A NOTE ON MAL’TSEV OBJECTS

. The aim of this work is to compare the distinct notions of Mal’tsev object in the sense of Weighill and in the sense of Montoli-Rodelo-Van der Linden.


Introduction
A variety of universal algebras V is called a Mal'tsev variety [21] when its theory admits a ternary operation p satisfying the equations ppx, y, yq " x and ppx, x, yq " y.
Such varieties were characterised in [21] by the fact that any pair of congruences R and S on a same algebra X is 2-permutable, i.e.RS " SR.It was shown in [19] that such varieties are also characterised by the fact that any homomorphic relation D from an algebra X to an algebra Z is difunctional: px 1 Dz 2 ^x2 Dz 2 ^x2 Dz 1 q ñ x 1 Dz 1 .
The notion of Mal'tsev variety was generalised to a categorical context in [8] (see also [7,9,3]).This was achieved by translating the above properties on homomorphic relations for a variety into similar properties on (internal) relations in a category.A regular category [2] C is called a Mal'tsev category when any pair of equivalence relations R and S in C on a same object X is such that RS " SR.Mal'tsev categories can also be characterised by the difunctionality of relations: Theorem 0.1.[8,9,7] A finitely complete category C is a Mal'tsev category if and only if any relation D Ñ X ˆZ in C is difunctional.
The main examples of Mal'tsev varieties are Grp of groups, Ab of abelian groups, R-Mod of modules over a commutative ring R, Rng of rings, and Heyt of Heyting algebras.More generally, any variety whose theory contains a group operation is a Mal'tsev variety; an example of a non-Mal'tsev variety is the variety Mon of monoids.As examples of Mal'tsev categories (that are not varietal) we have the category GrppTopq of topological groups, any abelian category or the dual of an elementary topos.Also, if C is a Mal'tsev category, then so are the (co)slice categories C{X and X{C, for any object X of C (see [3], for example).
There are also several other well-known characterisations of Mal'tsev varieties through nice properties on relations, such as the fact that every reflexive relation is necessarily a congruence.All of these characteristic properties have been generalised to the Mal'tsev categorical context (see [8,9,7,3,6]).This wide range of nice properties together with the long list of examples has contributed to the great amount of research developed on Mal'tsev varieties and categories over the past 70 years.They are also just the first instance of the family of n-permutable varieties [15], for n ě 2.An n-permutable variety is such that any pair of congruences R and S on a same algebra X is n-permutable, i.e. pR, Sq n " pS, Rq n , where pR, Sq " RSR ¨¨¨denotes the composite of R and S, n times; they have been generalised to n-permutable categories in [7].Concerning n-permutability, the reader may be interested in [18], to see how varietal proofs translate into categorical ones, and [24] for further properties on relations.
Weighill in [27] used the characterisation of a Mal'tsev category obtained through the difunctionality of all relations to introduce a definition of Mal'tsev object.To fully understand this definition, we must develop further on relations in C.
ÝÑ Z such that pr 1 , r 2 q is jointly monomorphic.We identify two relations X Ð R Ñ Z and X Ð R1 Ñ Z, when R factors through R 1 and vice-versa.When C admits binary products, The definition of a reflexive, symmetric, transitive and equivalence relation in C is obtained similarly.
Definition 0.2.[27] An object Y of C is called a W-Mal'tsev object 1 when for every relation X It follows from Theorem 0.1 that a finitely complete category C is a Mal'tsev category if and only all of its objects are W-Mal'tsev objects.
It is well-known that the category pSetq op is a Mal'tsev category [7], hence every set is a W-Mal'tsev object in pSetq op .Weighill's study of W-Mal'tsev objects led to the identification of interesting Mal'tsev subcategories of duals of categories of topological flavour.We will recall and generalise these results in Section 2.
Using a completely different approach, Bourn in [5] classified several categorical notions (including that of Mal'tsev category) through the fibration of points.A point pf : A Ñ B, s : B Ñ Aq in a category C is a split epimorphism f with a chosen splitting s.We may define the category of points in C, denoted by PtpCq: a morphism between points is a pair px, yq : pf, sq Ñ pf 1 , s 1 q of morphisms in C such that the following diagram commutes When C has pullbacks of split epimorphisms, the forgetful functor cod : PtpCq Ñ C, which associates with every split epimorphism its codomain, is a fibration called the fibration of points [4].
From the several classifying properties of the fibration of points cod studied in [5], we emphasize the following one.

Theorem 0.3. [5] A finitely complete category C is a Mal'tsev category if and only if cod is unital.
Unitality of cod means that the category Pt Y pCq of points in C over Y is a unital category, for every object Y of C. Recall from [5] that a pointed and finitely complete category is called a unital category when, for all objects A, C of C, the pair of morphisms px1 A , 0y : A Ñ AˆC, x0, 1 C y : C Ñ A ˆCq is jointly strongly epimorphic.The fact that Pt Y pCq is unital means: for every pullback of points over (which corresponds to a binary product in Pt Y pCq), the pair of morphisms px1 A , tf y, xsg, 1 C yq is jointly strongly epimorphic.
Since Pt 1 pCq -C, then every pointed Mal'tsev category C is necessarily unital.The converse is false.Indeed, Mon and SRng are examples of unital categories that are not Mal'tsev categories.
Inspired on the classification properties of the fibration of points cod studied in [5], the authors of [25] explored several algebraic categorical notions, such as those of (strongly) unital [5], subtractive [17], Mal'tsev [8,9,7] and protomodular categories [4], at an object-wise level.This led to the corresponding notions of (strongly) unital, subtractive, Mal'tsev and protomodular objects.This approach allows one to distinguish "good" objects, i.e. with stronger algebraic properties, in a setting with weaker algebraic properties.The goal of this work was to obtain a categorical-algebraic characterisation of groups amongst monoids and of rings amongst semirings.It was shown in [25]  The notions of W-Mal'tsev object, studied in [27], and that of Mal'tsev object, in the sense of [25], were obtained independently and developed with different goals in mind.Although their definitions are very different in nature, there is an obvious common property to both: a finitely complete category C is a Mal'tsev category if and only if all of its objects are W-Mal'tsev objects if and only if all of its objects are Mal'tsev objects.There are also several other properties shared by both notions, which we will recall in Section 1. Altogether, these observations led us to the natural question: To what extent are W-Mal'tsev and Mal'tsev objects comparable?
1. W-Mal'tsev object vs. Mal'tsev objects In this section we shall compare the notion of W-Mal'tsev object [27] and that of Mal'tsev object [25] in a base category C, which is finitely complete and may, eventually, admit some extra structure.We fix a finitely complete category C and denote by WpCq (resp.MpCq), the full subcategory of C determined by the W-Mal'tsev (resp.Mal'tsev) objects of C.
We begin by combining Theorems 0. The challenge of the comparison process is when the category C is not a Mal'tsev category, but admits (W-)Mal'tsev objects.Note that both definitions depend heavily on the surrounding category C. To have a Mal'tsev object one must check a property for all pullbacks of points over that object; to have a W-Mal'tsev object one must check a property concerning all relations in C.However, when C is a regular category [2] with binary coproducts, the result stated below in (W5) gives an independent (of all relations in C) way to check that an object Y is a W-Mal'tsev object.Consequently, one would expect that the notion of a Mal'tsev object is stronger than that of a W-Mal'tsev object.Indeed, that is the case as stated in Proposition 1.4.Additionally, the greater demand on Mal'tsev objects may lead to "trivial" cases.This happens when C " V -Cat (and V is not a cartesian quantale), where the only Mal'tsev object is the empty set H, while a W-Mal'tsev object is precisely a symmetric V ^-category (Theorem 2.4).See also Example 1.6 concerning (W-)Mal'tsev objects in OrdGrp, the category of preordered groups.In other contexts, such as Mon, the notions of W-Mal'tsev object and that of Mal'tsev object both coincide with a group (see Proposition 1.5 and Theorem 6.14 in [25]).
In what follows it will be useful to consider properties on relations in a category using generalised elements (see [7], for example).Let R be a relation from X to Z given by the subobject xr 1 , r 2 y : R Ñ X ˆZ.Let x : : A Ñ X and z : A Ñ Z be morphisms, which can be considered as generalised elements of X and Z, respectively.We write px, zq P A R when the morphism xx, zy factors through xr 1 , r 2 y Using this notation, an object Y is a W-Mal'tsev object in C when: for any relation xr 1 , r 2 y : R Ñ X ˆZ P C and morphisms (1.i) We extract from [27] the main results which are used in the comparison process.Note that, in [27] a Mal'tsev category need not be finitely complete by definition.Consequently, we adapted the result stated in (W4) to our finitely complete request.
(W1) WpCq " C if and only if C is a Mal'tsev category.(W2) WpCq is closed under colimits and (regular) quotients in C (Proposition 2.1 in [27]).(W3) When C is a well-powered regular category with coproducts, then WpCq is a coreflective subcategory of C; thus, WpCq is (finitely) complete whenever C is (Corollary 2.2 in [27]).(W4) Let C be a well-powered regular category with binary coproducts.Suppose that any relation in WpCq is also a relation in C. Then WpCq is the largest full subcategory of C which is a Mal'tsev category and which is closed under binary coproducts and quotients in C (Corollary 2.5 in [27]).(W5) Let C be a regular category with binary coproducts.An object Y is a W-Mal'tsev object in C if and only if, given the (regular epimorphism, monomorphism) factorisation in As usual, we write Y `Y " 2Y , Y `Y `Y " 3Y , and ι j : Y Ñ kY for the j-th coproduct coprojection (Proposition 2.3 in [27]).
The main results we wish to emphasize from [25] are the following ones.
(M1) MpCq " C if and only if C is a Mal'tsev category (Proposition 6.10 in [25]).(M2) If MpCq is closed under finite limits in C, then MpCq is a Mal'tsev category (Corollary 6.11 in [25]).(M3) When C is a pointed category, then C is a unital category if and only if 0 is a Mal'tsev object (Proposition 6.4 in [25]).(M4) When C is a regular category, then MpCq is closed under quotients in C (Proposition 6.12 in [25]).
(M5) When C is a regular category, an object Y in C is a Mal'tsev object if and only if every double split epimorphism over Y D (meaning that the four "obvious" squares commute) is a regular pushout, i.e. the comparison morphism xg 1 , f 1 y : D Ñ A ˆY C is a regular epimorphism.As a direct consequence of (M3), we have a trivial comparison result in a pointed context.Proposition 1.3.Let C be a pointed finitely complete category.Then the zero object is a W-Mal'tsev object, but it is not necessarily a Mal'tsev object.
Proof.The zero object 0 is always a W-Mal'tsev object in any pointed category.Indeed, given any relation R Ñ X ˆZ, the only generalised elements we can use are the zero morphisms 0 X : 0 Ñ X and 0 Z : 0 Ñ Z. So, the implication in (1.i) obviously holds when Y " 0. On the other hand, 0 is only a Mal'tsev object when C is a unital category by (M3).
When C is a regular category, then both WpCq and MpCq are closed under quotients in C -(M4) and (W2), where the later actually holds for any finitely complete C. If C also has binary coproducts, we may use (W5) to conclude that the notion of Mal'tsev object is stronger than that of W-Mal'tsev object.

Proof. Consider the double split epimorphism over
which is a regular pushout by assumption.We then get a (regular epimorphism, monomorphism) factorisation as in (1.ii), where R " Eqp∇q is the kernel pair of ∇.It easily follows that pι 1 , ι 1 q P Y R, since R " Eqp∇q and ∇ι 1 " ∇ι 1 .This shows that Y is a W-Mal'tsev object by (W5).
An interesting question is whether WpCq and MpCq are themselves Mal'tsev categories.From (W3) we know that, if C is a well-powered regular category with coproducts, then WpCq is finitely complete.Also, if every relation in WpCq is also a relation in C, then the full subcategory WpCq is a Mal'tsev category; actually, it is the largest one which is closed under binary coproducts and quotients in C (see (W4)).On the other hand, there are no "obvious" conditions on C from which we could deduce finite completeness for MpCq.So, we can only conclude that MpCq is a Mal'tsev category when it is closed under finite limits in C; this is (M2).When this is the case, we still do not know whether MpCq is the largest full subcategory of C which is a Mal'tsev category.
In [25] is was shown that the Mal'tsev objects in Mon are precisely the groups (Theorem 6.14).This is also the case with respect to W-Mal'tsev objects in Mon.It follows from the above and the next proposition that Grp is the largest full subcategory of Mon which is a Mal'tsev category and is closed under binary coproducts and quotients in Mon.
Proof.If Y is a group, then it is a Mal'tsev object in Mon; thus, it is a W-Mal'tsev object in Mon by Proposition 1.4.For the converse, suppose that pY, `, 0q is a W-Mal'tsev object in Mon.We use additive notation although Y is not necessarily an abelian monoid.Since Mon is a regular category with binary coproducts, we can apply (W5) to conclude that pι 1 , ι 1 q P Y R, where R is as in diagram (1.ii).So, for any x P Y , x ‰ 0, we have prxs, rxsq P R. (We use the notations ι 1 pxq " r x s, ι 2 pxq " r x s, for any x P Y , for the coprojections ι 1 , ι 2 : Y Ñ 2Y , and ι 1 pxq " r x s, ι 2 pxq " r x s, ι 3 pxq " rr xs, for any x P Y , for the coprojections ι 1 , ι 2 , ι 3 : Y Ñ 3Y .)Since e : 3Y Ñ R is surjective, there exists an element r u consequently " v 1 `w1 " 0, ¨¨¨, v k `wk " 0, and u 1 `¨¨¨`u k " x u 1 `v1 " 0, ¨¨¨, u k `vk " 0, and w 1 `¨¨¨`w k " x.
We may define the element y " v k `¨¨¨`v 1 of Y which is the inverse of x: x `y " u 1 `¨¨¨`u k´1 `pu k `vk q `vk´1 ¨¨¨`v 1 " u 1 `¨¨¨`pu k´1 `vk´1 q ¨¨¨`v 1 " ¨¨¨" 0, y `x " v k `¨¨¨`v 2 `pv 1 `w1 q `w2 `¨¨¨`w k " v k `¨¨¨`pv 2 `w2 q `¨¨¨`w k " ¨¨¨" 0.
We finish this section with the example of (W-)Mal'tsev objects in OrdGrp, the category of preordered groups.We denote a preordered group by pY, `, ďq, even though the associated group pY, `, 0q is not necessarily abelian.Recall from [13] that the positive cone of Y , P Y " tx P Y : 0 ď xu, is always a submonoid of Y which is closed under conjugation in Y .It was shown in [13] that a preordered group pY, `, ďq is a Mal'tsev object if and only if the preorder relation ď is an equivalence relation if and only if P Y is a group.We follow a similar argument as that of the proof of Proposition 1.5 to analyse, in the next example, properties of the positive cone of a W-Mal'tsev object in OrdGrp.
Example 1.6.Let pY, `, ďq be a W-Mal'tsev object in OrdGrp.Since OrdGrp is a regular category with binary coproducts, we can apply (W5) to conclude that pι 1 , ι 1 q P Y R, where R is as in diagram (1.ii).So, for any x P P Y , x ‰ 0, we have prxs, rxsq P P R .Since e is a regular epimorphism, there exists a positive element r u consequently " v 1 `w1 " 0, ¨¨¨, v k `wk " 0, and u 1 `¨¨¨`u k " x u 1 `v1 " 0, ¨¨¨, u k `vk " 0, and w 1 `¨¨¨`w k " x.
We conclude that u 1 " w 1 , ¨¨¨, u k " w k and the above positive element of 3Y has the shape takes positive elements of 3Y to positive elements of Y ; thus ´u1 ´¨¨¨´u k P P Y .
If Y is an abelian group, then from x " u 1 `¨¨¨`u k P P Y , we deduce ´x " ´u1 ´¨¨¨´u k P P Y , i.e.P Y is a group.In this case W-Mal'tsev objects and Mal'tsev objects coincide in OrdGrp.We do not know whether the two notions coincide for the non-abelian case.

V -categories and (W-)Mal'tsev objects
In this section we will generalise the characterisations of W-Mal'tsev objects in the duals of the category of metric spaces and of the category of topological spaces obtained in [27].For that we will make use of the concepts of V -category and of pU, V q-category.Here, as in [20], the notion of V -category will play the role of metric space, while a pU, V q-category (as introduced in [14]) will play the role of a topological space.We start by presenting the basic tools of this approach.
Throughout V is a unital and integral quantale; that is, V is a complete lattice equipped with a tensor product b, with unit k " J ‰ K, that distributes over arbitrary joins.As a category, V is a monoidal closed category.
When more than one tensor product may be considered, we use the notation V b to indicate that we are using the tensor product b in V .Definition 2.1.A V-category is a set X together with a map X ˆX Ñ V , whose image of px, x 1 q we denote by Xpx, x 1 q, such that, for each x, x 1 , x 2 P X, (R) k ď Xpx, xq; (T) Xpx, x 1 q b Xpx 1 , x 2 q ď Xpx, x 2 q.A V -category is said to be symmetric if, for every x, x 1 P X, Xpx, x 1 q " Xpx 1 , xq.A V -functor f : X Ñ Y is a map such that, for all x, x 1 P X, Xpx, x 1 q ď Y pf pxq, f px 1 qq.
The two axioms of a V -category express the existence of identities and the categorical composition law, but they may also be seen as a reflexivity and a transitivity condition, or even as two conditions usually imposed to metric structures: if in the complete half real line r0, 8s we consider the order relation ě and the tensor product `, (R) and (T) read as (R') 0 ě Xpx, xq; (T') Xpx, x 1 q `Xpx 1 , x 2 q ě Xpx, x 2 q, with (R) meaning that the distance from a point to itself is 0, and (T) the usual triangular inequality.
We denote by V -Cat the category of V -categories and V -functors, and by V -Cat sym its full subcategory of symmetric V -categories.Remark 2.2.When V is a complete lattice which is a frame, so that finite meets distribute over arbitrary joins, then V ^" pV, ď, ^, Jq is a (unital and integral) quantale.We call such quantales cartesian.It is well-known that, if V ^is a cartesian quantale, then the category V ^-Cat has special features, like being cartesian closed [20].Here we will show that it also has a key role in the study of W-Mal'tsev objects of pV ^-Catq op .
Moreover, even if the quantale V b is not a frame, the V b -categories X which, in addition, verify Xpx, x 1 q ^Xpx 1 , x 2 q ď Xpx, x 2 q will be specially relevant, and we will call them also V ^-categories.We note that, since k " J, they also satisfy J ď Xpx, xq for every x P X; moreover, u b v ď u ^v always holds, so the above condition guarantees immediately condition (T) of the above definition.
(1) When V " 2 " pt0 ă 1u, ^, 1q, a V -category is a preordered set and a V -functor is a monotone map.Hence V -Cat is the well-known category Ord of preordered sets and monotone maps.
(2) When V " r0, 8s `" pr0, 8s, ě, `, 0q, that is V is the complete lattice r0, 8s, ordered by ě, with tensor product b " `, a V -category is a (generalised) Lawvere metric space [20] (not necessarily separated nor symmetric, with 8 as a possible distance), since the two conditions above mean that Xpx, xq " 0 and Xpx, yq `Xpy, zq ě Xpx, zq, for x, y, z P X, and a V -functor is a non-expansive map.When V is the cartesian quantale r0, 8s max " pr0, 8s, ě, max, 0q, with tensor product b " max, then V max -Cat is the category of (generalised) ultrametric spaces.(3) The complete lattice pr0, 1s, ďq can be equipped with several tensor products -usually called t-norms -including the Lukasiewicz sum, which lead to interesting instances of categories of the form V -Cat, like Lawvere metric spaces, ultrametric spaces, and bounded metric spaces.
The forgetful functor V -Cat Ñ Set is topological, hence V -Cat is complete and cocomplete, with limits and colimits formed as in Set and equipped with the corresponding initial and final V -category structures, respectively.In [22,Corollary 8] it is shown that, under suitable conditions, V -Cat is an extensive category.In V -Cat epimorphisms are pullback-stable, since they are exactly surjective V -functors and pullbacks are formed as in Set.Therefore, as shown in [23, Proposition 3], pV -Catq op is a weakly Mal'tsev category.In addition, in [16,Theorem 4.6] it is shown that pV -Catq op is a quasi-variety, so in particular it is a regular category.Still, we think it is worth to prove here directly that pV -Catq op is a regular category.Indeed, in V -Cat regular monomorphisms coincide with extremal monomorphisms, and are exactly the injective maps f : X Ñ Y such that Xpx, x 1 q " Y pf pxq, f px 1 qq, for all x, x 1 P X.Moreover, V -Cat has the stable orthogonal factorisation system (epimorphism, regular monomorphism), which factors every V -functor f : X Ñ Y as where e is the corestriction of f to Z " f pXq and Zpy, y 1 q " Y py, y 1 q for all y, y 1 P Z.Given a pushout in V -Cat with m a regular monomorphism, for simplicity we assume that m is an inclusion.We know that, since pushouts are formed as in Set, we may consider W " pY `Zq{ ", where, for y P Y and z P Z, y " z exactly when z P X and f pzq " y, and n an inclusion.Then, for every y, y 1 P Y , W prys, ry 1 sq " Y py, y 1 q _ ł x"y, x 1 "y 1 Zpx, x 1 q " Y py, y 1 q _ ł x"y, x 1 "y 1 Xpx, x 1 q (because m is a regular monomorphism) and therefore the inclusion n : Y Ñ W is a regular monomorphism.Hence, since pV -Catq op is a regular category with binary coproducts we know by (W5) that a V -category Y is a W-Mal'tsev object if, and only if, given the where π i : Y 2 Ñ Y , i " 1, 2, are the product projections, and its (epimorphism, regular monomorphism)-factorisation in V -Cat there is a unique V -functor g : X Ñ Y such that the following diagram commutes (2.ii) As a set, X is the image of the map f , that is X " X 1 Y X 2 , with X 1 " tpx, y, yq ; x, y P Y u, X 2 " tpx, x, yq ; x, y P Y u; its V -category structure is inherited from Y 3 , that is Xppx, y, zq, px 1 , y 1 , z 1 qq " Y px, x 1 q ^Y py, y 1 q ^Y pz, z 1 q.
The map g : X Ñ Y must assign x both to each px, y, yq and each py, y, xq.(iii) ñ (i): We want to prove that g is a V -functor; that is, for each px, y, zq, px 1 , y 1 , z 1 q in X, Xppx, y, zq, px 1 , y 1 , z 1 qq " Y px, x 1 q ^Y py, y 1 q ^Y pz, z 1 q ď Y pgpx, y, zq, px 1 , y 1 , z 1 qq.
In case x " y and x 1 " y 1 , or in case y " z and y 1 " z 1 , the inequality is trivially satisfied; when x " y and y 1 " z 1 , using (iii), one obtains Y px, x 1 q ^Y px, y 1 q ^Y pz, y 1 q " Y pz, y 1 q ^Y py 1 , xq ^Y px, x 1 q ď Y pz, x 1 q, and in case y " z and x 1 " y 1 one gets Y px, x 1 q ^Y py, x 1 q ^Y py, z 1 q " Y px, x 1 q ^Y px 1 , yq ^Y py, z 1 q ď Y px, z 1 q.
(2) Since pV -Catq op is regular we may use the characterisation of Mal'tsev object of (M5).First we point out that a double split epimorphism (1.iii) in pV -Catq op is also a double split epimorphism in V -Cat, with the role of split epimorphisms and split monomorphisms interchanged: (this diagram is just the dual of (1.iii)).
Let V be a non-cartesian quantale.To show that the only Mal'tsev object in pV -Catq op is the empty set it is enough to check that Y " t1u, with Y p1, 1q " k, is not a Mal'tsev object, since by (M4) Mal'tsev objects are closed under quotients and, for any non-empty V -category Z, any map Y Ñ Z is a split monomorphism in V -Cat.
We will build a double split epimorphism in V -Cat as above so that it is not a regular pullback, i.e. the comparison morphism xg 1 , f 1 y : A `Y C Ñ D is not a regular monomorphism.If V is not cartesian then there exist u, v P V such that u b v ă u ^v.Consider in (2.iii) the symmetric V -categories A " t0, 1u, C " t1, 2u, and D " t0, 1, 2u, with Ap0, 1q " u, Cp1, 2q " v, and Dp0, 2q " u ^v, with the obvious V -functors (with s 1 p0q " 1 and t 1 p2q " 1).Then (2.iii) is a double split epimorphism but the comparison morphism xg 1 , f 1 y : A `Y C Ñ D, which is in fact a bijection, is not a regular monomorphism because in the pushout pA `Y Cqp0, 2q must be u b v. Remark 2.5.We point out that condition (ii) in case b " ^is the separation axiom (R1) studied in [10, Section 2.2].
The two statements of Theorem 2.4 have an immediate consequence, showing how drastically different may be the two notions of Mal'tsev object in the context of pV -Catq op .For instance, if V " r0, 8s `, then a Lawvere metric space is a W-Mal'tsev object exactly when it is a symmetric ultrametric space, while it is a Mal'tsev object only if it is empty.
(1) If V is a cartesian quantale, then pV -Cat sym q op is a Mal'tsev category.(2) For every unital and integral quantale V , the largest Mal'tsev subcategory of pV -Catq op closed under binary coproducts and regular epimorphisms is pV ^-Cat sym q op .
It is straightforward to check that, if we restrict our study to symmetric V b -categories, in case the tensor product b is commutative the double split epimorphism built above also works.Hence we may also conclude that Mal'tsev objects trivialise is this more restrictive setting.
Corollary 2.7.If V is a commutative unital and integral non-cartesian quantale, then the only Mal'tsev object in pV -Cat sym q op is the empty V -category.
In order to generalise Weighill's result for topological spaces we need to introduce an extra ingredient, the ultrafilter monad on Set, and its extension to V -Rel (see [14] for details).As Barr showed in [1], a topological space can be described via its ultrafilter convergence, meaning that a topological space can be given by a set X, together with a relation between ultrafilters on X and points of X, U XÝÑ Þ X such that, for every x P X, x P U X, and X P U 2 X, (1) 9 x Ñ x (where 9 x is the principal ultrafilter defined by x); (2) X Ñ x and x Ñ x ùñ µpXq Ñ x (where µpXq is the Kowalski sum of X).Given topological spaces X and Y , a map f : X Ñ Y is continuous if, for all x P U X and x P X, U f pxq Ñ f pxq whenever x Ñ x.
This approach can be generalised making use of the ultrafilter monad U , a unital and integral quantale V , and a lax extension of U to V -Rel.Definition 2.8.A pU,Vq-category is a set X together with a map a : U X ˆX Ñ V whose image of px, xq we denote by Xpx, xq, such that, for each x P X, x P U X, X P U 2 X, (R) k ď Xp 9 x, xq; (T) XpX, xq b Xpx, xq ď XpµpXq, xq, where by XpX, xq we mean the image, under U , of the V -relation a : U XÝÑ Þ X.A pU, V q-functor f : X Ñ Y is a map such that Xpx, xq ď Y pU f pxq, f pxqq, for all x P X, x P U X.
We denote by pU, V q-Cat the category of pU, V q-categories and pU, V q-functors.
(1) As shown by Barr [1], if V " 2, then pU, V q-Cat is isomorphic to the category Top of topological spaces and continuous maps.
(2) As shown in [11], if V " r0, 8s `pU, V q-Cat is isomorphic to the category App of Lowen's approach spaces and non-expansive maps.
Proof.As in V -Cat, in pU, V q-Cat (epimorphisms, regular monomorphisms) form a stable factorisation system, and a morphism m : X Ñ Z is a regular monomorphism if, and only if, it is an injective map and Xpx, xq " ZpU mpxq, mpxqq, for all x P U X and x P X.Given a pushout in pU, V q-Cat with m a regular monomorphism, which we assume, for simplicity, to be an inclusion, as for V -Cat we may consider W " pY `Zq{ ", where, for y P Y and z P Z, y " z exactly when z P X and f pzq " y.Then n, and therefore also U n, is an injective map, and, for every y P U Y and y P Y , W pU npyq, rysq " Y py, yq _ ł U gpzq"U npyq, x"y Zpz, xq.
Any z P U Z with U gpzq " U npyq must contain X, and consequently it is the image under U m of an ultrafilter x in X; hence Zpz, xq " Xpx, xq ď Y pU f pxq, f pxqq " Y py, yq.This implies that Y py, yq " W pU npyq, rysq, and therefore n : Y Ñ W is a regular monomorphism.
(1) For a pU, V q-category Y the following conditions are equivalent: (i) Y is a W-Mal'tsev object in ppU, V q-Catq op ; (ii) For all z P U Y , x, y P Y , (a) Y pz, xq ^Y pz, yq ď Y p 9 x, yq; (b) Y pz, xq ^Y p 9 x, yq ď Y pz, yq.(2) If V is not a cartesian quantale, then Y is Mal'tsev object in ppU, V q-Catq op if, and only if, Y " H. Proof.
For simplicity, we will denote the i-th projection Y k Ñ Y (k " 2, 3) by π i and U π i pwq by w i .Let h : Y ˆY Ñ X with hpx, yq " hpx, y, yq.
(i) ñ (ii): Fix z P U Y and x P Y .Since the canonical map U pY 2 q Ñ U pY q 2 is surjective, there exists r z P U pY 2 q such that r z 1 " 9 x and r z 2 " z.Then w " U hpr zq P U X and, since g ¨h " π 1 , w 1 " 9 x, w 2 " z " w 3 , and U gpwq " 9 x. Then we have Xpw, px, x, yqq ď Y p 9 x, yq ñ Y p 9 x, xq ^Y pz, xq ^Y pz, yq ď Y p 9 x, yq, that is (a) holds.Analogously, let p z P U pY 2 q be such that p z 1 " z and p z 2 " 9 x, and let x " U hpp zq, so that U gpxq " z.Then, for every y P Y , Xpx, px, x, yqq ď Y pz, yq ñ Y pz, xq ^Y p 9 x, xq ^Y p 9 x, yq ď Y pz, yq, that is (b) holds.

Proposition 1 . 4 .
Let C be a regular category with binary coproducts.If Y is a Mal'tsev object in C, then it is a W-Mal'tsev object in C.
that a monoid Y is a group if and only if Y is a Mal'tsev object in Mon if and only if Y is a protomodular object in Mon.An object Y of a finitely complete category C is called a Mal'tsev object if the category Pt Y pCq of points over Y is a unital category.

for any pullback of points over an arbitrary object
Y in C as in (0.ii), the pair of morphisms px1 A , tf y, xsg, 1 C yq is jointly strongly epimorphic.An immediate consequence of Definitions 0.2 and 0.4 is the following one (see (W1) and (M1) below).Proposition 1.2.Let C be a finitely complete category.The following statements are equivalent:(i) C is a Mal'tsev category;(ii) all objects of C are W-Mal'tsev objects;(iii) all objects of C are Mal'tsev objects.