Theory and Applications of Categories, Vol. 21, No. 6, 2008, pp. 91–117.

WEAKLY MAL’CEV CATEGORIES

N. MARTINS-FERREIRA

Abstract. We introduce a notion of weakly Mal’cev category, and show that: (a) every internal reflexive graph in a weakly Mal’tsev category admits at most one multiplicative graph structure in the sense of [10] (see also [11]), and such a structure always makes it an internal category; (b) (unlike the special case of Mal’tsev categories) there are weakly Mal’tsev categories in which not every internal category is an internal groupoid. We also give a simplified characterization of internal groupoids among internal categories in this context.

1. Introduction

A weakly Mal’cev category (WMC) is defined by the following two axioms:

1. Existence of pullbacks of split epis along split epis.

2. Every induced canonical pair of morphisms into a pullback (see Definition 2.3), is jointly epimorphic.

These two simple axioms are exactly what one needs in order to have a unique multiplicative structure, provided it exists, over a reflexive graph, and it then follows that this unique multiplicative structure is in fact an internal category (see [10] and [11]).

The name weakly Mal’cev category is motivated as follows. A Mal’cev category has finite limits (see [1] Definition 2.2.3, p.142) and the induced

canonical pair of morphisms into the pullback (see [1] Lemma 2.3.1, p.151) is strongly epimorphic.

Hence every Mal’cev category is an example of a weakly Mal’cev category. Examples of weakly Mal’cev categories that are not Mal’cev, are due to G. Janelidze,

and are the following:

Commutative monoids with cancelation.

The author thanks: Professor G. Janelidze for much appreciated help of various kinds; Zurab for the interesting discussions during the lunch time in CT2007.

Received by the editors 2008-02-22 and, in revised form, 2008-09-12. Published on 2008-09-25 in the Tholen Festschrift. 2000 Mathematics Subject Classification: Primary 18E05; Secondary 18B40.. Key words and phrases: Admissible reflexive graph, multiplicative graph, internal category, inter-

nal groupoid, weakly Mal’cev category, naturally weakly Mal’cev category, Mal’cev variety of universal algebras.

\begin{matrix}c \\ ◯ \end{matrix} N. Martins-Ferreira, 2008. Permission to copy for private use granted.

A category with objects (A, p, e) where A is a set, p a ternary operation, e a unary operation and where the following axioms are satisfied

p (x, y, y) = e (x) , p (x, x, y) = e (y)
e (x) = e (y) ⟹ x = y.

Note that p becomes a Mal’cev operation when e is the identity.

The setting of a weakly Mal’cev category seems to be the most appropriate to study internal categories (see final note for further discussion).

The main purpose of this paper is to introduce the concept of weakly Mal’cev category and describe some of its properties, establishing a convenient notation for ad hoc calculations.

In order to stress the significance of the proposed notion, we compare some of its properties with analogous and well known properties in the context of Mal’cev categories. They are the following (see the references, in particular [1],[8],[2],[4],[5],[6],[7] and [9]).

In the context of a Mal’cev category:

1. every reflexive graph admits at most one multiplication;

2. every multiplicative graph is an internal category;

3. every internal category is a groupoid.

In the context of a weakly Mal’cev category:

1. every reflexive graph admits at most one multiplication (here denoted by admissible, in that case);

2. every multiplicative graph (or admissible reflexive graph) is already an internal category;

3. not every admissible reflexive graph (or multiplicative graph, or internal category) is an internal groupoid, nevertheless there is an intrinsic description of the admissible reflexive graphs with the property of being a groupoid.

In commutative monoids with cancelation, an example of a internal category that is not a internal groupoid is the less or equal relation in the natural numbers considered as a preorder.

In a weakly Mal’cev category, given a diagram of the form

\xymatrix{ & B \ar[d]_{g} \\ A \ar[r]^{f} & C \ar[l]^{r} \ar[u]_{s}}

with fr = 1_{C} = gs, we may form the pullback (of a split epi along a split epi)

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

with projections π_{1} and π_{2}, and where e_{1}, e_{2} are the canonical induced morphisms, that is, they are such that

π_{1}e_{1} = 1_{A} , π_{2}e_{1} = sf
π_{1}e_{2} = rg , π_{2}e_{2} = 1_{B}.

The pair (e_{1}, e_{2}) is jointly epimorphic by definition. Then, for every triple of morphisms (h, l, k)

\xymatrix{A \ar[dr]_{h} & C \ar[l]_{r} \ar[r]^{s} \ar[d]^{l} & B \ar[dl]^{k} \\  & D}

such that hr = l = ks, there is at most one morphism

α : A ×_{C} B ⟶ D

such that

αe_{1} = h
αe_{2} = k

which is denoted by

α = [h l k]

when it exists. It is also convenient to specify the morphisms f and g; so that in general we will say: the triple of morphisms (h, l, k), as above, has the property (or not) that the morphism

[h l k]

exists, with respect to

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}} .

In the case [h l k] exists we will say that the triple (h, l, k) is admissible.

With this notation, the notion of admissible reflexive graph (that is an internal category) is the following:

In a weakly Mal’cev category, a reflexive graph

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

is said to be admissible when the triple

(1_{C_{1}}, e, 1_{C_{1}})

is admissible with respect to

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

It is then a multiplicative graph with multiplication

\xymatrix{C_{1} ×_{C_{0}} C_{1} \ar[r]^{[1 e 1]} & C_{1}}

which automatically satisfies the axioms of an internal category, and furthermore, it has the property of being a groupoid if and only if the triple

(π_{2}, 1_{C_{1}}, π_{1})

is admissible with respect to

\xymatrix{C_{1} ×_{C_{0}} C_{1} \ar[r]^{[1 e 1]} & C_{1} \ar[l]^{e_{2}} \ar[r]_{e_{1}} & C_{1} ×_{C_{0}} C_{1} \ar[l]_{[1 e 1]}} .

In the presence of a Mal’cev operation, p(x, y, z) written formally as in the case of Groups, that is p(x, y, z) = x − y + z, a general morphism

[h l k] : A ×_{C} B ⟶ D,

in case of existence, is given by

[h l k] (a, c, b) = h (a) − l (c) + k (b) ,

so that in particular the multiplication [1 e 1], in case of existence, is given by

[1 e 1] (· ⟵^{f} x ⟵^{g} ·) = f − 1_{x} + g,

while inverses (assuming [π_{2} 1_{C_{1}} π_{1}] exists) are given by

f^{−1} = [π_{2} 1_{C_{1}} π_{1}] [\begin{matrix}e_{1} \\ 1_{C_{1}} \\ e_{2} \end{matrix}] (f)
= (π_{2}e_{1} − 1 + π_{1}e_{2}) (f)
= (ed − 1 + ec) (f)
= ed (f) − f + ec (f)
= 1_{x} − f + 1_{y}

for arrows x ⟶^{f} y in C_{1}.

This paper is organized as follows: First we introduce the notion and deduce some properties of weakly Mal’cev categories; next we prove the equivalence between internal categories and admissible reflexive graphs; later we show the connection with Mal’cev categories; at the end we describe internal groupoids in weakly Mal’cev categories.

2. The notion of a weakly Mal’cev category

Let C be a given category.

2.1. Definition. [split span] A split span is a diagram in C of the form

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}}

such that

fr = 1_{C} = gs.

2.2. Definition. [split square] A split square is a diagram in C of the form

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

such that

fr = 1_{C} = gs
gp_{2} = fp_{1}
e_{2}s = e_{1}r
p_{2}e_{2} = 1_{B}
p_{2}e_{1} = sf
p_{1}e_{1} = 1_{A}
p_{1}e_{2} = rg,

in other words, it is a double split epi, in the sense that it is a split epi in the category of split epis in C.

The term split pullback will be used to refer to a split square as above, such that

\xymatrix{P \ar[r]^{p_{2}} \ar[d]_{p_{1}} & B \ar[d]^{g} \\ A \ar[r]^{f} & C}

is a pullback diagram.

2.3. Definition. [weakly Mal’cev category] A category C is weakly Mal’cev when:

1. It has pullbacks of split epis along split epis;

2. For every split square

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

if (P, p_{1}, p_{2}) is a pullback, then the pair (e_{1}, e_{2}) is jointly epimorphic, that is, given u, v : P ⟶ D,

\{ \begin{matrix}ue_{2} = ve_{2} \\ ue_{1} = ve_{1} \end{matrix} ⟹ u = v.

2.4. Proposition. In a weakly Mal’cev category, given a split span

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}}  (1)

for every object D, and a triple of morphisms (h, l, k)

\xymatrix{A \ar[dr]_{h} & C \ar[l]_{r} \ar[r]^{s} \ar[d]^{l} & B \ar[dl]^{k} \\  & D}

such that hr = l = ks, there exists at most one morphism, denoted by [h l k] when it exists, from the pullback

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

to the object D,

[h l k] : A ×_{C} B ⟶ D,

with the property that

[h l k] e_{1} = h
[h l k] e_{2} = k,

where e_{1} = ⟨1, sf⟩ = [\begin{matrix}1 \\ f \\ sf \end{matrix}] : A ⟶ A ×_{C} B, and e_{2} = ⟨rg, 1⟩ = [\begin{matrix}rg \\ g \\ 1 \end{matrix}] : B ⟶ A ×_{C} B

are the induced morphisms into the pullback.

Proof. The pullback A ×_{C} B, being a pullback of a split epi (g, s) along a split epi (f, r), exists in a weakly Mal’cev category, and e_{1}, e_{2}, the induced morphisms into the pullback, make the diagram

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

a split square; to prove that [h l k] is unique if it exists, suppose the existence of

p, q : A ×_{C} B ⟶ D

satisfying

pe_{1} = h, pe_{2} = k
qe_{1} = h, qe_{2} = k,

by definition of weakly Mal’cev, the pair (e_{1}, e_{2}) is jointly epimorphic and hence p = q.

Note that the morphism l, being determined by either h or k, is explicitly used to avoid always having to choose between hr and ks. Also, if h and k do not satisfy hr = ks then there is no morphism p : A ×_{C} B ⟶ D satisfying pe_{1} = h and pe_{2} = k because if it existed it would imply that hr = ks since e_{1}r = e_{2}s.

Relative to a split span

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}} ,  (2)

the notation

β = [\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}]

for a morphism into the pullback

\xymatrix{D \ar[drr]^{β_{2}} \ar[dr]^{β} \ar[ddr]_{β_{1}} \\  & A ×_{C} B \ar[r]^{π_{2}} \ar[d]_{π_{1}} & B \ar[d]^{g} \\  & A \ar[r]^{f} & C}

induced by β_{1}, β_{0}, β_{2} with

fβ_{1} = β_{0} = gβ_{2};

and

α = [α_{1} α_{0} α_{2}]

for a morphism from the pullback

\xymatrix{ & C \ar[dl]_{r} \ar[dr]^{s} \\ A \ar[r]^{e_{1}} \ar[dr]_{α_{1}} & A ×_{C} B \ar@{-->}[d]^{α}_{?} & B \ar[l]_{e_{2}} \ar[dl]^{α_{2}} \\  & D′}

induced (when it exists) by α_{1}, α_{0}, α_{2} with

α_{1}r = α_{0} = α_{2}s,

seems to be appropriate due to the following facts:

• every split span

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}}

determines a split pullback

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

where

π_{1} = [1 r rg]
π_{0} = [f 1 g]
π_{2} = [sf s 1]
e_{1} = [\begin{matrix}1 \\ f \\ sf \end{matrix}] , e_{0} = [\begin{matrix}r \\ 1 \\ s \end{matrix}] , e_{2} = [\begin{matrix}rg \\ g \\ 1 \end{matrix}]

1_{A×_{C}B} = [\begin{matrix}π_{1} \\ π_{0} \\ π_{2} \end{matrix}] = [e_{1} e_{0} e_{2}] = [\begin{matrix}1 & r & rg \\ f & 1 & g \\ sf & s & 1 \end{matrix}];

• for every u : D′ ⟶ D′′, the composite uα : A ×_{C} B ⟶ D′′ is given by the formula

u [α_{1} α_{0} α_{2}] = [uα_{1} uα_{0} uα_{2}] ,

whenever both sides are defined, in the sense that from [α_{1} α_{0} α_{2}] we can deduce the existence of [uα_{1} uα_{0} uα_{2}] , but given [uα_{1} uα_{0} uα_{2}] we can only write u [α_{1} α_{0} α_{2}] provided that the existence of [α_{1} α_{0} α_{2}] is already ensured.

• for every v : \overline{D} ⟶ D the composite βv : \overline{D} ⟶ A ×_{C} B is given by the formula

[\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] v = [\begin{matrix}β_{1}v \\ β_{0}v \\ β_{2}v \end{matrix}];

• it is sometimes useful to write the composite αβ : D ⟶ D′ as a formal formula

[α_{1} α_{0} α_{2}] [\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] = α_{1}β_{1} − α_{0}β_{0} + α_{2}β_{2},

it is not the case that it defines a Mal’cev operation, but for the following special cases one has

α_{1} = αe_{1} = [α_{1} α_{0} α_{2}] [\begin{matrix}1 \\ f \\ sf \end{matrix}] = α_{1} − α_{0}f + α_{2}sf = α_{1} − α_{0}f + α_{0}f

α_{0} = αe_{0} = [α_{1} α_{0} α_{2}] [\begin{matrix}r \\ 1 \\ s \end{matrix}] = α_{1}r − α_{0} + α_{2}s = α_{0} − α_{0} + α_{0}

α_{2} = αe_{2} = [α_{1} α_{0} α_{2}] [\begin{matrix}rg \\ g \\ 1 \end{matrix}] = α_{1}rg − α_{0}g + α_{2} = α_{0}g − α_{0}g + α_{2}

β_{1} = π_{1}β = [1 r rg] [\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] = β_{1} − rβ_{0} + rgβ_{2} = β_{1} − rβ_{0} + rβ_{0}

β_{0} = π_{0}β = [f 1 g] [\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] = fβ_{1} − β_{0} + gβ_{2} = β_{0} − β_{0} + β_{0}

β_{2} = π_{2}β = [sf s 1] [\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] = sfβ_{1} − sβ_{0} + β_{2} = sβ_{0} − sβ_{0} + β_{2},

and in general, for a triple of morphisms

x, y, z : D ⟶ D′

such that there exists

β : D ⟶ A ×_{C} B and α : A ×_{C} B ⟶ D′

with

α_{1}β_{1} = x , α_{0}β_{0} = y , α_{2}β_{2} = z,

and writing

αβ = [α_{1} α_{0} α_{2}] [\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] = α_{1}β_{1} − α_{0}β_{0} + α_{2}β_{2} = x − y + z

it is clear that there is a partially defined (relative to the split span (2)) ternary operation in hom (D, D′),

(x, y, z) ⟼ x − y + z,

but in general there is no reason for this to satisfy the Mal’cev axioms x − z + z = x and x − x + z = z; however, it does so if the category C is a Mal’cev variety of universal algebras, since in that case

[α_{1} α_{0} α_{2}] (a, c, b) = p (α_{1} (a) , α_{0} (c) , α_{2} (b)) ,

with p the Mal’cev operation on D′;

• also, given another split span

\xymatrix{A′ \ar[r]^{f′} & C′ \ar[l]^{r′} \ar[r]_{s′} & B′ \ar[l]_{g′}}

and a morphism γ = [γ_{1} γ_{0} γ_{2}] : A′ ×_{C′} B′ ⟶ D, the composite βγ,

\xymatrix{A′ \ar[dr]_{γ_{1}} & C′ \ar[l]_{r′} \ar[r]^{s′} \ar[d]^{γ_{0}} & B′ \ar[dl]^{γ_{2}} \\  & D \ar[dl]_{β_{1}} \ar[d]^{β_{0}} \ar[dr]^{β_{2}} \\ A \ar[r]^{f} & C & B \ar[l]_{g}}  (3)

is given by

[\begin{matrix}β_{1} \\ β_{0} \\ β_{2} \end{matrix}] [γ_{1} γ_{0} γ_{2}] = [\begin{matrix}β_{1}γ_{1} & β_{1}γ_{0} & β_{1}γ_{2} \\ β_{0}γ_{1} & β_{0}γ_{0} & β_{0}γ_{2} \\ β_{2}γ_{1} & β_{2}γ_{0} & β_{2}γ_{2} \end{matrix}] : A′ ×_{C′} B′ ⟶ A ×_{C} B

with components relative to the specified split spans, as displayed in (3).

Moreover, the following proposition describes the form of the morphisms between pullbacks, relative to the specified split spans.

2.5. Proposition. In a WMC (Weakly Mal’cev Category), given two split spans

\xymatrix{A′ \ar[r]^{f′} & C′ \ar[l]^{r′} \ar[r]_{s′} & B′ \ar[l]_{g′}}

and

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

a morphism φ : A′ ×_{C′} B′ ⟶ A ×_{C} B is of the form

φ = [\begin{matrix}π_{1}φe′_{1} & π_{1}φe′_{0} & π_{1φ}e′_{2} \\ π_{0}φe′_{1} & π_{0}φe′_{0} & π_{0}φe′_{2} \\ π_{2}φe′_{1} & π_{2}φe′_{0} & π_{2}φe′_{2} \end{matrix}] = [\begin{matrix}φ_{11} & φ_{10} & φ_{12} \\ φ_{01} & φ_{00} & φ_{02} \\ φ_{21} & φ_{20} & φ_{22} \end{matrix}]

and it determines the following commutative diagram

[figure: A′; φ_{11}; φ_{01}; φ_{21}; C′; φ_{10}; φ_{00}; φ_{20}; r′; s′; B′; φ_{12}; φ_{02}; φ_{22}; A; f; C; B .; g]  (4)

Conversely, given a commutative diagram as above, it determines a morphism φ : A′ ×_{C′} B′ ⟶ A ×_{C} B of the form

φ = [\begin{matrix}φ_{11} & φ_{10} & φ_{12} \\ φ_{01} & φ_{00} & φ_{02} \\ φ_{21} & φ_{20} & φ_{22} \end{matrix}]

if and only if the morphisms [φ_{11} φ_{10} φ_{12}] , [φ_{01} φ_{00} φ_{02}] and [φ_{21} φ_{20} φ_{22}] exist. In particular, given a commutative diagram

\xymatrix{A′ \ar[d]_{h} \ar[r]^{f′} & C′ \ar[d]^{l} & B′ \ar[l]_{g′} \ar[d]^{k} \\ A \ar[r]_{f} & C & B \ar[l]^{g}}

the induced morphism h ×_{l} k : A′ ×_{C′} B′ ⟶ A ×_{C} B is given by

h ×_{l} k = [\begin{matrix}hπ′_{1} \\ lπ′_{0} \\ kπ′_{2} \end{matrix}] = [\begin{matrix}h & hr′ & hr′g′ \\ fh & l & gk \\ ks′f′ & ks′ & k \end{matrix}] .

Proof. A given morphism φ : A′ ×_{C′} B′ ⟶ A ×_{C} B is always determined as a morphism into the pullback

\xymatrix{A ×_{C} B \ar[r]^{π_{2}} \ar[dr]^{π_{0}} \ar[d]_{π_{1}} & B \ar[d]^{g} \\ A \ar[r]_{f} & C}

by the components

φ = [\begin{matrix}π_{1}φ \\ π_{0}φ \\ π_{2}φ \end{matrix}] .

Since each one of the components is a morphism from the pullback A′ ×_{C′} B′, they are determined by the canonical morphisms e′_{1}, e′_{0}, e′_{2}

\xymatrix{A′ ×′_{C} B′ & B′ \ar[l]_{e′_{2}} \\ A′ \ar[u]^{e′_{1}} & C′ \ar[l]^{r′} \ar[u]_{s′} \ar[ul]_{e′_{0}}}

and

φ = [\begin{matrix}π_{1}φ \\ π_{0}φ \\ π_{2}φ \end{matrix}] = [\begin{matrix}[ & π_{1}φe′_{1} π_{1}φe′_{0} π_{1}φe′_{2} & ] \\ [ & π_{0}φe′_{1} π_{0}φe′_{0} π_{0}φe′_{2} & ] \\ [ & π_{2}φe′_{1} π_{2}φe′_{0} π_{2}φe′_{2} & ] \end{matrix}] .

In the same way one obtains

φ = [φe′_{1} φe′_{0} φe′_{2}] = [[\begin{matrix}π_{1}φe′_{1} \\ π_{0}φe′_{1} \\ π_{2}φe′_{1} \end{matrix}] [\begin{matrix}π_{1}φe′_{0} \\ π_{0}φe′_{0} \\ π_{2}φe′_{0} \end{matrix}] [\begin{matrix}π_{1}φe′_{2} \\ π_{0}φe′_{2} \\ π_{2}φe′_{2} \end{matrix}]]

so that one can simply write

φ = [\begin{matrix}π_{1}φe′_{1} & π_{1}φe′_{0} & π_{1φ}e′_{2} \\ π_{0}φe′_{1} & π_{0}φe′_{0} & π_{0}φe′_{2} \\ π_{2}φe′_{1} & π_{2}φe′_{0} & π_{2}φe′_{2} \end{matrix}] .

To prove that

φ = [\begin{matrix}φ_{11} & φ_{10} & φ_{12} \\ φ_{01} & φ_{00} & φ_{02} \\ φ_{21} & φ_{20} & φ_{22} \end{matrix}]

with the components φ_{ij} as in the commutative diagram (4), exists if and only if

[φ_{11} φ_{10} φ_{12}] , [φ_{01} φ_{00} φ_{02}] , [φ_{21} φ_{20} φ_{22}]

exists, observe that given φ, they are respectively π_{1}φ, π_{0}φ, π_{2}φ, conversely, given such morphisms, there is φ and it is given by

φ = [\begin{matrix}[ & φ_{11} φ_{10} φ_{12} & ] \\ [ & φ_{01} φ_{00} φ_{02} & ] \\ [ & φ_{21} φ_{20} φ_{22} & ] \end{matrix}] .

Finally, given h ×_{l} k : A′ ×_{C′} B′ ⟶ A ×_{C} B, by the previous argument, it is of the form

h ×_{l} k = [\begin{matrix}π_{1} (h ×_{l} k) \\ π_{0} (h ×_{l} k) \\ π_{2} (h ×_{l} k) \end{matrix}]

and by the properties of h ×_{l} k one has

[\begin{matrix}π_{1} (h ×_{l} k) \\ π_{0} (h ×_{l} k) \\ π_{2} (h ×_{l} k) \end{matrix}] = [\begin{matrix}hπ′_{1} \\ lπ′_{0} \\ kπ′_{2} \end{matrix}] = [\begin{matrix}h [1 r′ r′g′] \\ l [f′ 1 g′] \\ k [s′f′ s′ 1] \end{matrix}]

and since fh = lf′ and lg′ = gk one has

h ×_{l} k = [\begin{matrix}h & hr′ & hr′g′ \\ fh & l & gk \\ ks′f′ & ks′ & k \end{matrix}] .

3. Internal categories in weakly Mal’cev categories

The abbreviation WMC stands for Weakly Mal’cev Category. A triple of morphisms (h, l, k) as in Proposition 2.4 is said to be admissible with respect to the split span (1) if the morphism [h l k] exists. By abuse of notation we will also say that a reflexive graph is admissible when the triple (1, e, 1) is admissible.

3.1. Definition. [Admissible Reflexive Graph] In a WMC, a reflexive graph

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

is said to be admissible if the triple (1, e, 1) is admissible, that is, the morphism

[1 e 1]

exists relative to the split span

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

3.2. Theorem. In a WMC, every admissible reflexive graph is an internal category. More specifically, given the admissible reflexive graph

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

it is possible to construct the internal category

\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}}

where

1_{C_{2}} = [\begin{matrix}1_{C_{1}} & e & ec \\ d & 1_{C_{0}} & c \\ ed & e & 1_{C_{1}} \end{matrix} ]
π_{1} = [1 e ec]
π_{2} = [ed e 1]
e_{1} = [\begin{matrix}1 \\ d \\ ed \end{matrix}] , e_{2} = [\begin{matrix}ec \\ c \\ 1 \end{matrix}]
m = [1 e 1] .

Furthermore, every internal category is of this form.

Proof. The pullback C_{2} always exists in a WMC, since d and c are split epis. In a WMC, every split pullback is of the form presented above, and m = [1 e 1] is well defined because the reflexive graph is admissible by hypothesis. In order to be an internal category we have to prove that the following conditions hold

dm = dπ_{2}
cm = cπ_{1}
me_{1} = 1
me_{2} = 1
m (1 ×_{C_{0}} m) = m (m ×_{C_{0}} 1)

and we observe:

dm = d [1 e 1]
= [d 1 d]
= [ded de d]
= d [ed e 1]
= dπ_{2};

cm = c [1 e 1]
= [c 1 c]
= [c ce cec]
= c [1 e ec]
= cπ_{1};

me_{1} = [1 e 1] e_{1} = 1
me_{2} = [1 e 1] e_{2} = 1.

In order to compare m (1 × m) and m (m × 1), from the split span

\xymatrix{C_{2} \ar[r]^{π_{2}} & C_{1} \ar[l]^{e_{2}} \ar[r]_{e_{1}} & C_{2} \ar[l]_{π_{1}}}

construct the split pullback

\xymatrix{C_{3} \ar[r]^{p′_{2}} \ar[d]_{p′_{1}} & C_{2} \ar[l]^{e′_{2}} \ar[d]_{π_{1}} \\ C_{2} \ar[r]^{π_{2}} \ar[u]_{e′_{1}} & C_{1} \ar[l]^{e_{2}} \ar[u]_{e_{1}}}

and observe that from Proposition 2.5 and the commutativity of the following diagram

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

we have

(1 ×_{C_{0}} m) = π_{1} ×_{c} m = [\begin{matrix}π_{1}p′_{1} \\ cπ_{2}p′_{1} \\ mp′_{2} \end{matrix}]

= [\begin{matrix}π_{1} & ec & π_{1}e_{2}π_{1} \\ dπ_{1} & c & cm \\ me_{1}π_{2} & me_{1} & m \end{matrix}]

= [\begin{matrix}π_{1} & ec & ecm \\ π_{0} & c & cm \\ π_{2} & 1 & m \end{matrix}]
= [1_{C_{2}} e_{2} e_{2}m]

and similarly form the commutativity of the diagram

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

we have

(m ×_{C_{0}} 1) = m ×_{d} π_{2} = [e_{1}m e_{1} 1_{C_{2}}]

so that

m [1_{C_{2}} e_{2} e_{2}m] = [m me_{2} me_{2}m] = [m 1 m]

and

m [e_{1}m e_{1} 1_{C_{2}}] = [me_{1}m me_{1} m] = [m 1 m] .

To see that every internal category is obtained in this way simply observe that the morphism m is determined by me_{1} and me_{2}.

Next we show that the forgetful functor from the category of admissible reflexive graphs to the category of reflexive graphs is full, as it is the case in a Mal’cev category (see [11]).

3.3. Proposition. In a WMC, a morphism of admissible reflexive graphs is also a morphism of internal categories. More specifically, given a morphism of admissible reflexive graphs

\xymatrix{C_{1} \ar[r]^{d} \ar[r]_{c} \ar[d]_{f_{1}} & C_{0} \ar[l]|{e} \ar[d]^{f_{0}} \\ C′_{1} \ar[r]^{d} \ar[r]_{c} & C′_{0} \ar[l]|{e}}

then

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

is a morphism of internal categories, where

f_{2} = [\begin{matrix}f_{1} & f_{1}e & f_{1}ec \\ f_{0}d & f_{0} & f_{0}c \\ f_{1}ed & f_{1}e & f_{1} \end{matrix}] .

Proof. The morphism f_{2} : C_{2} ⟶ C′_{2}, being a morphism between pullbacks, by Proposition 2.5, it is of the form

f_{2} = [\begin{matrix}f_{1} & f_{1}e & f_{1}ec \\ f_{0}d & f_{0} & f_{0}c \\ f_{1}ed & f_{1}e & f_{1} \end{matrix}]
= [e_{1}f_{1} e_{1}ef_{0} e_{2}f_{1}]

and hence we have f_{1}m = mf_{2} :

f_{1}m = f_{1} [1 e 1] = [f_{1} f_{1}e f_{1}]
mf_{2} = m [e_{1}f_{1} e_{1}ef_{0} e_{2}f_{1}]
= [me_{1}f_{1} me_{1}ef_{0} me_{2}f_{1}]
= [f_{1} ef_{0} f_{1}] .

3.4. Theorem. In a WMC, C, the following is an equivalence of categories:

Cat (C) ∼ AdmRGrph (C) .

Proof. The equivalence is established by the previous results.

3.5. Corollary. In a WMC, a internal natural transformation t : f ⟶ g, corresponds to a morphism t : C_{0} ⟶ C′_{1}, as displayed in the following picture

[figure: C_{1}; d; c; f_{1}; g_{1}; C_{0}; e; t; f_{0}; g_{0}; C′_{1}; d; c; C′_{0}; e]

such that

dt = f_{0} , ct = g_{0}

and in addition

[1 e 1] [\begin{matrix}g_{1} \\ ectd \\ td \end{matrix}] = [1 e 1] [\begin{matrix}tc \\ edtc \\ f_{1} \end{matrix}] .

Proof. It is simply the interpretation of the condition

m ⟨tc, f_{1}⟩ = m ⟨g_{1}td⟩

for a internal natural transformation t : f ⟶ g in the weakly Mal’cev context.

3.6. Corollary. If B is a WMC then Cat(B) is a WMC .

We observe that Cat(B) is in fact a weakly Mal’cev sesquicategory (as introduced in the author’s PhD thesis) with internal transformations as 2-cells.

4. The connection with Mal’cev categories

The following is a result from [1] p.151, adapted to correspond to the present notation. This result was first established by D. Bourn in [8].

4.1. Lemma. Let C be a Mal’cev category. Consider the following diagram

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

where:

fr = 1_{C} = gs;

the up and left square is a pullback; e_{1} = ⟨1, ed⟩ , e_{2} = ⟨ec, 1⟩ .

Then the pair (e_{1}, e_{2}) is jointly strongly epimorphic in C.

From here one concludes that every Mal’cev category is a weakly Mal’cev category. An interesting particular case of a weakly Mal’cev category is obtained when the

pullback A ×_{C} B is also a pushout (of r and s). The result is very close to the concept of naturally Mal’cev category (see [13]) and it has the property that the morphism

[h l k]

always exists relative to a split span

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}}

and hence, we have the following equivalence of categories

Cat (B) ∼ RGrph (B) .

4.2. Proposition. In a Mal’cev variety of universal algebras, with Mal’cev operation

p : X × X × X ⟶ X
p (x, x, z) = z , p (x, z, z) = x,

the morphism

[h l k]

exists relative to the split span

\xymatrix{A \ar[r]^{f} & C \ar[l]^{r} \ar[r]_{s} & B \ar[l]_{g}}

if and only if

p (θ (h (a_{1}) , ..., h (a_{n})) , θ (l (c_{1}) , ..., l (c_{n})) , θ (k (b_{1}) , ..., k (b_{n}))) = = θ (p (h (a_{1}) , l (c_{1}) , k (b_{1})) , ...., p (h (a_{n}) , l (c_{n}) , k (b_{n})))

for all n-ary operation θ, for all n ∈ ℕ_{0}, and for all

a_{1}, ..., a_{n} ∈ A
b_{1}, ..., b_{n} ∈ B
c_{1}, ..., c_{n} ∈ C

with

f (a_{i}) = c_{i} = g (b_{i}) .

WEAKLY MAL’CEV CATEGORIES  109

Proof. First observe that if [h l k] : A ×_{C} B ⟶ D exists, it is given by

[h l k] (a, c, b) = p (h (a) , l (c) , k (b)) .

In fact we have

(a, c, b) = (p (a, r (c) , rg (b)) , p (f (a) , c, g (b)) , p (sf (a) , s (c) , b))
= p ((a, f (a) , sf (a)) , (r (c) , c, s (c)) , (rg (b) , g (b) , b))
= p (e_{1} (a) , e_{1}r (c) , e_{2} (b))

so that

[h l k] (a, c, b) = [h l k] p (e_{1} (a) , e_{1}r (c) , e_{2} (b))
= p (h (a) , l (c) , k (b)) .

In order to be an homomorphism of universal algebras one also has

[h l k] (θ (a_{1}, ..., a_{n}) , θ (c_{1}, ..., c_{n}) , θ (b_{1}, ..., b_{n})) =

= θ ([h l k] (a_{1}, c_{1}, b_{1}) , ..., [h l k] (a_{n}, c_{n}, b_{n}))

and by definition of [h l k] it becomes

p (hθ (a_{1}, ..., a_{n}) , lθ (c_{1}, ..., c_{n}) , kθ (b_{1}, ..., b_{n})) =

= θ (p (ha_{1}, lc_{1}, kb_{1}) , ..., p (ha_{n}, lc_{n}, kb_{n}))

and since h, l, k are homomorphisms of universal algebra one obtains the result.

As a simple observation one concludes that an internal category in a Mal’cev variety of universal algebras is a reflexive graph

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

such that

p (θ (a_{1}, ..., a_{n}) , θ (ed (a_{1}) , ..., ed (a_{n})) , θ (b_{1}, ..., b_{n})) =

= θ (p (a_{1}, ed (a_{1}) , b_{1}) , ...., p (a_{n}, ed (a_{n}) , b_{n}))

for all n-ary operation θ, for all n ∈ ℕ_{0}, and for all

a_{1}, ..., a_{n} ∈ C_{1}
b_{1}, ..., b_{n} ∈ C_{1}

with

d (a_{i}) = c (b_{i}) .

5. Internal groupoids in weakly Mal’cev categories

See [1] p.420 and [2].

5.1. Definition. Let C be a category with pullbacks of split epis along split epis. An internal groupoid is an internal category

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

together with a morphism t : C_{1} ⟶ C_{1}, satisfying the following conditions

dt = c
ct = d
m [ \begin{matrix}t \\ c \\ 1 \end{matrix} ] = ed , m [ \begin{matrix}1 \\ d \\ t \end{matrix} ] = ec.

It is well known that, for an internal category, being a groupoid is a property, not an additional structure (see for instance [1], p.149).

In a WMC, given an admissible reflexive graph (C_{1}, C_{0}, d, e, c) one may try to find the morphism

t : C_{1} ⟶ C_{1}

provided it exists.

In the case of a Mal’cev variety it would be of the form (see [2])

t (x) = ed (x) − x + ec (x)

which suggests us, in a WMC, to look for something of the form

t = [π_{2} 1 π_{1}] [\begin{matrix}e_{1} \\ 1 \\ e_{2} \end{matrix}] = π_{2}e_{1} − 1 + π_{1}e_{2} = ed − 1 + ec

and a suitable configuration where it makes sense.

The challenge is to find an appropriate split span that agrees with the following diagram

\xymatrix{ & C_{1} \ar[dl]_{e_{1}} \ar[d]^{1} \ar[dr]^{e_{2}} \\ C_{2} \ar[dr]_{π_{2}} & C_{1} \ar[d]^{1} & C_{2} \ar[dl]^{π_{1}} \\  & C_{1} & .}

The answer is

\xymatrix{C_{2} \ar[r]^{m} & C_{1} \ar[l]^{e_{2}} \ar[r]_{e_{1}} & C_{2} \ar[l]_{m}}  (5)

with the respective pullback denoted by

\xymatrix{C_{m} \ar[r]^{p′_{2}} \ar[d]_{p′_{1}} & C_{2} \ar[l]^{e′_{2}} \ar[d]_{m} \\ C_{2} \ar[r]^{m} \ar[u]_{e′_{1}} & C_{1} . \ar[l]^{e_{2}} \ar[u]_{e_{1}}}

Observe that in Sets the object C_{m} is exactly the set of commutative squares. The only question is whether or not the morphism

[π_{2} 1 π_{1}]

exists with respect to (5). The answer is given by the following proposition.

5.2. Proposition. In a WMC, for an internal category

\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}}

the following are equivalent:

1. It is a groupoid.

2. There is a morphism t : C_{1} ⟶ C_{1} such that

ct = d
m [\begin{matrix}1 \\ d \\ t \end{matrix}] = ec.

3. The morphism

[π_{2} 1 π_{1}]

exists with respect to the split span

\xymatrix{C_{2} \ar[r]^{m} & C_{1} \ar[l]^{e_{2}} \ar[r]_{e_{1}} & C_{2} \ar[l]_{m}} .

Proof. We will prove (1) ⟹ (2) ⟹ (3) ⟹ (1).

(1) ⟹ (2) is trivial by definition of groupoid.

(2) ⟹ (3) First observe that conditions

ct = d , m [\begin{matrix}1 \\ d \\ t \end{matrix}] = ec , me_{2} = 1 and de = 1 = ce

imply te = e:

me_{2} = 1 ⇔ m [\begin{matrix}ec \\ c \\ 1 \end{matrix}] = 1 ⟹ m [\begin{matrix}ec \\ c \\ 1 \end{matrix}] te = te ⇔ m [\begin{matrix}ecte \\ cte \\ te \end{matrix}] = te ⇔

⇔ m [\begin{matrix}e \\ de \\ te \end{matrix}] = te ⇔ m [\begin{matrix}1 \\ d \\ t \end{matrix}] e = te ⇔ ece = te ⇔ e = te;

Now, the commutativity of the following diagram

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

induces the morphism

π_{2} ×_{d} tπ_{2} : C_{m} ⟶ C_{2}

and we have

m (π_{2} ×_{d} tπ_{2}) = [π_{2} 1 π_{1}] ,

since by Proposition 2.5

(π_{2} ×_{d} tπ_{2}) = [\begin{matrix}π_{2} & π_{2}e_{2} & π_{2}e_{2}m \\ dπ_{2} & d & ctπ_{2} \\ tπ_{2}e_{1}m & tπ_{2}e_{1} & tπ_{2} \end{matrix}]

= [\begin{matrix}π_{2} & 1 & m \\ dπ_{2} & d & dπ_{2} \\ tedm & ted & tπ_{2} \end{matrix}] = [\begin{matrix}π_{2} & 1 & m \\ dπ_{2} & d & dπ_{2} \\ edπ_{2} & ed & tπ_{2} \end{matrix}]

= [e_{1}π_{2} e_{1} [\begin{matrix}1 & e & 1 \\ d & 1 & d \\ ted & te & t \end{matrix}]]

= [e_{1}π_{2} e_{1} [\begin{matrix}1 & e & 1 \\ d & 1 & d \\ ed & e & t \end{matrix}]]

= [e_{1}π_{2} e_{1} [e_{1} e_{1}e [\begin{matrix}1 \\ d \\ t \end{matrix}]]]

and composing with m gives the result

m (π_{2} ×_{d} tπ_{2}) = m [e_{1}π_{2} e_{1} [e_{1} e_{1}e [\begin{matrix}1 \\ d \\ t \end{matrix}]]]
= [π_{2} 1 [1 e ec]] = [π_{2} 1 π_{1}] .

(3) ⟹ (1) Given [π_{2} 1 π_{1}] define

t = [π_{2} 1 π_{1}] [\begin{matrix}e_{1} \\ 1 \\ e_{2} \end{matrix}]

and we have to prove

dt = c , ct = d , m [\begin{matrix}t \\ c \\ 1 \end{matrix}] = ed , m [\begin{matrix}1 \\ d \\ t \end{matrix}] = ec.

First observe that m = [1 e 1] and

1_{C_{m}} = [\begin{matrix}1_{C_{2}} & e_{2} & e_{2}m \\ m & 1_{C_{1}} & m \\ e_{1}m & e_{1} & 1_{C_{2}} \end{matrix}] = [ \begin{matrix}1_{C_{1}} & e & ec & ec & ec & e & ec \\ d & 1_{C_{0}} & c & e & c & 1 & c \\ ed & e & 1_{C_{1}} & 1 & 1 & e & 1 \\ 1 & e & 1 & 1_{C_{0}} & 1 & e & 1 \\ 1 & e & 1 & 1 & 1_{C_{1}} & e & ec \\ d & 1 & d & d & d & 1_{C_{0}} & c \\ ed & e & ed & ed & ed & e & 1_{C_{1}} \end{matrix} ] .

Denote by l_{i} the i^{th} line in the (7 × 7) identity of C_{m} and observe the following

l_{i} [\begin{matrix}e_{1} \\ 1 \\ e_{2} \end{matrix}] = l_{i} [ \begin{matrix}1_{C_{1}} \\ d \\ ed \\ 1_{C_{1}} \\ ec \\ c \\ 1_{C_{1}} \end{matrix} ] = \{ \begin{matrix}1_{C_{1}} \\ d \\ ed \\ 1_{C_{1}} \\ ec \\ c \\ 1_{C_{1}} \end{matrix} if i=1 if i=2 if i=3 if i=4 if i=5 if i=6 if i=7 .

Also let

l = [π_{2} 1 π_{1}] = [ ed e 1 1 1 e ec ]
ε = [\begin{matrix}e_{1} \\ 1 \\ e_{2} \end{matrix}]

so that

t = lε.

To show that dt = c we have

dt = dlε = [ d 1 d d d 1_{C_{0}} c ] ε = l_{6}ε = c; To show that ct = d we have

ct = clε = [ d 1_{C_{0}} c e c 1 c ] ε = l_{2}ε = d.

To prove that m [\begin{matrix}t \\ dt \\ 1 \end{matrix}] = ed first observe that

[\begin{matrix}t \\ dt \\ 1 \end{matrix}] = [\begin{matrix}lε \\ dlε \\ l_{7}ε \end{matrix}]

(note that we could also try to choose 1_{C_{1}} = l_{1}ε or 1_{C_{1}} = l_{4}ε but then

[\begin{matrix}l \\ dl \\ l_{1} \end{matrix}] and [\begin{matrix}l \\ dl \\ l_{4} \end{matrix}]

would not be defined), and then

[\begin{matrix}t \\ dt \\ 1 \end{matrix}] = [\begin{matrix}l \\ dl \\ l_{7} \end{matrix}] ε = [ \begin{matrix}ed & e & 1 & 1 & 1_{C_{1}} & e & ec \\ d & 1 & d & d & d & 1_{C_{0}} & c \\ ed & e & ed & ed & ed & e & 1_{C_{1}} \end{matrix} ] ε
= [e_{1}π_{2} e_{1} 1_{C_{2}}] ε

so that

m [\begin{matrix}t \\ dt \\ 1 \end{matrix}] = m [e_{1}π_{2} e_{1} 1_{C_{2}}] ε
= [π_{2} 1 m] ε
= [ ed e 1_{C_{1}} 1 1 e 1 ] ε
= l_{3}ε = ed.

A similar calculation shows m [\begin{matrix}1 \\ d \\ t \end{matrix}] = ec, in fact

[\begin{matrix}1 \\ d \\ t \end{matrix}] = [\begin{matrix}1 \\ ct \\ t \end{matrix}] = [\begin{matrix}l_{1}ε \\ clε \\ lε \end{matrix}] = [\begin{matrix}l_{1} \\ cl \\ l \end{matrix}] ε

= [ \begin{matrix}1_{C_{1}} & e & ec & ec & ec & e & ec \\ d & 1_{C_{0}} & c & e & c & 1 & c \\ ed & e & 1_{C_{1}} & 1 & 1 & e & ec \end{matrix} ] ε
= [1_{C_{2}} e_{2} e_{2}π_{1}] ε

and hence

m [\begin{matrix}1 \\ d \\ t \end{matrix}] = m [1_{C_{2}} e_{2} e_{2}π_{1}] ε
= [m 1 π_{1}] ε
= [ 1 e 1 1 1_{C_{1}} e ec ] ε
= l_{5}ε = ec.

6. Conclusion

We conclude by saying once again that the notion of weakly Mal’cev category is introduced with the unique purpose to have a setting (easy to handle) where a multiplicative graph, that is a diagram of the form

\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}}

where

\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}}

is a split pullback and me_{1} = 1_{C_{1}} = me_{2}, is already an internal category, that is, the conditions

dm = dπ_{2}
cm = cπ_{1}
m (1 × m) = m (m × 1)

are automatically satisfied.

However, the original motivation was to have a setting where a reflexive graph would admit at most one multiplication, so that the two axioms in the definition of a WMC are thus explained: − the existence of pullbacks of split epis along split epis is used to construct the pullback C_{2} of the split epi (c, e) along the split epi (d, e); − the requirement that the pair (e_{1}, e_{2}) is jointly epimorphic is used to uniquely determine the morphism m, provided it exists, from the two components me_{1} and me_{2}.

It was a happy surprise to observe that preservation of domain and codomain as well as associativity would automatically follow.

There are still many comparisons to be made in order to decide if this is in fact a good notion for a category and if it does not coincide with something already known. For example, it would be interesting to find out what other conditions are needed in order to have that every internal category is an internal groupoid. We leave this and other questions for a future work.

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Polytechnic Institute of Leiria, Leiria, Portugal

Email: nelsonmf@estg.ipleiria.pt

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THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

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Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, McGill University: gavin seal@fastmail.fm

Transmitting editors.

Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Cuill Inc.: valeria@cuill.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca