(he

) · Ue∗ · ηX+Y · m1 = U(hee∗Fm1

)ηX = U(he)ηX = ho.

In other words, for the interpretationho we obtain a unique extension to an interpretatX + Y → UK such that all the equationssv(xi, yj ) = sv′(xi, yj ) for v ∈ V hold. Thisproves thatK satisfies (4). �4.4. Corollary. For every uncountable regular cardinalλ and every(small) λ-ary signatureΣ a class ofΣ-algebras is aλ-ary prevariety iff it is closed inAlg Σ under limits andλ-filtered colimits.

Proof. It is obvious that everyλ-ary prevariety is closed under limits andλ-filteredcolimits. The converse follows from the result of Hébert and Rosický [7] thatsubcategories closed under limits andλ-filtered colimits areλ-orthogonality classes; se[3, 5.18], for a description of aλ-ary preequation (πh) characterizing orthogonality to ahomomorphismh :A → A′ havingλ-presentable domain and codomain.�4.5. Example (see[15], the present formulation due to[6]). A class of unary algebrawhich is closed under limits and filtered colimits but is not a finitary prevariety.Σ = {α,a} with α unary anda nullary. Denote byA the class of all algebras which


.3,

on ofnsome

d”

ains

(1) have a unique sequencea = y0, y1, y2, . . . of elements withαyn+1 = yn for alln = 1,2, . . . , and

(2) fulfil (α2z = yn) ⇒ (αz = yn+1) for all elementsz and alln = 0,1,2, . . . .

This class is easily seen to be closed under limits—in fact it is anω1-ary prevarietypresented by the preequation

∃!(y0, y1, y2, . . .)

[(a = y0) ∧

∧n∈ω

(αyn+1 = yn)

]

and the following implications, one for everyk = 0,1,2, . . .

∀(z, y0, y1, y2, . . .)

([(a = y0) ∧

∧n∈ω

(αyn+1 = yn) ∧ (α2z = yk

)] → (αz = yk+1)

).

It has been proved in [6] thatA is not anω-orthogonality class, thus, by Proposition 4A cannot be presented by finitary preequations.

5. Finitary prevarieties and ω-orthogonality classes in general

5.1. In the present section we characterize finitary prevarieties, i.e.,ω-orthogonalityclassesof the category AlgΣ , see Proposition 4.3. In fact, we present a new characterizatiω-orthogonality classes in any locally finitely presentable categoryK. This solves an opeproblem in a realm where all “natural” related characterizations have been known fortime already. Let us mention these first.

Recall that for a classM of morphisms inK we have two full subcategories “presenteby M:

M- Inj,

the injectivity class ofM, consists of all objectsK injective w.r.t. members ofM, i.e.,such that hom(−,K) sends every member ofM to an epimorphism inSet; and

M⊥,

the orthogonality class ofM, consists of all objectsK orthogonal to the members ofM,i.e., such that hom(−,K) sends every member ofM to an isomorphism.

By an ω-injectivity or ω-orthogonality classin K is meant a full subcategoryA forwhich there exists a setM of morphisms with finitely presentable domains and codomsuch that

A =M- Inj or A =M⊥,

respectively. The former concept has beencharacterized in [13] using the following


ism

y pure

ed31]) is.

e].

re

5.2. Definition. A morphism m :B → A is said to bepure provided that for everycommutative square

Xf

u

Y

v

Bm

A

with X andY finitely presentable the morphismu factorizes throughf (i.e.,u = u′f forsomeu′ :Y → B).

5.3. Remark. (a) Let K be a locally finitely presentable category. Then a morphm is pure iff it is, as an object of the arrow categoryK→, a filtered colimit ofsplit monomorphisms. Consequently, every split monomorphism is pure, and evermorphism is a strong monomorphism; see [3].

(b) More generally,m is calledλ-pure if the above conditions holds wheneverX andY

areλ-presentable.

5.4. Theorem (see [13]).A full subcategoryA of K is anω-injectivity class iff it is closedin K under

(i) products,(ii) filtered colimits, and(iii) pure subobjects.

5.5. Remark. The “expected” characterization ofω-orthogonality classes as classes closunder limits and filtered colimits (and thus closed under pure subobjects, see [3, 2.not true, see Example 4.5. This is all the more surprising sinceω is the only exceptionThat is, letλ be a cardinal with uncountable cofinality. Then theλ-orthogonalityclasses(i.e.,A =M⊥ where domains and codomains of morphisms ofM areλ-presentable) arprecisely the classes closed under limits,λ-filtered colimits andλ-pure subobjects; see [7

To find a remedy for this lack ofλ = ω, we introduce the following new concept, whea morphismf :X → Y in K is called anA-epimorphismprovided that the implication

uf = vf implies u = v

holds for all pairsu,v :Y → A with A ∈A.

5.6. Definition. Let A be a full subcategory ofK. A morphismm :B → A in K is calledA-pureprovided that in every commutative square


e

,

erty

Xf

u

Y

v

Bm

A

with X andY finitely presentable andf an A-epimorphism the morphismu factorizesthroughf .

5.7. Lemma. In every locally finitely presentable category allA-pure morphisms aremonomorphisms.

Proof. Let m :A → B beA-pure. It is sufficient to prove that for every finitely presentablobject Y every pairu1, u2 :Y → A with mu1 = mu2 = v fulfils u1 = u2. In fact, thefollowing square

Y + Y∇

[u1,u2]Y

v

Am

B

commutes. SinceY + Y is finitely presentable and the codiagonal∇ is an epimorphismwe conclude that[u1, u2] factorizes through∇—thus,u1 = u2. �5.8. Examples. Let K be a locally finitely presentable category and letA be a fullsubcategory ofK.

(1) Every pure morphism isA-pure.(2) Every equalizer of morphismsg,h :A → A′ with A′ ∈ A is A-pure. In fact, letm, in

the above square, be an equalizer ofg andh. Sincef is anA-epimorphism,gv = hv,thus, there isw with v = mw. Frommu = mwf it follows thatu = wf .

(3) Let K have the property that every epimorphism is strong (e.g.,K = Alg Σ forany finitary signatureΣ , see [3, Exercise 3.b]). LetA be cogenerating, i.e., givenmorphismsu1, u2 :K → L in K with u1 �= u2 there existsf :L → A, A ∈ A, withf u1 �= fu2. ThenA-epimorphisms are epimorphisms. Therefore

A -pure ⇔ monomorphism.

In fact, one implication is 5.7, and the reverse follows from the diagonal fill-in propbetween strong epimorphisms(=A-epimorphisms) and monomorphisms.

5.9. Theorem. In every locally finitely presentable category theω-orthogonality classesare precisely the full subcategoriesA closed under

(i) products(ii) filtered colimits, and(iii) A-pure subobjects.


.

em

Proof. (I) Sufficiency. LetA be a full subcategory ofK which fulfils (i)–(iii). Due to 5.8(2)we can strengthen (i) to

(i∗) closed under limits.

Denote byM the set of allK-morphismsf :X → Y such thatX and Y are finitelypresentable, and all objects ofA are orthogonal tof . We proveA=M⊥. Recall from [3]that (i∗) and (ii) imply thatA is a reflective subcategory whose reflectorR :K → Apreserves filtered colimits; we denote byrK :K → RK the reflection maps.

Given an objectB ∈ M⊥ we prove thatB ∈ A, thus establishing thatA = M⊥. It issufficient to verify that the reflectionrB of B is A-pure. Thus, let

Xf

u

Y

v

BrB

RB

be a commutative square wheref is an A-epimorphism andX and Y are finitelypresentable. ExpressB as a filtered colimit(bi :Bi → B)i∈I of finitely presentable objectsThe reflection arrowsrBi form a filtered diagram inK→ with the colimit(bi,Rbi) : rBi →rB (i ∈ I ). This follows easily from (ii) and fromR preserving filtered colimits. Sincf is a finitely presentable object ofK→ (see 1.55 of [3]), it follows that the morphis(u, v) :f → rB factorizes through one of the colimit morphisms(bi,Rbi) : rBi → rB . Thatis, there existu′, v′ such that the following diagram

Xf

u′Y

v′

BirBi

bi

RBi

Rbi

BrB

RB

commutes. Let us form a pushoutP of u′ andf , and denote byt the obvious factorizationmorphism:

Xf

u′

Y

v′

u

P

t

BirBi

f

RBi


limits

stse

e

ed

,

er,

rs

The morphismf lies in M. (In fact, sinceBi , X andY are finitely presentable, so isP .Sincef is anA-epimorphism, so isf . And for every morphismp :Bi → A, whereA ∈ A,there exists a factorization throughf : we have a uniquep′ :RBi → A with p = p′ · rBi

thus,p = (p′t)f .) SinceB ∈ M⊥, we conclude thatbi factorizes throughf , say,

bi = qf for q :P → B.

Thenu factorizes throughf , as requested:

u = biu′ = qf u′ = quf.

This proves theA-purity of rB , thus,B ∈A.(II) Necessity. It is easy to see that everyω-orthogonality classM⊥ (where all

morphisms inM have finitely presentable domains and codomains) is closed underand filtered colimits. Let us prove that for everyM⊥-pure subobjectm :B → A withA ∈ M⊥ we haveB ∈ M⊥. Given f :X → Y in M, for everyu :X → B there existsv :Y → A with mu = vf . Now f ∈ M is clearly anM⊥-epimorphism, therefore, the laequality implies thatu factorizes throughf . To prove that the factorization is unique, uthe fact thatA is orthogonal tof , andm is a monomorphism (by Lemma 5.7).�5.10. Corollary. Finitary prevarieties are precisely the classesA of Σ-algebras closed inAlg Σ under products, filtered colimits andA-pure subobjects.

In fact, we know that finitary prevarieties are precisely theω-orthogonality classes (seexample (2) of 4.2 and Proposition 4.3).

5.11. Corollary. Every finitary prevarietyA which is cogenerating inAlg Σ is a finitaryquasivariety.

Acknowledgments

The authors are grateful to Michel Hébert and Jirí Rosický whose suggestions improvthe presentation.

References

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