Theory and Applications of Categories, Vol. 21, No. 8, 2008, pp. 152–171.

APPROXIMATE MAL’TSEV OPERATIONS

Dedicated to Walter Tholen on the occasion of his 60th birthday

DOMINIQUE BOURN AND ZURAB JANELIDZE

Abstract. Let X and A be sets and α : X → A a map between them. We call a map μ : X × X × X → A an approximate Mal’tsev operation with approximation α, if it satisfies μ(x, y, y) = α(x) = μ(y, y, x) for all x, y ∈ X. Note that if A = X and the approximation α is an identity map, then μ becomes an ordinary Mal’tsev operation. We prove the following two characterization theorems: a category 𝕏 is a Mal’tsev category if and only if in the functor category Set^{𝕏^{op}×𝕏} there exists an internal approximate Mal’tsev operation hom_{𝕏}×hom_{𝕏}×hom_{𝕏} → A whose approximation α satisfies a suitable condition; a regular category 𝕏 with finite coproducts is a Mal’tsev category, if and only if in the functor category 𝕏^{𝕏} there exists an internal approximate Mal’tsev cooperation A → 1_{𝕏} + 1_{𝕏} + 1_{𝕏} whose approximation α is a natural transformation with every component a regular epimorphism in 𝕏. Note that in both of these characterization theorems, if require further the approximation α to be an identity morphism, then the conditions there involving α become equivalent to 𝕏 being a naturally Mal’tsev category.

1. Introduction

A Mal’tsev category is usually defined as a category 𝕏 with finite limits such that

(C1) every reflexive internal relation in 𝕏 is an equivalence relation,

although its original definition in [3] required in addition 𝕏 to be exact in the sense of M. Barr [1]. The roots of this concept go back to the work [14] of A. I. Mal’tsev, where it was shown that for a variety 𝕏 of universal algebras,

(C2) the composition of congruences on any object in 𝕏 is commutative

if and only if

(C3) the algebraic theory of 𝕏 contains a ternary term μ satisfying the term equations

μ(x, y, y) = x = μ(y, y, x).  (1)

The second author acknowledges the support of South African Claude Leon Foundation, INTAS (06-1000017-8609), and Georgian National Science Foundation (GNSF/ST06/3-004).

Received by the editors 2008-02-02 and, in revised form, 2008-11-11. Published on 2008-11-14 in the Tholen Festschrift. 2000 Mathematics Subject Classification: 18C99, 08B05. Key words and phrases: Mal’tsev category, Mal’tsev operation, unital category, strongly unital cat-

egory, subtractive category.

© Dominique Bourn and Zurab Janelidze, 2008. Permission to copy for private use granted.

Such varieties were later called Mal’tsev varieties (e.g. see [17]). In [6] G. D. Findlay showed that for a variety 𝕏, (C3) implies (C1), and the converse implication was obtained by H. Werner in [19]. The following similar “relational reformulation” of (C3) is due to J. Lambek [11]:

(C4) every internal relation R in 𝕏 is difunctional, i.e. it satisfies

(x_{1}Ry_{2} ∧ x_{2}Ry_{2} ∧ x_{2}Ry_{1}) ⇒ x_{1}Ry_{1}.

An advantage of conditions (C1) and (C4), to (C2) and (C3), is that (C1) and (C4) can be formulated in any category; indeed, an internal relation in a category is simply a diagram

\xymatrix{X & R \ar[l]_{r_{1}} \ar[r]^{r_{2}} & Y}

where r_{1} and r_{2} are jointly monomorphic, i.e. for any two parallel morphisms f, g with codomain R, we have

(r_{1}f = r_{1}g ∧ r_{2}f = r_{2}g) ⇒ f = g

(if the category has products, then this, of course, amounts to saying that the induced morphism (r_{1}, r_{2}) : R → X × Y is a monomorphism). Then “reflexivity”, “symmetry”, “transitivity” and “difunctionality” of R can be expressed using the Yoneda embedding; that is, we say that R is reflexive, symmetric, transitive or difunctional (assuming X = Y for the first three cases), if for every object S in the given category, the relation

\xymatrix{hom(S, X) & hom(S, R) \ar[l]_{hom(1_{S},r_{1})} \ar[r]^{hom(1_{S},r_{2})} & hom(S, Y)}

between sets is, respectively, reflexive, symmetric, transitive or difunctional in the usual sense. So, (C1) and (C4) make sense when 𝕏 is an arbitrary category. In [4] the equivalence of (C1) and (C4) was shown for arbitrary categories with finite limits (for general categories we only have (C4)⇒(C1) — an observation which is clear but never explicitly written, as far as we know). There are also many other relational conditions equivalent to (C1) and (C4) for finitely complete categories (see [8] for a unified exposition of many of those relational conditions).

Condition (C2) is also categorical, but it can be formulated only in special categories such as exact categories.

For exact categories, the equivalence of (C1), (C2) and (C4) was obtained by J. Meisen in [15]; in [5] T. Fay generalized this result to categories equipped with a suitable factorization system, and two decades later in [3] the same was proved in the more special case of regular categories.

Condition (C3) cannot be formulated even for exact categories, since we do not have “terms” there. However, we can still give a “categorical form” to (C3) by using F. W. Lawvere’s observation [12] that an n-ary term in the algebraic theory of a variety 𝕏 is the same as an internal n-ary operation U^{n} → U in the functor category Set^{𝕏}, where U is the forgetful functor U : 𝕏 → Set. Then (C3) can be rephrased as follows:

(C3′) The object U in the functor category Set^{𝕏} admits an internal Mal’tsev operation, i.e. a ternary operation μ satisfying (1).

Now if we take 𝕏 to be an abstract category, then the above condition makes sense as long as we have a specified functor U : 𝕏 → Set. In [16] M. C. Pedicchio showed the equivalence of (C1) and (C3′) in the case when U is a monadic functor.

In the present paper we add two new conditions to the list of conditions above, which are analogous to (C3′) but they have the advantage that the functor U is determined uniquely by 𝕏, so that those conditions are on a category 𝕏, rather than a pair (𝕏, U), just as conditions (C1), (C2) and (C4). The main “trick” is to replace Mal’tsev operations with “approximate Mal’tsev operations”:

Let X, A be sets and consider a map μ : X × X × X → A and a map α : X → A. We say that μ is an approximate Mal’tsev operation on X with approximation α, if

μ(x, y, y) = α(x) = μ(y, y, x)  (2)

holds true for all x, y ∈ X. In particular, an approximate Mal’tsev operation with approximation 1_{X} is a Mal’tsev operation in the usual sense. Our first new condition states that

(C5) the object hom_{𝕏} in the functor category Set^{𝕏×𝕏^{op}} admits an internal approximate Mal’tsev operation μ : hom_{𝕏}× hom_{𝕏}× hom_{𝕏} → A whose approximation α : hom → A satisfies a suitable condition — the condition (*) in Section 3 below.

We show that (C4) and (C5) are equivalent to each other for any category 𝕏 (Theorem 3.2). The important aspect of this result is that the condition (*) itself is not related to the Mal’tsev property (see Section 5).

Notice that if we require α in (C5) to be an identity natural transformation (in which case the condition (*) would be satisfied trivially), then we get a condition which is equivalent, in the presence of binary products, to the following condition:

(C6) The identity functor 1_{𝕏} admits an internal Mal’tsev operation μ in the functor category 𝕏^{𝕏}.

Categories satisfying (C6) were called naturally Mal’tsev categories by P. T. Johnstone in [9]. Note however that condition (C6) is much stronger than conditions (C1)-(C5), even in the case of varieties. Indeed, for instance, a pointed variety (that is, a variety which admits unique constant) is a naturally Mal’tsev variety if and only if it is abelian, i.e. it is the variety of R-modules for some ring R. Thus, for instance, the variety of groups is a Mal’tsev variety, but it is not a naturally Mal’tsev variety.

Our second condition is on a category 𝕏 with binary coproducts, and it states that

(C7) the identity functor 1_{𝕏} admits an internal approximate Mal’tsev co-operation μ : A → 1_{𝕏} + 1_{𝕏} + 1_{𝕏} in the functor category 𝕏^{𝕏} whose approximation α : A → 1_{𝕏} is a natural transformation with every component a strong epimorphism in 𝕏.

This condition turns out to be equivalent to (C5) with the following additional requirement on A in (C5): for every object X in 𝕏 the functor A(X, −) : 𝕏 → Set is representable (see Proposition 6.8 below).

We show that (C7) is equivalent to (C4) (and hence to (C1), (C2) and (C5)) for any regular category 𝕏 with binary coproducts (Theorem 4.2). And again, if α in (C7) is an identity morphism, then (under the presence of binary products in 𝕏) we obtain the condition (C6) (since when 𝕏 has binary products and coproducts, the condition (C6) is self-dual) which defines a naturally Mal’tsev category.

2. Preliminaries

We will work in a category ℂ with binary products and finite colimits. For two objects X and Y in ℂ, by X × Y and X + Y we denote the product and coproduct, respectively, of X and Y. If X = Y then we also write X^{2} = X × X and 2X = X + X; more generally, for a natural number n, we write

X^{n} = \underbrace{X × X × ... × X}_{n} and nX = \underbrace{X + X + ... + X}_{n} .

For an n-tuple X_{1}, ..., X_{n} of objects in ℂ, and for i ∈ {1, ..., n}, we write π_{i} and ι_{i} for the i-th product projection X_{1}×...×X_{n} → X_{i} and i-th coproduct injection X_{i} → X_{1}+...+X_{n}, respectively.

2.1. Definition. Let μ : X^{3} → A and α : X → A be morphisms in ℂ. We say that μ is an approximate Mal’tsev operation with approximation α if the diagram

\xymatrix{X^{3} \ar[r]^{μ} & A \\ X^{2} + X^{2} \ar[u]^{_{(\begin{matrix}π_{1} & π_{2} & π_{2} \\ π_{2} & π_{2} & π_{1} \end{matrix})}} \ar[r]_{_{(\begin{matrix}π_{1} \\ π_{1} \end{matrix})}} & X \ar[u]_{α}}  (3)

commutes. Note that in this case α = μ(1_{X}, 1_{X}, 1_{X}), so that α is uniquely determined by μ. We say that μ is a universal approximate Mal’tsev operation when the diagram (3) is a pushout. This is the case if and only if μ is the following coequalizer:

\xymatrix{X^{2} \ar[r]^{(π_{1},π_{2},π_{2})} \ar[r]|{(π_{1},π_{1},π_{1})} \ar[r]_{(π_{2},π_{2},π_{1})} & X^{3} \ar[r]^{μ} & A}

If A = X and α = 1_{X} then μ is said to be a Mal’tsev operation on X.

We will also consider approximate Mal’tsev operations in a category ℂ with binary products, which may not have finite colimits (and in particular, binary coproducts). In this case, instead of requiring the diagram (3) to commute, we should require the equalities μ(π_{1}, π_{2}, π_{2}) = απ_{1} = μ(π_{2}, π_{2}, π_{1}) to hold, i.e. the commutativity of the two diagrams

\xymatrix{X^{3} \ar[r]^{μ} & A & X^{3} \ar[r]^{μ} & A \\ X^{2} \ar[u]^{(π_{1},π_{2},π_{2})} \ar[r]_{π_{1}} & X \ar[u]_{α} & X^{2} \ar[u]^{(π_{2},π_{2},π_{1})} \ar[r]_{π_{1}} & X \ar[u]_{α}}

2.2. Observation. A morphism μ : X^{3} → A in ℂ is an approximate Mal’tsev operation with approximation α : X → A if and only if for any object C in ℂ and for any two morphisms x, y : C → X we have μ(x, y, y) = αx = μ(y, y, x).

Note that there is a bijection between Mal’tsev operations on X and left inverses of the approximation α of the universal approximate Mal’tsev operation on X. In particular, this gives that α is a split monomorphism if and only if there exists a Mal’tsev operation on X. When ℂ = Set, every X has a Mal’tsev operation and hence α is always an injective map.

2.3. Lemma. Let X be a set and μ a universal approximate Mal’tsev operation on X with approximation α. Then for any four elements x, y, z, t ∈ X we have

μ(x, y, z) = α(t) ⇒ (x = t ∧ y = z) ∨ (x = y ∧ z = t).

Proof. This follows from the general fact that for a pushout

\xymatrix{Y \ar[r]^{m} & A \\ Z \ar[u]^{f} \ar[r]_{g} & X \ar[u]_{a}}

in Set, if a is a monomorphism then for any y ∈ Y and x ∈ X we have

m(y) = a(x) ⇒ ∃_{z∈Z}((f(z) = y) ∧ (g(z) = x)).  (4)

This general fact is well known in the case when f is also a monomorphism, since then it simply states that the diagram above is a pullback (e.g. see Corollary 1.28 in [10]). In our case f may not be a monomorphism, but then we can reduce this more general case to the case of a monomorphic f as follows: decompose f as f = f′e with f′ an injection and e a surjection. Then we can obtain the above pushout square by pasting together two pushout squares, as indicated in the diagram

\xymatrix{Y \ar[r]^{m} & A \\ • \ar[u]^{f′} \ar[r] & • \ar[u]^{f′′} \\ Z \ar[u]^{e} \ar[r]_{g} & X \ar[u]^{e′} \ar[uu]_{a}}

Since e′ is a pushout of a surjection, it is itself surjective and since a is injective and f′′e′ = a we obtain that e′ is also injective, and hence bijective. Then, without loss of generality we can assume e′ = 1_{X}, so that f′′ = a. Now the top square in the above diagram is a pullback since it is a pushout and f′ is injective. After this, the fact that e is surjective immediately gives (4) for any y ∈ Y and x ∈ X.

Suppose ℂ is a functor category ℂ = 𝔻^{𝔼}, where 𝔻 is a category with binary products, and 𝔼 is an arbitrary category.

2.4. Observations.

(a) A morphism μ : X^{3} → A in 𝔻^{𝔼} is an approximate Mal’tsev operation with approximation α : X → A, if and only if for each object E in 𝔼 the E-component μ_{E} of μ is an approximate Mal’tsev operation in 𝔻 with approximation the E-component α_{E} of α. Further, if 𝔻 has finite colimits, then μ is universal if and only if so is every μ_{E}.

(b) Suppose 𝔻 has finite colimits. If a functor X : 𝔼 → 𝔻 is given, then, taking a universal approximate Mal’tsev operation μ_{E} : (X(E))^{3} → A(E) in 𝔻 for each object E in 𝔼, results in a unique approximate Mal’tsev operation μ : X^{3} → A in 𝔻^{𝔼} such that each E-component of μ is μ_{E}, and, moreover, by (a) above, the μ so constructed is universal.

In the present paper, by a Mal’tsev category we mean a category 𝕏 satisfying (C4) from Introduction. Note that we do not assume the existence of finite limits, as this is usually the case.^{1} Rephrasing (C4) as a suitable “diagram condition”, we obtain the following 2.5. Definition. A category 𝕏 is said to be a Mal’tsev category if in 𝕏, for any commutative diagram of solid arrows

1The authors thank the Referee for reminding us to say a few words about examples of Mal’tsev categories (in the sense of Definition 2.5) without finite limits: In any non-trivial Mal’tsev variety ℂ, take any full subcategory 𝕏 which contains at least one algebra free over a non-empty set of generators, and which is not closed under certain finite limits in ℂ. Then, 𝕏 is a Mal’tsev category without finite limits, since the inclusion 𝕏 ↪ ℂ preserves internal relations and all limits which exist in 𝕏. Thus, for instance, the categories of free/infinite groups, rings, and other similar group-like structures are examples.

[figure: S; z_{1}; x_{1}; y_{2}; X; Y; S; x_{1}; y_{1}; z_{4}; R; r_{1}; r_{2}; r_{1}; r_{2}; S; z_{2}; y_{2}; x_{2}; Y; X; S; x_{2}; y_{1}; z_{3}]  (5)

with r_{1}, r_{2} jointly monomorphic, there exists the dotted arrow z_{4} making the diagram commute.

3. The first characterization theorem

In this section ℂ is the functor category ℂ = Set^{𝕏×𝕏^{op}}, where 𝕏 is an arbitrary category. We are interested in approximate Mal’tsev operations μ : X^{3} → A in ℂ, where X = hom_{𝕏}. We will consider the following condition on the approximation α : hom_{𝕏} → A of μ:

(*) In 𝕏, for any diagram of solid arrows

[figure: R; r_{1}; r_{2}; X; Y; S; q; s_{1}; s_{2}]  (6)

with r_{1}, r_{2} jointly monomorphic, the dotted arrow q exists, making the diagram commutative, if and only if there exists an element a ∈ A(S, R) such that A(1_{S}, r_{1})(a) = α_{S,X}(s_{1}) and A(1_{S}, r_{2})(a) = α_{S,Y} (s_{2}).

Notice that the “only if” part in (*) is always true (indeed, for each q there we can take a = α_{S,R}(q)).

3.1. Lemma. In a Mal’tsev category 𝕏, consider a diagram of solid arrows (6) with r_{1}, r_{2} jointly monomorphic. The following conditions are equivalent to each other:

(a) There exists the dotted arrow q making the diagram (6) commute.

APPROXIMATE MAL’TSEV OPERATIONS  159

(b) There exist morphisms z_{1}, z_{2}, z_{3} : S → R such that μ_{S,X}(r_{1}z_{1}, r_{1}z_{2}, r_{1}z_{3}) = α_{S,X}(s_{1}) and μ_{S,Y} (r_{2}z_{1}, r_{2}z_{2}, r_{2}z_{3}) = α_{S,Y} (s_{2}) where μ is the universal approximate Mal’tsev operation on hom_{𝕏} with approximation α.

Proof. (a)⇒(b) is obvious (take z_{1} = z_{2} = z_{3} = q).

(b)⇒(a): Suppose we have μ_{S,X}(r_{1}z_{1}, r_{1}z_{2}, r_{1}z_{3}) = α_{S,X}(s_{1}),

μ_{S,Y} (r_{2}z_{1}, r_{2}z_{2}, r_{2}z_{3}) = α_{S,Y} (s_{2}).

By Observations 2.4(a) we have:

• μ_{S,X} is a universal approximate Mal’tsev operation on hom_{𝕏}(S, X) with approximation α_{S,X},

• μ_{S,Y} is a universal approximate Mal’tsev operation on hom_{𝕏}(S, Y) with approxima-

tion α_{S,Y}. Now we apply Lemma 2.3 (two times) to obtain from the above equalities that one of the following four systems of equalities holds true:

(I) \{ \begin{matrix}r_{1}z_{1} & = & s_{1}, \\ r_{1}z_{2} & = & r_{1}z_{3}, \\ r_{2}z_{3} & = & s_{2}, \\ r_{2}z_{1} & = & r_{2}z_{2}; \end{matrix} (II) \{ \begin{matrix}r_{1}z_{1} & = & s_{1}, \\ r_{1}z_{2} & = & r_{1}z_{3}, \\ r_{2}z_{1} & = & s_{2}, \\ r_{2}z_{2} & = & r_{2}z_{3}; \end{matrix}

(III) \{ \begin{matrix}r_{1}z_{3} & = & s_{1}, \\ r_{1}z_{1} & = & r_{1}z_{2}, \\ r_{2}z_{3} & = & s_{2}, \\ r_{2}z_{1} & = & r_{2}z_{2}; \end{matrix} (IV) \{ \begin{matrix}r_{1}z_{3} & = & s_{1}, \\ r_{1}z_{1} & = & r_{1}z_{2}, \\ r_{2}z_{1} & = & s_{2}, \\ r_{2}z_{2} & = & r_{2}z_{3}. \end{matrix}

In each case, we obtain the desired q as follows: (I): For x_{1} = s_{1}, y_{1} = s_{2} form a commutative diagram (5), and then take q = z_{4}. (II): Take q = z_{1}. (III): Take q = z_{3}. (IV): For x_{1} = s_{2}, y_{1} = s_{1} form a commutative diagram (5), with the places of z_{1} and z_{3}

interchanged, and then take q = z_{4}.

3.2. Theorem. For a category 𝕏 the following conditions are equivalent to each other:

(a) The approximation α : hom_{𝕏} → A of the universal approximate Mal’tsev operation on hom_{𝕏} satisfies (*).

(b) There exists an approximate Mal’tsev operation on hom_{𝕏} whose approximation α satisfies (*).

(c) 𝕏 is a Mal’tsev category.

Proof. (a)⇒(b) is trivially true.

(b)⇒(c): Let μ be an approximate Mal’tsev operation on hom_{𝕏} whose approximation α : hom_{𝕏} → A satisfies (*). Consider a commutative diagram of solid arrows (5), with r_{1}, r_{2} jointly monomorphic. We want to show that there exists the dotted arrow z_{4} making the diagram (5) commute. Consider the element a = μ_{S,R}(z_{1}, z_{2}, z_{3}) in A(S, R). We have

A(1_{S}, r_{1})(a) = μ_{S,X}(r_{1}z_{1}, r_{1}z_{2}, r_{1}z_{3}) = μ_{S,X}(x_{1}, x_{2}, x_{2}) = α_{S,X}(x_{1}),
A(1_{S}, r_{2})(a) = μ_{S,Y} (r_{2}z_{1}, r_{2}z_{2}, r_{2}z_{3}) = μ_{S,Y} (y_{2}, y_{2}, y_{1}) = α_{S,Y} (y_{1}).

Now, applying (*) we get the existence of the desired z_{4}.

(c)⇒(a): Let μ : hom_{𝕏} × hom_{𝕏} × hom_{𝕏} → A be a universal approximate Mal’tsev operation on hom_{𝕏} with approximation α. Consider a diagram (6) of solid arrows with r_{1}, r_{2} jointly monomorphic. Suppose for an element a ∈ A(S, R) we have A(1_{S}, r_{1})(a) = α_{S,X}(s_{1}) and A(1_{S}, r_{2})(a) = α_{S,Y} (s_{2}). We want to show that there exists the dotted arrow q in the diagram (6) making it commutative. The fact that the approximate Mal’tsev operation μ is universal implies that its every component is a surjective map, and in particular, μ_{S,R} is surjective. So there exist morphisms z_{1}, z_{2}, z_{3} : S → R such that μ_{S,R}(z_{1}, z_{2}, z_{3}) = a. Now, we have

α_{S,X}(s_{1}) = A(1_{S}, r_{1})(a) = μ_{S,X}(r_{1}z_{1}, r_{1}z_{2}, r_{1}z_{3}),
α_{S,Y} (s_{2}) = A(1_{S}, r_{2})(a) = μ_{S,Y} (r_{2}z_{1}, r_{2}z_{2}, r_{2}z_{3}).

Applying Lemma 3.1 we obtain the existence of the desired q.

4. The second characterization theorem

In this section ℂ is the category ℂ = (𝕏^{𝕏})^{op}, where 𝕏 is a category with finite limits and binary coproducts. But we will work in the dual category ℂ^{op} = 𝕏^{𝕏}; so a (universal) approximate Mal’tsev operation μ : X^{3} → A in ℂ with approximation α : X → A now becomes a (universal) approximate Mal’tsev co-operation μ : A → 3X (in ℂ^{op}) with approximation α : A → X. In particular, we are interested in the case when X = 1_{𝕏}.

4.1. Proposition. For a finitely complete category 𝕏 with binary coproducts the following conditions are equivalent to each other:

(a) 𝕏 is a Mal’tsev category.

(b) In 𝕏, for any diagram of solid arrows

[figure: S; k_{1}; ι_{1}; ι_{2}; 2S; 2S; S; ι_{1}; ι_{1}; k_{4}; R′; r′_{1}; r′_{2}; r′_{1}; r′_{2}; S; k_{2}; ι_{2}; ι_{2}; 2S; 2S; S; ι_{2}; ι_{1}; k_{3}]  (7)

with r′_{1}, r′_{2} jointly monomorphic, there exists the dotted arrow k_{4} making the diagram (7) commute.

Proof. (a)⇒(b): The diagram (7) is a special case of the diagram (5).

(b)⇒(a): Consider a diagram of solid arrows (5). Take the pullback

\xymatrix{R′ \ar[r]^{p} \ar[d]_{r′=(r′_{1},r′_{2})} & R \ar[d]^{r=(r_{1},r_{2})} \\ 2S × 2S \ar[r]_{(\begin{matrix}x_{1} \\ x_{2} \end{matrix})×(\begin{matrix}y_{1} \\ y_{2} \end{matrix})} & X × Y}  (8)

Commutativity of the diagrams

\xymatrix{S \ar[r]^{z_{1}} \ar[d]_{(ι_{1},ι_{2})} & R \ar[d]^{r} & S \ar[r]^{z_{2}} \ar[d]_{(ι_{2},ι_{2})} & R \ar[d]^{r} \\ 2S × 2S \ar[r]_{(\begin{matrix}x_{1} \\ x_{2} \end{matrix})×(\begin{matrix}y_{1} \\ y_{2} \end{matrix})} & X × Y & 2S × 2S \ar[r]_{(\begin{matrix}x_{1} \\ x_{2} \end{matrix})×(\begin{matrix}y_{1} \\ y_{2} \end{matrix})} & X × Y}

\xymatrix{S \ar[r]^{z_{3}} \ar[d]_{(ι_{2},ι_{1})} & R \ar[d]^{r} \\ 2S × 2S \ar[r]_{(\begin{matrix}x_{1} \\ x_{2} \end{matrix})×(\begin{matrix}y_{1} \\ y_{2} \end{matrix})} & X × Y}

implies the existence of morphisms k_{1}, k_{2}, k_{3} : S → R′ such that the diagram of solid arrows (7) commutes. Now, take z_{4} = pk_{4}, where k_{4} is the dotted arrow in (7), and p is taken from the pullback (8).

4.2. Theorem. For a regular category 𝕏 with binary coproducts the following conditions are equivalent to each other:

(a) The approximation α : A → 1_{𝕏} of the universal approximate Mal’tsev co-operation on 1_{𝕏} is a natural transformation with every component a regular epimorphism in 𝕏.

(b) There exists an approximate Mal’tsev co-operation on 1_{𝕏} whose approximation α has every component a regular epimorphism in 𝕏.

(c) 𝕏 is a Mal’tsev category.

Proof. (a)⇒(b) is trivially true.

(b)⇒(c): Suppose (b) is satisfied. We show that then 4.1(b) is satisfied. Consider a diagram (7) as in 4.1(b). Let μ : A → 1_{𝕏} + 1_{𝕏} + 1_{𝕏} be an approximate Mal’tsev cooperation with approximation α whose every component is a regular epimorphism. Now, in the diagram

\xymatrix{A(S) \ar[r]^{μ_{S}} \ar[dd]_{α_{S}} & 3S \ar[dd]_{_{(\begin{matrix}ι_{1} & ι_{2} \\ ι_{2} & ι_{2} \\ ι_{2} & ι_{1} \end{matrix})}} \ar[dr]^{(\begin{matrix}k_{1} \\ k_{2} \\ k_{3} \end{matrix})} \\  &  & R′ \ar[dl]^{r′=(r′_{1},r′_{2})} \\ S \ar[r]_{(ι_{1},ι_{1})} & 2S × 2S}  (9)

the rectangle commutes because μ_{S} is an approximate Mal’tsev co-operation with approximation α_{S}, and the triangle commutes because the solid arrow part in (7) commutes. Since α_{S} is a regular epimorphism, it is also a strong epimorphism, and hence there exists f : S → R′ with r′f = (ι_{1}, ι_{1}). We then take k_{4} = f in (7).

(c)⇒(a): Let μ be the universal approximate Mal’tsev operation on 1_{𝕏} with approximation α : A → 1_{𝕏}. Then for any object S in 𝕏 the rectangle in (9) is a pullback. Form the commutative diagram (9) by taking the arrows 3S → R′ and R′ → 2S × 2S in it to be a regular epimorphism and monomorphism decomposition of the arrow 3S → 2S ×2S. Then we get a commutative diagram of solid arrows (7). If 𝕏 is a Mal’tsev category, then we get an arrow k_{4} : S → R′ with r′k_{4} = (ι_{1}, ι_{1}). Now, the fact that the rectangle in (9) is a pullback implies that the rectangle

\xymatrix{A(S) \ar[d]_{α_{S}} \ar[r]^{μ_{S}} & 3S \ar[d]^{_{(\begin{matrix}k_{1} \\ k_{2} \\ k_{3} \end{matrix})}} \\ S \ar[r]_{k_{4}} & R′}

is a pullback. Since the morphism 3S → R′ here is a regular epimorphism, we conclude that α_{S} is also a regular epimorphism.

5. Beyond Mal’tsev categories

The class of Mal’tsev categories, as well as the classes of unital and strongly unital categories in the sense of [2], and the class of subtractive categories^{2} in the sense of [7], are particular instances of a class of categories with closed relations in the sense of [8]. All results obtained in this paper (including the Appendix) can be straightforwardly extended to those general classes of categories. In particular, this means that analogous results also hold true for unital, strongly unital and subtractive categories. For them, the condition (*) remains the same; however, instead of the diagram (3) we should work with the diagrams displayed in the following table:

Unital categories  \xymatrix{X^{2} \ar[r] & A \\ X + X \ar[u]^{_{(\begin{matrix}1_{X} & 0 \\ 0 & 1_{X} \end{matrix})}} \ar[r]_{_{(\begin{matrix}1_{X} \\ 1_{X} \end{matrix})}} & X \ar[u]}  ^{μ(x,0)=x=μ(0,x)}

Strongly unital categories \xymatrix{X^{3} \ar[r] & A \\ X^{2} + X^{2} \ar[u]^{_{(\begin{matrix}π_{1} & 0 & 0 \\ π_{2} & π_{2} & π_{1} \end{matrix})}} \ar[r]_{_{(\begin{matrix}π_{1} \\ π_{1} \end{matrix})}} & X \ar[u]} ^{μ(x,0,0)=x=μ(y,y,x)}

Subtractive categories  \xymatrix{X^{2} \ar[r] & A \\ X + X \ar[u]^{_{(\begin{matrix}1_{X} & 0 \\ 1_{X} & 1_{X} \end{matrix})}} \ar[r]_{_{(\begin{matrix}1_{X} \\ 0 \end{matrix})}} & X \ar[u]}  ^{μ(x,0)=x, μ(x,x)=0}

These diagrams, in each case, naturally correspond to the term identities (displayed righthand side in the table) that characterize the corresponding varieties of universal algebras.

2The notion of a subtractive category introduced in [7] is a pointed categorical version of the notion of a subtractive variety due to A. Ursini [18].

Note also that when we replace the diagram (3) with the diagrams in the above table, we have to replace difunctionality in (C4) with M-closedness in the sense of [8], where M is the extended matrix associated with the corresponding term identities (see [8]). Note: unital, strongly unital and subtractive categories are all pointed categories, and the morphisms 0 in the diagrams in the table represent zero morphisms in a pointed category; a variety is a pointed category if and only if its theory contains a unique constant — the symbol 0 in the term equalities given in the third column represents this unique constant.

6. Appendix

In this more technical section we investigate how much (C5) is related to (C7). In particular (C7) turns out to be equivalent to (C5) when for any object X, A(X, −) is representable.

Let 𝔻 and 𝔼 be categories. Let n be a natural number n ⩾ 1 and suppose that 𝔻 has n-fold products. Then for any two functors U, V : 𝔼 → 𝔻 there is a map

Φ_{n} : Nat(U^{n}, V) → Nat((hom_{𝔼})^{n}, hom_{𝔻} ∘ (U^{op} × V)),  (10)

which assigns to each natural transformation ν : U^{n} → V the natural transformation

Φ_{n}(ν) : (hom_{𝔼})^{n} → hom_{𝔻} ∘ (U^{op} × V),
(Φ_{n}(ν))_{X,Y} : (f_{1}, ..., f_{n} : X → Y) ↦ (ν_{Y} (U(f_{1}), ..., U(f_{n})) : U(X) → V (Y)).

If, further, 𝔼 has n-fold products and U preserves them, then Φ_{n} is a bijection whose inverse (Φ_{n})^{−1} is constructed as described below:

Suppose 𝔼 has n-fold products. Note: to say that U preserves n-fold products is the same as to say that for each object Y in 𝔼, the canonical morphism

(U(π_{1}), ..., U(π_{n})) : U(Y^{n}) → (U(Y))^{n}

(where π_{i} denotes i-th product projection π_{i} : Y^{n} → Y) is an isomorphism. For each natural transformation μ : (hom_{𝔼})^{n} → hom_{𝔻} ∘ (U^{op} × V) and for each object Y in 𝔼, the morphism

((Φ_{n})^{−1}(μ))_{Y} : (U(Y))^{3} → V (Y)

is defined as the unique morphism making the triangle

\xymatrix{(U(Y))^{n} \ar[r]^{((Φ_{n})^{−1}(μ))_{Y}} & V (Y) \\ U(Y^{n}) \ar[u]^{(U(π_{1}),...,U(π_{n}))} \ar[ur]_{μ_{Y^{n},Y} (π_{1},...,π_{n})}}

commute.

6.1. Remark. The bijection (Φ_{n})^{−1} can be obtained from the bijection in Yoneda Lemma (e.g. see [13]). Let A be an arbitrary functor A : 𝔼^{op} × 𝔼 → Set. By the Yoneda Lemma for 𝔼^{op}, we have a bijection

Nat(hom_{𝔼}(−, E), K) → K(E), E ∈ 𝔼^{op}, K ∈ Set^{𝔼^{op}},

which is natural in E and in K. This gives a bijection

Nat(hom_{𝔼}(−, E^{n}), A(−, E′)) → A(E^{n}, E′), E ∈ 𝔼^{op}, E′ ∈ 𝔼,

natural in E and E′, where the product E^{n} is taken in 𝔼. This itself gives a bijection

∫_{E} Nat(hom_{𝔼}(−, E^{n}), A(−, E)) → ∫_{E} A(E^{n}, E).

Note that we also have a bijection

Nat((hom_{𝔼})^{n}, A) → ∫_{E} Nat(hom_{𝔼}(−, E^{n}), A(−, E)).

Composing these two bijections we obtain a bijection

Nat((hom_{𝔼})^{n}, A) → ∫_{E} A(E^{n}, E).

Now take A = hom_{𝔻} ∘ (U^{op} × V). Then

∫_{E} A(E^{n}, E) = Nat(U((1_{𝔼})^{n}), V),

so the last bijection above in this special case becomes a bijection

Nat((hom_{𝔼})^{n}, hom_{𝔻} ∘ (U^{op} × V)) → Nat(U((1_{𝔼})^{n}), V).

Since U preserves n-fold products, there is a canonical isomorphism U((1_{𝔼})^{n}) → U^{n} which induces a bijection

Nat(U((1_{𝔼})^{n}), V) → Nat(U^{n}, V).

Composing the last two bijections above we obtain precisely (Φ_{n})^{−1}.

6.2. Proposition. If 𝔻 has binary products then for any natural transformations ω : U^{3} → V and ν : U → V, the condition (a) below implies (b). If, further, 𝔼 has binary products and U preserves them, then (a) is equivalent to (b).

(a) ω is an approximate Mal’tsev operation on U with approximation ν.

(b) Φ_{3}(ω) is an approximate Mal’tsev operation on hom_{𝔼} with approximation Φ_{1}(ν).

Proof. (a)⇒(b): Suppose ω is an approximate Mal’tsev operation on U with approximation ν. To see that Φ_{3}(ω) is an approximate Mal’tsev operation on hom_{𝔼} with approximation Φ_{1}(ν), we take any two objects X and Y in 𝔼 and show that (Φ_{3}(ω))_{X,Y} is an approximate Mal’tsev operation on hom_{𝔼}(X, Y) with approximation (Φ_{1}(ν))_{X,Y}. Take any two elements f, g ∈ hom_{𝔼}(X, Y). We must show

(Φ_{3}(ω))_{X,Y} (f, g, g) = (Φ_{1}(ν))_{X,Y} (f) = (Φ_{3}(ω))_{X,Y} (g, g, f).

But this follows straightforwardly from the constructions of Φ_{3}(ω) and Φ_{1}(ν) and the fact that ω_{Y} is an approximate Mal’tsev operation on U(Y) with approximation ν_{Y}.

Next, assuming that 𝔼 has binary products and U preserves them, we prove (b)⇒(a): Suppose Φ_{3}(ω) is an approximate Mal’tsev operation on hom_{𝔼} with approximation Φ_{1}(ν). Then, we have to show that for each object Y in 𝔼, the morphisms ω_{Y} and ν_{Y} satisfy ω_{Y} (\overline{π}_{1}, \overline{π}_{2}, \overline{π}_{2}) = ν_{Y} \overline{π}_{1} = ω_{Y} (\overline{π}_{2}, \overline{π}_{2}, \overline{π}_{1}), where \overline{π}_{1} and \overline{π}_{2} denote product projections (U(Y))^{2} → U(Y). To show ω_{Y} (\overline{π}_{1}, \overline{π}_{2}, \overline{π}_{2}) = ν_{Y} \overline{π}_{1}, consider the diagram

\xymatrix{(U(Y))^{3} \ar[rr]^{ω_{Y}} &  & V (Y) \\  & U(Y^{2}) \ar[dl]^{(U(π_{1}),U(π_{2}))} \ar[ul]^{(U(π_{1}),U(π_{2}),U(π_{2}))} \ar[ur]^{(Φ_{3}(ω))_{Y^{2},Y} (π_{1},π_{2},π_{2})=(Φ_{1}(ν))_{Y^{2},Y} (π_{1})} \ar[dr]^{U(π_{1})} \\ (U(Y))^{2} \ar[uu]^{(\overline{π}_{1},\overline{π}_{2},\overline{π}_{2})} \ar[rr]_{_{\overline{π}_{1}}} &  & U(Y) \ar[uu]_{ν_{Y}}}

Since every triangle in this diagram commutes, and since (U(π_{1}), U(π_{2})) is an isomorphism, we obtain that the rectangle commutes. The proof of the second equality

ω_{Y} (\overline{π}_{2}, \overline{π}_{2}, \overline{π}_{1}) = ν_{Y} \overline{π}_{1}

is analogous.

We now consider a dual setting, i.e. when

𝔻 = 𝕎^{op}, 𝔼 = 𝕏^{op}, U = G^{op}, V = F^{op},

where F, G are functors F, G : 𝕏 → 𝕎 between categories 𝕏 and 𝕎. Then, for each n, the map Φ_{n} gives rise to the map

Ψ_{n} : Nat(F, nG) → Nat((hom_{𝕏})^{n}, hom_{𝕎} ∘ (F^{op} × G)),  (11)

which assigns to each natural transformation ν : F → nG the natural transformation

Ψ_{n}(ν) : (hom_{𝕏})^{n} → hom_{𝕎} ∘ (F^{op} × G),

(Ψ_{n}(ν))_{X,Y} : (f_{1}, ..., f_{n} : X → Y) ↦ ((G(f_{1}) ; ⋮ ; G(f_{n})) ν_{X} : F (X) → G(Y)).

In particular, Ψ_{n} is obtained from Φ_{n} as the composite indicated in the diagram

\xymatrix{Nat((G^{op})^{n}, F^{op}) \ar[r]^{Φ_{n}} & Nat((hom_{𝕏^{op}})^{n}, hom_{𝕎^{op}} ∘ (G × F^{op})) \ar[d]^{Σ_{n}} \\ Nat(F, nG) \ar@{.>}[r]_{Ψ_{n}} \ar[u]^{Θ_{n}} & Nat((hom_{𝕏})^{n}, hom_{𝕎} ∘ (F^{op} × G))}

where Θ_{n} and Σ_{n} are the obvious bijections.

In this dual setting we have, dually: the map Ψ_{n} is defined when n-fold coproducts exist in 𝕎, and it is a bijection when 𝕏 also has n-fold coproducts and G preserves them.

Consider natural transformations

α : hom_{𝕏} → hom_{𝕎} ∘ (F^{op} × G), μ : (hom_{𝕏})^{n} → hom_{𝕎} ∘ (F^{op} × G).

It is easy to see that μ is an approximate Mal’tsev operation on hom_{𝕏} with approximation α if and only if Σ_{n}^{−1}(μ) is an approximate Mal’tsev operation on hom_{𝕏^{op}} with approximation Σ_{1}^{−1}(α). So from Proposition 6.2 we get:

6.3. Proposition. [A dual of Proposition 6.2] If 𝕎 has binary coproducts then for any natural transformations ω : F → 3G and ν : F → G, the condition (a) below implies (b). If, further, 𝕏 has binary coproducts and G preserves them, then (a) is equivalent to (b).

(a) ω is an approximate Mal’tsev co-operation on G with approximation ν.

(b) Ψ_{3}(ω) is an approximate Mal’tsev operation on hom_{𝕏} with approximation Ψ_{1}(ν).

Note that for n = 1 the map Ψ_{1} is the same as the map Φ_{1} in (10) where this time U = F and V = G. That is, we have:

Ψ_{1} = Φ_{1} : Nat(F, G) → Nat(hom_{𝕏}, hom_{𝕎} ∘ (F^{op} × G)).

In this case we adopt the following special notation: for a natural transformation ν : F → G we denote

ν̃ = Ψ_{1}(ν) = Φ_{1}(ν).

We can now make the following

6.4. Conclusion. Let 𝕏 and 𝔻 be categories, and let U, G be functors U, G : 𝕏 → 𝔻. The map

Nat(U, G) → Nat(hom_{𝕏}, hom_{𝔻} ∘ (U^{op} × G)), ν ↦ ν̃,

is a bijection. Further, for each ν ∈ Nat(U, G) we have:

(a) If 𝕏 and 𝔻 have binary products and U preserves binary products, then there is a bijection between approximate Mal’tsev operations on U with approximation ν, and approximate Mal’tsev operations on hom_{𝕏} with approximation ν̃.

(b) If 𝕏 and 𝔻 have binary coproducts and G preserves binary coproducts, then there is a bijection between approximate Mal’tsev co-operations on G with approximation ν, and approximate Mal’tsev operations on hom_{𝕏} with approximation ν̃.

Let 𝕏 and 𝔻 be categories and let ν be a natural transformation ν : U → G between functors U, G : 𝕏 → 𝔻. The condition (*) for α = ν̃ becomes:

(*′) In 𝕏, for any diagram (6) of solid arrows with r_{1}, r_{2} jointly monomorphic, the dotted arrow q exists, making the diagram (6) commute in 𝕏, if (and only if) there exists the dotted arrow a in the diagram

[figure: G(R); G(r_{1}); G(r_{2}); G(X); G(Y); U(X); ν_{X}; U(S); a; U(s_{1}); U(s_{2}); U(Y); ν_{Y}]  (12)

making the diagram (12) commute in 𝔻.

Note that by naturality of ν, commutativity of (12) is equivalent to commutativity of the following diagram:

[figure: G(R); G(r_{1}); G(r_{2}); G(X); G(Y); G(S); G(s_{1}); U(S); a; ν_{S}; ν_{S}; G(S); G(s_{2})]  (13)

If 𝔻 has binary products, then we can further replace the above diagram with the following one:

\xymatrix{U(S) \ar[d]_{ν_{S}} \ar@{.>}[r]^{a} & G(R) \ar[d]^{(G(r_{1}),G(r_{2}))} \\ G(S) \ar[r]_{(G(s_{1}),G(s_{2}))} & G(X) × G(Y)}  (14)

After these observations we easily get the following two lemmas:

6.5. Lemma. Suppose both 𝕏 and 𝔻 have finite limits and G preserves finite limits and reflects isomorphisms. If every component of ν is a strong epimorphism in 𝔻, then α = ν̃ satisfies (*). In particular, α = ν̃ satisfies (*) when U = G and ν = 1_{U}.

6.6. Lemma. Suppose 𝕏 = 𝔻 and 𝕏 has finite products, and G = 1_{𝕏}. Then, α = ν̃ satisfies (*) if and only if every component of ν is a strong epimorphism.

Lemma 6.5 allows to deduce the following known result from Theorem 3.2:

6.7. Corollary. [5], [16] Let 𝕏 and 𝔻 be categories having finite limits and U a functor U : 𝕏 → 𝔻 which preserves finite limits and reflects isomorphisms. If there exists a Mal’tsev operation on U then 𝕏 is a Mal’tsev category.

Proof. According to Conclusion 6.4(a) there exists an approximate Mal’tsev operation on hom_{𝕏} with approximation ˜1_{U}. By Lemma 6.5, α = ˜1_{U} satisfies (*). Therefore, by Theorem 3.2, 𝕏 is a Mal’tsev category.

6.8. Proposition. For a category 𝕏 with binary coproducts the following conditions are equivalent to each other:

(a) There exists an approximate Mal’tsev operation on hom_{𝕏} whose approximation α : hom_{𝕏} → A satisfies (*) and for each object X in 𝕏 the functor A(X, −) : 𝕏 → Set is representable.

(b) There exists an approximate Mal’tsev co-operation μ on 1_{𝕏} such that every component of the approximation of μ is a strong epimorphism.

Proof. For a functor A : 𝕏^{op} × 𝕏 → Set the following conditions are equivalent to each other:

• For every object X in 𝕏 the functor A(X, −) : 𝕏 → Set is representable.

• A is naturally isomorphic to a functor of the form hom_{𝕏} ∘ (U^{op} × 1_{𝕏}), for some functor U : 𝕏 → 𝕏.

After this observation, the equivalence of (a) and (b) follows easily from Conclusion 6.4 and Lemma 6.6.

6.9. Remark. In a regular category, regular epimorphisms coincide with strong epimorphisms. So when 𝕏 is regular and has binary coproducts, the conditions 4.2(b) and 6.8(b) are identical, and from Theorem 4.2 and Proposition 6.8 we get: 𝕏 is a Mal’tsev category if and only if it satisfies 6.8(a).

6.10. Remark. Recall that to prove (b)⇒(c) in Theorem 4.2 we used Proposition 4.1. Alternatively, (b)⇒(c) in Theorem 4.2 follows from (b)⇒(a) of Proposition 6.8 and (b)⇒(c) of Theorem 3.2.

170  DOMINIQUE BOURN AND ZURAB JANELIDZE

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Laboratoire de Mathématiques Pures et Appliqués Joseph Liouville, Université du Littoral BP 699, 62228 Calais Cedex, France Department of Mathematics and Applied Mathematics, University of Cape Town Rondebosch 7701, Cape Town, South Africa

Email: Dominique.Bourn@lmpa.univ-littoral.fr
      zurabj@gmail.com

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