n-PERMUTABLE LOCALLY FINITELY PRESENTABLE CATEGORIES

We characterize n-permutable locally finitely presentable categories Lex(C op ,S et) by a condition on the dual of the essentially algebraic theory C op .W e apply these results to exact Maltsev categories as well as to n-permutable quasivarieties and varieties.


Introduction
In recent years there has been a considerable interest in expressing properties of an essentially algebraic category (i.e. a locally finitely presentable category) in terms of properties of the corresponding essentially algebraic theory.From this point of view, complete answers have been given with respect to basic properties such as regularity, exactness, extensivity, cartesian closedness and so on (see [8], [9], [4] and [7]).In this note we analyse the condition of n-permutability of the composition of equivalence relations for a regular locally finitely presentable category It is known [5] that n-permutability can be equivalently stated by saying that, for any reflexive relation R, the corresponding generated equivalence relation R is given by a finite construction, more precisely So, if we want to characterize n-permutability of K in terms of a corresponding condition on the essentially algebraic theory C op , or more simply on its dual C (where C is considered as a dense subcategory of K), we must interpret such a finiteness condition in C. The answer to this problem is given in theorem 2.5, where we show that K is (n + 1)-permutable (n ≥ 1) if and only if C is weakly regular and certain relations R X,C n , defined for any reflexive graph X in C and C ∈ C, are transitive.The formal definition of R X,C n simply corresponds to the syntactic interpretation in C of the following relation: if n -related if and only if (f 0 , f 1 ) factorizes through the n-iterated Notice that in C we must now consider reflexive graphs instead of reflexive relations; in fact conditions on reflexive relations do not suffice to force permutability in K.This theorem admits interesting applications in the case

Preliminaries on regular n-permutable categories
In this section we fix the notations and recall some properties of regular categories.A category A is regular [3] if it is finitely complete, every kernel pair has a coequalizer and regular epimorphisms are stable under pullbacks.If A is regular, any arrow f : A → B can be factored as f = i • p with p a regular epimorphism and i a monomorphism.A regular category A is exact when any equivalence relation is effective (a kernel pair).
A relation R from A to B will be denoted by (r 0 , r 1 ) : R A × B; for a relation R on an object A we shall also write R In this section we always assume that the category A is regular: this assumption will assure that the composition of relations is associative.If R and S are equivalence relations on an object A, we have the increasing sequence The smallest equivalence relation containing both R and S, denoted by R ∨ S (when it exists), contains all the terms of this sequence, and these terms are all different in general.If there is an n ≥ 2 for which the relation (R, S) n is an equivalence relation, then R ∨ S = (R, S) n .

Definition. A regular category
Of course, if n = 2 we get the notion of Maltsev category: this notion was introduced in [6] as a weakening of the notion of abelian category.Since the Maltsev property can be expressed without the assumption of regularity of the category A, we shall adopt the simpler and classical The equivalence between the Maltsev axiom, the 2-permutability of the composition of equivalence relations and two other nice properties is recalled in the following 1.3.Theorem.[6] Let A be a regular category.The following statements are equivalent: [12] asserts that a finitary variety is Maltsev precisely when its theory contains a ternary operation p(x, y, z) satisfying the axioms p(x, y, y) = x, p(x, x, y) = y; for instance in the variety of groups such a term p(x, y, z) is given by xy −1 z.Among Maltsev varieties are then those of groups, abelian groups, modules over a fixed ring, rings, commutative rings, associative algebras and Lie algebras.The variety of quasi-groups is also Maltsev, as is the variety of Heyting algebras.There are non-varietal examples of exact Maltsev categories: any abelian category is exact Maltsev, as is the dual of the category of sets and, more generally, the dual of any topos.Finally, the category of topological groups is regular Maltsev [5].

for any A ∈ A any reflexive relation R on A is transitive 4. for any A ∈ A and for any R, S ∈ Eq
It is interesting to know that the properties in theorem 1.3 remain equivalent in the n-permutable case (n ≥ 2): indeed, we have the following 1.5.Theorem.[5] Let A be a regular category.The following statements are equivalent:

for any reflexive relation R on an object
A is n-permutable

for any reflexive relation R on an object
for any A ∈ A and for any R, S ∈ Eq(A), we have (R, S) n = R ∨ S 1.6.Examples.Hagemann and Mitschke [11] proved that a finitary variety is n-permutable if and only if there exist n + 1 ternary terms p 0 (x, y, z), p 1 (x, y, z), . . ., p n (x, y, z) satisfying This result clearly includes Maltsev theorem, this latter being the special case where n = 2.The property of (n+1)-permutability can be shown to be strictly weaker than the one of npermutability for each n ≥ 2. In particular there are examples of 3-permutable varieties that fail to be Maltsev as, for instance, the variety of generalized right complemented semigroups: these algebras have two binary operations • and * satisfying In this case the theorem of Hagemann and Mitschke can be applied by choosing We recall that 3-permutable categories are called Goursat categories [5].The property of 3-permutability, unlike 4-permutability, is strong enough to force the modularity of the lattice Eq(A) of equivalence relations on any object A of the category.

Regular locally finitely presentable categories
A locally finitely presentable category K (see [10] or [2]) is a cocomplete category which admits a small set S of finitely presentable objects such that any object K ∈ K is a filtered colimit of objects of S. Any such category is equivalent to a category Lex[C op , Set] of finite limit preserving functors from a small finitely complete category C op to the category of sets.Via the Yoneda embedding sending an object C ∈ C to the functor Hom(−, C) the category C is a full subcategory of Lex[C op , Set] and the objects of C form a family of dense generators.The dual category of C is called the essentially algebraic theory, while Lex[C op , Set] is the category of models of the theory.Many properties of a locally finitely presentable category can be expressed just in terms of its essentially algebraic theory C and various results in this direction can be found in the literature, for instance in [7], [8], [9] and [4].In this paper we are interested in the property of n-permutability of the composition of the equivalence relations; the regularity of Lex[C op , Set] will be always required in order to express this kind of property.
Regular locally finitely presentable categories have been characterized in [7] as categories of finite limit preserving functors from the dual of a finitely cocomplete "weakly regular" category to the category of sets.We recall the definition:

Definition. [7] A category C is weakly regular if every commutative diagram
where f is a regular epi.
Observe that any regular category is weakly regular: moreover one has the following 2.2.Theorem.[7] Let C be a category with finite colimits.The following conditions are equivalent:

C is weakly regular
In order to express our main results we now introduce two important notions.The first one is the notion of n-iterated graph: The conditions above can be expressed by the commutativity of the diagram below (in which we assume that n is odd) By means of the notion of n-iterated graph, we now define a relation on the set Hom(C, X 0 ): be a reflexive graph and n ≥ 1.Two arrows n , and we write f 0 R X,C n f 1 , if there exists a factorisation of the graph C and β is a regular epi.
Remark that the relation R X,C n is always reflexive: indeed, for any arrow f n f 0 .2.5.Theorem.Let C be a category with finite colimits.The following conditions are equivalent (n ≥ 1): For this, we consider (f 0 , f 1 ) in R X,C n and we're going to show that (f 0 , f 1 ) belongs to Hom(C, (I, I o ) n ).By assumption there is a factorisation as in the diagram (in the diagrams above we have assumed that n is odd).This limit can be clearly obtained by iterated pullbacks in Lex[C op , Set].There is then an arrow η : The regularity of Lex[C op , Set] implies that the induced arrow from (L; l 0 , l 1 , . . ., l n−1 ) to the limit (M ; m 0 , m 1 , . . ., m n−1 ) is a regular epi.By recalling the definition of the composite of relations in a regular category, one then has a regular epi q • : L (I, The arrow q • • η : K → (I, I o ) n is such that Let us then prove that (f 0 , f 1 ) ∈ Hom(C, (I, Keeping in mind that the regular epi q • : L → (I, I o ) n is a directed colimit of regular epis (q • ) j : L j → [(I, I o ) n ] j in C (see for instance [1]) and that C is finitely presentable, we obtain an arrow α j as in the diagram y y α q q q q q q q q q q q q q q q Since any reflexive relation in Lex[C op , Set] can be written as a filtered colimit of reflexive graphs in C and a filtered colimit of transitive relations is a transitive relation, then it suffices to check the property for a reflexive graph in C. Let X 1 G G X 0 be a reflexive graph and let G G X 0 is transitive.We recall that the category C is a dense generator in Set]: this means that the inclusion of K into Set C op (via the restriction of the Yoneda embedding) is fully faithful.From this it follows that the relation (I, 2.6.Corollary.Let C be a category with finite colimits.The following conditions are equivalent: where both If q : X 0 Q denotes the coequalizer of x 0 and x 1 , then Proof.By corollary 2.6 the category Lex[C op , Set] is exact Maltsev.With the same notations as in the lemma 3.2, if there are two arrows f 0 , f 1 : C → X 0 such that q • f 0 = q • f 1 , these must factorize through the regular image We then get the following 3.4.Proposition.Let C be a category with finite colimits.The following conditions are equivalent:

C is weakly regular and pro-maltsev
Proof. 1. ⇒ 2. It follows by corollary 2.6 and lemma 3.3.2. ⇒ 1.By corollary 2.6 and lemma 3.2 one knows that Lex[C op , Set] is regular Maltsev.By using the pro-maltsev property it is possible to show that, given a reflexive graph X 1 More generally, one can define the notion of pro-n-permutable category: and for any pair of arrows f 0 , f 1 : C → X 0 with q • f 0 = q • f 1 , where q is the coequalizer of x 0 and x 1 , (f 0 , f 1 ) is in the relation R X,C n−1 .By adopting the same technique as above, theorem 2.5 allows to generalize these results to the n-permutable case (n ≥ 2): 3.6.Proposition.Let C be a category with finite colimits.The following conditions are equivalent: C is weakly regular and pro-n-permutable 3.7.Remark.The notion of weakly regular pro-n-permutable category is clearly stronger than the one of pro-exact category in the sense of [7]: to see it, one just needs to remark that any n-iterated graph (in our sense) of a reflexive and symmetric graph X is an "iteration" of X as defined in that paper.Observe that pro-exactness corresponds, with the same notations as in definition 3.5, to n .The pro-n-permutability of a weakly regular category C can be accordingly thought as a synthetic way to express at the same time the "exactness" of the category Lex[C op , Set] and the fact that the join of two equivalence relations in Lex[C op , Set] can be obtained in a "finite number of steps".This last condition precisely expresses the n-permutability of the composition of equivalence relations.

Quasivarieties and varieties
By a quasivariety is meant a class of many-sorted finitary algebras that can be defined by implications of the form where n ∈ ω and α i and β are equations (with both sides of the same sort).Any quasivariety is a locally finitely presentable category; in [1] Adámek and Porst characterized (the dual of) those essentially algebraic theories whose models form a quasivariety as being the finitely cocomplete categories which have enough regular projectives: 4.1.Theorem.[1] Let C be a category with finite colimits.The following conditions are equivalent: 1. Lex[C op , Set] is a quasivariety

C has enough regular projectives
Any quasivariety is a regular category: it seems then natural to investigate the condition of n-permutability for quasivarieties.For this, we begin with the following 4.2.Definition.Let A be a regular category and n ≥ Of course, if A is regular Maltsev, any reflexive graph is a 1-pseudo transitive relation, since I = (I, I o ) 1 is an equivalence relation.We then define a notion of n-permutable object: 4.3.Definition.An object P in a category C is n-permutable (n ≥ 2) if the functor Hom(P, −) sends reflexive graphs to (n − 1)-pseudo transitive relations (in Set).
We can then give our characterization of n-permutable quasivarieties: 2. ⇒ 1.By theorems 2.5 and 4.1 we just have to prove that the relation R X,C n−1 is transitive for any reflexive graph X in C and C ∈ C. Since C has a regular projective cover by assumption, it suffices to check that R X,P n−1 is transitive for P regular projective.This follows by R X,P n−1 = Hom(P, (I, I o ) n−1 ) = (J, J o ) n−1 and P is n-permutable.As a corollary of this result, we now give a characterization of n-permutable finitary varieties.For this, we first recall the notion of effective projective object, due to Pedicchio and Wood: 4.5.Definition.An object P in a category C is an effective projective object if the functor Hom(P, −) preserves coequalizers of reflexive graphs.
This notion plays an essential role in a recent work [14] by Pedicchio and Wood.In this paper the authors characterized (the dual of) those essentially algebraic theories whose models form a finitary variety of algebras: these are precisely the finitely cocomplete categories which have enough effective regular projectives: 4.6.Theorem.[14] Let C be a category with finite colimits.The following conditions are equivalent:

|
The set of equivalence relations on an object A ∈ A is denoted by Eq(A).For any relation (r 0 , r 1 ) : R A × B we can consider the opposite relation R o given by (r 1 , r 0 ) : R B × A. Given two relations (r 0 , r 1 ) : R A × B and (s 0 , s 1 ) : S B × C in a regular category A, the composite S • R is defined as the image of the morphism (r 0 • u 0 , s 1 • u 1 ) : R × B S → A × C: | y y y y y y y y u 1

2 .
C is weakly regular and the relation R X,C n is transitive for any reflexive graph X in C and C ∈ C. Proof.We first prove that R X,C n = Hom(C, (I, I o ) n ) for any C ∈ C and any reflexive graph X 1

1 G
Set] is an effective equivalence relation.Indeed, one can easily prove that the pro-maltsev property implies that the kernel pair in Lex[C op , Set] of the coequalizer of the arrows x 0 and x 1 factorizes throughI x 0 G G x G X 0 :this certainly suffices to conclude that I is a kernel pair in Lex[C op , Set].Now, for any equivalence relationA d G G c G G A 0 in Lex[C op , Set],write it as a filtered colimit of reflexive graphs in C: since the regular image of any of these reflexive graphs is an effective equivalence relation, we get that A d G G c G G A 0 is an effective equivalence relation.

4. 4 . 1 G 1 G 1 G
Proposition.Let C be a category with finite colimits and n ≥ 2. The following conditions are equivalent:1.Lex[C op , Set] is a n-permutable quasivariety2.C has enough n-permutable regular projectivesProof.1. ⇒ 2. By theorem 4.1 the category C has enough regular projectives.It suffices to prove that, for any regular projective P and for any reflexive graph X : X 1x 0 G G x G X 0 , the relation (J, J o ) n−1 Hom(P, X 0 ) × Hom(P, X 0 )is transitive (whereJ j 0 G G j G Hom(P, X 0 ) is the image factorisation in Set of the reflexive graph Hom(P, X 1 ) G G G G Hom(P, X 0 ) ).Since P is regular projective then (J, J o ) n−1 =Hom(P, (I, I o ) n−1 ), whereI x 0 G G x G X 0 is the image of X in Lex[C op , Set].The result then follows from theorem 2.5 and R X,P n−1 = Hom(P, (I, I o ) n−1 ).

1 . 1 .
Lex[C op , Set] is a variety 2. C has enough effective regular projectives This theorem, together with proposition 4.4, gives the following corollary: 4.7.Corollary.Let C be a category with finite colimits.The following conditions are equivalent: Lex[C op , Set] is a n-permutable variety 2. C has enough n-permutable effective regular projectives By theorem 2.2 we know that C is weakly regular.The relation (I, I o ) n is transitive in Lex[C op , Set] by assumption (and by theorem 1.5): this implies that Hom(C, (I, I o ) n ) = R X,C n is a transitive relation in the category of sets.2. ⇒ 1.By theorem 2.2, the category Lex[C op , Set] is regular.By theorem 1.5 we just need to show that any reflexive relation