Theory and Applications of Categories, Vol. 14, No. 11, 2005, pp. 244–286.

ON THE REPRESENTABILITY OF ACTIONS IN A SEMI-ABELIAN CATEGORY

F. BORCEUX, G. JANELIDZE, AND G.M. KELLY

Abstract. We consider a semi-abelian category V and we write Act(G, X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(−, X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove first that the representability of Act(−, X) reduces to the preservation of binary coproducts. Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary coproducts by the functor Act(−, X) in a general semi-abelian category. Finally, we exhibit the precise form of the more involved “if and only if” amalgamation property corresponding to the representability of actions: this condition is in particular related to a new notion of “normalization of a morphism”. We provide also a wide supply of algebraic examples and counter-examples, giving in particular evidence of the relevance of the object representing Act(−, X), when it turns out to exist.

1. Actions and split exact sequences

A semi-abelian category is a Barr-exact, Bourn-protomodular, finitely complete and finitely cocomplete category with a zero object 0. The existence of finite limits and a zero object implies that Bourn-protomodularity is equivalent to, and so can be replaced with, the following split version of the short five lemma:

[figure: K; k_{1}; A_{1}; s_{1}; q_{1}; Q; α; K; k_{2}; A_{2}; s_{2}; q_{2}; Q]

given a commutative diagram of “kernels of split epimorphisms”

q_{i}s_{i} = 1_{Q}, k_{i} = Ker q_{i}, i = 1, 2

The first named author was supported by FNRS grant 1.5.168.05F; the second was partially supported by Australian Research Council and by INTAS-97-31961; the third is grateful to the Australian Research Council, a grant of whom made possible Janelidze’s visit to Sydney

Received by the editors 2005-02-24 and, in revised form, 2005-06-28. Transmitted by W. Tholen. Published on 2005-08-25. 2000 Mathematics Subject Classification: 18C10, 18D35, 18G15. Key words and phrases: semi-abelian category, variety, semi-direct product, action. \begin{matrix}c \\ ◯ \end{matrix} F. Borceux, G. Janelidze, and G.M. Kelly, 2005. Permission to copy for private use granted.

ON THE REPRESENTABILITY OF ACTIONS IN A SEMI-ABELIAN CATEGORY 245

the morphism α is an isomorphism (see [25], originally from [13]).

This implies the more precise formulation of the short five lemma, where as usual a sequence of morphisms is called exact when the image of each morphism is the kernel of the next one.

1.1. Lemma. [Short five lemma] In a semi-abelian category, let us consider a commutative diagram of short exact sequences.

[figure: 0; Y; l; B; q; H; 0; h; f; (∗); g; 0; X; k; A; p; G; 0]

1. One has always p = Coker k (and analogously, q = Coker l).

2. If g and h are isomorphisms, f is an isomorphism.

3. If g and h are monomorphisms, f is a monomorphism.

4. If g and h are regular epimorphisms, f is a regular epimorphism.

5. h is an isomorphism if and only if the square (*) is a pullback.

Proof. See e.g. [7] 4.6 and [8], 4.2.4 and 4.2.5.^{1}

The algebraic theories 𝕋 giving rise to a semi-abelian variety Set^{𝕋} of set-theoretical models have been characterized in [14]: they are the theories containing, for some natural number n ∈ ℕ

• exactly one constant 0;

• n binary operations α_{i} satisfying α_{i}(x, x) = 0;

• a (n + 1)-ary operation θ satisfying θ(α_{1}(x, y), …, α_{n}(x, y), y) = x.

For example, a theory 𝕋 with a unique constant 0 and binary operations + and − satisfying the group axioms is semi-abelian: simply put

n = 1, α(x, y) = x − y, θ(x, y) = x + y.

Now let V be an arbitrary semi-abelian category. A point over an object G of V is a triple (A, p, s), where p: A [figure]G and s: G [figure]A are morphisms in V with ps = 1_{G}. The points over G form a category Pt(G) when we define a morphism f: (A, p, s) [figure](B, q, t) to be a morphism f: A [figure]B in V for which qf = p and fs = t (see [10]). Upon choosing for each point (A, p, s) a definite kernel κ: Ker p [figure]A of p, we get a functor K: Pt(G) [figure]V sending (A, p, s) to Ker p; this functor has the left adjoint sending X to

1For the facility of the reader, we refer often to [8] with precise references, instead of sending him back to a wide number of original papers.

(G + X, (1, 0): G + X [figure]G, i: G [figure]G + X))

(where i is the coprojection), and it is monadic (see [13]). The corresponding monad on V is written as G♭−, its value at X being the (chosen) kernel G♭X of (1, 0): G + X [figure]G. It is shown in [9] that G ↦ G♭− is a functor from V to the category of monads on V.

Given a (G♭−)-algebra (X, ξ), the corresponding action ξ: G♭X [figure]X of the monad G♭− on the object X of V will also be called an action of the object G on X, or simply a G-action on X; we write Act(G, X) for the set of such actions. A morphism f: G [figure]H in V gives a morphism f♭−: G♭ − [figure]H♭− of monads, composition with which gives a morphism Act(f, X): Act(H, X) [figure]Act(G, X) of sets; so that Act(−, X) constitutes a contravariant functor from V to the category Set of sets. Our concern in this paper is with the representability of this functor; that is, with the existence of an object [X] of V and a natural isomorphism Act(G, X) ≅ V(G, [X]).

We first need an alternative description of Act(G, X) in terms of split extensions. This description, given in lemma 1.3 below, goes back to [13] and was given in more details in [9], although as part of wider calculations; so as to keep the present paper self-contained, we give here the following direct argument.

Let us call an algebra (X, ξ) for the monad G♭− simply a G-algebra, writing G-Alg for the category of these, with U: G-Alg [figure]V for the forgetful functor sending (X, ξ) to X, and with W: Pt(G) [figure]G-Alg for the canonical comparison functor having UW = K: Pt(G) [figure]V. To say that K is monadic is to say that W is an equivalence. We may of course denote a G-algebra (X, ξ) by a single letter such as C.

Given a G-algebra (Y, η) and an isomorphism f: X [figure]Y in V, there is a unique action ξ of G on X for which f: (X, ξ) [figure](Y, η) is a morphism – in fact an isomorphism – of G-algebras; we are forced to take for ξ the composite

G♭X [figure: G♭f]G♭Y [figure: η]Y [figure: f^{−1}]X.

We say that the G-action ξ – the G-structure of the algebra (X, ξ) – has been obtained by transporting along the isomorphism f the G-structure on (Y, η).

Act(G, X) is in effect the set of G-algebras with underlying object X. Write ACT(G, X) for the set whose elements are pairs (C, c) consisting of a G-algebra C together with an isomorphism c: X [figure]UC in V. There is a function ACT(G, X) [figure]Act(G, X) sending (C, c) to the G-action on X obtained by transporting along c the action of G on C; and clearly Act(G, X) is isomorphic to the quotient of ACT(G, X) by the equivalence relation ∼, where (C, c) ∼ (D, d) whenever dc^{−1} : UC [figure]UD is a morphism C [figure]D of G-algebras – that is, whenever dc^{−1} : UC [figure]UD is Uf for some f: C [figure]D (necessarily unique, and necessarily invertible) in G-Alg.

We can imitate the formation of ACT(G, X), of the equivalence relation ∼, and of the quotient set Act(G, X) = ACT(G, X)/ ∼, with any faithful and conservative functor into V in place of U. In particular, write SPLEXT(G, X) for the analogue of ACT(G, X) when U is replaced by K: Pt(G) [figure]V. An object (E, e) of SPLEXT(G, X) is an object E of Pt(G) together with an isomorphism e: X [figure]KE; we have (E, e) ∼ (H, h) when he^{−1} : KE [figure]KH is Kg for some (necessarily unique and invertible) g: E [figure]H in Pt(G); and we define SplExt(G, X) as the quotient set SPLEXT(G, X)/ ∼. Since UW = K, there is a function SPLEXT(G, X) [figure]ACT(G, X) sending (E, e) to (W E, e), which respects the equivalence relations ∼, and hence induces a function SplExt(G, X) [figure]Act(G, X; which is easily seen to be a bijection because W is an equivalence.

An object of SPLEXT(G, X) consists of an object E = (A, p, s) of Pt(G) and an isomorphism e: X [figure]KE = Ker p; equivalently, it consists of a short exact sequence

0  [figure]X [figure: k]A s ; [figure: p]G  [figure]0  (1)

where ps = 1 and where k (= κe) is some kernel of p (as distinct from the chosen kernel κ: Ker p [figure]A).

1.2. Definition. In a semi-abelian category a short exact sequence with split quotient part as in (1) is said to be a split exact sequence, and to constitute a split extension of G by X. We call a monomorphism k: X [figure]A protosplit if it forms the kernel part of such a sequence. (Note that in the abelian case, “protosplit” reduces to “split”.)

It is immediate that the elements of SPLEXT(G, X) corresponding to two such sequences (k, A, p, s) and (k′, A′, p′, s′) are equivalent under the relation ∼ precisely when there is a morphism f: A [figure]A′ of V (necessarily invertible by Lemma 1.1) satisfying fk = k′, p′f = p, and fs = s′. When this is so, the two split extensions are said to be isomorphic; thus SplExt(G, X) is the set of isomorphism classes of split extensions of G by X. Summing up, we have established:

1.3. Lemma. For objects G and X in a semi-abelian category, the comparison functor W: Pt(G) [figure]G-Alg induces a bijection

τ_{G}: SplExt(G, X) ≅ Act(G, X)  (2)

between the set of isomorphism classes of split extensions of G by X and the set of Gactions on X.

The right side here is a contravariant functor of G; we now make the left side into such a functor. Given a split extension (k, A, p, s) of G by X as in (1) and a morphism g: H [figure]G, let the pullback of p and g be given by q: B [figure]H and f: B [figure]A, let t: H [figure]B be the unique morphism with ft = sg and qt = 1, and let l: X [figure]B be the unique morphism with fl = k and ql = 0. In fact the monomorphism l is a kernel of q; for if qx = 0 we have pfx = gqx = 0, so that fx = ky for some y; whereupon fx = ky = fly while qx = 0 = qly, giving x = ly. Thus (l, B, q, t) is a split extension of H by X.

The isomorphism class of the split extension (l, B, q, t) is independent of the choice of the pullback, and depends only on the isomorphism class of (k, A, p, s); so the process gives a function

SplExt(g, X): SplExt(G, X) [figure]SplExt(H, X),

which clearly makes SplExt(−, X) into a contravariant functor from V to Set.

In proving the following proposition, we use the explicit description of the equivalence W: Pt(G) [figure]G-Alg, as given in Section 6 of [9]: W (A, p, s) is K(A, p, s) = Ker p with the G-action ζ: G♭(Ker p) [figure]Ker p where ζ is the unique morphism with kζ equal to the composite

G♭(Ker p) [figure: λ]G + (Ker p) [figure: (s, k)]A,

in which λ is the (chosen) kernel of (1, 0): G + (Ker p) [figure]G.

1.4. Proposition. The bijection τ_{G} of (2) above extends to an isomorphism

τ: SplExt(−, X) ≅ Act(−, X)  (3)

of functors.

Proof. The function SPLEXT(G, X) [figure]ACT(G, X) sends (E, e) to (W E, e), and the surjection ACT(G, X) [figure]Act(G, X) transports the structure of W F along e to obtain an action on X. Accordingly the bijection τ_{G} takes the isomorphism class of (k, A, p, s) to the action ξ: G♭X [figure]X, where kξ is the composite

G♭X [figure: λ]G + X [figure: (s, k)]A,

where λ is the kernel of (1, 0): G + X [figure]G. Now let g: H [figure]G, and let SplExt(g, X) take the isomorphism class of (k, A, p, s) to that of (l, B, q, t); as above, the image of this under τ_{H} is the action η: H♭X [figure]X where lη is the composite

H♭X [figure: λ]H + X [figure: (t, l)]B.

It follows that η is the composite

H♭X [figure: g♭X]G♭X [figure: ξ]X;

for

kξ(g♭X) = (s, k)λ(g♭X)

= (s, k)(g + X)λ by the naturality of λ = f(t, l)λ = flη = kη.

where f: A [figure]B is the morphism used above when describing the functoriality of SplExt(−, X). That is to say, η = Act(g, X)ξ, as desired.

As we said, our concern in this paper is with the representability of the functor Act(−, X); that is, with the existence of an object [X] of V and a natural isomorphism

Act(G, X) ≅ V(G, [X]);

this is a very strong property. In fact, from now on, we shall always work with the isomorphic – but more handy – functor SplExt(−, X) (see proposition 1.4).

Let us give at once examples of such situations.

1.5. Proposition.

1. When V is the semi-abelian category of groups, each functor Act(−, X) is representable by the group Aut(−, X) of automorphisms of X.

2. When V is the semi-abelian category of Lie algebras on a ring R, each functor Act(−, X) is representable by the Lie algebra Der(X) of derivations of X.

3. When E is a cartesian closed category and V is the corresponding category of internal groups (respectively, internal Lie algebras), each functor SplExt(−, X) is still representable.

4. When E is a topos with Natural Number Object and V is the corresponding category of internal groups (respectively, internal Lie rings), V is semi-abelian and each functor Act(−, X) is representable.

Proof. Statements 1, 2, 3 are reformulations of well-known results, as explained in [9]. Notice that in condition 3 of proposition 1.5, the category V is generally not semi-abelian: thus the functor Act(−, X) does not exist in general, while the functor SplExt(−, X) still makes sense.

In statement 4, the theory 𝕋 of internal groups (resp. internal Lie rings) admits a finite presentation. Therefore, the corresponding category E^{𝕋} of models in a topos E with Natural Number Object is finitely cocomplete (see [31]). Trivially, E^{𝕋} is pointed. It is exact since so is E (see [2] 5.11). It is protomodular by [8] 3.1.16. It is thus semi-abelian. One concludes by statement 3 and proposition 1.4.

In this paper, we consider first a certain number of other basic examples, where the functor Act(−, X) is representable by an easily describable object. And next we switch to the main concern of the paper, namely, the proof of a general representability theorem for Act(−, X).

2. Associative algebras

The developments in this section have non-trivial intersections with several considerations in [32] and [3].

We fix once for all a base ring R, which is commutative and unital. Every “algebra” considered in this section is an associative R-algebra, not necessarily commutative, not necessarily unital; every morphism is a morphism of such algebras. Analogously, given such an algebra A, the term “ left A-module” will always mean an A-R-bimodule, and analogously on the right.

We write simply Alg for the category of R-algebras. This category is semi-abelian (see [14]), thus it is equipped with a notion of semi-direct product and a notion of action of an algebra G on an algebra X. Of course when R = ℤ, the category Alg reduces to the category Rg of rings.

2.1. Proposition. For a fixed algebra G, there is an equivalence of categories between

1. the category of G-bialgebras;

2. the category Pt(G) of points over G in Alg.

Proof. By a G-bialgebra X, we mean an algebra X equipped with the structure of a G-bimodule and satisfying the additional algebra axioms

g(xx′) = (gx)x′, (xg)x′ = x(gx′), (xx′)g = x(x′g)

for g ∈ G and x, x′ ∈ X.

Given a G-bialgebra X, define A to be the semi-direct product G ⋉ X, which is the cartesian product of the corresponding R-algebras, equipped with the multiplication

(g, x)(g′, x′) = (gg′, gx′ + xg′ + xx′).

We obtain a point p, s: G ⋉ X [figure]G by defining

p(g, x) = g, s(g) = (g, 0).

Notice that X = Ker p.

Conversely, a split epimorphism p, s: A[figure]G of R-algebras is in particular a split epimorphism of R-modules, thus A ≅ G × X as an R-module, with X = Ker p. Notice that given x ∈ X and g ∈ G,

p(s(g)x) = ps(g)p(x) = g0 = 0

thus s(g)x ∈ X = Ker p. Analogously, xs(g) ∈ X. The actions of G on X are then given by

gx = s(g) · x, xg = x · s(g).

Proposition 2.1 shows thus that the notion of algebra action, in the sense of the theory of semi-abelian categories, is exactly given by the notion of G-bialgebra structure on an algebra X. In order to study the representability of the functor Act(−, X) for an algebra X, we prove first the following lemma:

2.2. Lemma. Let X be an algebra. Write LEnd(X) and REnd(X) for, respectively, the algebras of left-X-linear and right-X-linear endomorphisms of X, with the composition as multiplication. Then

[X] = {(f, g)|∀x, x′ ∈ X f(x) · x′ = x · g(x′)} ⊆ LEnd(X)^{op} × REnd(X)

is a subalgebra of the product.

Proof. This is routine calculation.

2.3. Proposition. Given an algebra X, the functor

Act(−, X): Alg [figure]Set

is representable by the algebra [X] of lemma 2.2 as soon as

∀f ∈ LEnd(X) ∀g ∈ REnd(X) fg = gf.

Proof. It is immediate to observe that a G-bialgebra structure on X is the same thing as two algebra homomorphisms

λ: G  [figure]LEnd(X, X)^{op}, ρ: G  [figure]REnd(X, X)

satisfying the additional conditions

1. ∀g ∈ G ∀x, x′ ∈ X λ(g)(x) · x′ = x · ρ(g)(x′);

2. ∀g, g′ ∈ G ∀x ∈ X λ(g)(ρ(g′)(x)) = ρ(g′)(λ(g)(x)).

The first condition means simply that the pair (λ, ρ) factors through the subalgebra [X]. The second condition holds by assumption on X.

Writing IJ for the usual multiplication of two-sided ideals in an algebra, we get an easy sufficient condition for the representability of the functor Act(−, X):

2.4. Proposition. Let X be an algebra such that XX = X. Then the functor

Act(−, X): Alg [figure]Set

is representable.

Proof. The assumption means that every element x ∈ X can be written as x = ∑_{i=1}^{n} y_{i}z_{i}, with n ∈ ℕ and y_{i}, z_{i} ∈ X. Then given f ∈ LEnd(X) and g ∈ REnd(X), we get

f(g(x)) = f (g (∑_{i=1}^{n} y_{i}z_{i})) = f (∑_{i=1}^{n} g(y_{i})z_{i}) = ∑_{i=1}^{n} g(y_{i})f(z_{i})

= g (∑_{i=1}^{n} y_{i}f(z_{i})) = g (f (∑_{i=1}^{n} y_{i}z_{i})) = g(f(x))

thus fg = gf and one concludes by proposition 2.3.

Things become more simple in the category ComAlg of commutative algebras (without necessarily a unit). The results could be deduced from those for arbitrary algebras, but a direct argument is almost as short and more enlightening.

2.5. Proposition. For a fixed commutative algebra G, there is an equivalence of categories between

1. the category of commutative G-algebras;

2. the category Pt(G) of points over G in ComAlg.

Proof. Analogous to that of proposition 2.1.

Proposition 2.5 shows thus that the notion of commutative algebra action, in the sense of the theory of semi-abelian categories, is exactly given by the notion of G-algebra structure on a commutative algebra X.

2.6. Theorem. Given a commutative algebra X, the following conditions are equivalent:

1. the functor

Act(−, X): ComAlg [figure]Set

is representable;

2. the algebra End(X) of X-linear endomorphisms of X is commutative.

In these conditions, the functor Act(−, X) is represented by End(X).

Proof. (2 ⇒ 1). It is immediate to observe that a G-algebra structure on X is the same thing as an algebra homomorphism G [figure]End(X), where End(X) is equipped with the pointwise R-module structure and the composition as multiplication. By proposition 2.5, the algebra End(X) represents the functor Act(−, X) as soon as this algebra is commutative.

Conversely, suppose that the functor Act(−, X) is representable by a commutative algebra [X]. Proposition 2.5 and the observation at the beginning of this proof show now the existence of natural isomorphisms of functors

ComAlg(−, [X]) ≅ Act(−, X) ≅ Alg(−, End(X)).

In particular, the identity on [X] corresponds by these bijections to an algebra homomorphism u: [X] [figure]End(X). For every commutative algebra G, composition with u induces thus a bijection

Alg(G, [X]) = ComAlg(G, [X]) [figure: ≅]Alg(G, End(X)).

The free non necessarily commutative algebra on one generator is the algebra R^{∗}[t] of polynomials with coefficients in R and a zero constant term. But this algebra is commutative, thus can be chosen as algebra G in the bijection above. And since it is a strong generator in the category Alg of all algebras, u is an isomorphism.

ON THE REPRESENTABILITY OF ACTIONS IN A SEMI-ABELIAN CATEGORY 253

Again we deduce: 2.7. Proposition. Let X be a commutative algebra such that XX = X. Then the algebra End(X) is commutative and represents the functor

Act(−, X): ComAlg [figure]Set.

Proof. See the proof of proposition 2.4.

Let us now consider some straightforward examples of interest. 2.8. Proposition. When the commutative algebra X has one of the following properties:

• X is Taylor-regular;

• X is pure;

• X is von Neumann-regular;

• X is Boolean;

• X is unital; the algebra End(X) is commutative and represents the functor

Act(−, X): ComAlg [figure]Set.

Proof. An X-module M is Taylor-regular (see [42]) when the scalar multiplication X ⊗_{X} M [figure]M is an isomorphism. Putting M = X, one concludes by proposition 2.7 since the image of the multiplication is precisely the ideal XX.

An ideal I ◁ X is pure (see [5]) when

∀x ∈ I ∃ε ∈ I x = xε.

Putting I = X, we get a special case of a Taylor-regular algebra.

The algebra X is von Neumann regular when

∀x ∈ X ∃y ∈ X x = xyx.

This is a special case of a pure algebra: put ε = yx.

The algebra X is Boolean when

∀x ∈ X xx = x.

This is a special case of a von Neumann regular algebra: put y = x.

Finally every unital algebra is pure: simply choose ε = 1.

3. Boolean rings

This section studies the case of the category BooRg of Boolean rings. We know already that plain ring actions on a Boolean ring are representable (see proposition 2.8). One has also:

3.1. Proposition. Given a Boolean ring X, the ring End(X) is Boolean and still represents the functor

Act(−, X): BooRg [figure]Set.

Proof. The ring End(X) is Boolean, simply because its multiplication coincides with the pointwise multiplication. Indeed, given f, h ∈ End(X)

f(h(x)) = f(h(xx)) = f(xh(x)) = f(x)h(x).

Notice moreover that computing (x + x)^{2} yields at once x + x = 0.

We know by proposition 2.5 that every split exact sequence of rings has the form

0  [figure]X [figure: k]G ⋉ X s ; [figure: p]G  [figure]0

for some G-algebra structure on X. When G and X are Boolean, then G ⋉ X is Boolean as well since

(g, x) = (g, x)^{2} = (g^{2}, gx + gx + x^{2}) = (g, gx + gx + x) = (g, 0 + x) = (g, x).

Thus every split exact sequence in ComRg with G and X Boolean is a split exact sequence in BooRg. This proves that the functor Act(−, X) on BooRg is the restriction of the functor Act(−, X) on ComRg. One concludes by proposition 2.6, since End(X) is Boolean.

Let us also mention here a useful result, which will turn out to have close connections with our general representability theorem (see proposition 6.2).

3.2. Proposition. The category BooRg of Boolean rings satisfies the amalgamation property (see definition 6.1).

Proof. A Boolean algebra can be defined as a Boolean ring with unit (see [4]); the correspondence between the various operations is given by

x ∨ y = x + xy + y, x ∧ y = xy, x + y = (x ∧ ¬y) ∨ (¬x ∧ y).

The category Bool of Boolean algebras is thus a subcategory of BooRg: the subcategory of Boolean rings with a unit and morphisms preserving that unit.

Notice further that writing 2 = {0, 1} for the two-element Boolean algebra, BooRg is equivalent to the slice category Bool/2. The morphisms f: B [figure]2 of Boolean algebras correspond bijectively with the maximal ideals f^{−1}(0) ⊆ B; and each (maximal) ideal is a Boolean ring. Conversely, given a Boolean ring R without necessarily a unit, R is a maximal ideal in the following Boolean ring \overline{R} = R × \{0, 1\}, admitting (0, 1) as a unit:

(r, n) + (s, m) = (r + s, n + m), (r, n) × (s, m) = (rs + nr + ms, nm).

The category Bool of Boolean algebras satisfies the amalgamation property (see [20] or [30]), from which each slice category Bool/B does, thus in particular the category BooRg of Boolean rings.

4. Commutative von Neumann regular rings

A ring X (without necessarily a unit) is von Neumann regular when

∀x ∈ X ∃y ∈ X xyx = x.

Putting x^{∗} = yxy one gets further

x = xx^{∗}x, x^{∗} = x^{∗}xx^{∗}.

In the commutative case, an element x^{∗} with those properties is necessarily unique (see for example [27], V.2.6). Indeed if x′ is another such element

x^{∗} = x^{∗}xx^{∗} = x^{∗}xx′xx^{∗} = x^{∗}xx^{∗}x′x = x^{∗}x′x

and analogously starting from x′. This proves that the theory of commutative von Neumann regular rings is algebraic and can be obtained from the theory of commutative rings by adding a unary operation ( )^{∗} satisfying the two axioms above. This is of course a semi-abelian theory, since it contains a group operation and has a unique constant 0 (see [14]).

The uniqueness of x^{∗} forces every ring homomorphism between commutative regular rings to preserves the operation ( )^{∗}. Thus the category ComRegRg of commutative regular rings is a full subcategory of the category ComRg of commutative rings.

Let us first summarize several well-known facts (see, e.g.[38]).

4.1. Lemma. Writing a, a_{i}, b, c, e for elements of a commutative von Neumann regular ring R:

1. (ab)^{∗} = a^{∗}b^{∗};

2. ∀a ∃e e^{2} = e, e = e^{∗}, a = ae;

3. ∀a_{1}, …, a_{n} ∃e e^{2} = e, e = e^{∗}, a_{1}e = a_{1}, …, a_{n}e = a_{n};

4. (a = b) ⇔ (∀c ac = bc);

Proof. (1) follows at once from the uniqueness of (ab)^{∗}. For (2), choose e = a^{∗}a. Given a, b with corresponding idempotents e, e′ as in condition 2, put e = e + e′ − ee′; iterate the process to get condition 3; put c = e to get condition 4.

4.2. Corollary. Every finitely generated object R of the category of commutative von Neumann regular rings is a unital ring.

Proof. Write a_{1}, …, a_{n} for a family of generators; choose e as in condition 3 of lemma 4.1. It suffices to prove that e remains a unit for every element constructed form the generators a_{i} and the operations +, −, × and ( )^{∗}. Only the last case requires a comment: if xe = e, then

x^{∗}e = x^{∗}xx^{∗}e = x^{∗}xex^{∗} = x^{∗}xx^{∗} = x^{∗}.

Proposition 2.8 gives us a first bit of information concerning the representability of actions for von Neumann regular rings. Let us observe further that:

4.3. Lemma. When X is a commutative von Neumann regular ring, the ring End(X) of X-linear endomorphisms of X is still a commutative von Neumann regular ring.

Proof. By proposition 2.8, End(X) is commutative. Given f ∈ End(X), define f^{∗}(x) = (f(x^{∗}))^{∗}; let us prove that this makes End(X) a von Neumann regular ring.

First,

f^{∗}(ab) = (f((ab)^{∗}))^{∗} = (f(a^{∗}b^{∗}))^{∗} = (a^{∗}f(b^{∗}))^{∗} = a^{∗∗}(f(b^{∗}))^{∗} = af^{∗}(b).

Next

f^{∗}(a + b)c = f^{∗}((a + b)c) = (a + b)f^{∗}(c) = af^{∗}(c) + bf^{∗}(c)
= f^{∗}(ac) + f^{∗}(bc) = f^{∗}(a)c + f^{∗}(b)c = (f^{∗}(a) + f^{∗}(b))c

and so by lemma 4.1.4, f^{∗} ∈ End(X).

It remains to observe that ff^{∗}f = f. Given a ∈ R and e as in lemma 4.1.2, we have

f^{∗}(e) = (f(e^{∗}))^{∗} = (f(e))^{∗}.

Given two endomorphisms f, g ∈ End(X) we have also

fg(e) = fg(ee) = f(eg(e)) = f(e)g(e)

and therefore

ff^{∗}f(e) = f(e)f^{∗}(e)f(e) = f(e)(f(e))^{∗}f(e) = f(e).

Finally

ff^{∗}f(a) = ff^{∗}f(ae) = aff^{∗}f(e) = af(e) = f(ae) = f(a).

4.4. Lemma. Consider a split exact sequence

0  [figure]X [figure: k]A s ; [figure: q]G  [figure]0

in the category of commutative rings. When X and G are von Neumann regular rings, A is a von Neumann regular ring as well.

Proof. In the locally finitely presentable category ComRegRg of commutative von Neumann regular rings, every object G is the filtered colimit (σ_{i}: G_{i} [figure]G)_{i∈I} of its finitely generated subobjects. But filtered colimits are computed as in the category of sets, thus are also filtered colimits in the category of all commutative rings.

Pulling back the split exact sequence of the statement along each morphism σ_{i} yields a filtered diagram of split exact sequences, still with the kernel X (see lemma 1.1.5).

0  [figure]X [figure: k_{i}]A_{i} s_{i} ; [figure: q_{i}]G_{i}  [figure]0.

By universality of filtered colimits, A ≅ colim_{i∈I} A_{i}. If we can prove that each A_{i} is a von Neumann regular ring, the same will hold for A, as a filtered colimit of von Neumann regular rings. But each ring G_{i} is unital by corollary 4.2. So it suffices to prove the statement in the special case where the ring G is unital.

Let us prove next that we can reduce further the problem to the case where both G and X are unital. So we suppose already that G is unital and, for simplicity, we view both s and k as canonical inclusions.

Write Xe for the ideal of X generated by an idempotent element e. If e′ is another idempotent element such that ee′ = e, then Xe ⊆ Xe′. By lemma 4.1.3, the family of ideals Xe, with e = e^{2} ∈ R, is thus filtered. But still by lemma 4.1.2, the ring X is generated by its idempotent elements. Thus finally X is the filtered union of its principal ideals Xe with e idempotent. Of course, each of these ideals is a unital ring: the unit is simply e.

For each e = e^{2} ∈ X, we consider further

A_{e} = {xe + g|x ∈ X, g ∈ G} ⊆ A.

We observe that:

1. Since X is an ideal in A, each A_{e} is a subring of A.

2. Each ring A_{e} still contains G, so that the pair (q, s) restricts as a split epimorphism

q_{e}, s_{e}: A_{e} [figure]G.

3. Since q(xe + g) = g,

Ker q_{e} = Xe = {xe|x ∈ X} = {x|x ∈ X, xe = x}.

4. Since A ≅ G × X as abelian groups (see propositions 2.5, 2.1) and X is the filtered union of the various Xe, A is the filtered union of the various A_{e} ≅ G × Xe.

By this last observation, it suffices to prove that each A_{e} is a von Neumann regular ring. And this time we have a split exact sequence

0  [figure]Xe  [figure]A_{e} s_{e} ; [figure: q_{e}]G  [figure]0

with both Xe and G commutative von Neumann regular rings with a unit.

So we have reduced the problem to the case where both X and G admit a unit. In that case we shall prove that the ring A is isomorphic to the ring G × X. And since the product of two von Neumann regular rings is trivially a von Neumann regular ring, the proof will be complete.

We know already that A and G × X are isomorphic as abelian groups (see propositions 2.5, 2.1). We still view k and s as canonical inclusions and we write u ∈ X, v ∈ G for the units of these two rings. Since each element of A can be written as x + g with x ∈ X and g ∈ G, it follows at once that e = u + v − uv ∈ A is a unit for the ring A:

(x + g)(u + v − uv) = x + xv − xv + gu + g − gu = x + g.

Now since u and e − u are idempotent elements of A, the morphism

A [figure]A(e − u) × Au, a ↦ (a(e − u), au)

is an isomorphism of rings, with inverse

A(e − u) × Au [figure]A, (a, b) ↦ a + b.

It remains to prove that we have ring isomorphisms

A(e − u) ≅ G, Au ≅ X.

The second isomorphism is easy: u is the unit of X and X is an ideal of A, thus

X = Xu ⊆ Au ⊆ X.

To prove the first isomorphism, notice first that q(u) = 0 implies q(e) = q(v) = v. Consider then the mapping

G [figure]A(e − u), g ↦ g(e − u)

which is a ring homomorphism, since e − u is idempotent. This mapping is injective because g(e − u) = 0 forces ge = gu ∈ X and thus

g = gv = q(g)q(e) = q(ge) = 0.

The mapping is also surjective because every a ∈ A can be written a = g + x, with g ∈ G and x ∈ X, and

a(e − u) = (g + x)(e − u) = g(e − u) + xe − xu = g(e − u) + x − x = g(e − u).

4.5. Proposition. Let V be the semi-abelian category of commutative von Neumann regular rings. For every object X ∈ V, the functor Act(−, X) is representable by the commutative von Neumann regular ring End(X) of X-linear endomorphisms of X.

Proof. By lemma 4.4, the functor Act(−, X) of the statement is the restriction of the corresponding functor defined on the category of all commutative rings. Since End(X) is a von Neumann regular ring by lemma 4.3, we conclude by proposition 2.8.

As for Boolean rings, let us conclude this section with proving the amalgamation property in the category of commutative von Neumann regular rings.

4.6. Lemma. The category of (not necessarily unital) commutative von Neumann regular rings satisfies the amalgamation property.

Proof. In a locally finitely presentable category, every finite diagram can be presented as the filtered colimit of a family of diagrams of the same shape, whose all objects are finitely presentable (see [19], the uniformization lemma). Taking the images of the various canonical morphisms, we conclude that every finite diagram is the filtered colimit of a family of diagrams with the same shape, whose all objects are finitely generated subobjects of the original ones.

Consider now a pushout δα = γβ of commutative von Neumann regular rings, with α and β injective. Applying the argument above to the diagram (α, β), write it as a filtered colimit of diagrams (α_{i}, β_{i}), with each X_{i} a finitely generated subobject of X and analogously for A_{i} and B_{i}.

[figure: X; α; A; X_{i}; α_{i}; A_{i}; β; δ; β_{i}; δ_{i}; B; γ; C; B_{i}; γ_{i}; C_{i}]

The morphisms α_{i} and β_{i} are still injective, as restrictions of α and β; define (C_{i}, δ_{i}, γ_{i}) to be their pushout (which of course is still finitely generated). The pushout δα = γβ is the filtered colimit of the pushouts δ_{i}α_{i} = γ_{i}β_{i}; thus if γ_{i} and δ_{i} turn out to be monomorphisms, so are δ and γ. So, it suffices to prove the amalgamation property for finitely generated commutative regular rings. By corollary 4.2, we have reduced the problem to the case where the rings are unital.

Let us thus assume that X, A, B have a unit. Of course, α and β have no reason to preserve the unit. Writing Ra for the principal ideal generated by an element a of a commutative ring R, we consider the following squares, where the various morphisms are the restrictions of α, β, γ, δ.

[figure: X; α_{1}; Aα(1); 0; A(1 − α(1)); β_{1}; δ_{1}; δ_{2}; Bβ(1); γ_{1}; Cδα(1); B(1 − β(1)); γ_{2}; C(1 − δα(1))]

Now given an idempotent element e in a commutative ring R, the ideal Re is always a unital ring (with unit e) and is also a retract of R, with the multiplication by e as a retraction. Moreover when R is regular, so is every ideal I ◁ R, since i ∈ I implies i^{∗} = i^{∗}ii^{∗} ∈ I. Thus the two squares above are still pushouts of commutative regular rings, as retracts of the pushout δα = γβ.

But the left hand square is now a pushout in the category of unital commutative von Neumann regular rings and morphisms preserving the unit. This category satisfies the amalgamation property (see [18] or [30]), thus δ_{1} and γ_{1} are injective. And the right hand pushout is in fact a coproduct: therefore δ_{2} and γ_{2} are injective as well, with retractions (id, 0) and (0, id).

Finally, if R is a commutative ring with unit and e ∈ R is idempotent, as already observed in the proof of lemma 4.4, the morphism

R [figure]Re × R(1 − e), r ↦ (re, r(1 − e))

is an isomorphism: it is trivially injective and the pair (ue, v(1 − e)) is the image of ue + v(1 − e). Via such isomorphisms, we conclude that δ ≅ δ_{1} × δ_{2} and γ ≅ γ_{1} × γ_{2}, thus δ and γ are injective.

5. Locally well-presentable semi-abelian categories

We switch now to the proof of a general representability theorem for the functors SplExt(−, X). We shall prove such a theorem for a very wide class of semi-abelian categories V: the locally well-presentable ones. For such categories, we reduce first the representability of the functors

SplExt(−, X): V [figure]Set

to the preservation of binary coproducts.

5.1. Definition. A category V is locally well-presentable when

1. V is locally presentable;

2. in V, finite limits commute with filtered colimits.

Of course every locally finitely presentable category is locally well-presentable. But also all Grothendieck toposes are locally well-presentable (see [22], or [6] 3.4.16) and these are generally not locally finitely presentable. In fact, the models of a semi-abelian algebraic theory in a Grothendieck topos E constitute always a semi-abelian locally wellpresentable category (see proposition 5.2). Putting E = Set, this contains in particular the case of the semi-abelian varieties of universal algebra.

5.2. Proposition. The category E^{𝕋} of models of a semi-abelian theory 𝕋 in a Grothendieck topos E is locally well-presentable.

Proof. Trivially, E^{𝕋} is pointed. It is exact since so is E (see [2] 5.11). It is protomodular by [8] 3.1.16. It is also semi-abelian locally presentable (see [22] or [6] 3.4.16 and [1] 2.63); thus in particular it is complete and cocomplete. So E^{𝕋} is already semi-abelian by [25] 2.5. Finally if E is the topos of sheaves on a site (C, T ), then E^{𝕋} is a localization of the category of 𝕋-models in the topos of presheaves Set^{C^{op}}. In the case of presheaves, finite limits and filtered colimits of 𝕋-models are computed pointwise, thus commute. And the reflection to the category E^{𝕋} preserves finite limits and filtered colimits.

The following two results are essentially part of the “folklore”, but we did not find an explicit reference for them. Of course when we say that a contravariant functor preserves some colimits, we clearly mean that it transforms these colimits in limits.

5.3. Proposition. Let V be a locally presentable category. A contravariant functor F: V [figure]Set is representable if and only if it preserves small colimits.

Proof. The category V is cocomplete and has a generating set; it is also co-well-powered (see [1], 1.58). One concludes by [29], 4.90.

Let us recall that a finitely complete category V is a Mal’tsev category when every reflexive relation in V is at once an equivalence relation (see [15], [16], [17], [35]. Semiabelian categories are Mal’tsev categories (see [25]).

5.4. Proposition. Let V be a finitely cocomplete Barr-exact Mal’tsev category. A contravariant functor F: V [figure]Set preserves finite colimits if and only if it preserves

1. the initial object;

2. binary coproducts;

3. coequalizers of kernel pairs.

When the category V is also locally presentable, the functor F is representable when, moreover, it preserves

4. filtered colimits.

Proof. Conditions 1 and 2 take care of all finite coproducts. But in a category with finite coproducts, every finite colimit can be presented as the coequalizer of a pair of morphisms with a common section (see [33], exercise V-2-1). Given such a pair (u, v) with common section r as in diagram 1, consider the image factorization (u, v) = ρπ. By assumption, the composite (u, v)r is the diagonal of B × B. Thus R is a reflexive relation on B and by the Mal’tsev property, an equivalence relation. By Barr-exactness of V (see [2]), R is a kernel pair relation. Since π is an epimorphism, one has still q = Coker (p_{1}ρ, p_{2}ρ) and thus (p_{1}ρ, p_{2}ρ) is the kernel pair of q.

Consider now for simplicity the covariant functor

F: V [figure]Set^{op}.

[figure: A; u; v; r; B; q; Q; π; (u, v); p_{1}; p_{2}; R; ρ; B × B] Diagram 1

By condition 4 in the statement, F preserves regular epimorphisms, but also the coequalizer q = Coker (p_{1}ρ, p_{2}ρ). Thus F (π) is an epimorphism and therefore

F (q) = Coker (F (p_{1}ρ), F (p_{2}ρ))
= Coker (F (p_{1}ρ)F (π), F (p_{2}ρ)F (π))
= Coker (F (u), F (v)).

This concludes the proof that F preserves all finite colimits.

Suppose next that V is also locally presentable. An arbitrary colimit is the filtered colimit of its finite subcolimits, thus condition 4 forces the preservation of all small colimits. By proposition 5.3, F is then representable.

We now switch back to the functor SplExt(−, X).

5.5. Lemma. Let V be a semi-abelian category. The functor SplExt(−, X) preserves the initial object.

Proof. When G = 0 in the split exact sequence of definition 1.2, k is an isomorphism and thus SplExt(0, X) is a singleton.

5.6. Lemma. Let V be a semi-abelian category. The functor SplExt(−, X) preserves the coequalizers of kernel pairs.

Proof. In V, let us consider a regular epimorphism γ and its kernel pair (u, v). We must prove to have an equalizer in Set

SplExt(G, X)[figure: SplExt(γ, X)]SplExt(H, X)SplExt(u, X) [figure: SplExt(v, X)]SplExt(P, X).

We prove first the injectivity of SplExt(γ, X). Given a point (A, p, s) over G, consider its pullback (B, q, t) over H, as in the following diagram.

[figure: C; u′; v′; B; γ′; A; w; r; q; t; p; s; P; u; v; H; γ; G]

Since γu = γv, pulling back (B, q, t) along u or v yields the same point (C, w, r) over P. Now γ′ is a strong epimorphism since so is γ, while (u′, v′) is the kernel pair of γ′ since (u, v) is the kernel pair of γ. Therefore γ′ = Coker (u′, v′). As a consequence, (A, p, s) is indeed entirely determined via a coequalizer process from the left hand square. Thus if two points over G have the same pullback over H, they are isomorphic.

For the surjectivity, still using the same diagram, consider now a split exact sequence (l, q, t) ∈ SplExt(H, X) whose pullbacks along u and v are the same: let us write (m, w, r) ∈ SplExt(P, X) for this pullback. We put further γ′ = Coker (u′, v′) and p, s are the factorizations of q, t through the cokernels. Trivially ps = 1_{G}.

By a well-known Barr–Kock result (see [2], 6.10), the square pγ′ = γq is a pullback since so are the squares qu′ = uw and qv′ = vw. By lemma 1.1, the kernel k = Ker p is isomorphic to l = Ker q, thus (k, p, l) ∈ SplExt(G, X). So (l, q, t) is the image of (k, p, s) under SplExt(γ, X).

5.7. Lemma. Let V be a locally well-presentable semi-abelian category. The functor SplExt(−, X) preserves filtered colimits.

Proof. This is an immediate consequence of the commutation between filtered colimits and kernels in V. Notice that a filtered colimit of a constant diagram on X is again X.

As a conclusion of the various results of this section, we get:

5.8. Theorem. Let V be a semi-abelian category. For a fixed object X ∈ V, the following conditions are equivalent:

1. the functor SplExt(−, X) preserves binary coproducts;

2. the functor SplExt(−, X) preserves finite colimits.

When V is also locally well-presentable, those conditions are further equivalent to:

3. the functor SplExt(−, X) is representable.

6. A necessary condition and some sufficient conditions

Let us first recall the standard form of the amalgamation property.

6.1. Definition. In a category with pushouts, two monomorphisms l, m with the same domain X satisfy the amalgamation property when in the pushout square

[figure: X; n_{1}; S_{1}; n_{2}; σ_{1}; S_{2}; σ_{2}; S_{1} +_{X} S_{2}]

the morphisms σ_{1} and σ_{2} are still monomorphisms.

Here is at once a necessary condition for the representability of actions.

6.2. Proposition. Let X be an object of a semi-abelian category V. If the functor SplExt(−, X) preserves binary coproducts (in particular, when it is representable), the amalgamation property holds in V for protosplit monomorphisms with domain X (see definition 1.2).

Proof. Start with two split exact sequences (l, q, t) and (m, r, w) as in diagram 2. Since the functor SplExt(−, X) preserves binary coproducts, there is a unique split exact

[figure: B; t; q; G; l; s_{B}; σ_{G}; X; k; A; s; p; G + H; m; s_{C}; σ_{H}; C; r; w; H] Diagram 2

sequence (k, p, s) such that the squares σ_{G}q = ps_{B} and σ_{H}w = ps_{C} are pullbacks. The morphisms σ_{G} and σ_{H} are monomorphisms, with retractions (1_{G}, 0) and (0, 1_{H}). So s_{B} and s_{C} are monomorphisms and the square s_{B}l = k = s_{C}m is commutative. Thus s_{B} and s_{C} factor through the pushout of l, m and, since s_{B}, s_{C} are monomorphisms, so are the morphisms of this pushout.

Let us immediately exhibit a sufficient condition for the representability of actions: this result is in fact a corollary of the more general considerations of section 4. We use the notation of definition 6.1.

6.3. Proposition. Let V be a semi-abelian category. Suppose that given two protosplit monomorphisms n_{1}, n_{2} with domain X, the diagonal n = σ_{i}n_{i} of their pushout is a normal monomorphism. Then the functor SplExt(−, X) preserves finite colimits. Moreover, when V is also locally well-presentable, the functor SplExt(−, X) is representable.

Proof. Anticipating on definition 7.1 and lemma 7.2, the morphism n is its own universal normalization. Thus the conditions of theorem 8.5 are satisfied and the functors SplExt(−, X) are representable.

Notice that proposition 6.3 does not require the morphisms σ_{i} to be monomorphisms. But of course, propositions 6.3 and 6.2 force this to be the case. It is interesting to notice moreover that with the notation of proposition 6.3:

• n being a monomorphism is a necessary condition for the representability of SplExt(−, X) (proposition 6.2);

• n being a normal monomorphism is a sufficient condition (proposition 6.3).

Thus the necessary and sufficient condition is “squeezed” between these two properties.

In the applications, it is often more convenient to use the following variation on proposition 6.3.

6.4. Definition. A semi-abelian category V satisfies the axiom of normality of unions when given a commutative square of subobjects

[figure: X; l; B; m; s_{B}; C; s_{C}; A]

if X is normal in both B and C, then it is also normal in their union B ∨ C.

6.5. Theorem. Let V be a semi-abelian category. If V satisfies

1. the amalgamation property for normal monomorphisms;

2. the normality of unions;

then every functor SplExt(−, X) preserves finite limits. When moreover V is locally wellpresentable, every functor SplExt(−, X) is representable.

Proof. In the pushout of definition 6.1, σ_{1} and σ_{2} are now monomorphisms, so that the pushout square is a union. One concludes by proposition 6.3.

Let us now exhibit the link between our axiom of normality of unions and D. Bourn’s axiom of strong protomodularity. We thank D. Bourn for fruitful exchanges of messages concerning this.

6.6. Proposition. In a semi-abelian category V, consider a short exact sequence

0  [figure]K [figure: k]A [figure: q]Q  [figure]0.

For a subobject s: S [figure]A, the union K ∨ S [figure]A is the subobject q^{−1}(q(S)).

Proof. Trivially,

K = q^{−1}(0) ⊆ q^{−1}(q(S)), S ⊆ q^{−1}(q(S)).

Next, if T ⊆ A contains K and S, T is saturated for the quotient (see [8], 4.3.8) and thus

q^{−1}(q(S)) ⊆ q^{−1}(q(T)) = T.

A semi-abelian category is strongly protomodular in the sense of Dominique Bourn when the inverse image functors of the fibration of points reflect normality. In terms of short exact sequences this means that given a diagram of split exact sequences,

[figure: 0; A; k; B; s; p; C; 0; u; v; 0; D; l; E; t; q; C; 0]

if u is a normal monomorphism, then lu is a normal monomorphism as well. This notion goes back to the work of Gerstenhaber on Moore categories (see [23], [12], [39]).

6.7. Proposition. If a semi-abelian category V satisfies the axiom of normality of unions, it is strongly protomodular.

Proof. In the diagram above, q(B) = C thus by proposition 6.6, E = D ∨ B. An alternative proof consists in observing that D ∨ C = E by [25] 2.4, while D ∨ B ⊇ D ∨ C follows from the equality t = vs. One concludes by the normality of unions.

Eventually, here is an abstract and rather general setting forcing the normality of unions.

6.8. Example. Let 𝕋 be a one-sorted pointed algebraic theory containing a set N of binary terms such that

1. every normal subalgebra X ⊆ B is such that

∀x ∈ X ∀b ∈ B ∀t ∈ N t(x, b) ∈ X;

2. a subalgebra X ⊆ S is normal as soon as there exists a subset U ⊆ S that generates S and has the following property

∀t ∈ N ∀x ∈ X ∀u ∈ U t(x, u) ∈ X.

The category Set^{𝕋} of 𝕋-algebras satisfies the normality of unions.

Proof. In the conditions of definition 6.4, choose for U the set theoretical union U = B ∪ C and take S to be the union of B and C as subobjects of A. Clearly U generates S and condition 1 of the statement implies that t(x, u) ∈ X for all x ∈ X and u ∈ U.

The reader should observe that conditions 1 and 2 in example 6.8 are inherited by every subvariety (i.e. by all theories obtained from 𝕋 by adding axioms). The categories of interest (see [34]) are examples of situations as in example 6.8 and these cover the cases of groups, Lie algebras, rings, and so on.

7. The normalization of a morphism

This section introduces a new notion: the normalization of a morphism. This will be the key ingredient to express the necessary and sufficient condition forcing the representability of the functors SplExt(−, X).

7.1. Definition. Let n be a morphism in a semi-abelian category V. By a normalization of n we mean a commutative diagram

[figure: X; n; S; q; Q; p; 0; X; ñ; S̃; q̃; Q; 0]

where q = Coker n and the bottom line is a short exact sequence.

Of course, the choice of a specific cokernel of n is unessential in this definition. Let us first make some easy observations.

7.2. Lemma. In the situation of definition 7.1:

1. n is a monomorphism;

2. p is a regular (thus normal) epimorphism;

3. if q is a split epimorphism, the bottom exact sequence is split;

4. n is a normal monomorphism if and only if p is an isomorphism; 5. every normal monomorphism n admits a universal normalization

(namely, itself together with its cokernel);

6. the morphism n admits a universal normalization if and only if n admits a unique normalization (up to isomorphism).

Proof. The morphism n is a monomorphism since so is ñ.

The normal closure of n – the smallest normal subobject of S containing X – is the kernel \overline{n} = Ker q of q. This yields at once the various factorizations as in diagram 3. In

[figure: X; s; n; \overline{X}; \overline{n}; S; q; Q; r; p; X; ñ; S̃; q̃; Q] Diagram 3

particular r is a regular epimorphism and by the short five lemma for regular epimorphisms, p is a regular epimorphism as well.

If q admits a section t, t̃= pt is a section of q̃.

When n is a normal monomorphism, (n, q) is a short exact sequence and by the short five lemma for isomorphisms, p is an isomorphism. And of course if p is an isomorphism, n is normal because so is ñ. This implies trivially condition 5.

Finally by the short five lemma for isomorphisms, every morphism between two normalizations of n is necessarily an isomorphism. This implies at once the last affirmation.

Let us comment a little bit more that notion of “normalization”. In the conditions of definition 7.1, n is thus a monomorphism and p is a normal epimorphism (lemma 7.2). Thus S̃ is the quotient of S by the kernel m: Y [figure]S of p and clearly, the whole situation is entirely determined by the knowledge of n and m. Let us thus translate the notion of “normalization of n” in terms of this normal subobject m of S.

7.3. Proposition. In a semi-abelian category V, consider a monomorphism n: X [figure]S with normal closure \overline{n}: \overline{X} [figure]S (i.e. \overline{n} = Ker Coker n). Up to isomorphisms, there is a bijection between

1. the normalizations of n;

2. the normal subobjects m: Y [figure]S such that

Y ⊆ \overline{X}, X ∧ Y = 0, X ∨ Y = \overline{X}.

Proof. Consider diagram 4 with m a normal subobject. Put q = Coker n, p = Coker m, ñ = pn, q̃ = Coker ñ. Let χ be the factorization through the cokernels, and s the inclusion of X in \overline{X}.

[figure: X; Y; s; n; t; m; \overline{X}; \overline{n}; S; q; Q; r; p; τ; χ; θ; X; ñ; S̃; q̃; Q̃] Diagram 4

The construction above determines a normalization of n precisely when

1. there exists a morphism τ such that q = τp;

2. ñ is a monomorphism;

3. that monomorphism ñ is normal.

Indeed, in the case of a normalization, χ is an isomorphism and it suffices to put τ = χ^{−1}q̃; conditions 2 and 3 are then trivial. Conversely, by conditions 2 and 3, ñ = Ker q̃, from which we obtain the factorization r through the kernels. Since ñrs = p\overline{n}s = pn = ñ, we get rs = 1_{X}. Thus τñ = τñrs = τp\overline{n}s = qn = 0, from which we obtain a factorization θ of τ through q̃ = Coker ñ. The equalities

θχq = θq̃p = τp = q, χθq̃p = χτp = χq = q̃p

prove finally that χ and θ are inverse isomorphisms.

The three conditions above are respectively equivalent to:

1. Y ⊆ \overline{X};

2. X ∧ Y = 0; indeed ñ is a monomorphism if and only if its kernel is zero (see [8] 3.1.21); but the kernel of ñ = pn is trivially the kernel Y of p intersected with X;

3. the monomorphism ñ is normal;

and it remains to see that we can replace the third condition by X ∨ Y = \overline{X}.

We know by proposition 6.6 that Y ∨X = p^{−1}(p(X)). But since ñ = pn is a monomorphism by condition 2, p(X) is simply the subobject ñ. When ñ is normal, the factorization r exists and so \overline{X} factors through p^{−1}(p(X)) = Y ∨ X, proving the equality \overline{X} = X ∨ Y. Conversely when \overline{X} = X ∨ Y = p^{−1}(p(X)), then \overline{X} is the pullback of ñ along p, yielding again the existence of the epimorphism r. But then ñ is the image of the normal monomorphism \overline{n} along the regular epimorphism p: thus ñ is normal (see [7] 3.9.3).

Having a normalization is certainly a strong property. For example:

7.4. Counterexample. In the category of groups, consider a simple group S and a subgroup n: X [figure]S; let q = Coker n. The morphism n admits a normalization if and only if X = 0 or X = S.

Proof. With the notation of definition 7.1, p is a quotient map by lemma 7.2; thus by simplicity of S, S̃ = 0 or S̃ = S. If S̃ = 0, then X = 0. If S̃ = S, then n = ñ and X is normal in S, thus X = 0 or X = S.

But nevertheless there are highly interesting examples of normalizations:

7.5. Example. In a semi-abelian category V, consider an object G and the corresponding monad G♭− defining the semi-direct product. The G-algebra structures (X, ξ) on a fixed object X ∈ V are in bijection with the (isomorphism classes of) normalizations of the canonical morphism σ_{X}: X [figure]G + X.

Proof. It is routine to observe that (1_{G}, 0) is the cokernel of σ_{X}.

[figure: X; σ_{X}; G + X; σ_{G}; (1_{G}, 0); G; p; 0; X; s_{X}; G ⋉ (X, ξ); s; p; G; 0]

This cokernel is a split epimorphism with section the canonical inclusion σ_{G}. The kernel of (1_{G}, 0) is precisely the object G♭X (see [13]).

By lemma 7.2, every normalization of σ_{X} yields a bottom split exact sequence, thus also a factorization ξ: T_{G}(X) [figure]X through the kernels.

Conversely every (G♭−)-algebra structure (X, ξ) on X yields a normalization diagram as above (see [13] again). The equivalence between the category of G-algebras and the category of points over G forces the conclusion.

8. The representability theorem

This section presents the central result of this paper: a necessary and sufficient condition for the representability of the functors SplExt(−, X). Here is that condition:

8.1. Definition. Let V be a semi-abelian category. Consider the pushout of two protosplit monomorphisms n_{1}, n_{2} with common domain X and the corresponding diagonal morphism n.

[figure: X; n_{1}; S_{1}; n_{2}; n; σ_{1}; S_{2}; σ_{2}; S_{1} +_{X} S_{2}]

We say that the category V satisfies the protonormalization of pushouts over X when each such morphism n admits a universal normalization.

First of all, we recall a well-known fact.

8.2. Lemma. Let F: C [figure]Set be a functor defined on a category C with binary products. The following conditions are equivalent:

1. F preserves finite products;

2. the category Elts(F) of elements of F has binary products preserved by the forgetful functor ϕ: Elts(F) [figure]C.

Proof. When F preserves binary products, (C, x) × (D, y) ≅ (C × D, (x, y)) in Elts(F). Conversely if (C × D, z) = (C, x) × (D, y), the two elements x ∈ F (C) and y ∈ F (D) are the images of z ∈ F (C × D) by F (p_{C}) and F (p_{D}). If z′ ∈ F (C × D) is another element mapped on x and y by F (p_{C}) and F (p_{D}), the corresponding factorization (C × D, z′) [figure](C × D, z) commutes in C with the projections, thus is the identity on C × D. Therefore z = z′.

As an intermediate step in our arguments, we consider the category SplPsExt[X] with objects the triples (n, q, t) where X is fixed, q = Coker n and qt = 1_{Q}.

[figure: X; n; S; t; q; Q; φ; ψ; X; n′; S′; t′; q′; Q′]

The morphisms are the triples (1_{X}, φ, ψ) making the diagram commutative. This is thus the category of “split-pseudo-right-exact sequences”.

8.3. Lemma. Let V be a semi-abelian category. The category SplPsExt[X] admits binary coproducts preserved by the “cokernel part” functor

SplPsExt[X] [figure]V, (X → S ⇆ Q) ↦ Q.

Proof. Given two objects of SplPsExt[X]

X [figure: n_{i}]S_{i} t_{i} ; [figure: q_{i}]Q_{i}, i = 1, 2

their coproduct is simply

X [figure: (n_{1}, n_{2})]S_{1} +_{X} S_{2} t_{1} + t_{2} ; [figure: q_{1} + q_{2}]Q_{1} + Q_{2}.

8.4. Proposition. Let X be an object of a semi-abelian category V. The following conditions are equivalent:

1. the functor SplExt(−, X) preserves binary coproducts;

2. the category V satisfies the protonormalization of pushouts over X.

Proof. Given X ∈ V, consider the category of elements of the functor SplExt(−, X). It is equivalent to the subcategory SplExt[X] of SplPsExt[X] whose objects are split exact sequences and whose morphisms yield a pullback diagram at the level of cokernels (see the proof of proposition 1.4). But by lemma 1.1, the pullback requirement holds always since the morphisms have identities in the component X. This shows that we have a full subcategory

i: SplExt[X] ↪ SplPsExt[X].

We use the notation of definition 7.1. Let (n, q, t) be an object of the category SplPsExt[X] such that n has a universal normalization (ñ, q̃). By lemma 7.2, this normalization is unique and q̃ admits a unique section t̃ making the diagram commutative. Let us prove that

(1_{X}, p, 1_{Q}): (n, q, t) [figure](ñ, q̃, t̃)

is the universal reflection of (n, q, t) along the inclusion functor i.

Consider for this another morphism

(1_{X}, p′, p′′): (n, q, t) [figure](n′, q′, t′)

in SplPsExt[X], with thus

0  [figure]X [figure: n′]S′ t′ ; [figure: q′]Q′  [figure]0

a split exact sequence. By lemma 1.1, pulling q′ back along p′′: Q [figure]Q′ and taking the kernel yields a split exact sequence (n′′, q′′, t′′) with kernel object X; due to the pullback construction, the morphism (1_{X}, p′, p′′) factors through this sequence.

[figure: (n, q, t); (1_{X}, p, 1_{Q}); (ñ, q̃, t̃); (1_{X}, p′, p′′); ≅; (n′, q′, t′); (n′′, q′′, t′′)]

Thus (n′′, q′′) is another normalization of n and by uniqueness of such a normalization, this is simply (ñ, q̃) (up to an isomorphism). So (1_{X}, p′, p′′) factors indeed through (1_{X}, p, 1_{Q}); this factorization is unique since, by lemma 7.2, p is an epimorphism.

Finally, since SplPsExt[X] has binary coproducts, a standard argument on universal constructions shows that the coproduct of two objects (n_{i}, q_{i}, t_{i}) of SplExt[X] exists if and only if the corresponding coproduct in SplPsExt[X] (which exists by lemma 8.3) admits a universal reflection along the inclusion functor i. By the form of binary coproducts in SplPsExt[X], this is exactly the protonormalization of pushouts. Moreover, the construction of the universal morphism (1_{X}, p, 1_{Q}) above shows that universal reflections preserve the “cokernel part”. Thus the “cokernel part functor” on SplExt[X], which is the forgetful functor of this category of elements, preserves binary coproducts since this is the case for SplPsExt[X], by lemma 8.3.

And finally, theorem 5.8 and proposition 8.4 yield the expected representability theorem:

8.5. Theorem. Let X be an object of a semi-abelian category V. The following conditions are equivalent:

1. the category V satisfies the protonormalization of pushouts over X;

2. the functor SplExt(−, X) preserves finite colimits.

When moreover, the category V is locally well-presentable, these conditions are also equivalent to

3. the functor SplExt(−, X) is representable.

It should again be observed that the protonormalization of pushouts in definition 8.1 does not explicitly require σ_{1} and σ_{2} to be monomorphisms (= the amalgamation property for n_{1} and n_{2}): but we know by theorem 8.5 and proposition 6.2 that this is nevertheless the case.

9. A topos theoretic example

Examples of the representability of functors Act(−, X) in a topos theoretic context have already been given in proposition 1.5. And the same type of arguments extends at once to Boolean rings or commutative von Neumann regular rings, in a topos with Natural Number Object.

Let us also observe that the sufficient conditions in theorem 6.5 transfer easily from Set to a Grothendieck topos.

9.1. Lemma. Let 𝕋 be a semi-abelian category whose category Set^{𝕋} of models satisfies the amalgamation property for normal monomorphisms and the normality of unions. The same properties hold in the semi-abelian category E^{𝕋} of 𝕋-models in a Grothendieck topos E. Then for every object X ∈ E^{𝕋}, the functor Act(−, X) is representable.

Proof. If E is the topos of sheaves on a site (C, T ), the announced properties hold in the category of 𝕋-models in the topos Set^{C^{op}} of presheaves, simply because they hold pointwise. But E^{𝕋} is a localization of this category, thus the corresponding reflection preserves all the ingredients involved in the properties indicated. One concludes by theorem 6.5.

Of course, the topos case calls for a more intrinsic approach of the representability theorem. Some routine work shows that the category E^{𝕋} and the functor Act(−, X) can be enriched in E, viewed as a cartesian closed category. One could then ask for the representability of Act(−, X) as a functor enriched in E. One could even generalize further the situation and consider an internal algebraic theory 𝕋 yielding a semi-abelian category of models. But all these considerations escape the scope of the present paper, devoted to the Set-valued representability of actions.

Let us conclude this discussion of the topos theoretic case with an example of highly non-algebraic nature.

9.2. Proposition. Let E_{∗}^{op} be the dual of the category of pointed objects of an elementary topos E. This category is semi-abelian and when E is boolean, the actions on an object (X, ∗) are representable by (X, ∗) itself.

Proof. It is known that the dual category of the category of pointed objects of a topos is semi-abelian (see [8] 5.1.8). To avoid any confusion, let us work in the category E_{∗} of pointed objects, not in its dual.

We consider first a co-split co-exact sequence

(1, ∗)  [figure](G, ∗) q ; [figure: s](A, ∗) [figure: p](X, ∗)  [figure](1, ∗).

Since the topos E is boolean, the base point ∗: 1 [figure]X is a complemented subobject, thus X ≅ 1 ⨿ Y in E (to avoid confusion, we write ⨿ for the coproduct in E and + for the coproduct in E_{∗}). Then in E

A ≅ p^{−1}(1) ⨿ p^{−1}(Y) ≅ G ⨿ p^{−1}(Y).

Since p is an epimorphism in E, its restriction

p_{Y} : p^{−1}(Y) [figure]Y

is an epimorphism as well. The kernel pair of p_{Y} is the restriction to p^{−1}(Y) × p^{−1}(Y) of the kernel pair of p, which is

(G × G) ∨ Δ_{A} = (G × G) ∨ Δ_{G} ∨ Δ_{p^{−1}(Y)} = (G × G) ∨ Δ_{p^{−1}(Y)}.

Since G is disjoint from p^{−1}(Y) in A, the kernel pair of p_{Y} is thus simply Δ_{p^{−1}(Y)}, proving that p_{Y} is a monomorphism as well. Thus p_{Y} is an isomorphism. This yields a corresponding morphism

∗ ⨿ p_{Y}^{−1} : 1 ⨿ Y [figure]G ⨿ p^{−1}(Y) ≅ A

whose composite with q yields a morphism (X, ∗) [figure](G, ∗).

Conversely given a morphism τ: (X, ∗) [figure](G, ∗), we get at once a co-split co-exact sequence

(1, ∗)  [figure](G, ∗) (id, τ) ; [figure: s_{(G,∗)}](G, ∗) + (X, ∗) [figure: (0, id)](X, ∗)  [figure](1, ∗).

It is routine to check that we have constructed inverse bijections (up to isomorphisms of co-split co-exact sequences).

10. Some counter-examples

This paper introduces three conditions related to the representability of actions: a necessary condition, a sufficient condition, a necessary and sufficient condition. This section presents various counter-examples distinguishing between these conditions, thus proving the pertinence of all of them.

First of all, let us make clear that the representability of actions is by no means a general property of semi-abelian categories, even of semi-abelian varieties. As the following counter-examples will show, this is even a rather exceptional property.

10.1. Counterexample. Actions are generally not representable on the category ComRg of commutative rings.

Proof. By theorem 2.6, it suffices to exhibit a ring X such that End(X) is not commutative. For example, the additive abelian group ℤ^{2} with the zero multiplication, for which End(X) is the ring of 2 × 2-matrices with entries in ℤ.

Second, let us exhibit a counter-example showing that the necessary condition of proposition 6.2 is not sufficient. By a non-associative ring, we mean clearly an additive abelian group equipped with a multiplication which distributes over the addition in each variable. Analogous arguments can be developed in the commutative case.

10.2. Counterexample. In the variety of non-associative rings, which is semiabelian:

1. the amalgamation property holds for protosplit monomorphisms;

2. the actions on an object X are representable if and only if X is the zero ring.

Proof. The arguments developed in the proofs of propositions 2.1 and 2.3 apply as such to prove that, for non associative rings, there is a bijection between

• the (isomorphism classes of) split exact sequences of the form

0  [figure]X [figure: k]A s ; [figure: p]G  [figure]0;

• the pairs of additive abelian group homomorphisms

λ: G  [figure]Ab(X, X), ρ: G  [figure]Ab(X, X).

This proves thus the isomorphism

Act(G, X) ≅ Ab(G, Ab(X, X) × Ab(X, X)).

Given a second split exact sequence

0  [figure]X [figure: k′]A′ s′ ; [figure: p′]G′  [figure]0;

with corresponding pair (λ′, ρ′), we get at once abelian group homomorphisms

(λ, λ′): G ⊕ G′  [figure]Ab(X, X), (ρ, ρ′): G ⊕ G′  [figure]Ab(X, X)

and therefore a corresponding split exact sequence of non-associative rings

0  [figure]X [figure: k′′]A′′ s′′ ; [figure: p′′]G × G′  [figure]0.

And the two canonical inclusions of G, G′ in G × G′ are ring homomorphisms making the following diagram commutative:

[figure: G; (id, 0); G × G′; (0, id); G′; λ; (λ, λ′); λ′; Ab(X, X)]

and analogously with (ρ, ρ′). But we have

A ≅ G ⋉ (X, λ, ρ), A′′ ≅ (G × G′) ⋉ (X, (λ, λ′), (ρ, ρ′)), A′ ≅ G′ ⋉ (X, λ′, ρ′)

and therefore the morphisms (id, 0), (0, id) induce corresponding morphisms of nonassociative rings

A [figure: α]A′′ [figure: α′] A′.

But at the level of underlying additive abelian groups, the square αk = α′k′ is simply

[figure: X; k; G × X; k′; α; G′ × X; α′; G × G′ × X]

k(x) = (0, x), k′(x) = (0, x), α(g, x) = (g, 0, x), α′(g′, x) = (0, g′, x).

Thus the ring homomorphisms α, α′ are injective and by factorization, also the canonical morphisms of the pushout of k and k′ are injective. This proves the amalgamation property for protosplit monomorphisms.

Now suppose that X has representable actions. In particular, the functor Act(−, X) transforms binary coproducts in binary products, yielding the canonical bijection

Act(G + G, X) ≅ Act(G, X) × Act(G, X)

for every non-associative ring G. The form of Act(−, X), established earlier in the proof, yields thus the canonical bijections

Ab(G + G, Ab(X, X)^{2}) ≅ Ab(G, Ab(X, X)^{2}) × Ab(G′, Ab(X, X)^{2}).  (∗)

Write now 𝕋 for the theory of non-associative rings and 𝕄 for the theory of magmas (a set equipped with a binary operation, without any axiom). The forgetful functor U: Set^{𝕋} [figure]Set^{𝕄} mapping a non-associative ring on its underlying multiplicative magma is algebraic, thus has a left adjoint L. Trivially, L(M, ·) is the free abelian group on the set M epuipped with the multiplication induced by distributivity from the multiplication on the magma (M, ·) of generators. The terminal magma – the singleton {x} – is the magma with a single generator x satisfying the relation xx = x. The corresponding ring L{x} is simply the ring (ℤ, +, ×) of integers. Therefore, since L preserves colimits,

L({x} + {y}) ≅ L{x} + L{y} ≅ ℤ + ℤ.

And the magma {x}+{y} is the quotient of the magma of all bracketed words in x and y, by the congruence generated by the relations xx = x and yy = y. This is thus an infinite countable set and the non-associative ring ℤ + ℤ admits as underlying additive group a free abelian group on infinitely many generators.

Writing ω for the infinite countable cardinal, the canonical bijection (∗) become thus, choosing G = ℤ,

(p_{{x}}, p_{{y}}): (Ab(X, X)^{2})^{ω} ≅ Ab(X, X)^{2} × Ab(X, X)^{2}.

But the pair of projections (p_{{x}}, p_{{y}}) is a bijection only when Ab(X, X)^{2} is the zero group, that is, when (X, +) itself is the zero group.

Finally, let us provide a counter-example showing that the sufficient conditions in proposition 6.3 or theorem 6.5 are not necessary. The theory 𝕋 used in this counterexample can already be found in [13] and [12].

10.3. Counterexample. Let 𝕋 be the theory having two abelian group operations with the same neutral element. The variety Set^{𝕋} is semi-abelian and there exist objects X ∈ Set^{𝕋} such that:

1. the functor Act(−, X) is representable;

2. the sufficient conditions of proposition 6.3 and theorem 6.5 are not satisfied.

Proof. A split exact sequence of 𝕋-algebras is in particular a split exact sequence of abelian groups for the first abelian group operations, thus has set theoretically the form

0  [figure]X [figure: s_{X}]G × X s_{G} ; [figure: p_{G}]G  [figure]0
s_{X}(x) = (x, 0), p_{G}(g, x) = g, s_{G}(g) = (g, 0)

with the first abelian group structure on G×X defined pointwise. And this is a split exact sequence of 𝕋-algebras when G × X is equipped with a second abelian group structure, with neutral element (0, 0), and making s_{X}, p_{G}, s_{G} group homomorphisms for the second abelian group operations. Let us use the symbol + to denote these second abelian group operations.

Since a split exact sequence of 𝕋-algebras is also a split exact sequence of abelian groups for the second abelian group operations, we know that

∀(g, x) ∈ G × X ∃!g′ ∈ G ∃!x′ ∈ X (g, x) = (g′, 0) + (0, x′).

But p_{G} is a +-homomorphism, thus we have necessarily in G × X

(g′, 0) + (0, x′) = (g′, g′ ∗ x′)

for some element g′ ∗ x′ ∈ X. Therefore g = g′ and the condition above can be rephrased as

∀(g, x) ∈ G × X ∃!x′ ∈ X x = g ∗ x′.

Still in other words, this means that the mapping

g ∗ −: X [figure]X, x ↦ g ∗ x

is bijective. For simplicity, we write (g ∗ −)^{−1}(x) = x_{g} and we have thus

g ∗ x_{g} = x, (g ∗ x)_{g} = x.

Let us also write Perm(X) for the set of permutations of X. The mapping

ξ: (G, 0) [figure](Perm(X), 1_{X}), g ↦ g ∗ −

is a morphism of pointed sets, since

(g, 0) + (0, 0) = (g, 0), (0, 0) + (0, x) = (0, x)

that is

g ∗ 0 = 0, 0 ∗ x = x.

Moreover this mapping ξ determines entirely the second abelian group operation of G×X. Indeed, since s_{X} and s_{G} are +-homomorphisms, we have

(g, x) + (h, y) = (g, g ∗ x_{g}) + (h, h ∗ y_{h})
= (g, 0) + (0, x_{g}) + (h, 0) + (0, y_{h})
= (g + h, 0) + (0, x_{g} + x_{h})
= (g + h, (g + h) ∗ (x_{g} + y_{h})).

Conversely, given a morphism of pointed sets

ξ: (G, 0) [figure](Perm(X), 1_{X}), g ↦ g ∗ −

and putting as before x_{g} = (g ∗ −)^{−1}(x), we define an addition on G × X by the formula

(g, x) + (h, y) = (g + h, (g + h) ∗ (x_{g} + y_{h})).

It is trivial that this addition is commutative. It admits (0, 0) as neutral element because ξ is a morphism of pointed sets. It is associative because every sum of (g, x), (h, y) and (k, z) yields the result

(g + h + k, (g + h + k) ∗ (x_{g} + y_{h} + z_{k})).

Finally the opposite of the element (g, x) is given by

−(g, x) = (−g, (−g) ∗ (−x_{g})).

We conclude that the functor Act(−, X) is isomorphic to

Act(−, X): Set^{𝕋} [figure]Set, G ↦ Set_{∗}((G, 0), (Perm(X), 1_{X})

where Set_{∗} denotes the category of pointed sets.

Now if X = {0, x} has exactly two elements (that is, X ≅ ℤ/2ℤ with both group operations necessarily equal), the only second abelian group operation of G × X is the pointwise one. Indeed for every g ∈ G, g ∗ 0 = 0 and therefore g ∗ x ≠ g ∗ 0 = 0, thus g ∗ x = x. Thus every permutation g ∗ − is the identity on X and the second abelian group operation of G × X is pointwise. But then each Act(G, X) is a singleton and the functor Act(−, X) is representable by the zero 𝕋-algebra.

It remains to show that the sufficient conditions of proposition 6.3 and theorem 6.5 are not satisfied for this 𝕋-algebra X = ℤ_{2}. For this consider the square of injections

[figure: ℤ_{2}; α; ℤ × ℤ_{2}; β; δ; ℤ × ℤ_{2}; γ; ℤ × ℤ × ℤ_{2}]

α(x) = (0, x), β(x) = (0, x), γ(z, x) = (z, 0, x), δ(z, x) = (0, z, x).

Equip each of the two occurrences of ℤ × ℤ_{2} with twice the pointwise addition inherited form the usual additions of ℤ and ℤ_{2}. And as first addition on the object ℤ × ℤ × ℤ_{2}, take again the pointwise one.

It is trivial that given an abelian group (A, +_{1}), every permutation φ of A yields another group operation on the set A, namely

a +_{2} b = φ^{−1}(φ(a) + φ(b)).

and both operations have the same zero as soon as φ(0) = 0. Define the second group operation on ℤ × ℤ × ℤ_{2} in that way, choosing for φ the permutation which interchanges the two elements

(1, 1, 0), (−1, 1, 0)

and fixes all the other elements. This completes the description of the square above, where α, β, γ, δ are trivially morphisms of 𝕋-algebras.

We have a split exact sequence

0  [figure]ℤ_{2}  [figure: α]ℤ × ℤ_{2} s_{ℤ} ; [figure: p_{ℤ}]ℤ  [figure]0

thus α is a protosplit monomorphism, and analogously for β. Moreover, already for the first additions, every element of ℤ × ℤ × ℤ_{2} is the sum of two elements in the two copies of ℤ × ℤ_{2}, thus the square γα = δβ is certainly a union of 𝕋-algebras. Let us now observe that ℤ_{2} is not a normal subobject of ℤ × ℤ × ℤ_{2}.

For this we consider the cokernel q of that inclusion:

ℤ_{2} [figure: δα]ℤ × ℤ × ℤ_{2} [figure: q]Q.

We keep using the notation +_{1} for the pointwise addition on ℤ × ℤ × ℤ_{2} and +_{2} for the second addition, that is, the addition +_{1} “twisted” by the permutation φ. Since

(1, 1, 0) +_{1} (0, 0, 1) = (1, 1, 1)

with (0, 0, 1) ∈ ℤ_{2}, we have

[1, 1, 0] = [1, 1, 1] ∈ Q.

And since

(1, 1, 0) +_{2} (0, 0, 1) = φ^{−1}((−1, 1, 0) +_{1} (0, 0, 1)) = φ^{−1}(−1, 1, 1) = (−1, 1, 1)

with again (0, 0, 1) ∈ ℤ_{2}, we have also

[1, 1, 0] = [−1, 1, 1] ∈ Q.

By transitivity this implies

[1, 1, 1] = [−1, 1, 1] ∈ Q

and thus

[2, 0, 0] = [1, 1, 1] −_{1} [−1, 1, 1] = 0 ∈ Q.

But (2, 0, 0) ∉ ℤ_{2}, thus ℤ_{2} is not a normal subobject of ℤ × ℤ × ℤ_{2}. This proves that the sufficient condition in theorem 6.5 is not satisfied.

Finally let us observe that the protosplit monomorphisms α, β do not satisfy either the sufficient condition in proposition 6.3. Writing S for their pushout, the union above (as every union) is a quotient of that pushout and we get a commutative square

[figure: ℤ_{2}; s; S; p; ℤ_{2}; δα; ℤ × ℤ × ℤ_{2}]

where s is the diagonal morphism of the pushout, which is thus a monomorphism. Since p is a quotient morphism, the normality of s would imply that of its image δα along p (see [7] 3.9.3), which is not the case.

11. Open problems

In this section, V denotes a semi-abelian variety of universal algebras, although in many cases one could also consider much wider contexts. Let us recall:

(A) Representability of actions in V, for which a necessary and sufficient condition is provided by Theorem 8.5, implies amalgamation property for protosplit monomorphisms (by Proposition 6.2) in V.

(B) Whenever V satisfies the normality of unions (Definition 6.4), the representability of actions in V is equivalent to the amalgamation property for protosplit monomorphisms (by Propositions 6.2 and 6.3).

(C) If V is a category of interest in the sense of M. Barr (see G. Orzech [34]), then it satisfies the normality of unions (see Example 6.8 and the last paragraph of Section 6); therefore – we repeat from (B) – the representability of actions in V is equivalent to the amalgamation property for protosplit monomorphisms. In particular, this applies to every subvariety of the following varieties:

a. Groups;

b. Associative algebras over an arbitrary fixed commutative unital ring;

c. Lie algebras over an arbitrary fixed commutative unital ring. d. Any of the structures above equipped with any fixed set of unary linear operators; this includes – up to a category equivalence – crossed modules and other internal categorical structures in the categories of those structures.

Note also that this applies not only to subvarieties, but also to all full subcategories closed under finite limits and quotients that are varieties – such as von Neumann regular rings among all rings.

We would like to propose the following open problems:

Problem 1. Give a syntactical characterization of semi-abelian varieties satisfying all/any of the conditions mentioned in (B).

Problem 2. Using (C), find new examples of categories of interest with representable actions and/or just individual objects in a category of interest with representable actions, and, if possible, describe the representing objects in each case. Note that the amalgamation property of all monomorphisms have been studied by many authors (see [30]), but the amalgamation property of protosplit ones is new, and it is a much weaker condition. For example the amalgamation property of all monomorphisms is known for Lie algebras over a field, but its protosplit version holds for Lie algebras over an arbitrary unital commutative ring – simply because the object actions are representable (by derivations). We expect a kind of negative result for varieties of groups, since the only non-abelian variety of groups where we are able to prove the representability of actions is the variety of all groups.

Problem 3. As we know from Counterexample 10.3, the representability of actions does not imply the normality of unions (even for protosplit monomorphisms). However, we do not have any example of a semi-abelian variety in which every object has representable actions, but the normality of unions does not hold. Does such an example exist?

Before formulating our next problems, let us recall:

a. An object X in, say, a semi-abelian category, is said to be abelian if the codiagonal X + X [figure]X factors through the canonical morphism X + X [figure]X × X. This notion goes back to S.A. Huq [24], who observes that it is a reformulation of S. Mac Lane’s observation for groups.

b. An abelian object in a semi-abelian category is the same as a group object; the group structure is unique, as observed by many authors independently in various contexts many years ago.

c. An abelian object in a semi-abelian variety is the same as what universal algebraists call an abelian algebra (in the pointed case).

d. An object X is abelian if and only if its largest commutator [X, X] is trivial; in the case of a semi-abelian variety, all known universal-algebraic and categorical definitions of the largest commutator are equivalent (see e.g. [24], [41], [21], [36], [26]). In particular [X, X] is the usual commutator in the cases of groups and of Lie algebras, and [X, X] = XX for rings (commutative or not).

e. A group is an abelian object in the category of groups if and only if it is abelian in the usual sense. A ring is an abelian object in the category of rings if and only if it has zero multiplication; the same is true for non-associative rings, for commutative rings, for Lie algebras, etc.

Problem 4. Characterize and/or give new examples of semi-abelian varieties in which every abelian object has representable actions.

Problem 5. Characterize and/or give new examples of semi-abelian varieties in which every perfect object has representable actions. Of course we call an object X perfect if it has trivial abelianization, i.e. if [X, X] = X. Note that, as follows from Propositions 2.4 and 2.7, in the categories of rings/algebras, commutative or not, every perfect object has representable actions.

Problem 6. For an object X in V, define the conjugation action of X on itself as the action corresponding to the split epimorphism (first projection, diagonal): X×X [figure]X. If the representing object [X] does exist, this determines a morphism X [figure][X]. Investigate the cases where this morphism is a

a. monomorphism;

b. (normal) epimorphism;

c. isomorphism.

Note (as easily follows from Proposition 2.7; see also proposition 2.8) that X [figure][X] is an isomorphism for Taylor-regular rings but not, say, for all Boolean rings. For finite Boolean rings, this however follows independently from two results: they are unital and their dual category is equivalent to the category of pointed objects in the topos of finite sets.

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Université Catholique de Louvain, Belgium

Mathematical Institute of the Georgian Academy of Sciences, Tbilisi, Georgia and University of Cape Town, South-Africa

University of Sydney, Australia

Email: borceux@math.ucl.ac.be
      janelidg@maths.uct.ac.za
      maxk@maths.usyd.edu.au

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