Theory and Applications of Categories, Vol. 23, No. 9, 2010, pp. 170–198.

STAR-MULTIPLICATIVE GRAPHS IN POINTED PROTOMODULAR CATEGORIES

Dedicated to Dominique Bourn on the occasion of his sixtieth birthday

N. MARTINS-FERREIRA

Abstract. Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in [10] in the semiabelian context.

When the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of [10].

1. Introduction

In a category B, pointed with kernels of split epimorphisms, we consider the following three conditions:

(I) Split extensions are jointly epic;

(II) The Split Short Five Lemma holds;

(III) Admissibility is reflected by the Kernel Functor.

Condition (II) is well known (see [1] and references there, see also [9]). Condition (I) states that in every split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}} , αβ = 1 , k = ker α ,

This work was done during the Post-Doctoral position held by the author at CMUC, supported by the FCT grant SFRH/BPD/4321/2008.

Received by the editors 2009-04-15 and, in revised form, 2009-10-22. Published on 2010-02-05 in the Bourn Festschrift. 2000 Mathematics Subject Classification: 18D35. Key words and phrases: Internal category, internal groupoid, reflexive graph, multiplicative graph,

star-multiplicative graph, jointly epic pair, admissible pair, jointly epic split extension, split short five lemma, pointed protomodular.

\begin{matrix}c \\ ◯ \end{matrix} N. Martins-Ferreira, 2010. Permission to copy for private use granted.

STAR-MULTIPLICATIVE GRAPHS IN POINTED PROTOMODULAR CATEGORIES 171

the pair (k, β) is jointly epic. This condition was originally one of the axioms considered by M. Gerstenhaber, in 1970, in the definition of Moore categories ([18]).

For a given jointly epic pair (k, β), a pair of morphisms (g, h), with common codomain,

\xymatrix{X \ar[r]^{k} \ar[dr]_{g} & A \ar@{-->}[d] & B \ar[l]_{β} \ar[dl]^{h} \\  & A′}

is said to be admissible, w.r.t. (k, β), if there is a (necessarily unique) morphism φ : A ⟶ A′, such that φk = g and φβ = h. By condition (III) we mean that the kernel functor Ker : Pt_{B} (B) ⟶ B, reflects admissibility in the following sense: a pair of morphisms in Pt_{B} (B) is admissible w.r.t. a jointly epic pair, provided this holds for their images by Ker. Details are presented in Section 2.

Condition (II) may also be interpreted as saying that the kernel functor reflects isomorphisms.

The reason for considering these particular three conditions, either individually or together, is the following.

Let the base category B, be pointed with pullbacks along split epimorphisms and so, in particular, with products and kernels of split epimorphisms.

We consider the following chain of forgetful functors

Grpd (B) ⟶^{(4)} Cat (B) ⟶^{(3)} MG (B) ⟶^{(2)} SMG (B) ⟶^{(1)} RG (B) ,

from internal structures of groupoids to categories (forgets inverses), to multiplicative graphs (forgets associativity and codomains, but not domains!, see footnote at the end of page 174), to star-multiplicative graphs (keeps only composition around the origin), to reflexive graphs (forgets composition). We prove that

(I) ⟹ (3) is an isomorphism,
(II) ⟹ (4) is an isomorphism,
(III) ⟹ (2) is an isomorphism.

We also observe that the following implications hold

(I) ⟹ weakly Mal’cev (as defined in [17]),
(II) + equalizers ⟹ (I),
(III) + products ⟹ (I).

If B is abelian all categories in the above chain coincide.

If B is either the category of groups or the one of rings we have that Grpd (B) ∼ SMG (B). This is not true in arbitrary protomodular categories. Indeed, in the category of digroups (sets with two group operations sharing the same unit), which is a semiabelian category, the functor (2) is not an isomorphism, as I learned from G. Janelidze. Hence, condition (III) arises as an attempt to fill in this gap.

Another approach to this problem, involving strong protomodularity, was presented in [15]. We remark that if in (I) the pair (k, β) is required to be jointly strongly epic, then we obtain (II) (see [1] and references there).

The classical notion of crossed module may be extended to the more general context of a category satisfying (I): it is a split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}} , αβ = 1 , k = ker α ,

together with a morphism

h : X ⟶ B ,

such that the dashed arrows in the following diagram can be inserted in order to make it commutative

[figure: X; ⟨1,0⟩; X × X; π_{2}; (a); X; ⟨1,1⟩; h; X; k; h; A; α; (b); B; β; B; ⟨1,0⟩; B × B; π_{2}; B .; ⟨1,1⟩]  (1.1)

Since the horizontal rows are split extensions, the morphisms represented by the dashed arrows are unique, provided they exist.

If B also has binary coproducts and coequalizers of reflexive pairs then we can define the category

Act (B)

of internal actions in B (see [10] and [16]). If, as in a semi-abelian category [12], there is an equivalence

Act (B) ∼ Pt (B)

then the above definition coincides with the notion of internal crossed module, in the sense of Janelidze.

In this work we show that if our base category B (pointed with pullbacks along split epimorphisms) satisfies (I),(II) and (III), we have the following equivalences of categories

Grpd (B) ∼ SMG (B) ∼ Xmod (B) .

In the case of groups (see Proposition 3.2), to give a internal groupoid is to give a split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}}

together with a morphism h : X ⟶ B satisfying (with additive notation though the groups are not assumed to be commutative)

h (β (b) + k (x) − β (b)) = b + h (x) − b

which is equivalent to insert the arrow with label (b) in (1.1), and satisfying

k ((x′, x′) + (x, 0) − (x′, x′)) = βh (x′) + k (x) − βh (x′)

which is equivalent to insert the arrow with label (a) in (1.1) and simplifies to

x′ + x − x′ = βh (x′) + k (x) − βh (x′) ∈ X .

The above equations look more familiar if we introduce the usual notation for actions, writing

b · x = β (b) + k (x) − β (b) ∈ X
h (x′) · x = βh (x′) + k (x) − βh (x′) ∈ X ,

so that the above conditions become the familiar axioms for a crossed module in Groups

h (b · x) = b + h (x) − b
h (x′) · x = x′ + x − x′.

For the case of rings (see Proposition 3.1) we have a split extension as above, together with a morphism h : X ⟶ B satisfying, for every x, y ∈ X ⊆ A and b ∈ B,

h(β(b)x) = bh(x) , h(xβ(b)) = h(x)b
xβh(y) = xy = βh(x)y .

In Section 4, assuming condition (I), we characterize, for a fixed split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}} , αβ = 1 , k = ker α ,  (1.2)

the morphisms

h : X ⟶ B

that give rise to a reflexive graph, a star-multiplicative graph, a multiplicative graph, an internal category, or an internal groupoid.

Acknowledgement. Thanks are due to G. Janelidze and M. Sobral. Very useful conversations with G. Gutierres and S. Mantovani are also gratefully acknowledged. Thanks are due to the referee for the additional remark on condition (III).

2. Definitions and Technicalities

Let B be a category with pullbacks along split epis. Therefore, for every morphism h : C ⟶ B and split epi f : A ⟶ B with section r : B ⟶ A, the pullback of h along f exists

\xymatrix{A ×_{B} C \ar[r]^{π_{2}} \ar[d]_{π_{1}} & C \ar[l]^{⟨rh,1⟩} \ar[d]^{h} \\ A \ar[r]^{f} & B ; \ar[l]^{r}}

and the second projection, π_{2}, is always a split epi with splitting ⟨rh, 1⟩.

We assume that B is also pointed, so that, in particular, we have binary products and kernels of split epis. It is easy to see that the kernel of the split epi π_{2} is given by the induced morphism ⟨k, 0⟩ into the pullback, where k : X ⟶ A is the kernel of f:

[figure: X; ⟨k,0⟩; A ×_{B} C; π_{2}; π_{1}; C; ⟨rh,1⟩; h; X; k; A; f; B .; r]

A reflexive graph in B is a diagram

\xymatrix{C_{1} \ar[r]^{d} \ar[r]_{c} & C_{0} \ar[l]|{e}} , de = 1_{C_{0}} = ce .  (2.1)

We recall that a multiplicative graph^{1} (see [13], [8] and [11]) is a reflexive graph, as above, together with a multiplication m : C_{2} ⟶ C_{1} such that me_{1} = 1_{C_{1}} = me_{2}, where C_{2} is the pullback of c along the split epi d

\xymatrix{C_{2} \ar[r]^{π_{2}} \ar[d]_{π_{1}} & C_{1} \ar[l]^{e_{2}} \ar[d]_{c} \\ C_{1} \ar[r]^{d} \ar[u]_{e_{1}} & C_{0} \ar[l]^{e} \ar[u]_{e}}

and

e_{1} = ⟨1, ed⟩
e_{2} = ⟨ec, 1⟩

are the two canonical induced morphisms into the pullback.

1Instead of the original definition for a multiplicative graph, and for the purpose of this paper only, it is convenient to assume the extra condition dm = dπ_{2}; otherwise the forgetful functor from multiplicative to star-multiplicative graphs would not be well defined. Also note that this extra condition is trivial once we assume (I).

An internal category is a multiplicative graph satisfying the additional requirements

dm = dπ_{2} , cm = cπ_{1}

and the associativity condition (see also the note at the end of page 188).

An internal groupoid is an internal category where “every arrow is invertible” (see for example the Appendix of [1] for a precise definition).

A star-multiplicative graph is a reflexive graph, as in (2.1), together with a “starmultiplication”, that is, a morphism

s : C_{∗} ⟶ X

where C_{∗} is obtained by pulling back ck along d, and k : X ⟶ C_{1} is the kernel of the split epi d, as displayed below

[figure: X; ⟨k,0⟩; C_{∗}; p_{2}; p_{1}; X; ⟨eck,1⟩; ck; X; k; C_{1}; d; C_{0} ,; e]

satisfying the following condition

s ⟨k, 0⟩ = 1_{X} = s ⟨eck, 1⟩ .

When B is the category of pointed sets, C_{∗} is the set of composable pairs of arrows in C_{1}, starting at zero, in the following sense

C_{∗} = {(f, g) ∈ C_{1} × C_{1} | d (f) = c (g) and d (g) = 0} ,

a ⟵^{f} b ⟵^{g} ∗

meaning that we can only compose two (appropriate) arrows f ∘ g in C_{1}, if the second one, g, starts at the origin (or zero, or star, hence the name star-multiplicative).

Every multiplicative graph (with dm = dπ_{2}) is, in particular, a star-multiplicative graph, by restricting C_{2} to C_{∗} and C_{1} to X:

⟨p_{1}, kp_{2}⟩ : C_{∗} ⟶ C_{2} , k : X ⟶ C_{1}

and defining s as the unique morphism such that ks = m ⟨p_{1}, kp_{2}⟩ (which is well defined if dm = dπ_{2}). Observe also that ⟨p_{1}, kp_{2}⟩ is the kernel of the split epi dπ_{2}.

This notion of star-multiplicative graph was introduced by G. Janelidze in [10] in order to describe internal crossed modules in semi-abelian categories. However it is not true that the star-multiplicative graphs are multiplicative in an arbitrary semi-abelian category. In [10], G. Janelidze asked for a description of semi-abelian categories with the property that every star-multiplication (uniquely) extends to an internal category structure.

Instead of semi-abelian categories we consider the problem in the context of pointed protomodular categories and conclude that under condition (III) we obtain the desired result.

Let B be a pointed category with kernels of split epis, and consider the category Pt (B) of points in B, that is, objects are split epis

\xymatrix{A \ar[r]^{α} & B \ar[l]^{β}} , αβ = 1

and morphisms are pairs (f, g), making the obvious following squares commutative

\xymatrix{A \ar[r]^{α} \ar[d]_{f} & B \ar[l]^{β} \ar[d]^{g} \\ A′ \ar[r]^{α′} & B′ . \ar[l]^{β′}}  (2.2)

For every B ∈ B, since B has kernels of split epis, we may consider the category of points over B, denoted Pt_{B} (B) and the Kernel Functor, denoted by Ker

Ker : Pt_{B} (B) ⟶ B

sending a morphism f in Pt_{B} (B)

[figure: X; k; f_{0}; A; α; f; B; β; X′; k′; A′; α′; B; β′]

to the morphism f_{0} : X ⟶ X′ such that fk = k′f_{0}.

A pair of morphisms (e_{1}, e_{2}) with a common codomain

e_{1} : A ⟶ E ⟵ A′ : e_{2}

is said to be jointly epic if, for every two parallel morphisms

f, g : E ⟶ D ,

we have

\{ \begin{matrix}fe_{1} = ge_{1} \\ fe_{2} = ge_{2} \end{matrix} ⟹ f = g .

Let (e_{1}, e_{2}) be a jointly epic pair

e_{1} : A ⟶ E ⟵ A′ : e_{2} .

A pair (f, g) of morphisms with a common codomain

f : A ⟶ D ⟵ A′ : g

is said to be admissible w.r.t. the pair (e_{1}, e_{2}), if there is a (necessarily unique) morphism

φ : E ⟶ D

such that

φe_{1} = f , φe_{2} = g .

In this case we write

[f g] : E ⟶ D

to denote the morphism φ.

In a pointed category, a diagram of the form

\xymatrix{X \ar[r]^{k} & A \ar[r]^{f} & B \ar[l]^{r}} , fr = 1 , fk = 0

will be called a split chain and, if k is a kernel of f, a split extension. If, furthermore the pair (k, r) is jointly epic we say that it is a jointly epic split chain or jointly epic split extension, respectively.

Let A and B be pointed categories. A functor F : A ⟶ B is said to reflect admissibility if for every split chain in A

\xymatrix{A \ar[r]^{e_{1}} & E \ar[r]^{π} & A′ \ar[l]^{e_{2}}} , πe_{2} = 1 , πe_{1} = 0_{A} ,

whose image by F is a split extension in B

\xymatrix{F A \ar[r]^{F (e_{1})} & F E \ar[r]^{F (π)} & F A′ \ar[l]^{F (e_{2})}} ,

we have that:

– the pair (F (e_{1}) , F (e_{2})) is jointly epic in B,

– the pair (e_{1}, e_{2}) is jointly epic in A,

– and, a pair of morphisms (f, g) with common codomain

f : A ⟶ D ⟵ A′ : g

in A, is admissible w.r.t. (e_{1}, e_{2}) whenever (F (f) , F (g)) is admissible with respect to (F (e_{1}) , F (e_{2})).

We are now in position to state the three properties (I), (II) and (III), mentioned in the Introduction.

Throughout B will denote a pointed category with kernels of split epis. We say that B satisfies:

(I) (the Jointly Epic Split Extension condition) if every split extension is jointly epic, that is, for every diagram of the form

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}} , αβ = 1 , αk = 0 ,

if k = ker α then the pair (k, β) is jointly epic;

(II) (the Split Short Five Lemma) if given any diagram of the form

\xymatrix{X \ar[r]^{k} \ar[d]_{f} & A \ar[r]^{α} \ar[d]_{h} & B \ar[l]^{β} \ar[d]^{g} \\ X′ \ar[r]^{k′} & A′ \ar[r]^{α′} & B′ , \ar[l]^{β′}}  (2.3)

where the rows are split extensions and the vertical arrows constitute a morphism of split extensions, if f and g are isomorphisms then h is an isomorphism as well;

(III) (the Reflected Admissibility property) if the Kernel Functor Ker : Pt_{B} (B) ⟶ B reflects admissibility. In detail, given any diagram of the form

[figure: X; i_{1}; k; Y; p; k̄; X′; i_{2}; k′; A; e_{1}; α; E; π; ᾱ; A′; e_{2}; α′; B; β; B; β̄; B; β′]  (2.4)

where the columns are split extensions and the appropriate horizontal arrows constitute morphisms of split extensions, with πe_{2} = 1_{A′}, πe_{1} = β′α, if the top row is a split extension, then

(i_{1}, i_{2})  is jointly epic in B
(e_{1}, e_{2}) is jointly epic in Pt_{B} (B)

and furthermore, a pair (f, g) of morphisms in Pt_{B} (B)

[figure: X; f_{0}; k; D_{0}; k_{0}; X′; g_{0}; k′; A; f; α; D; α_{0}; A′; g; α′; B; β; B; β_{0}; B; β′]

is admissible (in Pt_{B} (B) ) w.r.t. (e_{1}, e_{2}) whenever the pair (f_{0}, g_{0}) is admissible w.r.t. (i_{1}, i_{2}) .

2.1. Remark. In order to better understand condition (III) we should be able to characterize the split chains in Pt_{B} (B), whose image by Ker is a split extension in B. Note that we are not assuming the existence of kernels in Pt_{B} (B); that would correspond to the existence of split pullbacks in the sense of [7], p. 43. Nevertheless, a split chain in Pt_{B} (B), say (e_{1}, π, e_{2}) as above, whose image by Ker is a split extension, always satisfies (upper left square in (2.4))

e_{1}k = ker π .

In fact, the square πk̄ = k′p is a pullback because 1 : B → B is a monomorphism and k̄, k′ are kernels; hence if i_{1} = ker p then e_{1}k = ker π.

Later on we will assume the existence of pullbacks along split epimorphisms so that the kernel functor will preserve split extensions. Also in this case, the split extensions in Pt_{B} (B) are precisely the split chains, say (e_{1}, π, e_{2}) as above, such that the square (see diagram (2.4))

\xymatrix{A \ar[r]^{α} \ar[d]_{e_{1}} & B \ar[d]^{β′} \\ E \ar[r]^{π} & A′}

is a pullback.

Furthermore, if the category B is protomodular, then every split chain in Pt_{B} (B), whose image by Ker is a split extension in B, is itself a split extension in Pt_{B} (B). As it is well known in this context the above square is a pullback if and only if e_{1}k = ker π (with k = ker α as in diagram (2.4)).

We deduce immediately:

2.2. Proposition. Let B be a pointed category with kernels of split epis and equalizers. If B satisfies (II) then it also satisfies (I).

Proof. For a given split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}} , αβ = 1 , k = ker α

the equalizer, q, of any two parallel morphisms

f, g : A ⟶ D

satisfying fk = gk and fβ = gβ is in fact an isomorphism, because it fits in the following diagram

[figure: X; k′; Y; αq; q; B; β′; X; k; A; α; B; β]

where k′ and β′ are the unique morphisms such that qk′ = k and qβ′ = β. Obviously we have αqβ′ = 1 and k′ = ker (αq) so by the Split Short Five Lemma we conclude that q is an isomorphism and hence that f = g.

It is perhaps useful to remark that this result can also be deduced from the fact that property (II) is equivalent to split extensions being jointly strongly epic and the fact that, in a category with equalizers, any jointly strongly epic pair of arrows is jointly epic [1].

2.3. Proposition. Let B be a pointed category with kernels of split epis and binary products. If B satisfies (III) then it also satisfies (I).

Proof. We simply observe that any split extension is the image under Ker of some split chain in Pt_{B} (B), namely

[figure: X; i_{1}; Y; p; X′; i_{2}; X × B; i_{1}×1; Y × B; p×1; X′ × B; i_{2}×1; B; B; B .]

2.4. Corollary. Let B be a pointed category with finite limits. If B satisfies (II) or (III) then B also satisfies (I).

2.5. Proposition. Let B be pointed, with kernels of split epis and satisfying (I). Then B satisfies (II) if and only if given any diagram of the form (2.3), with f and g isomorphisms, the pair (kf^{−1}, βg^{−1}) is admissible w.r.t. (k′, β′).

Proof. The fact that (k′, β′) and (k, β) are jointly epic forces the two compositions hh′ and h′h to be, respectively, the identity on A′ and A (with h′ = [kf^{−1}, βg^{−1}]).

2.6. Proposition. Let B be pointed, with kernels of split epis and satisfying (I). Then B satisfies (III) if and only if, given any diagram of the form

[figure: X; i_{1}; k; Y; p; k̄; X′; i_{2}; k′; A; e_{1}; α; E; π; ᾱ; A′; e_{2}; α′; B; β; B; β̄; B; β′]  (2.5)

where the columns and the top row are split extensions and e_{1}, π, e_{2} are morphisms of split extensions, with πe_{2} = 1, πe_{1} = β′α, i.e.,

k = ker α ,  \overline{k} = ker \overline{α} ,  k′ = ker α′ ,  i_{1} = ker p ,
αβ = 1 ,  \overline{α}\overline{β} = 1 ,  α′β′ = 1 ,  pi_{2} = 1 ,
\overline{α}e_{1} = α ,  e_{1}β = \overline{β} ,  e_{1}k = \overline{k}i_{1} ,
\overline{α}e_{2} = α′ ,  e_{2}β′ = \overline{β} ,  e_{2}k′ = \overline{k}i_{2} ,
α′π = \overline{α} ,  π\overline{β} = β ,  π\overline{k} = k′p ,
πe_{2} = 1 ,  πe_{1} = β′α ,

if the pair (f_{0}, g_{0}) is admissible w.r.t (i_{1}, i_{2}) then the pair (f, g) is admissible w.r.t (e_{1}, e_{2}) where f and g , displayed as follows

[figure: X; f_{0}; k; D_{0}; k_{0}; X′; g_{0}; k′; A; f; α; D; α_{0}; A′; g; α′; B; β; B; β_{0}; B , k_{0} = ker α_{0} , α_{0}β_{0} = 1 ,; β′]

are morphisms of split extensions, that is, satisfying

fβ = β_{0} ,  α_{0}f = α ,  fk = k_{0}f_{0} ,
gβ′ = β_{0} ,  α_{0}g = α′ ,  gk′ = k_{0}g_{0} .

Proof. Since B satisfies (I), we simply have to prove that the pair (e_{1}, e_{2}) is jointly epic.

Indeed, in diagram 2.5, the right hand upper square k′p = πk̄ is a pullback because the lower horizontal morphism 1 : B ⟶ B is a monomorphism. Since the top row is a split extension it follows that e_{1}k = ker π. And since e_{1}k = ker π, we have that e_{1}k and e_{2} are jointly epic by condition (I), and therefore also e_{1} and e_{2} are jointly epic.

2.7. Proposition. In the context of Proposition 2.6, above, in order to have (III), it is sufficient to check that if the pair (f_{0}, g_{0}) is admissible w.r.t (i_{1}, i_{2}) then the pair (k_{0}f_{0}, g) is admissible w.r.t (e_{1}k, e_{2}).

Proof. Note that from 2.6 we know that e_{1}k = ker π. If (k_{0}f_{0}, g) is admissible w.r.t (e_{1}k, e_{2}), then we have a morphism (in Pt_{B} (B))

γ : E ⟶ D

such that γe_{1}k = k_{0}f_{0} and γe_{2} = g. We have to prove that γe_{1} = f, and in fact we have

\{ \begin{matrix}γe_{1}k = k_{0}f_{0} = fk \\ γe_{1}β = γβ̄ = γe_{2}β′ = gβ′ = β_{0} = fβ \end{matrix} ⟹ γe_{1} = f

because (k, β) is jointly epic (since we have (I)).

One last result relates condition (I) with weakly Mal’cev categories [17].

2.8. Proposition. Let B be a pointed category with pullbacks of split epis. If B satisfies (I), then B is a weakly Mal’cev category.

Proof. Given a diagram of the form

\xymatrix{A \ar[r]^{f} & B \ar[l]^{r} \ar[r]_{s} & C \ar[l]_{g}} , fr = 1_{B} = gs ,

we may construct the pullback of g along the split epi f, and since g itself is a split epi, we obtain induced morphisms into the pullback e_{1} = ⟨1, sf⟩ , e_{2} = ⟨rg, 1⟩, as follows

\xymatrix{A ×_{B} C \ar[r]^{π_{2}} \ar[d]_{π_{1}} & C \ar[l]^{e_{2}} \ar[d]_{g} \\ A \ar[r]^{f} \ar[u]_{e_{1}} & B . \ar[l]^{r} \ar[u]_{s}}

We have to prove that the pair (e_{1}, e_{2}) is jointly epic.

Since we have kernels of split epis, and every split extension is jointly epic, the above diagram becomes

[figure: X; ⟨k,0⟩; A ×_{B} C; π_{2}; π_{1}; C; e_{2}; g; X; k; A; f; e_{1}; B; r; s]

where k = ker f, and hence the pair (⟨k, 0⟩ , e_{2}) is jointly epic (because ⟨k, 0⟩ = ker π_{2}), but ⟨k, 0⟩ is also equal to e_{1}k,

e_{1}k = ⟨1, sf⟩ k = ⟨k, sfk⟩ = ⟨k, 0⟩ .

Now, (e_{1}k, e_{2}) is jointly epic, so (e_{1}, e_{2}) is also jointly epic.

3. Examples

The following are some examples of categories satisfying one or more of the above properties (I),(II),(III).

The dual of the category of topological pointed spaces satisfies (I) but not (II). The following example, due to G. Gutierres, shows why condition (II) fails.

Consider the diagram of sets and maps

[figure: X; A; k; s; f; B; p; X; A′; k′; s′; B; p′]

where X = B = {0, 1}, A = A′ = {0, 1, 2}, k(0) = k(1) = 0, k(2) = 1, p(0) = 0, p(1) = 1, s(0) = 0, s(1) = s(2) = 1, and similarly for k′, p′, s′. The map f is the identity map. Clearly, k, p, s and k′, p′, s′ are continuous if considering the following topologies on the sets above: X is indiscrete; B has {1} as the only non trivial open subset; A has {1, 2} and {1} as nontrivial open sets; while A′ has {1, 2} as the only non trivial open set. Now, it is also clear that f is continuous while its inverse is not.

By definition, all pointed protomodular categories satisfy (II).

In the case of syntactical examples we have that any quasivariety of universal algebras containing a unique constant 0 and a ternary term

p (x, y, z)

satisfying the following axioms

p (x, y, y) = p (p (x, y, 0) , 0, y) = p (y, y, x)  (3.1)
p (0, y, y) = 0  (3.2)
p (x, y, y) = p (x′, y, y) ⟹ x = x′  (3.3)

also satisfy property I. In fact, given a split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}}  (3.4)

and two morphisms

f : X ⟶ D ⟵ B : g

there is at most one morphism γ : A ⟶ D with γk = f and γβ = g. The morphism γ can be defined as a function from the set A to the set D, if and only if the following equation has a (necessarily unique) solution γ_{a} (in D) for every a in A

p_{D} (fp_{A} (a, βα (a) , 0) , 0, gα (a)) = p_{D} (γ_{a}, gα (a) , gα (a)) .

We observe that p_{A} (a, βα (a) , 0) is considered as an element in X, as αp_{A} (a, βα (a) , 0) = 0; even if the assignment a ⟼ γ_{a} defines a map from the set A to the set D, some further restrictions are necessary in order to have a homomorphism of such structures.

If replacing the axiom p (0, y, y) = 0 by the stronger one p (x, y, y) = x then the variety is protomodular (see [1], p.234) and we also have (II). An example as above, where p (x, y, y) = x does not hold for every x, may be constructed in the set of natural numbers, with zero, as follows

p (0, 0, 0) = 0
p (0, x, y) = \{ \begin{matrix}0 & if & x = y \\ f_{1} (x, y) & if & x ≠ y \end{matrix}
p (x, 0, y) = \{ \begin{matrix}0 & if & x = y \\ nx & if & x ≠ y \end{matrix}
p (x, y, 0) = \{ \begin{matrix}0 & if & x = y \\ x & if & x ≠ y \end{matrix}
p (x, 0, 0) = p (0, x, 0) = p (0, 0, x) = x

p (x, y, z) = \{ \begin{matrix}x & if & x = y = z \\ nx & if & x ≠ y = z \\ nz & if & x = y ≠ z \\ f_{2} (x, y) & if & x = z ≠ y \\ f_{3} (x, y, z) & if & x ≠ y ≠ z ≠ x \end{matrix}

where x, y, z are non zero, n is any natural number other than 1 and 0, and f_{1}, f_{2}, f_{3} are any maps on the natural numbers.

Another sort of examples, still in the varietal case, satisfying (I), are varieties of universal algebras containing a unique constant, 0, with n binary terms t_{i} and one (n + 1)- ary term t, for some n, satisfying

t_{i} (x, x) = 0
t (t_{1} (x, y) , ..., t_{n} (x, y) , y) = (t_{1} (x′, y) , ..., t_{n} (x′, y) , y) ⟹ x = x′.

Again, to see that it satisfies (I), we introduce a derived binary operation

e (x, y) = t (t_{1} (x, y) , ..., t_{n} (x, y) , y)

and observe that for any split extension (3.4) and morphisms f and g as above, there is at most one morphism γ : A ⟶ D with γk = f and γβ = g, and furthermore it is defined as a map if and only if the following equation has a (necessarily unique) solution γ_{a} in D for every a in A

e (γ_{a}, gα (a)) = t (ft_{1} (a, βα (a)) , ..., ft_{n} (a, βα (a)) , gα (a)) .

Note that we consider t_{i} (a, βα (a)) in X and not in A.

Of course that in the particular case where

e (x, y) = x

we have the characterization of pointed protomodular categories in the varietal case (see [6]).

Property (III) is still to be better understood. For the moment we just show that the categories of groups and rings do satisfy (III).

First we show that both categories satisfy (I) and give explicit formulas for a given pair of morphisms to be admissible w.r.t. a jointly epic pair obtained from a split extension. Then we use Proposition 2.7 to show that condition (III) is satisfied.

3.1. Proposition. In rings every split extension is jointly epic, that is, property (I) holds in rings. Furthermore, given any split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}}  (3.5)

the pair of morphisms (f, g)

f : X ⟶ D ⟵ B : g

is admissible w.r.t. (k, β) if and only if for every x ∈ X and b ∈ B,

f (β (b) k (x)) = g (b) f (x) , f(k(x)β(b)) = f(x)g(b)

and in that case, the morphism γ = [f, g] : A ⟶ D is given by

γ (a) = f (a − βα (a)) + gα (a) .

Proof. For every a in A we have a = k (a − βα (a)) + βα (a) (because a − βα (a) ∈ X) and since γ is such that γk = f and γβ = g we obtain

γ (a) = f (a − βα (a)) + gα (a) .

This proves (k, β) to be jointly epic.

Now, since f and g are ring homomorphisms we have obviously

γ (a + a′) = γ (a) + γ (a′) .

Assuming γ (aa′) = γ (a) γ (a′) in particular we have

γ (β (b) k (x)) = γ (β (b)) γ (k (x))

but by definition of γ and because β (b) k (x) ∈ X we have

f (β (b) k (x)) = g (b) f (x) .

A similar argument proves f(k(x)β(b)) = f(x)g(b).

Conversely, assuming f (β (b) k (x)) = g (b) f (x) and f(k(x)β(b)) = f(x)g(b), some standard algebraic manipulation shows that γ (aa′) = γ (a) γ (a′).

3.2. Proposition. In groups every split extension is jointly epic, and for every such

\xymatrix{X \ar[r]^{k} & A \ar[r]^{α} & B \ar[l]^{β}}  (3.6)

the pair of morphisms (f, g)

f : X ⟶ D ⟵ B : g

is admissible w.r.t. (k, β) if and only if

f (β (b) + k (x) − β (b)) = g (b) + f (x) − g (b) , ∀x ∈ X, ∀b ∈ B,

and in that case, the morphism γ = [f, g] : A ⟶ D is given by

γ (a) = f (a − βα (a)) + gα (a) .

Proof. This result is well known so we omit the proof.

3.3. Proposition. The condition (III) holds in the category of groups.

Proof. Since groups satisfies (I), by Proposition 2.7, it is sufficient to prove that, in the notation of Proposition 2.6, if (f_{0}, g_{0}) is admissible w.r.t. (i_{1}, i_{2}) then (k_{0}f_{0}, g) is admissible w.r.t. (e_{1}k, e_{2}).

Assume all the notations of Proposition 2.6. Suppose (f_{0}, g_{0}) is admissible w.r.t. (i_{1}, i_{2}), that is (Proposition 3.2) we have

f_{0} (i_{2} (x′) + i_{1} (x) − i_{2} (x′)) = g_{0} (x′) + f_{0} (x) − g_{0} (x′) , ∀x ∈ X, ∀x′ ∈ X′.  (3.7)

We will also need the fact that f : A ⟶ D is a group homomorphism and because (k, β) is jointly epic we have that the pair (fk, fβ) is admissible w.r.t. (k, β), i.e., (note fk = k_{0}f_{0})

k_{0}f_{0} (β (b) + k (x) − β (b)) = fβ (b) + k_{0}f_{0} (x) − fβ (b) , ∀b ∈ B, ∀x ∈ X.  (3.8)

We want to prove that (k_{0}f_{0}, g) is admissible w.r.t. (e_{1}k, e_{2}), that is,

k_{0}f_{0} (e_{2} (a′) + e_{1}k (x) − e_{2} (a′)) = g (a′) + k_{0}f_{0} (x) − g (a′) , ∀a′ ∈ A′, ∀x ∈ X.

First observe that a′ = k′ (x′) + β′ (b′), where x′ = a′ − β′α′ (a′) ∈ X′ and b′ = α′ (a′) ∈ B, and then the above equation becomes (note that e_{2}k′ = k̄i_{2}, e_{2}β′ = β̄, e_{1}k = k̄i_{1}, gk′ = k_{0}g_{0}, gβ′ = β_{0})

k_{0}f_{0} (k̄i_{2} (x′) + (β̄ (b′) + k̄i_{1} (x) − β̄ (b′)) − k̄i_{2} (x′)) = k_{0}g_{0} (x′) + β_{0} (b′) + k_{0}f_{0} (x) − β_{0} (b′) − k_{0}g_{0} (x′) ;

the argument of k_{0}f_{0} on the left side above is considered as an element in X (because it is in the kernel of ᾱ) but to evaluate it we have to do computations in E; now, because k̄ is an inclusion, we may also evaluate as in Y and hence it becomes (note that i_{1} is also an inclusion)

i_{2} (x′) + i_{1} (β̄ (b′) + k̄i_{1} (x) − β̄ (b′)) − i_{2} (x′) .

We may now use (3.7) to obtain

k_{0}g_{0} (x′) + k_{0}f_{0} (β̄ (b′) + k̄i_{1} (x) − β̄ (b′)) − k_{0}g_{0} (x′) = k_{0}g_{0} (x′) + β_{0} (b′) + k_{0}f_{0} (x) − β_{0} (b′) − k_{0}g_{0} (x′)

which simplifies to

k_{0}f_{0} (β̄ (b′) + k̄i_{1} (x) − β̄ (b′)) = β_{0} (b′) + k_{0}f_{0} (x) − β_{0} (b′) .

Again we observe that β̄ (b′) + k̄i_{1} (x) − β̄ (b′) is an element in X (it is in the kernel of ᾱ) but it is evaluated as in E; now, because (in Groups) e_{1} is a monomorphism and we have

β̄ (b′) + k̄i_{1} (x) − β̄ (b′) = e_{1}β (b′) + e_{1}k (x) − e_{1}β (b′)

we may also consider

β (b′) + k (x) − β (b′) ,

and using (3.8) we have the desired result since fβ = β_{0}.

Of course that a simpler proof is obtained if (since we are in groups), instead of diagram (2.5), we consider the one obtained up to isomorphism as follows

[figure: X; i_{1}; X ⋊ X′; [0,1]; X′; i_{2}; X ⋊ B; i_{1}⋊1; (X ⋊ X′) ⋊ B; p⋊1; X′ ⋊ B; i_{2}⋊1; B; B; B]

where i_{1} = \binom{1}{0}, i_{2} = \binom{0}{1} and it is completely determined by actions of B in X and X′, and an action of X′ in X, with the action of B on X ⋊ X′ given by b · (x, x′) = (b · x, b · x′). It is now routine calculations to check that

([\begin{matrix}f_{0} \\ 0 \end{matrix}] , [\begin{matrix}g_{0} & 0 \\ 0 & 1 \end{matrix}]) = (k_{0}f_{0}, g)

is admissible w.r.t.

([\begin{matrix}1 \\ 0 \\ 0 \end{matrix}] , [\begin{matrix}0 & 0 \\ 1 & 0 \\ 0 & 1 \end{matrix}]) = (e_{1}k, e_{2}) ,

whenever (f_{0}, g_{0}) is admissible w.r.t.

([\begin{matrix}1 \\ 0 \end{matrix}] , [\begin{matrix}0 \\ 1 \end{matrix}]) = (i_{1}, i_{2}) .

The reason why we choose to present the proof stated above, is because it may shed some light in the process of obtaining more general results than groups. For example rings.

3.4. Proposition. The category of rings satisfies (III), the Kernel Reflected Admissibility Property.

Proof. We will follow the argument used to prove the result in Groups.

Assume we have (f_{0}, g_{0}) admissible w.r.t. (i_{1}, i_{2}), that is (see Proposition 3.1 )

f_{0} (i_{2} (x′) i_{1} (x)) = g_{0} (x′) f (x) , ∀x ∈ X, ∀x′ ∈ X′.  (3.9)

We will also need the fact that (k, β) is jointly epic and that (fk, fβ) is admissible w.r.t. (k, β), i.e., (note fk = k_{0}f_{0})

k_{0}f_{0} (β (b) k (x)) = fβ (b) k_{0}f_{0} (x) , ∀b ∈ B, ∀x ∈ X.  (3.10)

We have to prove that (k_{0}f_{0}, g) is admissible w.r.t. (e_{1}k, e_{2}), i.e.,

k_{0}f_{0} (e_{2} (a′) e_{1}k (x)) = g (a′) k_{0}f_{0} (x) , ∀a′ ∈ A′, ∀x ∈ X.

We observe that

e_{2} (a′) = e_{2}k′ (x′) + e_{2}β′ (b′)
g (a′) = gk′ (x′) + gβ′ (b′)

with x′ = a′ − β′α′ (a′), b′ = α′ (a′) and so we have

k_{0}f_{0} (e_{2}k′ (x′) e_{1}k (x)) + k_{0}f_{0} (e_{2}β′ (b′) e_{1}k (x)) = gk′ (x′) k_{0}f_{0} (x) + gβ′ (b′) k_{0}f_{0} (x)

or equivalently (since e_{2}k′ = k̄i_{2}, e_{2}β′ = e_{1}β, e_{1}k = k̄i_{1}, gk′ = k_{0}g_{0}, gβ′ = β_{0})

k_{0}f_{0} (k̄i_{2} (x′) k̄i_{1} (x)) + k_{0}f_{0} (e_{1}β (b′) e_{1}k (x)) = k_{0}g_{0} (x′) k_{0}f_{0} (x) + β_{0} (b′) k_{0}f_{0} (x) ,

again, we have that the elements k̄i_{2} (x′) k̄i_{1} (x) and i_{2} (x′) i_{1} (x) are the same when considered in X and also the elements e_{1}β (b′) e_{1}k (x) and β (b′) k (x) are the same if considered in X, simply because k̄ is a monomorphism and e_{1}β (b′) e_{1}k (x) = e_{1}k (β (b′) k (x)) = k̄i_{1} (β (b′) k (x)), the resulting equation from above is

k_{0}f_{0} (i_{2} (x′) i_{1} (x)) + k_{0}f_{0} (β (b′) k (x)) = k_{0}g_{0} (x′) k_{0}f_{0} (x) + fβ (b′) k_{0}f_{0} (x) ,

and it follows directly from (3.9) and (3.10).

4. Results

From now on we assume that B is a pointed category with pullbacks along split epis, and so, in particular, it has binary products and kernels of split epis. Denoting by RG (B), SMG (B), MG (B), Cat (B), Grpd (B), respectively the categories of reflexive graphs, star-multiplicative graphs, multiplicative graphs, categories and groupoids, internal to B, we have the following obvious^{2} chain of forgetful functors

Grpd (B) ⟶^{(4)} Cat (B) ⟶^{(3)} MG (B) ⟶^{(2)} SMG (B) ⟶^{(1)} RG (B) .

2Note that the forgetful functor (2) is only defined for the multiplicative graphs satisfying dm = dπ_{2} , and so we restrict only to those; but please note that under condition (I), the equation dm = dπ_{2} is trivially satisfied.

4.1. Proposition. If B satisfies (II), the Split Short Five Lemma, then the forgetful functor (4) is an isomorphism.

Proof. We have to prove that an internal category in B is always an internal groupoid. Let be given an internal category in B, that is, a diagram

\xymatrix{C_{2} \ar[r]^{π_{2}} \ar[r]_{π_{1}} \ar[r]|{m} & C_{1} \ar[l]|{e_{2}} \ar[l]|{e_{1}} \ar[r]^{d} \ar[r]_{c} & C_{0} \ar[l]|{e}}

satisfying

de = 1 = ce me_{1} = 1 = me_{2}

dm = dπ_{2} , cm = cπ_{1}

associativity

where C_{2} is the object in the following (split) pullback diagram with projections π_{1}, π_{2} and e_{1} = ⟨1, ed⟩ , e_{2} = ⟨ec, 1⟩ the induced morphisms into the pullback

\xymatrix{C_{2} \ar[r]^{π_{2}} \ar[d]_{π_{1}} & C_{1} \ar[l]^{e_{2}} \ar[d]_{c} \\ C_{1} \ar[r]^{d} \ar[u]_{e_{1}} & C_{0} . \ar[l]^{e} \ar[u]_{e}}

It is a fact (see [3],[1]) that the above structure is an internal groupoid if and only if the following morphism

⟨m, π_{2}⟩ : C_{2} ⟶ C_{d}

is an isomorphism, with C_{d} obtained by pulling back d along the split epi d, as follows

\xymatrix{C_{2} \ar[drr]^{π_{2}} \ar[ddr]_{m} \ar@{-->}[dr] \\  & C_{d} \ar[r]^{π′_{2}} \ar[d]_{π′_{1}} & C_{1} \ar[d]^{d} \\  & C_{1} \ar[r]^{d} & C_{0} . \ar[l]^{e}}

We simply observe that ⟨m, π_{2}⟩ fits in the following diagram and hence it is a morphism of split extensions

[figure: X; ⟨k,0⟩; C_{2}; π_{2}; ⟨m,π_{2}⟩; C_{1}; e_{2}; X; ⟨k,0⟩; C_{d}; π′_{2}; C_{1} , k = ker d .; ⟨1,1⟩]

By the Split Short Five Lemma, the morphism ⟨m, π_{2}⟩ is in fact an isomorphism, and the given internal category is in fact an internal groupoid.

4.2. Proposition. If B satisfies (I), the Jointly Epic Split Extension condition, then the forgetful functor (3) is an isomorphism.

Proof. From Proposition 2.8 we conclude that B is a weakly Mal’cev category. In [17] it is proved that in a weakly Mal’cev category, every multiplicative graph is in fact an internal category.

4.3. Proposition. If B satisfies (III), the Reflected Admissibility property, then the forgetful functor (2) is an isomorphism.

Proof. We have to prove that, given a reflexive graph

\xymatrix{A \ar[r]^{d} \ar[r]_{c} & B \ar[l]|{e}} , de = 1 = ce

and a morphism

s : A ×_{B} X ⟶ X

where k : X ⟶ A is the kernel of the the split epi d , and A ×_{B} X is the object in the following pullback diagram (with kernels)

[figure: X; ⟨k,0⟩; A ×_{B} X; p_{2}; p_{1}; X; ⟨eck,1⟩; ck; X; k; A; d; B; e]

satisfying

s ⟨k, 0⟩ = 1_{X} = s ⟨eck, 1⟩ ,

it is always possible to define (and in a unique way) a morphism

m : A ×_{B} A ⟶ A

with A ×_{B} A as in the following (split) pullback diagram

\xymatrix{A ×_{B} A \ar[r]^{π_{2}} \ar[d]_{π_{1}} & A \ar[l]^{e_{2}} \ar[d]_{c} \\ A \ar[r]^{d} \ar[u]_{e_{1}} & B \ar[l]^{e} \ar[u]_{e}}

satisfying

me_{1} = 1_{A} = me_{2} .

Consider the following diagram

[figure: X; ⟨k,0⟩; k; A ×_{B} X; p_{2}; 1×_{B}k; X; ⟨eck,1⟩; k; A; e_{1}; d; A ×_{B} A; π_{2}; dπ_{2}; A; e_{2}; d; B; e; B; ⟨e,e⟩; B .; e]  (4.1)

Its columns are split extensions: to see that 1 ×_{B} k is the kernel of dπ_{2} simply observe the following diagram

\xymatrix{ & Y \ar[drr]^{kg′} \ar@{-->}[ddr]|{g′} \ar[d]|{<f,kg′>} \ar@{-->}[ddl]_{<f,g′>} \\  & A ×_{B} A \ar[rr]^{π_{2}} \ar[dd]_{π_{1}} &  & A \ar[r]^{d} \ar[dd]^{c} & B \\ A ×_{B} X \ar@{-->}[ur] \ar[dr]_{p_{1}} \ar[rr]^{p_{2}} &  & X \ar[ur]_{k} \ar[dr] \\  & A \ar[rr]^{d} &  & B .}

Since all the appropriate squares in (4.1) commute, the top horizontal arrows are the image under the Kernel Functor of e_{1}, e_{2} and π_{2}, considered as morphisms of spit extensions.

Now, the top row is a split extension, so that by the Kernel Reflected Admissibility Property (III) we may conclude the following assertions^{3}

• (⟨k, 0⟩ , ⟨eck, 1⟩) is jointly epic;

• (e_{1}, e_{2}) is jointly epic;

• for every pair of morphisms (f, g) in Pt_{B} (B)

[figure: A; f; d; D; α; A; g; d; B; e; B; β; B; e]

if (Ker (f) , Ker (g)) is admissible w.r.t. (⟨k, 0⟩ , ⟨eck, 1⟩) then (f, g) is admissible w.r.t. (e_{1}, e_{2}).

In particular the pair of morphisms (1_{A}, 1_{A}) in Pt_{B} (B)

[figure: X; 1_{X}; k; X; k; X; 1_{X}; k; A; 1_{A}; d; A; d; A; 1_{A}; d; B; e; B; e; B; e]

is such that the pair (1_{X}, 1_{X}) is admissible w.r.t. (⟨k, 0⟩ , ⟨eck, 1⟩), because we have given s : A ×_{B} X ⟶ X with s ⟨k, 0⟩ = 1_{X} and s ⟨eck, 1⟩ = 1_{X}, and hence we may conclude that the pair (1_{A}, 1_{A}) is admissible w.r.t. (e_{1}, e_{2}), which gives us the desired (unique) morphism

3In fact the pair (e_{1}, e_{2}) is jointly epic in Pt_{B} (B), but in the presence of binary products it is also jointly epic in the ground category B.

m : A ×_{B} A ⟶ A

with me_{1} = 1_{A} and me_{2} = 1_{A}.

This does not seem to be a necessary condition for (2) to be an isomorphism. Indeed, from the proof above, it is clear that in the presence of pullbacks along split epimorphisms, we could restrict condition (III) to those split extensions in Pt_{B} (B), that are induced by a reflexive graph in B; rather than asking the reflected admissibility condition for an arbitrary split chain in Pt_{B} (B), whose image by Ker is a split extension in B .

Next we show that under (I), the Jointly Epic Split Extension condition, all the forgetful functors in the chain above are injective on objects. This means that for a reflexive graph

\xymatrix{A \ar[r]^{d} \ar[r]_{c} & B \ar[l]|{e}} , de = 1 = ce

it is a property whether or not it admits a star-multiplication, a multiplication (which makes it automatically an internal category, by Proposition 4.2), a multiplication with inverses (making it a groupoid).

It is remarkable that property (I) provides a tool to characterize the different levels at which a given reflexive graph is.

Similar situations were already observed for example in the case of pointed protomodular categories where in fact every split extension is jointly strongly epic, and the notions of central morphism, connector, and cooperator are introduced, with very strong classifying properties (see for example [1], [4], [2], [5] and references there).

Under (I) the notion of reflexive graph itself may be decomposed into a split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{d} & B \ar[l]^{e}} , de = 1, k = ker d

together with a morphism

h : X ⟶ B

having the property that the pair (h, 1_{B}) is admissible w.r.t. (k, e), thus giving c : A ⟶ B as the (unique) morphism with

ck = h , ce = 1 .

We will start with a very general setting and then restrict to the present case.

Let B be any category and consider the category B^{D} where objects are diagrams in B of the form

\xymatrix{X \ar[r]^{k} \ar[rr]_{h} & A & B \ar[l]_{e}}

with the only requirement that the pair (k, e) is jointly epic.

Now let B be pointed with kernels of split epis, and restrict B^{D} to the category of diagrams as above such that (k, e) is jointly epic and the pairs (0_{X,B}, 1_{B}) and (h, 1_{B}) are admissible w.r.t. (k, e) and, furthermore, the morphism k is a kernel for

[0, 1] : A ⟶ B.

We have thus the following proposition.

4.4. Proposition. If B satisfies (I), the Jointly Epic Split Extension property, then the functor

B^{D} ⟶ RG (B)

which assigns to each object

\xymatrix{X \ar[r]^{k} \ar[rr]_{h} & A & B \ar[l]_{e}}

the reflexive graph

\xymatrix{A \ar[r]^{[0,1]} \ar[r]_{[h,1]} & B \ar[l]|{e}}

is an equivalence.

Proof. Given a reflexive graph

\xymatrix{A \ar[r]^{d} \ar[r]_{c} & B \ar[l]|{e}} , de = 1 = ce,

if considering k the kernel of the split epi d, we have that (k, e) is jointly epic and hence

d = [0 1]
c = [ck 1] .

Let again B be a pointed category with pullbacks along split epis satisfying property (I).

We consider a fixed jointly epic pair

\xymatrix{X \ar[r]^{k} & A & B \ar[l]_{e}}  (4.2)

such that the pair (0_{X,B}, 1_{B}) is admissible w.r.t. (k, e), and k = ker ([0, 1]), where we write [0, 1] : A ⟶ B for the induced morphism with [0, 1]k = 0_{X,B} and [0, 1]e = 1_{B}.

A given morphism h : X ⟶ B is said to be of

type 1 if the pair (h, 1_{B}) is admissible w.r.t. (k, e). In this case [h, 1] : A ⟶ B will denote the unique morphism with the property that [h, 1]k = h , [h, 1]e = 1; type 2 if the pair (1_{X}, 1_{X}) is admissible w.r.t. (⟨k, 0⟩ , ⟨eh, 1⟩).

Here A ×_{B} X is obtained by pulling back h along the split epi [0, 1] = d, that is

[figure: X; ⟨k,0⟩; A ×_{B} X; p_{2}; p_{1}; X; ⟨eh,1⟩; h; X; k; A; d; B .; e]

We write [1_{X}, 1_{X}] : A ×_{B} X ⟶ X for the unique morphism satisfying

[1, 1] ⟨k, 0⟩ = 1 = [1, 1] ⟨eh, 1⟩ .

type 3 if it is of type 1 and, in addition, the pair (k, 1_{A}) is admissible w.r.t. (e_{1}k, e_{2})

\xymatrix{X \ar[r]^{e_{1}k} & A_{2} & A \ar[l]_{e_{2}}} .

Here A_{2} is the pullback of [h 1] along [0 1], and e_{1} = ⟨1, ed⟩ , e_{2} = ⟨ec, 1⟩ are the induced morphisms into the pullback, as displayed in the following diagram

[figure: X; ⟨k,0⟩; A_{2}; π_{2}; π_{1}; A; e_{2}; [h 1]; X; k; A; [0 1]; e_{1}; B .; e; e]  (4.3)

We also write

m = [k 1_{A}] : A_{2} ⟶ A

and observe that

π_{2} = [0_{X,A} 1_{A}]
π_{1} = [k [eh e]]
e_{1}k = ⟨k, 0_{X,A}⟩ = ker π_{2} .

type 4 if it is of type 1 and, in addition, the pair (⟨k, 0_{X,A}⟩ , e_{2}) is admissible with respect to (⟨k, 0_{X,A}⟩ , ⟨1_{A}, 1_{A}⟩)

⟨k, 0_{X,A}⟩ : X ⟶ A_{d} ⟵ A : ⟨1_{A}, 1_{A}⟩ .

That is, we may insert the dashed arrow, denoted by [⟨k, 0_{X,A}⟩ e_{2}], in the following diagram of split extensions

[figure: X; ⟨k,0⟩; A_{d}; π′_{2}; A; ⟨1,1⟩; X; ⟨k,0⟩; A_{2}; π_{2}; A ;; e_{2}]

where A_{2}, π_{2}, e_{2} are as in (4.3) and A_{d} is the pullback of d = [0 1] along itself, as displayed in the following diagram

[figure: X; ⟨k,0⟩; A_{d}; π′_{2}; π′_{1}; A; <ed,1>; d; X; k; A; [0 1]; B .; e]

Note that we are replacing the induced section of of π′_{2}, ⟨ed, 1⟩, by the morphism ⟨1, 1⟩.

4.5. Proposition. Let B be a pointed category with pullbacks of split epis and satisfying (I). To give a reflexive graph in B is to give a split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{[0 1]} & B \ar[l]^{e}}

together with a morphism

h : X ⟶ B

of type 1. Furthermore:

• the reflexive graph is a star-multiplicative graph if and only if h is also of type 2 w.r.t. (k, e);

• the reflexive graph is a multiplicative graph (and hence an internal category) if and only if h is of type 3 (w.r.t. (k, e));

• the reflexive graph is an internal groupoid if and only if h is of type 3 and type 4.

Proof. That type 1 corresponds to reflexive graphs follows from Proposition 4.4. Type 1 and type 2, when combined, give

c = [h 1] : A ⟶ B

and

s = [1_{X} 1_{X}] : A ×_{B} X ⟶ X

satisfying the required conditions for a star-multiplicative graph. Of course, since (k, e) and (⟨k, 0⟩ , ⟨eh, 1⟩) are jointly epic, every star-multiplicative graph is obtained in this way.

Type 3 is the same with multiplicative graphs.

Type 3 and type 4 give in fact an internal groupoid, since we can fill in the following diagram

[figure: X; ⟨k,0⟩; A_{d}; π′_{2}; [<k,0> e_{2}]; A; <1,1>; X; ⟨k,0⟩; A_{2}; π_{2}; <m,π_{2}>; A; e_{2}]

with the two morphisms in the middle, with opposite directions. Since (⟨k, 0⟩ , ⟨1, 1⟩) and (⟨k, 0⟩ , e_{2}) are jointly epic it follows that they are inverse to each other.

Finally we conclude by presenting a simple characterization of internal groupoids (a similar result, but in a slightly different context, was presented in [15]).

4.6. Proposition. Let B be a pointed category with pullbacks of split epis and satisfying (I),(II) and (III). Giving an internal groupoid in B is to give a split extension

\xymatrix{X \ar[r]^{k} & A \ar[r]^{[0 1]} & B \ar[l]^{e}}

together with a morphism

h : X ⟶ B

such that the dashed arrows in the following diagram can be inserted in order to make it commutative

[figure: X; ⟨1,0⟩; X × X; π_{2}; (a); X; ⟨1,1⟩; h; X; k; h; A; [0 1]; (b); B; e; B; ⟨1,0⟩; B × B; π_{2}; B .; ⟨1,1⟩]

Proof. It is sufficient to prove that every star-multiplicative graph is of this form (since we are assuming (I), (II) and (III) and we have Propositions 4.3, 4.2 and 4.1).

Inserting (b) is equivalent to the fact that (h, 1) is admissible w.r.t. (k, e) (i.e., h is of type 1 above), and we have

(b) = ⟨[h 1] , [0 1]⟩ .

Inserting (a) is equivalent to say that the diagram

[figure: X; ⟨1,0⟩; X × X; π_{2}; X; ⟨1,1⟩; X; ⟨k,0⟩; A ×_{B} X; p_{2}; X; ⟨eh,1⟩]

can be completed, i.e., the dashed arrow can be inserted; because of (II), the dashed arrow, existing is an isomorphism, and hence it is also equivalent to say that h is of type 2, with the morphism [1_{X}, 1_{X}] given by

A ×_{B} X ⟶^{≅} X × X ⟶^{π_{1}} X .

5. Conclusion

In [10] it is said that in a semi-abelian category in general, not every internal groupoid is obtained from the description above, this means that we have to impose condition (III) in a semi-abelian category in order to have the desired result. This is a new condition (at least for the knowledge of the author) and some future work is needed in order to better understand it. We have only checked that it holds in the semi-abelian categories of groups and rings. It is also clear that it holds in every additive category, with kernels of split epimorphisms, since in that case every pair is admissible.

As for the future development of this topics we observe that two extreme cases may occur: either (III) is too restrictive and only a few semi-abelian categories (of interest) satisfy it, or the opposite; in the second case, the results here presented have a wide range of application, and certainly deserve to be further investigated; if, instead, the first is the case, then this condition may be a good approximation for an axiomatic treatment of the category of groups and rings [9].

References

[1] Borceux, F. and Bourn, D.: Mal’cev, Protomodular, Homological and Semi-Abelian Categories, Math. Appl. 566, Kluwer, 2004.

[2] Bourn, D.: “Intrinsic centrality and associated classifying properties”, J. of Algebra, 256, (2002), 126–145.

[3] Bourn, D.: “The Shift functor and the comprehensive factorization for internal groupoids”, Cahiers Top. Géom. Diff. Catégoriques, 28, (1987), 197–226.

[4] Bourn, D. and Gran, M.: “Centrality and normality in protomodular categories”, Theory and Applications of Categories, Vol. 9, (2002), 151–165.

[5] Bourn, D. and Gran, M.: “Centrality and connectors in Mal’cev categories”, Algebra Universalis, 48 (2002), 309–331.

[6] Bourn, D. and Janelidze, G.: “Characterization of protomodular varieties of universal algebra”, Theory and Applications of Categories, Vol. 11, (2003), 143–147.

[7] Bourn, D. and Janelidze, G.: “Protomodularity, descent, and semidirect products ”, Theory and Applications of Categories, Vol. 4, No. 2, (1998), 37–46.

[8] Carboni, A., Pedicchio, M.C. and Pirovano, N.: “Internal graphs and internal groupoids in Mal’cev categories”, Proc.Conf. Category Theory 1991, Montreal.

[9] Clementino, M.M.: “Towards categorical behaviour of groups”, Bulletin CIM, December 2007, No. 23.

[10] Janelidze, G.: “Internal Crossed Modules”, Georgian Math. J., 10, (2003), no. 1, 99–114.

[11] Janelidze, G.: “Internal categories in Malcev varieties”, preprint, Univ. North York, Toronto, 1990.

[12] Janelidze, G., Márki, L. and Tholen, W.: “Semiabelian categories”, J. Pure and Appl. Alg., 168, (2002), 367–386.

[13] Janelidze, G. and Pedicchio, M.C.: “Internal categories and groupoids in congruence modular categories”, Journal of Algebra, Vol. 193, No. 2, (1997), 552–570.

[14] MacLane, S.: Categories for the Working Mathematician, Springer-Verlag, 1998, 2^{nd} edition.

[15] Mantovani, S.: “Internal crossed modules in strongly protomodular categories”, talk presented at Categories in Algebra, Geometry and Logic conference, held in Brussels, Belgium, October 2008.

[16] Martins-Ferreira, N.: “Internal precategories relative to split epimorphisms”, arXiv:0903.0333v1

[17] Martins-Ferreira, N.: “Weakly Mal’cev categories”, Theory and Applications of Categories, Vol. 21, No. 6, (2008), 91–117.

[18] Rodelo, D.: “Moore categories”, Theory and Applications of Categories, Vol. 12, 2004, No. 6, pp 237–247.

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