Theory and Applications of Categories, Vol. 7, No. 9, pp. 206–218.

NORMAL FUNCTORS AND STRONG PROTOMODULARITY

DOMINIQUE BOURN

Transmitted by Robert Rosebrugh

ABSTRACT. The notion of normal subobject having an intrinsic meaning in any protomodular category, we introduce the notion of normal functor, namely left exact conservative functor which reflects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect to the fibration of pointed objects, are not only conservative (this is the definition of a protomodular category), but also normal. This leads to the notion of strongly protomodular category. Some of their properties are given, the main one being that this notion is inherited by the slice categories.

Introduction

There are four general types of example of protomodular categories:

1) the varieties of classical algebraic structures, such as the category of groups, the category of rings, the category of associative or Lie algebras over a given ring A, the varieties of Ω-groups

2) the categories of internal algebraic structures of the previous kind in a left exact category ℂ

3) the non syntactical examples, such as the dual of any elementary topos 𝔼

4) the constructible examples which inherit the property of being protomodular, such as the slice categories ℂ/Z or the fibres Pt_{Z}ℂ of the fibration π of pointed objects.

We showed in [2] that the notion of protomodular category allows intrinsic definition of the concept of normal subobject, without any right exactness condition, in a way which allows recovery of the classical one in the setting of the category Gp of groups, and, more generally, of the classical concept of ideal in the setting of examples of type 1. The importance of this concept in examples of type 1 motivates its exploration in the other types of examples. The first aim of this paper was an attempt of a characterization of the normal subobjects in examples of types 2 and 4. For instance (type 4), when the basic protomodular category ℂ is pointed or quasi-pointed (i.e. when the map 0 → 1 is monic), a subobject j : (X, f) → (Y, g) is normal in the slice category ℂ/Z if and only if the map j · kerf : K[f] → X → Y is normal in ℂ. The same result holds in the fibre Pt_{Z}ℂ.

But this attempt led us mainly to focus on the notion of normal functor, namely a left exact conservative functor which reflects the normal subobjects. This kind of functor has still a meaning when ℂ is only left exact and, even under this mild assumption, it has very nice properties. For instance, when a reflexive graph \underline{X}_{1} has its image F \underline{X}_{1} endowed with a groupoid structure, then \underline{X}_{1} is itself endowed with a groupoid structure above the one on F \underline{X}_{1}. More specifically, when ℂ is a left exact category and Gpℂ the category of internal groups in ℂ, then the extension of the Yoneda embedding to the category of groups in ℂ, still denoted by Y : Gpℂ → Gp^{ℂ^{op}}, is normal, which gives a characterization (type 2) of normal subobjects in Gpℂ. The same result and characterization hold for internal rings.

Received by the editors 1999 December 8 and, in revised form, 2000 June 19.

Published on 2000 July 7.

2000 Mathematics Subject Classification: 18D05,08B05,18G30,20L17.

Key words and phrases: abstract normal subobject, preservation and reflection of normal subobject, Mal’cev and protomodular categories.

\begin{matrix}c \\ ◯ \end{matrix} Dominique Bourn, 2000. Permission to copy for private use granted.

The main point here will be the observation that the category Gp has not only its change of base functors, with respect to the fibration π of pointed objects, conservative (this is the definition of a protomodular category), but also normal. It is proved that a protomodular category does not necessarily satisfy this property, and I am indebted to G. Janelidze from whom I learned of the category of digroups which is the setting of the counterexample. This leads naturally to the definition of strongly protomodular categories as those which have their change of base functors, with respect to π, normal. Some properties of these categories are given, mainly the inheritance of the notion by slice categories and a characterization of internal groupoids.

The unexpected role of normal functors explains the organization of the paper along the following line:

1) Normal functors

2) Protomodularity and strong protomodularity

3) A counterexample

4) Example of strong protomodularity

5) Some properties of strongly protomodular categories

1. Normal functors

We shall suppose ℂ a left exact category and denote Relℂ the category of equivalence relations in ℂ. Let us recall that a map f : R → R′ in Relℂ is cartesian with respect to the forgetful functor U : Relℂ → ℂ associating with any equivalence relation R its underlying object X, when the following diagram is a joint pullback (in this case we shall often denote R by f^{−1}(R′) and call it the inverse image of R′ by f). It is called fibrant when the square with the d_{0} (or equivalently with the d_{1}) is a pullback:

\xymatrix{R \ar[d]^{d_{1}} \ar[d]_{d_{0}} \ar[r]^{ϕ} & R′ \ar[d]^{d_{1}} \ar[d]_{d_{0}} \\ X \ar[r]_{f} & X′}

Thus a map f : X → X′ in ℂ determines a map in Relℂ if and only if it satisfies the property R ⊂ f^{−1}(R′). When it is the case, we shall use the same symbol f to denote the induced map f : R → R′ in Relℂ.

A left exact functor F : ℂ → ℂ′ between left exact categories extends to a functor F : Relℂ → Relℂ′ which preserves the cartesian and the fibrant maps. It reflects them as well, when it also reflects isomorphisms, since then F reflects the finite limits. In this situation, it also satisfies a nice property:

1.1. Proposition. When a left exact functor F : ℂ → ℂ′ between left exact categories reflects the isomorphisms, then each map in Relℂ is cocartesian (i.e. the dual of a cartesian map) with respect to the factorization functor Φ : Relℂ → ℂ ×ℂ Relℂ′.

Proof. Let us consider f : R → R′ and g : R → R′′ two maps in Relℂ such that there is a map k : X′ → X′′ in ℂ satisfying k · f = g and determining a map F k : F R′ → F R′′ in Relℂ′ such that F k · F f = F g.

Let us denote S = k^{−1}(R′′). Then F S = (F k)^{−1}(F R′′). We have F k : F R′ → F R′′ and thus F R′ ⊂ F S. In other words there is an isomorphism F R′ ∩ F S ≃ F R′. Consequently the inclusion R′ ∩ S ↪ R′ is mapped to an isomorphism. But F is conservative, whence an isomorphism R′ ∩ S ≃ R′ and the inclusion R′ ↪ S which determines a morphism k : R′ → R′′. That k·f = g in Relℂ is a consequence of the fact that F, being conservative, is faithful.

Let us recall that a map j : I → X in ℂ is normal to an equivalence relation R [2] when j^{−1}(R) is the coarse relation grI on I (i.e. the kernel equivalence of the terminal map I → 1) and the induced map grI → R in Relℂ is fibrant. This implies that j is necessarily a monomorphism. This definition gives an intrinsic way to express that I is an equivalence class of R. Clearly left exact functors preserve this kind of monomorphism.

1.2. Definition. We shall call normal a left exact functor F : ℂ → ℂ′ between left exact categories which reflects isomorphisms and normal monomorphisms: when j is such that F j is normal to some equivalence relation S on F X, then there exists an equivalence relation R on X such that j is normal to R and F R = S.

According to proposition 1.1, this R is unique up to isomorphism.

Examples.

1) Let Gp be the category of groups, Mag the category of magmas (sets endowed with a binary operation) and U : Gp → Mag the forgetful functor. Then U is normal.

2) Let ℂ be a left exact category and Gpℂ the category of internal groups in ℂ . Then the extension of the Yoneda embedding to the category of groups in ℂ, still denoted by Y : Gpℂ → Gp^{ℂ^{op}}, is normal, [3] Prop. 5 and Theorem 6. This gives immediately a characterization of normal subobjects in the category Gpℂ, since the normal subobjects in a category Gp𝔼 are those subobjects which are componentwise normal. The same result and characterization hold for internal rings.

3) Any conservative left exact functor between additive categories with kernels is normal.

1.3. Proposition. Suppose that F : ℂ → ℂ′ is normal and ℂ is quasi-pointed (i.e. it has an initial object 0 and the map 0 → 1 is a monomorphism), then the functor Φ : Relℂ → ℂ ×ℂ′ Relℂ′ is cofibrant on the fibrant morphisms of Relℂ′: i.e. given an object R in Relℂ and a map (f, F f) in ℂ×ℂ′ Relℂ′ with domain Φ(R) and such that the map F f is fibrant in Relℂ′, there is a cocartesian map with respect to Φ above (f, F f) in Relℂ.

Proof. According to proposition 1.1, it is sufficient to prove the existence of a map above (f, F f).

Let R be an equivalence relation on X in Relℂ and f : X → Y a map in ℂ such that there is an equivalence relation S on F Y which makes F f : F R → S a fibrant map in Relℂ′. The category ℂ being quasi-pointed, the map 0 → 1 is monic and consequently the kernel relation of any map Z → 0 is the coarse relation grZ. So we can associate with R the normal monomorphism j = d_{1} · k determined by the following pullback:

\xymatrix{I \ar[r]^{k} \ar[d] & R \ar[r]^{d_{1}} \ar[d]^{d_{0}} & X \\ 0 \ar[r] & X}

This map j is normal to R and consequently F j is normal to F R. But the morphism F f · F j : grF I → S is fibrant in Relℂ′ as a composition of fibrant maps, and cartesian as any morphism with domain of the form grZ. Consequently the morphism F f · F j in ℂ′ is normal to S. The functor F being normal, the map f · j is normal to some S^{∗} on Y in ℂ such that F S^{∗} = S. Proposition 1.1 asserts the existence of a factorization f : R → S^{∗} which is necessarily fibrant since F f is fibrant.

We have also the following result:

1.4. Proposition. Suppose ℂ and ℂ′ pointed, and F : ℂ → ℂ′ normal. When a split epimorphism (f, s) : X → Y in ℂ with kernel K[f] is such that (F f, F s) is the canonical split epimorphism given by the projection F K[f] × F Y → F Y, then X is isomorphic to K[f]×Y, and (f, s) is, up to isomorphism, the canonical split epimorphism K[f]×Y → Y.

Proof. For the sake of simplicity, let us denote by K the kernel K[f] of the map f. Clearly the canonical inclusion F s = i_{F Y} : F Y → F K × F Y is normal to the kernel relation R[p_{F K}] of the projection p_{F K} : F K × F Y → F K. The functor F being normal, let us denote by S the equivalence relation on X which makes s normal to S and such that F S = R[p_{F K}]. Now, if R[f] denotes the kernel relation of the map f, let us consider R[f]□S the double relation on X, determined by the inverse image of R[f] × R[f] by the map [d_{0}, d_{1}] : S → X × X. It corresponds to the subobject of X^{4} consisting of the quadruples (x, x′, y, y′) such that xSx′, ySy′, xRy, x′Ry′. This R[f]□S is sent by F to R[p_{F Y}]□R[p_{F K}]. Consequently the following diagram is a pullback since its image by F is clearly a pullback:

\xymatrix{R[f]□S \ar[r]^{d_{0}} \ar[d]_{d_{0}} & R[f] \ar[d]^{d_{0}} \\ S \ar[r]_{d_{0}} & X}

which, internally speaking, means that, for all x, x′, y satisfying xRy, xSx′,there is a unique y′ such that x′Ry′ and ySy′. Then according to [2], theorem 11, there is a map ϕ : K × Y → X such that ϕ.i_{K} = kerf and ϕ · i_{Y} = s. Its internally corresponds to the function which, in the set theoretical context, associates with (k, y) the unique x determined by the triple e, s(y) and k, where e denotes the distinguished point of any object Z, since we do have eRk and eSs(y) in X. But F ϕ is obviously an isomorphism and consequently ϕ is an isomorphism.

We shall specifically need later on the following definition:

1.5. Definition. A functor F : ℂ → ℂ′ is called strongly normal when it is left exact, conservative and such that Φ : Relℂ → ℂ ×ℂ′ Relℂ′ is cofibrant on the fibrant morphisms of Relℂ′.

2. Protomodularity and strong protomodularity

We denote by Ptℂ the category whose objects are the split epimorphisms in ℂ with a given splitting and morphisms the commutative squares between these data. We denote by π : Ptℂ → ℂ the functor associating its codomain with any split epimorphism. As soon as ℂ has pullbacks, the functor π is a fibration which is called the fibration of pointed objects.

Remark. A reflexive relation (d_{0}, d_{1}) : R ↪ X × X on an object X in a left exact category ℂ determines, in the fibre Pt_{X}ℂ above X, a subobject (d_{0}, s_{0}) of the object (p_{0}, s_{0}), p_{0} : X × X → X. The converse is true as well. Now, when R is an equivalence relation, X ×R determines an equivalence relation on the object (p_{0}, s_{0}) in Pt_{X}ℂ to which the inclusion (d_{0}, s_{0}) ↪ (p_{o}, s_{0}) is normal.

The category ℂ is said to be protomodular [1] when π has its change of base functors conservative. When ℂ is pointed, this condition is equivalent to the split short five lemma, which makes the category Gp of groups the leading example of this notion. As soon as a functor F : ℂ → ℂ′ preserves pullbacks and is conservative, ℂ′ protomodular implies ℂ protomodular. Accordingly any fibre Pt_{X}ℂ of π above an object X is protomodular, as well as any slice category ℂ/X.

Now in a protomodular category ℂ, when j : I → X is normal to some R, this R is unique up to isomorphism [2]. So the fact of being normal becomes an intrinsic property in ℂ. Of course, in the category Gp this notion coincides with the classical one, and in the category Rng of rings it coincides with the notion of two-sided ideals.

Remark. We know that a protomodular category is always Mal’cev (i.e. any reflexive relation is an equivalence relation), see [5] and [2]. The previous remark allows us to express the Mal’cev property in the following terms: When ℂ is protomodular, then any subobject of (p_{0}, s_{0}) in the fibre Pt_{X}ℂ above X is normal in this fibre.

We just saw that the fibres Pt_{X}ℂ and the slice categories ℂ/X are still protomodular. It is then natural to ask whether it is possible to characterize the normal monomorphisms of these categories in terms of normality in ℂ. There is a simple answer when ℂ is quasi-pointed.

2.1. Proposition. When ℂ is quasi-pointed and protomodular, a map j : (X, f) → (Y, g) is normal in ℂ/Z if and only if the map j · kerf : K[f] → X → Y is normal in ℂ. The same result holds in Pt_{Z}ℂ.

Proof. The map 0 → 1 is monic and consequently the kernel relation of any map Z → 0 is the coarse relation grZ. Let us consider the following diagram in ℂ:

\xymatrix{K[f] × K[f] \ar[r] \ar[d]^{p_{1}} \ar[d]_{p_{0}} & X ×_{Z} X \ar[r] \ar[d]^{p_{1}} \ar[d]_{p_{0}} & R \ar[d]^{d_{1}} \ar[d]_{d_{0}} \\ K[f] \ar[d] \ar[r]_{ker f} & X \ar[d]_{f} \ar[r]_{j} & Y \ar[d]^{g} \\ 0 \ar[r] & Z \ar[r] & Z}

where R is the equivalence relation in ℂ/Z to which j is normal and K[f] the kernel of f, i.e. the pullback of the map f along the initial monomorphism 0 → Z. Thus the upper left hand side diagram determines a fibrant map in Relℂ and makes j ·ker f : grK[f] → R a fibrant and cartesian map in Relℂ. So j · ker f is normal to R in ℂ.

Conversely, let us suppose the map j · ker f normal in ℂ, and denote by R the equivalence relation to which it is normal. We must show that R lies in ℂ/Z and that j is normal to R in this category. Let us denote by R[f] and R[g] the kernel equivalences of f and g. The map j : R[f] → R[g] is cartesian in Relℂ, as well as ker f : grK[f] → R[f]. Thus j · ker f : grK[f] → R[g] is cartesian, while j · ker f : grK[f] → R is cartesian and fibrant. Consequently the following square is a pullback in Relℂ:

\xymatrix{grK[f] \ar[r] \ar[d] & R ∩ R[g] \ar[d]^{t} \\ grK[f] \ar[r] & R}

Now the lower and thus the upper maps are fibrant. Consequently the change of base functor (j · ker f)^{∗} with respect to π : Ptℂ → ℂ maps t to an isomorphism. The category ℂ being protomodular, the map t is itself an isomorphism, we have R ⊂ R[g] and R belongs to ℂ/Z. Now let us consider the following pullbacks in Relℂ:

\xymatrix{grK[f] \ar[r] \ar[d] & j^{−1}R \ar[d]^{m} \ar[r] & R \ar[d] \\ grK[f] \ar[r] & R[f] \ar[r] & R[g]}

The left hand side square is a pullback since the right hand side square and the total square are pullbacks. But ker f : grK[f] → R[f] is fibrant, so the map grK[f] → j^{−1}R is fibrant too. Thus since (ker f)^{∗}(m) is an isomorphism in Pt_{K[f]}ℂ, the map m is itself an isomorphism. We have therefore j : R[f] → R which is certainly fibrant, since ℂ is protomodular, and both ker f : grK[f] → R[f] and j · ker f : grK[f] → R are fibrant. Thus j is normal to R in ℂ/Z.

Remark. We do know the following result in the category Gp of groups: given a morphism between two exact sequences, when f′′ is an isomorphism and f′ a normal monomorphism, then g′ · f′ is a normal monomorphism (see [6] for instance, axiom 1.2 for a Moore category):

\xymatrix{1 \ar[r] & H′ \ar[d]_{f′} \ar[r]^{h′} & H \ar[d]^{f} \ar[r]^{h} & H′′ \ar[r] \ar[d]^{f′′} & 1 \\ 1 \ar[r] & G′ \ar[r]_{g′} & G \ar[r]_{g} & G′′ \ar[r] & 1}

Thus f ·h′ = f ·ker h is normal in Gp. Suppose now f′′ an identity. This result means that, the map h being a surjection, the map f is normal in Gp/H′′ if and only if the map f′ is normal in Gp.

On the other hand, the homomorphisms h and g are certainly epimorphic when h is split. Whence the following theorem:

2.2. Theorem. In the category Gp of groups, the change of base functors with respect to π : PtGp → Gp are normal.

We shall study this specific property, so let us introduce:

2.3. Definition. A left exact category is said to be strongly protomodular when the change of base functors with respect to π : Ptℂ → ℂ are normal.

Each fibre being pointed, this implies that each change of base functor is strongly normal.

3. A counterexample

A protomodular category is not necessarily strongly protomodular. Let us denote by U : Gp → Set^{∗} the forgetful functor towards the category of pointed sets, associating with each group its underlying set pointed by the unit element. Let us call the category of digroups the category defined by the following pullback:

\xymatrix{DiGp \ar[r]^{p_{1}} \ar[d]_{p_{o}} & Gp \ar[d]^{U} \\ Gp \ar[r]_{U} & _{Set_{∗}}}

The functor p_{0} is clearly left exact and conservative. Thus the category DiGp is protomodular. On the other hand, it is clearly pointed. We are going to show that, however, it is not strongly protomodular.

In any pointed protomodular category, the diagonal s_{0} : X → X × X is normal if and only if X is an internal (abelian) group [2]. For that reason, given a digroup structure on a set G, its diagonal in DiGp is normal if and only if the two laws on G are abelian and coincide. Now let (G, ·, #) be a digroup. We shall denote respectively by x^{−1} and x^{∘} the inverse of an element x with respect to the two laws.

3.1. Proposition. A monomorphism j : (G′, ·, #) → (G, ·, #) is normal in DiGp if and only if:

1) (G′, ·) is normal in (G, ·)

2) (G′, #) is normal in (G, #)

3) z · x^{−1} ∈ G′ if and only if z#x^{∘} ∈ G′.

Proof. Let R be the equivalence relation in DiGp to which j is normal. Then p_{0}(j) and p_{1}(j) are normal to p_{0}(R) and p_{1}(R) in Gp and thus the two first statements are satisfied. The third is a consequence of the fact that: z · x^{−1} ∈ G′ ⇔ zRx ⇔ z#x^{∘} ∈ G′.

Conversely, let us suppose the three statements satisfied. Let us define zRx by z·x^{−1} ∈ G′. The subgroup (G′, ·) being normal, R defines a subgroup of (G × G, ·). Now z · x^{−1} is equivalent to z#x^{∘} ∈ G′ and then R defines a subgroup of (G × G, #).

Now let us produce the counterexample. Let A be an abelian group, such that there is an element a with a ≠ −a. Let us define θ : A × A → A × A in the following way:

if x ≠ a, then θ(z, x) = (z, x), if x = a and z ≠ a, z ≠ −a, then θ(z, a) = (z, a)

while θ(a, a) = (−a, a) and θ(−a, a) = (a, a).

We have then: θ ≠ Id, θ^{2} = Id, and p_{1} · θ = p_{1}. Now let # be the transform along θ of the ordinary product law on A × A. Whence (z, x)#(z′, x′) = θ(θ(z, x) + θ(z′, x′)). Now (A × A, +, #) is a digroup and p_{1} : (A × A, +, #) → (A, +, +) a digroup homomorphism which is split in DiGp by the homomorphism s, with s(z) = (0, z). Thus (p_{1}, s) is an object in PtDiGp above (A, +, +). Moreover the kernel of p_{1} is again (A, +, +).

Let ω_{A} : 1 → (A, +, +) denote the initial map in DiGp. So, if s_{0} denotes the diagonal of the object (p_{1}, s) in Pt_{A}DiGp, then (ω_{A})^{∗}(s_{0}) is the diagonal s_{0} : A → A × A, which is normal in DiGp since the two laws on A are abelian and coincide. We are now going to prove that, however, this diagonal s_{0} of (p_{1}, s) in Pt_{A}DiGp is not normal in this fibre and that, consequently, (ω_{A})^{∗} does not reflect the normal monomorphisms, which will mean that DiGp is not strongly protomodular. For that, according to proposition 2.1, we must check that k = s_{0} · ker p_{1} is not normal in DiGp. We have k(z) = (z, z, 0). Note that (−a, a)^{∘} = (−a, −a). Then if z ≠ a, z ≠ −a, we have (a, z, a)#(−a, z + 2a, a)^{∘} = (a, z, a)#(−a, −z − 2a, −a) = (−2a, −2a, 0) which belongs consequently to k(A), while (a, z, a) + (a, −z − 2a, −a) = (2a, −2a, 0) does not. The third condition of proposition 2.1 is not fulfilled by k(A), and k is not normal in DiGp.

4. Examples of strong protomodularity

Proposition 2.1 allows easily to check when a pointed or quasi-pointed protomodular category is strongly protomodular. So besides the category Gp of groups, the category Rng of rings, for instance, is strongly protomodular. In the same way, any presheaf category of groups Gp^{E^{op}} or of rings Rng^{Eop} is strongly protomodular, since a morphism j : F → F′ is normal in these categories if and only if, for every object X in 𝔼, the homomorphism j(X) : F (X) → F′(X) is normal.

A category ℂ is Naturally Mal’cev [7] when the fibres of π : Ptℂ → ℂ are additive. According to example 3 of normal functor, any protomodular Naturally Mal’cev category is strongly protomodular. As a particular case, any essentially affine category [1] (i.e. when π : Ptℂ → ℂ is trivial: any change of base functor with respect to π is an equivalence of categories), is strongly protomodular.

We have also a very helpful result:

4.1. Proposition. Let F : ℂ → ℂ′ be a strongly normal functor. Then:

1) the functors F_{Z} : ℂ/Z → ℂ′/F Z are normal, and consequently the extension PtF : Ptℂ → Ptℂ′ is fibrewise strongly normal

2) if ℂ′ is strongly protomodular, the category ℂ is itself strongly protomodular.

Proof. The functor F, being left exact and conservative, produces left exact conservative functors F_{Z} : ℂ/Z → ℂ′/F Z and F_{Z} : Pt_{Z}ℂ → Pt_{F Z}ℂ′.

1) Now let R be an equivalence relation on the object h in ℂ/Z. Let f : h → h′ be a map in ℂ/Z and F f : F (R) → S a fibrant map in Rel(ℂ′/F Z). It is fibrant in Relℂ′, and F being strongly normal, there is an equivalence relation S^{∗} in Relℂ above S, making f : R → S^{∗} a fibrant map in Relℂ. We must show that S^{∗} lies in ℂ/Z on the object h′. So let us consider S^{∗} ∩ R[h′]. It is mapped to S ∩ R[F h′] which is S since S is an equivalence relation on F h′ in ℂ′/F Z. Thus the inclusion S^{∗} ∩ R[h′] ↪ S^{∗} is mapped by F to an isomorphism and consequently is itself an isomorphism, which means that S^{∗} lies in ℂ/Z on the object h′. The same reasons hold for F_{Z} : Pt_{Z}ℂ → Pt_{F Z}ℂ′.

2) Clearly, when f : X → X′ is a map in ℂ, the following diagram commutes:

\xymatrix{Pt_{X′}C \ar[d]_{F_{X′}} \ar[r]^{f^{∗}} & Pt_{X}C \ar[d]^{F_{X}} \\ Pt_{F X′}C′ \ar[r]_{(F f)^{∗}} & Pt_{F X}C′}

Now when ℂ′ is strongly protomodular, the functor (F f)^{∗} is strongly normal and consequently so is F_{X} · f^{∗} = (F f)^{∗} · F_{X′} since F_{X′} is itself strongly normal. Now F_{X} is also strongly normal and thus f^{∗} is strongly normal.

As a corollary, we get:

4.2. Corollary. When ℂ is left exact, the category Gpℂ of internal groups in ℂ is strongly protomodular.

Proof. The extension of the Yoneda embedding Y : Gpℂ → Gp^{ℂ^{op}} is normal [3], but Gpℂ is pointed and consequently Y is strongly normal. The presheaf category Gp^{ℂ^{op}} is strongly protomodular, and then so is Gpℂ.

Suppose ℂ left exact and denote Grdℂ the category of internal groupoids in ℂ. Let ( )_{0} : Grdℂ → ℂ the functor associating with each groupoid its object of objects. It is a fibration. Clearly Grd_{1}ℂ, the fibre above 1, is just Gpℂ.

But any fibre Grd_{X}ℂ is quasi-pointed and shares with Grd_{1}ℂ the property of being protomodular [1].

Actually, we have more:

4.3. Proposition. Any fibre Grd_{X}ℂ is strongly protomodular.

Proof. Again the extension of the Yoneda embedding Y : Grdℂ → Grd^{ℂ^{op}}, where Grd denotes the category of ordinary groupoids, is fibrewise normal: Grd_{X}ℂ → (Grd^{ℂ^{op}})_{Y_{X}} [3]. But each fibre is quasi-pointed, thus Y is fibrewise strongly normal. On the other hand Grd^{ℂ^{op}} has its fibres above Set^{ℂ^{op}} strongly protomodular, consequently, Grdℂ is fibrewise (with respect to ( )_{0} : Grdℂ → ℂ) strongly protomodular.

5. Some properties of strongly protomodular categories

1) Abelian objects We recalled that in a protomodular category ℂ an object X has at most one internal

group structure, which is necessarily abelian. In this situation we call X abelian. This is the case if and only if it has a point 1 → X and its diagonal s_{0} : X → X × X is normal [2]. Then, given any map f : X → X′ in a strongly protomodular category, the functor f^{∗} : Pt_{X′}ℂ → Pt_{X}ℂ between the fibres reflects the abelian objects. In particular, when ℂ is moreover pointed, we recover the well known result for the category Gp of groups following which a split epimorphism (h, t) with codomain Z determines a group structure in Gp/Z if and only if the kernel of h is an abelian group.

2) Internal groupoids

When the category 𝔼 is left exact, an internal groupoid in 𝔼 is a reflexive graph \underline{X}_{1}:

\xymatrix{X_{0} & X_{1} \ar[l]^{d_{1}} \ar[l]_{d_{0}}}

endowed with a map d_{2} : R[d_{0}] → X_{1}, in such a way that all the simplicial identities hold as far as level 3 when the diagram is completed by the kernel pair of d_{0} : R[d_{0}] → X_{1}. Now the following identities: d_{0} · d_{2} = d_{1} · p_{0} and d_{2} · s_{0} · s_{0} = s_{0} · d_{1} · s_{0} = s_{0} make d_{2} : (d_{1} · p_{0}, s_{0} · s_{0}) → (d_{0}, s_{0}) a map in Pt_{X_{0}}𝔼 which furthermore makes the factorization [d_{2}, p_{0}] : (d_{1} · p_{0}, s_{0} · s_{0}) → (d_{0}, s_{0}) × (d_{1}, s_{0}) in Pt_{X_{0}}𝔼 an isomorphism. This remark has interesting consequences:

5.1. Proposition. Given a strongly normal functor F : 𝔼 → 𝔼′ between left exact categories and a reflexive graph \underline{X}_{1} in 𝔼. Then, when F (\underline{X}_{1}) is endowed with a groupoid structure, there is a unique, up to isomorphism, groupoid structure on \underline{X}_{1} above the one on F (\underline{X}_{1}).

Proof. The extension PtF : Pt𝔼 → Pt𝔼′ is fibrewise strongly normal and consequently F_{X_{0}} : Pt_{X_{0}}𝔼 → Pt_{F X_{0}}𝔼′ is strongly normal. Now the object (d_{1} · p_{0}, s_{0} · s_{0}) in Pt_{X_{0}}𝔼 is mapped by F_{X_{0}} to the product (F d_{0}, F s_{0}) × (F d_{1}, F s_{0}) in Pt_{F X_{0}}𝔼′, since F (\underline{X}_{1}) has a groupoid structure, and the map p_{0} : (d_{1} · p_{0}, s_{0} · s_{0}) → (d_{1}, s_{0}) is sent to the projection p : (F d_{0}, F s_{0})×(F d_{1}, F s_{0}) → (F d_{1}, F s_{0}). Then according to proposition 1.4, (d_{1}·p_{0}, s_{0}·s_{0}) is isomorphic, in Pt_{X_{0}}𝔼, to the product Kerp_{0}×(d_{1}, s_{0}) = (d_{0}, s_{0})×(d_{1}, s_{0}), which produces a map d_{2} : (d_{1} · p_{0}, s_{0} · s_{0}) → (d_{0}, s_{0}) in Pt_{X_{0}}𝔼. This map d_{2} in 𝔼 satisfies the axiom of groupoids since F (d_{2}) satisfies them in 𝔼′ and F is faithful.

On the other hand, the kernel of d_{2} : (d_{1} · p_{0}, s_{0} · s_{0}) → (d_{0}, s_{0}) in the pointed category Pt_{X_{0}}𝔼, is the map s_{0} : (d_{1}, s_{0}) → (d_{1} · p_{0}, s_{0} · s_{0}), which is necessarily normal in Pt_{X_{0}}𝔼 to the kernel equivalence R[d_{2}] of d_{2}. Now, when 𝔼 is protomodular, there is at most one groupoid structure on a given reflexive graph. In this situation the previous property is characteristic:

5.2. Proposition. Given a protomodular category ℂ, a reflexive graph \underline{X}_{1} is a groupoid if and only if the map s_{0} : (d_{1}, s_{0}) → (d_{1} · p_{0}, s_{0} · s_{0}) is normal in Pt_{X_{0}}ℂ.

Proof. Let us suppose this map normal. So we have an epimorphism p_{0} : (d_{1}·p_{0}, s_{0}·s_{0}) → (d_{1}, s_{0}), in the protomodular fibre Pt_{X_{0}}ℂ, split by the normal monomorphism s_{0}. Then, according to [2], proposition 12, the object (d_{1} · p_{0}, s_{0} · s_{0}) is isomorphic in Pt_{X_{0}}ℂ to the product Kerp_{0} × (d_{1}, s_{0}) = (d_{0}, s_{0}) × (d_{1}, s_{0}). Whence a map d_{2} : (d_{1} · p_{0}, s_{0} · s_{0}) → (d_{0}, s_{0}) in this category. The map d_{2}, in ℂ, is enough to make \underline{X}_{1} a groupoid since, thanks to [5], theorem 2.2, and the fact that a protomodular category is always Mal’cev, this map satisfies necessarily the axioms of a groupoid.

Two consequences follow, concerning the strongly protomodular categories:

A) If ℂ is a pointed strongly protomodular category, a reflexive graph \underline{X}_{1} is a groupoid if and only if the map ω^{∗}(s_{0}) : K[d_{1}] → K[d_{1} · p_{0}] is normal in ℂ, where ω denotes the initial map 1 → X_{0}.

Remark. In the category Gp of groups, K[d_{1} · p_{0}] is the group whose objects are the pairs (α, β) of arrows of the graph \underline{X}_{1} with same domain and satisfying d_{1}(α) = 1. To say that ω^{∗}(s_{0}) is normal means that, for each arrow γ with codomain 1, we have: α·γ ·α^{−1} = β ·γ ·β^{−1}. This is clearly equivalent to (β^{−1} ·α)·γ = γ ·(β^{−1} ·α) with β^{−1} ·α in K[d_{0}]. Thus to say that ω^{∗}(s_{0}) is normal is equivalent to saying that [K[d_{0}], K[d_{1}]] = 1, which is precisely the condition given in [8] to define categorical groups.

B) Now suppose we are given any strongly protomodular category ℂ and a morphism \underline{f}_{1} : \underline{X}_{1} → \underline{Y}_{1} of reflexive graphs such that the square with the d_{0} and the square with the d_{1} are pullbacks. Then if \underline{X}_{1} is a groupoid, \underline{Y}_{1} is itself a groupoid.

3) The slice categories ℂ/X

We are now going to prove that, when ℂ is strongly protomodular, the slice categories ℂ/X and the fibres Pt_{X}ℂ are still strongly protomodular. The forgetful functor ℂ/X → ℂ preserves the pullbacks, but no longer the products. However it satisfies the following definition:

5.3. Definition. A functor F : 𝔼 → 𝔼′ is paraexact when it preserves the pullbacks and is such that the factorization F (X × X′) → F (X) × F (X′) is a monomorphism and the following square a pullback for any pair of maps (f, f′) in 𝔼 × 𝔼:

\xymatrix{F (X × X′) \ar[r]^{F (f×f′)} \ar[d] & F (Y × Y′) \ar[d] \\ F X × F X′ \ar[r]_{F f×F f′} & F Y × F Y′}

Remark. Then F extends to a functor F : Rel𝔼 → Rel𝔼′ which preserves the cartesian and fibrant maps. It preserves also the intersection of two equivalence relations defined on the same object X. But it does not preserve the coarse relation any more, and consequently does not preserves the fact that a map j would be normal to some equivalence relation R. However, when F is furthermore conservative, as this is the case for the forgetful functor ℂ/X → ℂ, each map in Rel𝔼 is cocartesian with respect to the factorization functor Φ : Rel𝔼 → 𝔼 ×𝔼′ Rel𝔼′.

5.4. Definition. A paraexact functor F is said to be paranormal when it is conservative and such that the factorization functor Φ : Rel𝔼 → 𝔼 ×𝔼′ Rel𝔼′ is cofibrant on the fibrant morphisms of Rel𝔼′.

5.5. Proposition. Let F : ℂ → ℂ′ be a paranormal functor. Then:

1) the functors F_{Z} : ℂ/Z → ℂ′/F Z are normal, and consequently the extension PtF : Ptℂ → Ptℂ′ is fibrewise strongly normal

2) if ℂ′ is strongly protomodular, the category ℂ is itself strongly protomodular.

Proof. The proof mimics exactly the proof of Proposition 4.1.

5.6. Corollary. When ℂ is strongly protomodular, this is also the case for the slice categories ℂ/Z and the fibres Pt_{Z}ℂ.

Proof. When ℂ is protomodular, the forgetful functor ℂ/Z → ℂ is not only paraexact, but also paranormal. Thus, when ℂ is strongly protomodular, so is ℂ/Z. On the other hand, the forgetful functor Pt_{Z}ℂ → ℂ/Z is strongly normal as soon as ℂ is left exact.

References

[1] D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Springer Lecture Notes Math. 1488,1991, pp. 43-62.

[2] D. Bourn, Normal subobjects and abelian objects in protomodular categories, Journal of Algebra, 228, 2000, pp 143-164.

[3] D. Bourn, Aspherical Mal’cev groupoids and their directions, Cahiers du LAMPA, n^{∘}110, 1999, Université du Littoral.

[4] D. Bourn and G. Janelidze, Protomodularity, descent, and semi-direct products, Theory and Applications of Categories 4, n^{∘}2, 1998, pp. 37-46.

[5] A. Carboni, M.-C. Pedicchio and N. Pirovano, Internal graphs and internal groupoids in Mal’cev categories, Canadian Math. Soc. Conference proceedings, vol.13, 1992, pp. 97-109.

[6] M. Gerstenhaber, A categorical setting for Baer extension theory, Proceedings of Symposia in Pure Mathematics ,18, 1970, pp. 50-64.

[7] P. T. Johnstone, Affine categories and naturally Mal’cev categories, Journal of Pure and Applied Algebra 61, 1989, pp. 251-256.

[8] J. L. Loday, Spaces with finitely many non-trivial homotopy groups, Journal of Pure and Applied Algebra 24, 1982, pp. 179-202.

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: carboni@fis.unico.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk 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 Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz 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: walters@fis.unico.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca