# A note on Mal’tsev objects

Maria Manuel Clementino  CMUC, Department of Mathematics, University of
Coimbra, 3000-143 Coimbra, Portugal<EMAIL_ADDRESS>and  Diana Rodelo
Department de Mathematics, University of the Algarve, 8005-139 Faro, Portugal
and CMUC, Department of Mathematics, University of Coimbra, 3000-143 Coimbra,
Portugal<EMAIL_ADDRESS>
###### Abstract.

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.

###### Key words and phrases:

Unital categories, Mal’tsev categories, Mal’tsev objects, $V$-categories

###### 1991 Mathematics Subject Classification:

18E13, 20J15, 08C05, 54D10, 54E35

The authors acknowledge partial financial assistance by the Centre for
Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the
Portuguese Government through FCT/MCTES

## Introduction

A variety of universal algebras $\mathbb{V}$ is called a _Mal’tsev variety_
[20] when its theory admits a ternary operation $p$ satisfying the equations

$p(x,y,y)=x\;\;\mathrm{and}\;\;p(x,x,y)=y.$

Such varieties were characterised in [20] 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 [18] that such varieties are also characterised by
the fact that any homomorphic relation $D$ from an algebra $X$ to an algebra
$Z$ is _difunctional_ : $(x_{1}Dz_{2}\wedge x_{2}Dz_{2}\wedge
x_{2}Dz_{1})\Rightarrow x_{1}Dz_{1}$.

The notion of Mal’tsev variety was generalised to a categorical context in [7]
(see also [6, 8, 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] $\mathbb{C}$ is called a
_Mal’tsev category_ when any pair of equivalence relations $R$ and $S$ in
$\mathbb{C}$ 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.

[7, 8, 6] A finitely complete category $\mathbb{C}$ is a Mal’tsev category if
and only if any relation $D\to X\times Z$ in $\mathbb{C}$ is difunctional.

The main examples of Mal’tsev varieties are $\mathsf{Grp}$ of groups,
$\mathsf{Ab}$ of abelian groups, $R$-$\mathsf{Mod}$ of modules over a
commutative ring $R$, $\mathsf{Rng}$ of rings, and $\mathsf{Heyt}$ 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
$\mathsf{Mon}$ of monoids. As examples of Mal’tsev categories (that are not
varietal) we have the category $\mathsf{Grp}(\mathsf{Top})$ of topological
groups, any abelian category or the dual of an elementary topos. Also, if
$\mathbb{C}$ is a Mal’tsev category, then so are the (co)slice categories
$\mathbb{C}/X$ and $X/\mathbb{C}$, for any object $X$ of $\mathbb{C}$ (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,
again, [7, 8, 6, 3]). 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
[14], for $n\geq 2$. An $n$-permutable variety is such that any pair of
congruences $R$ and $S$ on a same algebra $X$ is $n$-permutable, i.e.
$(R,S)_{n}=(S,R)_{n}$, where $(R,S)=RSR\cdots$ denotes the composite of $R$
and $S$, $n$ times; they have been generalised to $n$-permutable categories in
[6]. Concerning $n$-permutability, the reader may be interested in [17], to
see how varietal proofs translate into categorical ones, and [23] for further
properties on relations.

Weighill in [26] 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 $\mathbb{C}$.

A (binary) _relation_ $R$ in a category $\mathbb{C}$ from an object $X$ to an
object $Z$ is a span $X\stackrel{{\scriptstyle
r_{1}}}{{\longleftarrow}}R\stackrel{{\scriptstyle r_{2}}}{{\longrightarrow}}Z$
such that $(r_{1},r_{2})$ is jointly monomorphic. We identify two relations
$X\leftarrow R\rightarrow Z$ and $X\leftarrow R^{\prime}\rightarrow Z$, when
$R$ factors through $R^{\prime}$ and vice-versa. When $\mathbb{C}$ admits
binary products, a relation $R$ from $X$ to $Z$ is a subobject of $X\times Z$,
denoted $\langle r_{1},r_{2}\rangle\colon R\to X\times Z$. A relation
$X\stackrel{{\scriptstyle r_{1}}}{{\longleftarrow}}R\stackrel{{\scriptstyle
r_{2}}}{{\longrightarrow}}Z$ in $\mathbb{C}$ is difunctional when the relation

$\textstyle{\mathbb{C}(Y,X)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{C}(Y,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{C}(Y,r_{1})}$$\scriptstyle{\mathbb{C}(Y,r_{2})}$$\textstyle{\mathbb{C}(Y,Z)}$
(0.i)

in $\mathsf{Set}$ is difunctional, for every object $Y$ of $\mathbb{C}$. The
definition of a reflexive, symmetric, transitive and equivalence relation in
$\mathbb{C}$ is obtained similarly.

######  Definition 0.2.

[26] An object $Y$ of $\mathbb{C}$ is called a W-Mal’tsev object111We added
the prefix “W-” to distinguish these from the Mal’tsev objects of Definition
0.4 when for every relation $X\stackrel{{\scriptstyle
r_{1}}}{{\longleftarrow}}R\stackrel{{\scriptstyle r_{2}}}{{\longrightarrow}}Z$
in $\mathbb{C}$, the $\mathsf{Set}$-relation (0.i) is difunctional.

It follows from Theorem 0.1 that a finitely complete category $\mathbb{C}$ is
a Mal’tsev category if and only all of its objects are W-Mal’tsev objects.

It is well-known that the category $(\mathsf{Set})^{\rm op}$ is a Mal’tsev
category [6], hence every set is a W-Mal’tsev object in $(\mathsf{Set})^{\rm
op}$. 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_ $(f\colon A\to B,s\colon B\to A)$ in a category
$\mathbb{C}$ is a split epimorphism $f$ with a chosen splitting $s$. We may
define the category of points in $\mathbb{C}$, denoted by
$\mathsf{Pt}(\mathbb{C})$: a morphism between points is a pair
$(x,y)\colon(f,s)\to(f^{\prime},s^{\prime})$ of morphisms in $\mathbb{C}$ such
that the following diagram commutes

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{y}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{x}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{B^{\prime}.}$

When $\mathbb{C}$ has pullbacks of split epimorphisms, the forgetful functor
$\mathsf{cod}\colon{\mathsf{Pt}(\mathbb{C})\to\mathbb{C}}$, 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
$\mathsf{cod}$ studied in [5], we emphasize the following one.

######  Theorem 0.3.

[5] A finitely complete category $\mathbb{C}$ is a Mal’tsev category if and
only if $\mathsf{cod}$ is unital.

Unitality of $\mathsf{cod}$ means that the category
$\mathsf{Pt}_{Y}(\mathbb{C})$ of points in $\mathbb{C}$ over $Y$ is a unital
category, for every object $Y$ of $\mathbb{C}$. Recall from [5] that a pointed
and finitely complete category is called a _unital category_ when the pair of
morphisms $(\langle 1_{A},0\rangle\colon A\to A\times C,\;\langle
0,1_{C}\rangle\colon C\to A\times C)$ is jointly strongly epimorphic, for all
objects $A,C$ of $\mathbb{C}$. The fact that $\mathsf{Pt}_{Y}(\mathbb{C})$ is
unital means: for every pullback of points over $Y$

$\textstyle{A\times_{Y}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\scriptstyle{\pi_{C}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\langle
sg,1_{C}\rangle}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
1_{A},tf\rangle}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{t}$
(0.ii)

(which corresponds to a binary product in $\mathsf{Pt}_{Y}(\mathbb{C})$), the
pair of morphisms $(\langle 1_{A},tf\rangle$, $\langle sg,1_{C}\rangle)$ is
jointly strongly epimorphic.

Since $\mathsf{Pt}_{1}(\mathbb{C})\cong\mathbb{C}$, then every pointed
Mal’tsev category $\mathbb{C}$ is necessarily unital. The converse is false.
Indeed, $\mathsf{Mon}$ and $\mathsf{SRng}$ are examples of unital categories
that are not Mal’tsev categories.

Inspired on the classification properties of the fibration of points
$\mathsf{cod}$ studied in [5], the authors of [24] explored several algebraic
categorical notions, such as those of (strongly) unital [5], subtractive [16],
Mal’tsev [7, 8, 6] 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 [24] that a monoid $Y$ is a group if and
only if $Y$ is a _Mal’tsev object_ in $\mathsf{Mon}$ if and only if $Y$ is a
_protomodular object_ in $\mathsf{Mon}$.

######  Definition 0.4.

An object $Y$ of a finitely complete category $\mathbb{C}$ is called a
Mal’tsev object if the category $\mathsf{Pt}_{Y}(\mathbb{C})$ of points over
$Y$ is a unital category.

It follows from Theorem 0.3 that a finitely complete category $\mathbb{C}$ is
a Mal’tsev category if and only all of its objects are Mal’tsev objects.

The notions of W-Mal’tsev object, studied in [26], and that of Mal’tsev
object, in the sense of [24], 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 $\mathbb{C}$ 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 [26] and that
of Mal’tsev object [24] in a base category $\mathbb{C}$, which is finitely
complete and may, eventually, admit some extra structure. We fix a finitely
complete category $\mathbb{C}$ and denote by $\mathsf{W}(\mathbb{C})$ (resp.
$\mathsf{M}(\mathbb{C})$), the full subcategory of $\mathbb{C}$ determined by
the W-Mal’tsev (resp. Mal’tsev) objects of $\mathbb{C}$.

We begin by combining Theorems 0.1 and 0.3 into one:

######  Theorem 1.1.

Let $\mathbb{C}$ be a finitely complete category. The following statements are
equivalent:

  * (i)

$\mathbb{C}$ is a Mal’tsev category;

  * (ii)

diagram (0.i) is a difunctional relation in $\mathsf{Set}$, for any object $Y$
in $\mathbb{C}$ and any relation $X\stackrel{{\scriptstyle
r_{1}}}{{\longleftarrow}}R\stackrel{{\scriptstyle r_{2}}}{{\longrightarrow}}Z$
in $\mathbb{C}$;

  * (iii)

for any pullback of points over an arbitrary object $Y$ in $\mathbb{C}$ as in
(0.ii), the pair of morphisms $(\langle 1_{A},tf\rangle$, $\langle
sg,1_{C}\rangle)$ 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 $\mathbb{C}$ be a finitely complete category. The following statements are
equivalent:

  * (i)

$\mathbb{C}$ is a Mal’tsev category;

  * (ii)

all objects of $\mathbb{C}$ are W-Mal’tsev objects;

  * (iii)

all objects of $\mathbb{C}$ are Mal’tsev objects.

The challenge of the comparison process is when the category $\mathbb{C}$ is
not a Mal’tsev category, but admits (W-)Mal’tsev objects. Note that both
definitions depend heavily on the surrounding category $\mathbb{C}$. 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 $\mathbb{C}$. However, when $\mathbb{C}$ is a regular
category [2] with binary coproducts, the result stated below in (W5) gives an
independent (of all relations in $\mathbb{C}$) 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
$\mathbb{C}=V\mbox{-}\mathsf{Cat}$ (and $V$ is not a cartesian quantale),
where the only Mal’tsev object is the empty set $\emptyset$, while a
W-Mal’tsev object is precisely a symmetric $V_{\wedge}$-category (Theorem
2.4). See also Example 1.6 concerning (W-)Mal’tsev objects in
$\mathsf{OrdGrp}$, the category of preordered groups. In other contexts, such
as $\mathsf{Mon}$, 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
[24]).

In what follows it will be useful to consider properties on relations in a
category using generalised elements (see [6], for example). Let $R$ be a
relation from $X$ to $Z$ given by the subobject $\langle
r_{1},r_{2}\rangle\colon R\to X\times Z$. Let $x\colon\colon A\to X$ and
$z\colon A\to Z$ be morphisms, which can be considered as _generalised
elements_ of $X$ and $Z$, respectively. We write $(x,z)\in_{A}R$ when the
morphism $\langle x,z\rangle$ factors through $\langle r_{1},r_{2}\rangle$

---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
x,z\rangle}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
r_{1},r_{2}\rangle}$$\textstyle{X\times Z.}$

Using this notation, an object $Y$ is a W-Mal’tsev object in $\mathbb{C}$
when: for any relation $\langle r_{1},r_{2}\rangle\colon R\to X\times
Z\in\mathbb{C}$ and morphisms $x_{1},x_{2}\colon Y\to X$ and
$z_{1},z_{2}\colon Y\to Z$,

$\left(\,(x_{1},z_{2})\in_{Y}R,(x_{2},z_{2})\in_{Y}R,(x_{2},z_{1})\in_{Y}R\,\right)\Rightarrow(x_{1},z_{1})\in_{Y}R.$
(1.i)

We extract from [26] the main results which are used in the comparison
process. Note that, in [26] 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)

$\mathsf{W}(\mathbb{C})=\mathbb{C}$ if and only if $\mathbb{C}$ is a Mal’tsev
category.

  * (W2)

$\mathsf{W}(\mathbb{C})$ is closed under colimits and (regular) quotients in
$\mathbb{C}$ (Proposition 2.1 in [26]).

  * (W3)

When $\mathbb{C}$ is a well-powered regular category with coproducts, then
$\mathsf{W}(\mathbb{C})$ is a coreflective subcategory of $\mathbb{C}$; thus,
$\mathsf{W}(\mathbb{C})$ is (finitely) complete whenever $\mathbb{C}$ is
(Corollary 2.2 in [26]).

  * (W4)

Let $\mathbb{C}$ be a well-powered regular category with binary coproducts.
Suppose that any relation in $\mathsf{W}(\mathbb{C})$ is also a relation in
$\mathbb{C}$. Then $\mathsf{W}(\mathbb{C})$ is the largest full subcategory of
$\mathbb{C}$ which is a Mal’tsev category and which is closed under binary
coproducts and quotients in $\mathbb{C}$ (Corollary 2.5 in [26]).

  * (W5)

Let $\mathbb{C}$ be a regular category with binary coproducts. An object $Y$
is a W-Mal’tsev object in $\mathbb{C}$ if and only if, given the (regular
epimorphism, monomorphism) factorisation in $\mathbb{C}$

---
$\textstyle{3Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{\left(\begin{smallmatrix}\iota_{1}&\iota_{2}\\\
\iota_{2}&\iota_{2}\\\
\iota_{2}&\iota_{1}\end{smallmatrix}\right)}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
r_{1},r_{2}\rangle}$$\textstyle{2Y\times 2Y}$ (1.ii)

(which guarantees that $(\iota_{1},\iota_{2})\in_{Y}R$,
$(\iota_{2},\iota_{2})\in_{Y}R$, $(\iota_{2},\iota_{1})\in_{Y}R$), we have
$(\iota_{1},\iota_{1})\in_{Y}R$. As usual, we write $Y+Y=2Y$, $Y+Y+Y=3Y$, and
$\iota_{j}\colon Y\to kY$ for the $j$-th coproduct coprojection (Proposition
2.3 in [26]).

The main results we wish to emphasize from [24] are the following ones.

  * (M1)

$\mathsf{M}(\mathbb{C})=\mathbb{C}$ if and only if $\mathbb{C}$ is a Mal’tsev
category (Proposition 6.10 in [24]).

  * (M2)

If $\mathsf{M}(\mathbb{C})$ is closed under finite limits in $\mathbb{C}$,
then $\mathsf{M}(\mathbb{C})$ is a Mal’tsev category (Corollary 6.11 in [24]).

  * (M3)

When $\mathbb{C}$ is a pointed category, then $\mathbb{C}$ is a unital
category if and only if $0$ is a Mal’tsev object (Proposition 6.4 in [24]).

  * (M4)

When $\mathbb{C}$ is a regular category, then $\mathsf{M}(\mathbb{C})$ is
closed under quotients in $\mathbb{C}$ (Proposition 6.12 in [24]).

  * (M5)

When $\mathbb{C}$ is a regular category, an object $Y$ in $\mathbb{C}$ is a
Mal’tsev object if and only if every double split epimorphism over $Y$

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{s^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{t}$
(1.iii)

(meaning that the four “obvious” squares commute) is a _regular pushout_ ,
i.e. the comparison morphism $\langle g^{\prime},f^{\prime}\rangle\colon D\to
A\times_{Y}C$ is a regular epimorphism.

As a direct consequence of (M3), we have a trivial comparison result in a
pointed context.

######  Proposition 1.3.

Let $\mathbb{C}$ 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 $R\to X\times Z$, the only generalised elements we
can use are the zero morphisms $0_{X}\colon 0\to X$ and $0_{Z}\colon 0\to Z$.
So, the implication in (1.i) obviously holds when $Y=0$. On the other hand, 0
is only a Mal’tsev object when $\mathbb{C}$ is a unital category by (M3). ∎

When $\mathbb{C}$ is a regular category, then both $\mathsf{W}(\mathbb{C})$
and $\mathsf{M}(\mathbb{C})$ are closed under quotients in $\mathbb{C}$ – (M4)
and (W2), where the later actually holds for any finitely complete
$\mathbb{C}$. If $\mathbb{C}$ 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.

######  Proposition 1.4.

Let $\mathbb{C}$ be a regular category with binary coproducts. If $Y$ is a
Mal’tsev object in $\mathbb{C}$, then it is a W-Mal’tsev object in
$\mathbb{C}$.

###### Proof.

Consider the double split epimorphism over $Y$

$\textstyle{3Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\left(\begin{smallmatrix}{\iota_{1}}\\\
{\iota_{2}}\\\
{\iota_{2}}\end{smallmatrix}\right)}}$$\scriptstyle{{\left(\begin{smallmatrix}{\iota_{2}}\\\
{\iota_{2}}\\\
{\iota_{1}}\end{smallmatrix}\right)}}$$\textstyle{2Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\scriptstyle{\iota_{32}={\left(\begin{smallmatrix}{\iota_{3}}\\\
{\iota_{2}}\\\
\end{smallmatrix}\right)}}$$\textstyle{2Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{12}={\left(\begin{smallmatrix}{\iota_{1}}\\\
{\iota_{2}}\\\
\end{smallmatrix}\right)}}$$\scriptstyle{\nabla={\left(\begin{smallmatrix}{1_{Y}}\\\
{1_{Y}}\\\
\end{smallmatrix}\right)}}$$\textstyle{Y,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\iota_{2}}$

which is a regular pushout by assumption. We then get a (regular epimorphism,
monomorphism) factorisation as in (1.ii), where $R=\mathsf{Eq}(\nabla)$ is the
kernel pair of $\nabla$. It easily follows that
$(\iota_{1},\iota_{1})\in_{Y}R$, since $R=\mathsf{Eq}(\nabla)$ and
$\nabla\iota_{1}=\nabla\iota_{1}$. This shows that $Y$ is a W-Mal’tsev object
by (W5). ∎

An interesting question is whether $\mathsf{W}(\mathbb{C})$ and
$\mathsf{M}(\mathbb{C})$ are themselves Mal’tsev categories. From (W3) we know
that, if $\mathbb{C}$ is a well-powered regular category with coproducts, then
$\mathsf{W}(\mathbb{C})$ is finitely complete. Also, if every relation in
$\mathsf{W}(\mathbb{C})$ is also a relation in $\mathbb{C}$, then the full
subcategory $\mathsf{W}(\mathbb{C})$ is a Mal’tsev category; actually, it is
the largest one which is closed under binary coproducts and quotients in
$\mathbb{C}$ (see (W4)). On the other hand, there are no “obvious” conditions
on $\mathbb{C}$ from which we could deduce finite completeness for
$\mathsf{M}(\mathbb{C})$. So, we can only conclude that
$\mathsf{M}(\mathbb{C})$ is a Mal’tsev category when it is closed under finite
limits in $\mathbb{C}$; this is (M2). When this is the case, we still do not
know whether $\mathsf{M}(\mathbb{C})$ is the largest full subcategory of
$\mathbb{C}$ which is a Mal’tsev category.

In [24] is was shown that the Mal’tsev objects in $\mathsf{Mon}$ are precisely
the groups (Theorem 6.14). This is also the case with respect to W-Mal’tsev
objects in $\mathsf{Mon}$. It follows from the above and the next proposition
that $\mathsf{Grp}$ is the largest full subcategory of $\mathsf{Mon}$ which is
a Mal’tsev category and is closed under binary coproducts and quotients in
$\mathsf{Mon}$.

######  Proposition 1.5.

A monoid $Y$ is a W-Mal’tsev object in $\mathsf{Mon}$ if and only if $Y$ is a
group.

###### Proof.

If $Y$ is a group, then it is a Mal’tsev object in $\mathsf{Mon}$; thus, it is
a W-Mal’tsev object in $\mathsf{Mon}$ by Proposition 1.4. For the converse,
suppose that $(Y,+,0)$ is a W-Mal’tsev object in $\mathsf{Mon}$. We use
additive notation although $Y$ is not necessarily an abelian monoid. Since
$\mathsf{Mon}$ is a regular category with binary coproducts, we can apply (W5)
to conclude that $(\iota_{1},\iota_{1})\in_{Y}R$, where $R$ is as in diagram
(1.ii). So, for any $x\in Y$, $x\neq 0$, we have
$([\underline{x}],[\underline{x}])\in R$. (We use the notations
$\iota_{1}(x)=[\,\underline{x}\,]$, $\iota_{2}(x)=[\,\overline{x}\,]$, for any
$x\in Y$, for the coprojections $\iota_{1},\iota_{2}\colon Y\to 2Y$, and
$\iota_{1}(x)=[\,\underline{x}\,]$, $\iota_{2}(x)=[\,\overline{x}\,]$,
$\iota_{3}(x)=[\widetilde{x}]$, for any $x\in Y$, for the coprojections
$\iota_{1},\iota_{2},\iota_{3}\colon Y\to 3Y$.) Since $e\colon 3Y\to R$ is
surjective, there exists an element
$[\,\underline{u_{1}}\overline{v_{1}}\widetilde{w_{1}}\cdots\underline{u_{k}}\overline{v_{k}}\widetilde{w_{k}}\,]\in
3Y$ such that
$e([\,\underline{u_{1}}\overline{v_{1}}\widetilde{w_{1}}\cdots\underline{u_{k}}\overline{v_{k}}\widetilde{w_{k}}\,])=([\underline{x}],[\underline{x}])$.
We deduce that

$\left\\{\begin{array}[]{l}\vspace{20pt}\end{array}\right.$
$[\,\underline{u_{1}}\overline{v_{1}+w_{1}}\cdots\underline{u_{k}}\overline{v_{k}+w_{k}}\,]=[\underline{x}]$
$[\,\overline{u_{1}+v_{1}}\underline{w_{1}}\cdots\overline{u_{k}+v_{k}}\underline{w_{k}}\,]=[\underline{x}]$;

consequently

$\left\\{\begin{array}[]{l}v_{1}+w_{1}=0,\cdots,v_{k}+w_{k}=0,\;\;\mathrm{and}\;\;u_{1}+\cdots+u_{k}=x\vspace{3pt}\\\
u_{1}+v_{1}=0,\cdots,u_{k}+v_{k}=0,\;\;\mathrm{and}\;\;w_{1}+\cdots+w_{k}=x.\end{array}\right.$

We may define the element $y=v_{k}+\cdots+v_{1}$ of $Y$ which is the inverse
of $x$:

$\begin{array}[]{l}x+y=u_{1}+\cdots+u_{k-1}+(u_{k}+v_{k})+v_{k-1}\cdots+v_{1}=u_{1}+\cdots+(u_{k-1}+v_{k-1})\cdots+v_{1}=\cdots=0,\\\
y+x=v_{k}+\cdots+v_{2}+(v_{1}+w_{1})+w_{2}+\cdots+w_{k}=v_{k}+\cdots+(v_{2}+w_{2})+\cdots+w_{k}=\cdots=0.\end{array}$

∎

We finish this section with the example of (W-)Mal’tsev objects in
$\mathsf{OrdGrp}$, the category of preordered groups. We denote a preordered
group by $(Y,+,\leq)$, even though the associated group $(Y,+,0)$ is not
necessarily abelian. Recall from [12] that the _positive cone_ of $Y$,
$P_{Y}=\\{x\in Y:0\leq x\\}$, is always a submonoid of $Y$ which is closed
under conjugation in $Y$. It was shown in [12] that a preordered group
$(Y,+,\leq)$ is a Mal’tsev object if and only if the preorder relation $\leq$
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
$\mathsf{OrdGrp}$.

######  Example 1.6.

Let $(Y,+,\leq)$ be a W-Mal’tsev object in $\mathsf{OrdGrp}$. Since
$\mathsf{OrdGrp}$ is a regular category with binary coproducts, we can apply
(W5) to conclude that $(\iota_{1},\iota_{1})\in_{Y}R$, where $R$ is as in
diagram (1.ii). So, for any $x\in P_{Y}$, $x\neq 0$, we have
$([\underline{x}],[\underline{x}])\in P_{R}$. Since $e$ is a regular
epimorphism, there exists a positive element
$[\,\underline{u_{1}}\overline{v_{1}}\widetilde{w_{1}}\cdots\underline{u_{k}}\overline{v_{k}}\widetilde{w_{k}}\,]\in
P_{3Y}$ such that
$e([\,\underline{u_{1}}\overline{v_{1}}\widetilde{w_{1}}\cdots\underline{u_{k}}\overline{v_{k}}\widetilde{w_{k}}\,])=([\underline{x}],[\underline{x}])$.
We deduce that

$\left\\{\begin{array}[]{l}\vspace{20pt}\end{array}\right.$
$[\,\underline{u_{1}}\overline{v_{1}+w_{1}}\cdots\underline{u_{k}}\overline{v_{k}+w_{k}}\,]=[\underline{x}]$
$[\,\overline{u_{1}+v_{1}}\underline{w_{1}}\cdots\overline{u_{k}+v_{k}}\underline{w_{k}}\,]=[\underline{x}]$;

consequently

$\left\\{\begin{array}[]{l}v_{1}+w_{1}=0,\cdots,v_{k}+w_{k}=0,\;\;\mathrm{and}\;\;u_{1}+\cdots+u_{k}=x\vspace{3pt}\\\
u_{1}+v_{1}=0,\cdots,u_{k}+v_{k}=0,\;\;\mathrm{and}\;\;w_{1}+\cdots+w_{k}=x.\end{array}\right.$

We conclude that $u_{1}=w_{1},\cdots,u_{k}=w_{k}$ and the above positive
element of $3Y$ has the shape
$[\,\underline{u_{1}}\overline{\mbox{-}u_{1}}\widetilde{u_{1}}\cdots\underline{u_{k}}\overline{\mbox{-}u_{k}}\widetilde{u_{k}}\,]$.
The monotone morphism

$\begin{array}[]{rccc}{\left(\begin{smallmatrix}{0}\\\ {1_{Y}}\\\
{0}\end{smallmatrix}\right)}\colon&3Y&\to&Y\\\
&[\,\underline{u_{1}}\overline{\mbox{-}u_{1}}\widetilde{u_{1}}\cdots\underline{u_{k}}\overline{\mbox{-}u_{k}}\widetilde{u_{k}}\,]&\mapsto&-u_{1}-\cdots-
u_{k}\end{array}$

takes positive elements of $3Y$ to positive elements of $Y$; consequently
$-u_{1}-\cdots-u_{k}\in P_{Y}$.

If $Y$ is an abelian group, then from $x=u_{1}+\cdots+u_{k}\in P_{Y}$, we
deduce $-x=-u_{1}-\cdots-u_{k}\in P_{Y}$, i.e. $P_{Y}$ is a group. In this
case W-Mal’tsev objects and Mal’tsev objects coincide in $\mathsf{OrdGrp}$. We
do not know whether the two notions coincide for the non-abelian case.

##  2\. $V$-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 [26]. For that we will make use of the concepts
of $V$-category and of $(U,V)$-category. Here, as in [19], the notion of
$V$-category will play the role of metric space, while a $(U,V)$-category (as
introduced in [13]) 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 $\otimes$, with unit $k=\top\neq\bot$,
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
$V_{\otimes}$ to indicate that we are using the tensor product $\otimes$ in
$V$.

######  Definition 2.1.

A V-category is a set $X$ together with a map $X\times X\to V$, whose image of
$(x,x^{\prime})$ we denote by $X(x,x^{\prime})$, such that, for each
$x,x^{\prime},x^{\prime\prime}\in X$,

  1. (R)

$k\leq X(x,x)$;

  2. (T)

$X(x,x^{\prime})\otimes X(x^{\prime},x^{\prime\prime})\leq
X(x,x^{\prime\prime})$.

A $V$-category is said to be _symmetric_ if, for every $x,x^{\prime}\in X$,
$X(x,x^{\prime})=X(x^{\prime},x)$.
A _$V$ -functor_ $f\colon X\to Y$ is a map such that, for all $x,x^{\prime}\in
X$, $X(x,x^{\prime})\leq Y(f(x),f(x^{\prime}))$.

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 $[0,\infty]$ we consider the
order relation $\geq$ and the tensor product $+$, (R) and (T) read as

  1. (R’)

$0\geq X(x,x)$;

  2. (T’)

$X(x,x^{\prime})+X(x^{\prime},x^{\prime\prime})\geq X(x,x^{\prime\prime})$,

with (R) meaning that the distance from a point to itself is $0$, and (T) the
usual triangular inequality.

We denote by $V\mbox{-}\mathsf{Cat}$ the category of $V$-categories and
$V$-functors, and by $V\mbox{-}\mathsf{Cat}_{\rm sym}$ 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 $V_{\wedge}=(V,\leq,\wedge,\top)$ is a
(unital and integral) quantale. We call such quantales _cartesian_. It is
well-known that, if $V_{\wedge}$ is a cartesian quantale, then the category
$V_{\wedge}\mbox{-}\mathsf{Cat}$ has special features, like being cartesian
closed [19]. Here we will show that it also has a key role in the study of
W-Mal’tsev objects of $(V_{\wedge}\mbox{-}\mathsf{Cat})^{\rm op}$.

Moreover, even if the quantale $V_{\otimes}$ is not a frame, the
$V_{\otimes}$-categories $X$ which, in addition, verify

$X(x,x^{\prime})\wedge X(x^{\prime},x^{\prime\prime})\leq
X(x,x^{\prime\prime})$

will be specially relevant, and we will call them also
$V_{\wedge}$-categories. We note that, since $k=\top$, they also satisfy
$\top\leq X(x,x)$ for every $x\in X$; moreover, $u\otimes v\leq u\wedge v$
always holds, so the above condition guarantees immediately condition (T) of
the above definition.

######  Examples 2.3.

  1. (1)

When $V=2=(\\{0<1\\},\wedge,1)$, a $V$-category is a preordered set and a
$V$-functor is a monotone map. Hence $V\mbox{-}\mathsf{Cat}$ is the well-known
category $\mathsf{Ord}$ of preordered sets and monotone maps.

  2. (2)

When $V=[0,\infty]_{+}=([0,\infty],\geq,+,0)$, that is $V$ is the complete
lattice $[0,\infty]$, ordered by $\geq$, with tensor product $\otimes=+$, a
$V$-category is a (_generalised_) _Lawvere metric space_ [19] (not necessarily
separated nor symmetric, with $\infty$ as a possible distance), since the two
conditions above mean that $X(x,x)=0$ and $X(x,y)+X(y,z)\geq X(x,z)$, for
$x,y,z\in X$, and a $V$-functor is a _non-expansive map_.
When $V$ is the cartesian quantale $[0,\infty]_{\rm max}=([0,\infty],\geq,{\rm
max},0)$, with tensor product $\otimes={\rm max}$, then $V_{\rm
max}\mbox{-}\mathsf{Cat}$ is the category of (_generalised_) _ultrametric
spaces_.

  3. (3)

The complete lattice $([0,1],\leq)$ 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
$V\mbox{-}\mathsf{Cat}$, like Lawvere metric spaces, ultrametric spaces, and
_bounded metric spaces_.

  4. (4)

The set of _distribution functions_

$\Delta=\\{\varphi\colon[0,\infty]\to[0,1]\,;\mbox{ for all
}\alpha\in[0,\infty]\;\;\varphi(\alpha)=\bigvee_{\beta<\alpha}\,\varphi(\beta)$

with the pointwise order is a complete lattice. The tensor is defined, for
each $\varphi,\psi\in\Delta$, by

$(\varphi\otimes\psi)(\alpha)=\bigvee_{\beta+\gamma\leq\alpha}\,\varphi(\beta)\times\psi(\gamma),$

having as unit the distribution function $\kappa\colon[0,\infty]\to[0,1]$,
with $\kappa(\alpha)=0$ if $\alpha=0$ and $\kappa(\alpha)=1$ otherwise. Then
$\Delta$-$\mathsf{Cat}$ is the category of _probabilistic metric spaces_ and
(_probabilistic_) _non-expansive_ maps.

For more details and examples see for instance [11].

The forgetful functor $V\mbox{-}\mathsf{Cat}\to\mathsf{Set}$ is topological,
hence $V\mbox{-}\mathsf{Cat}$ is complete and cocomplete, with limits and
colimits formed as in $\mathsf{Set}$ and equipped with the corresponding
initial and final $V$-category structures, respectively. In [21, Corollary 8]
it is shown that, under suitable conditions, $V\mbox{-}\mathsf{Cat}$ is an
extensive category. In $V\mbox{-}\mathsf{Cat}$ epimorphisms are pullback-
stable, since they are exactly surjective $V$-functors and pullbacks are
formed as in $\mathsf{Set}$. Therefore, as shown in [22, Proposition 3],
_$(V\mbox{-}\mathsf{Cat})^{\rm op}$ is a weakly Mal’tsev category_. In
addition, in [15, Theorem 4.6] it is shown that _$(V\mbox{-}\mathsf{Cat})^{\rm
op}$ is a quasi-variety_, so in particular it is a regular category. Still, we
think it is worth to prove here directly that _$(V\mbox{-}\mathsf{Cat})^{\rm
op}$ is a regular category_. Indeed, in $V\mbox{-}\mathsf{Cat}$ regular
monomorphisms coincide with extremal monomorphisms, and are exactly the
injective maps $f\colon X\to Y$ such that
$X(x,x^{\prime})=Y(f(x),f(x^{\prime}))$, for all $x,x^{\prime}\in X$.
Moreover, $V\mbox{-}\mathsf{Cat}$ has the stable orthogonal factorisation
system (epimorphism, regular monomorphism), which factors every $V$-functor
$f\colon X\to Y$ as

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{e}$$\textstyle{Y}$$\textstyle{Z=f(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$

where $e$ is the corestriction of $f$ to $Z=f(X)$ and
$Z(y,y^{\prime})=Y(y,y^{\prime})$ for all $y,y^{\prime}\in Z$. Given a pushout
in $V\mbox{-}\mathsf{Cat}$

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\scriptstyle{f}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{W}$

with $m$ a regular monomorphism, for simplicity we assume that $m$ is an
inclusion. We know that, since pushouts are formed as in $\mathsf{Set}$, we
may consider $W=(Y+Z)/\sim$, where, for $y\in Y$ and $z\in Z$, $y\sim z$
exactly when $z\in X$ and $f(z)=y$, and $n$ an inclusion. Then, for every
$y,y^{\prime}\in Y$,

$\begin{array}[]{rcll}W([y],[y^{\prime}])&=&\displaystyle
Y(y,y^{\prime})\vee\bigvee_{x\sim y,\,x^{\prime}\sim
y^{\prime}}\,Z(x,x^{\prime})\\\ &=&\displaystyle
Y(y,y^{\prime})\vee\bigvee_{x\sim y,\,x^{\prime}\sim
y^{\prime}}\,X(x,x^{\prime})&\mbox{(because $m$ is a regular monomorphism)}\\\
&=&Y(y,y^{\prime})&\mbox{(because $f$ is a $V$-functor)}\end{array}$

and therefore the inclusion $n\colon Y\to W$ is a regular monomorphism.

Hence, since $(V\mbox{-}\mathsf{Cat})^{\rm op}$ 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 $V$-functor $f\colon Y^{2}+Y^{2}\to Y^{3}$
defined by

|
---|---
$\textstyle{Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\pi_{1},\pi_{2},\pi_{2}\rangle}$$\scriptstyle{\iota_{1}}$$\textstyle{Y^{2}+Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\,f\,}$$\textstyle{Y^{3}}$$\textstyle{Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\langle\pi_{2},\pi_{2},\pi_{1}\rangle}$

where $\pi_{i}\colon Y^{2}\to Y$, $i=1,2$, are the product projections, and
its (epimorphism, regular monomorphism)-factorisation in
$V\mbox{-}\mathsf{Cat}$

$(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.82777pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y^{2}+Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
25.73749pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.82777pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.82777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Y^{3}}$}}}}}}}\ignorespaces}}}}\ignorespaces)=(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
18.82777pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y^{2}+Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.19809pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.82777pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.82777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
63.82416pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{m}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 81.8972pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
81.8972pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Y^{3}}$}}}}}}}\ignorespaces}}}}\ignorespaces)$ (2.i)

there is a unique $V$-functor $g\colon X\to Y$ such that the following diagram
commutes

|  |
---|---|---
$\textstyle{Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\iota_{1}}$$\textstyle{Y^{2}+Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\,e\,}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\,g\,}$$\textstyle{Y}$$\textstyle{Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\pi_{1}}$
(2.ii)

As a set, $X$ is the image of the map $f$, that is

$X=X_{1}\cup X_{2},\mbox{ with }X_{1}=\\{(x,y,y)\,;\,x,y\in
Y\\},\;X_{2}=\\{(x,x,y)\,;\,x,y\in Y\\};$

its $V$-category structure is inherited from $Y^{3}$, that is

$X((x,y,z),(x^{\prime},y^{\prime},z^{\prime}))=Y(x,x^{\prime})\wedge
Y(y,y^{\prime})\wedge Y(z,z^{\prime}).$

The map $g\colon X\to Y$ must assign $x$ both to each $(x,y,y)$ and each
$(y,y,x)$.

######  Theorem 2.4.

Let $Y$ be a $V$-category.

  1. (1)

The following conditions are equivalent:

    1. (i)

$Y$ is a W-Mal’tsev object in $(V\mbox{-}\mathsf{Cat})^{\rm op}$;

    2. (ii)

for all $x,y,z\in Y$, $Y(z,x)\wedge Y(z,y)\leq Y(x,y)$;

    3. (iii)

$Y$ is a symmetric $V_{\wedge}$-category.

  2. (2)

If $V$ is not a cartesian quantale, then $Y$ is a Mal’tsev object in
$(V\mbox{-}\mathsf{Cat})^{\rm op}$ if, and only if, $Y=\emptyset$.

###### Proof.

(1) (i) $\Rightarrow$ (ii): If the map $g\colon X\to Y$ from (2.ii) is a
$V$-functor, then in particular

$X((x,z,z),(x,x,y))=Y(x,x)\wedge Y(z,x)\wedge Y(z,y)\leq
Y(g(x,z,z),g(x,x,y))=Y(x,y).$

(ii) $\Rightarrow$ (iii): With $z=y$ in (ii) one gets

$Y(y,x)=Y(y,x)\wedge Y(y,y)\leq Y(x,y),$

hence $Y$ is symmetric. Then, for any $x,y,z\in Y$,

$Y(x,z)\wedge Y(z,y)=Y(z,x)\wedge Y(z,y)\leq Y(x,y).$

(iii) $\Rightarrow$ (i): We want to prove that $g$ is a $V$-functor; that is,
for each $(x,y,z),(x^{\prime},y^{\prime},z^{\prime})$ in $X$,

$X((x,y,z),(x^{\prime},y^{\prime},z^{\prime}))=Y(x,x^{\prime})\wedge
Y(y,y^{\prime})\wedge Y(z,z^{\prime})\leq
Y(g(x,y,z),(x^{\prime},y^{\prime},z^{\prime})).$

In case $x=y$ and $x^{\prime}=y^{\prime}$, or in case $y=z$ and
$y^{\prime}=z^{\prime}$, the inequality is trivially satisfied; when $x=y$ and
$y^{\prime}=z^{\prime}$, using (iii), one obtains

$Y(x,x^{\prime})\wedge Y(x,y^{\prime})\wedge
Y(z,y^{\prime})=Y(z,y^{\prime})\wedge Y(y^{\prime},x)\wedge
Y(x,x^{\prime})\leq Y(z,x^{\prime}),$

and in case $y=z$ and $x^{\prime}=y^{\prime}$ one gets

$Y(x,x^{\prime})\wedge Y(y,x^{\prime})\wedge
Y(y,z^{\prime})=Y(x,x^{\prime})\wedge Y(x^{\prime},y)\wedge
Y(y,z^{\prime})\leq Y(x,z^{\prime}).$

(2) Since $(V\mbox{-}\mathsf{Cat})^{\rm op}$ 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 $(V\mbox{-}\mathsf{Cat})^{\rm op}$ is also a
double split epimorphism in $V\mbox{-}\mathsf{Cat}$, with the role of split
epimorphisms and split monomorphisms interchanged:

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\scriptstyle{s^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{f^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{s}$$\textstyle{Y.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$
(2.iii)

(this diagram is just the dual of (1.iii)).

Let $V$ be a non-cartesian quantale. For each non-empty $V$-category $Y$ we
will build a double split epimorphism in $V\mbox{-}\mathsf{Cat}$ as above so
that it is not a _regular pullback_ , i.e. the comparison morphism $\langle
g^{\prime},f^{\prime}\rangle\colon A+_{Y}C\to D$ is not a regular
monomorphism. If $V$ is not cartesian then there exist $u,v\in V$ such that
$u\otimes v<u\wedge v$. Let $Y$ be a $V$-category and $y_{0}\in Y$. Consider
the $V$-categories $A=\\{0\\}\stackrel{{\scriptstyle\cdot}}{{\cup}}Y$,
$C=Y\stackrel{{\scriptstyle\cdot}}{{\cup}}\\{1\\}$, and
$D=\\{0\\}\stackrel{{\scriptstyle\cdot}}{{\cup}}Y\stackrel{{\scriptstyle\cdot}}{{\cup}}\\{1\\}$,
with $A(0,y)=u\otimes A(y_{0},y)$ and $A(y,0)=A(y,y_{0})\otimes u$,
$C(y,1)=Y(y,y_{0}\otimes v)$ and $C(1,y)=v\otimes C(y_{0},y)$, and
$D(0,1)=u\wedge v=D(1,0)$; the remaining structures are defined so that the
obvious inclusions are regular monomorphisms. Then (2.iii) is a double split
epimorphism but the comparison morphism $\langle
g^{\prime},f^{\prime}\rangle\colon A+_{Y}C\to D$, which is in fact a
bijection, is not a regular monomorphism because in the pushout
$(A+_{Y}C)(0,1)$ must be $u\otimes v$. ∎

######  Remark 2.5.

We point out that condition (ii) in case $\otimes=\wedge$ is the separation
axiom (R1) studied in [9, Section 2.2].

The two statements of our theorem have an immediate consequence, showing how
drastically different may be the two notions of Mal’tsev object in the context
of $(V\mbox{-}\mathsf{Cat})^{\rm op}$. For instance, if $V=[0,\infty]_{+}$,
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.

######  Corollary 2.6.

  1. (1)

If $V$ is a cartesian quantale, then $(V\mbox{-}\mathsf{Cat}_{\rm sym})^{\rm
op}$ is a Mal’tsev category.

  2. (2)

For every unital and integral quantale $V$, the largest Mal’tsev subcategory
of $(V\mbox{-}\mathsf{Cat})^{\rm op}$ closed under binary coproducts and
regular epimorphisms is $(V_{\wedge}$-$\mathsf{Cat}_{\rm sym})^{\rm op}$.

It is straightforward to check that, if we restrict our study to symmetric
$V_{\otimes}$-categories, in case the tensor product $\otimes$ 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 $(V\mbox{-}\mathsf{Cat}_{\rm sym})^{\rm op}$ 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 $\mathsf{Set}$, and
its extension to $V\mbox{-}\mathsf{Rel}$ (see [13] 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 $X$,
$UX{\longrightarrow{\mapstochar}\hskip 11.19443pt}X$ such that, for every
$x\in X$, $\mathfrak{x}\in UX$, and $\mathfrak{X}\in U^{2}X$,

  1. (1)

$\stackrel{{\scriptstyle\cdot}}{{x}}\,\to x$ (where
$\stackrel{{\scriptstyle\cdot}}{{x}}$ is the principal ultrafilter defined by
$x$);

  2. (2)

$\mathfrak{X}\to\mathfrak{x}$ and $\mathfrak{x}\to x$ $\implies$
$\mu(\mathfrak{X})\to x$ (where $\mu(\mathfrak{X})$ is the Kowalski sum of
$\mathfrak{X}$).

Given topological spaces $X$ and $Y$, a map $f\colon X\to Y$ is continuous if,
for all $\mathfrak{x}\in UX$ and $x\in X$, $Uf(\mathfrak{x})\to f(x)$ whenever
$\mathfrak{x}\to x$.

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
$V\mbox{-}\mathsf{Rel}$.

######  Definition 2.8.

A $\mathbf{(\emph{U,V})}$-category is a set $X$ together with a map $a\colon
UX\times X\to V$ whose image of $(\mathfrak{x},x)$ we denote by
$X(\mathfrak{x},x)$, such that, for each $x\in X$, $\mathfrak{x}\in UX$,
$\mathfrak{X}\in U^{2}X$,

  1. (R)

$k\leq X(\stackrel{{\scriptstyle\cdot}}{{x}},x)$;

  2. (T)

$X(\mathfrak{X},\mathfrak{x})\otimes X(\mathfrak{x},x)\leq
X(\mu(\mathfrak{X}),x)$,

where by $X(\mathfrak{X},\mathfrak{x})$ we mean the image, under $U$, of the
$V$-relation $a\colon UX{\longrightarrow{\mapstochar}\hskip 11.19443pt}X$.
A $(U,V)$-functor $f\colon X\to Y$ is a map such that, for all $x\in X$,
$\mathfrak{x}\in UX$, $X(\mathfrak{x},x)\leq Y(Uf(\mathfrak{x}),f(x))$.

We denote by $(U,V)\mbox{-}\mathsf{Cat}$ the category of $(U,V)$-categories
and $(U,V)$-functors.

######  Examples 2.9.

  1. (1)

As shown by Barr [1], if $V=2$, then $(U,V)\mbox{-}\mathsf{Cat}$ is isomorphic
to the category $\mathsf{Top}$ of topological spaces and continuous maps.

  2. (2)

As shown in [10], if $V=[0,\infty]_{+}$ $(U,V)\mbox{-}\mathsf{Cat}$ is
isomorphic to the category $\mathsf{App}$ of Lowen’s _approach spaces_ and
non-expansive maps.

######  Proposition 2.10.

The category $((U,V)\mbox{-}\mathsf{Cat})^{\rm op}$ is regular.

###### Proof.

As in $V\mbox{-}\mathsf{Cat}$, in $(U,V)\mbox{-}\mathsf{Cat}$ (epimorphisms,
regular monomorphisms) form a stable factorisation system, and a morphism
$m\colon X\to Z$ is a regular monomorphism if, and only if, it is an injective
map and $X(\mathfrak{x},x)=Z(Um(\mathfrak{x}),m(x))$, for all $\mathfrak{x}\in
UX$ and $x\in X$. Given a pushout in $(U,V)\mbox{-}\mathsf{Cat}$

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\scriptstyle{f}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{W}$

with $m$ a regular monomorphism, which we assume, for simplicity, to be an
inclusion, as for $V\mbox{-}\mathsf{Cat}$ we may consider $W=(Y+Z)/\sim$,
where, for $y\in Y$ and $z\in Z$, $y\sim z$ exactly when $z\in X$ and
$f(z)=y$. Then $n$, and therefore also $Un$, is an injective map, and, for
every $\mathfrak{y}\in UY$ and $y\in Y$,

$W(Un(\mathfrak{y}),[y])=\displaystyle
Y(\mathfrak{y},y)\vee\bigvee_{Ug(\mathfrak{z})=Un(\mathfrak{y}),\,x\sim
y}\,Z(\mathfrak{z},x).$

Any $\mathfrak{z}\in UZ$ with $Ug(\mathfrak{z})=Un(\mathfrak{y})$ must contain
$X$, and consequently it is the image under $Um$ of an ultrafilter
$\mathfrak{x}$ in $X$; hence $Z(\mathfrak{z},x)=X(\mathfrak{x},x)\leq
Y(Uf(\mathfrak{x}),f(x))=Y(\mathfrak{y},y)$. This implies that
$Y(\mathfrak{y},y)=W(Un(\mathfrak{y}),[y])$, and therefore $n\colon Y\to W$ is
a regular monomorphism. ∎

######  Proposition 2.11.

For a $(U,V)$-category $Y$ the following conditions are equivalent:

  1. (i)

$Y$ is a W-Mal’tsev object in $((U,V)\mbox{-}\mathsf{Cat})^{\rm op}$;

  2. (ii)

For all $\mathfrak{z}\in UY$, $x,y\in Y$,

    1. (1)

$Y(\mathfrak{z},x)\wedge Y(\mathfrak{z},y)\leq
Y(\stackrel{{\scriptstyle\cdot}}{{x}},y)$;

    2. (2)

$Y(\mathfrak{z},x)\wedge Y(\stackrel{{\scriptstyle\cdot}}{{x}},y)\leq
Y(\mathfrak{z},y)$.

###### Proof.

The map $g\colon X\to Y$ is a $(U,V)$-functor if, and only if, for every
$\mathfrak{w}\in UX$ and $(x,y,z)\in X$,

$X(\mathfrak{w},(x,y,z))=Y(U\pi_{1}(\mathfrak{w}),x)\wedge
Y(U\pi_{2}(\mathfrak{w}),y)\wedge Y(U\pi_{3}(\mathfrak{w}),z)\leq
Y(Ug(\mathfrak{w}),g(x,y,z)).$

For simplicity, we will denote the $i$-th projection $Y^{k}\to Y$ ($k=2,3$) by
$\pi_{i}$ and $U\pi_{i}(\mathfrak{w})$ by $\mathfrak{w}_{i}$.

Let $h\colon Y\times Y\to X$ with $h(x,y)=h(x,y,y)$.

(i) $\Rightarrow$ (ii): Fix $\mathfrak{z}\in UY$ and $x\in Y$. Let
$\widetilde{\mathfrak{z}}\in U(Y^{2})$ be such that
$\widetilde{z_{1}}=\stackrel{{\scriptstyle\cdot}}{{c}}$ and
$\widetilde{z_{2}}=\mathfrak{z}$. Then
$\mathfrak{w}=Uh(\widetilde{\mathfrak{z}})\in UX$ and, since $g\cdot
h=\pi_{1}$, $\mathfrak{w}_{1}=\stackrel{{\scriptstyle\cdot}}{{x}}$,
$\mathfrak{w}_{2}=\mathfrak{z}=\mathfrak{w}_{3}$, and
$Ug(\mathfrak{w})=\stackrel{{\scriptstyle\cdot}}{{x}}$. Then we have

$X(\mathfrak{w},(x,x,y))\leq
Y(\stackrel{{\scriptstyle\cdot}}{{x}},y)\;\Rightarrow\;Y(\stackrel{{\scriptstyle\cdot}}{{x}},x)\wedge
Y(\mathfrak{z},x)\wedge Y(\mathfrak{z},y)\leq
Y(\stackrel{{\scriptstyle\cdot}}{{x}},y),$

that is (1) holds.
Analogously, let $\widehat{\mathfrak{z}}\in U(Y^{2})$ be such that
$\widehat{z_{1}}=\mathfrak{z}$ and
$\widehat{z_{2}}=\stackrel{{\scriptstyle\cdot}}{{x}}$, and let
$\mathfrak{x}=Uh(\widehat{\mathfrak{z}})$, so that
$Ug(\mathfrak{x})=\mathfrak{z}$. Then, for every $y\in Y$,

$X(\mathfrak{x},(x,x,y))\leq
Y(\mathfrak{z},y)\;\Rightarrow\;Y(\mathfrak{z},x)\wedge
Y(\stackrel{{\scriptstyle\cdot}}{{x}},x)\wedge
Y(\stackrel{{\scriptstyle\cdot}}{{x}},y)\leq Y(\mathfrak{z},y),$

that is (2) holds.

(ii) $\Rightarrow$ (i): Conversely, assume (1) and (2). Let $\mathfrak{w}\in
UX$ and assume that $X_{1}\in\mathfrak{w}$ (for the case
$X_{2}\in\mathfrak{w}$ the proof is analogous). Then
$\mathfrak{w}=Uh(\mathfrak{y})$ for some $\mathfrak{y}\in U(Y^{2})$. Then

$X(\mathfrak{w},(x,y,y))=Y(\mathfrak{y}_{1},x)\wedge
Y(\mathfrak{y}_{2},y)\wedge Y(\mathfrak{y}_{2},y)\leq
Y(\mathfrak{y}_{1},x)=Y(Ug(\mathfrak{w}),x),$
$X(\mathfrak{w},(x,x,y))=Y(\mathfrak{y}_{1},x)\wedge
Y(\mathfrak{y}_{2},x)\wedge Y(\mathfrak{y}_{2},y)\leq
Y(\mathfrak{y}_{1},x)\wedge Y(\stackrel{{\scriptstyle\cdot}}{{x}},y)\leq
Y(\mathfrak{y}_{1},y);$

that is, $g$ is a $(U,V)$-functor. ∎

######  Remark 2.12.

The result of the proposition above applies immediately to $\mathsf{Top}$ and
$\mathsf{App}$, and it may be extended to other suitable
$\mathsf{Set}$-monads. For simplicity we chose to restrict ourselves to the
ultrafilter monad, recovering the result in $\mathsf{Top}$ and obtaining a
characterisation of W-Mal’tsev objects in $(\mathsf{App})^{\rm op}$.

At a first glance it may seem that here we need an extra condition to recover
the result in $\mathsf{Top}$. Indeed, in $(\mathsf{Top})^{\rm op}$ condition
(1) is the well-known separation axiom (R1) we have mentioned in Remark 2.5,
while condition (2) follows from axiom (T) of the definition of
$(U,V)$-category in case $V$ is cartesian, which is the case of
$\mathsf{Top}$. When $V=[0,\infty]_{+}$ condition (1) is stronger than (R1),
which reads as $Y(\mathfrak{z},x)+Y(\mathfrak{z},y)\geq
Y(\stackrel{{\scriptstyle\cdot}}{{x}},y)$ for every $\mathfrak{z}\in UY$,
$x,y\in Y$, and (2) does not follow from axiom (T).

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