INTERNAL PROFUNCTORS AND COMMUTATOR THEORY; APPLICATIONS TO EXTENSIONS CLASSIFICATION AND CATEGORICAL GALOIS THEORY

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1 Theory and Applications of Categories, Vol. 24, No. 7, 200, pp INTERNAL PROFUNCTORS AND COMMUTATOR THEORY; APPLICATIONS TO EXTENSIONS CLASSIFICATION AND CATEGORICAL GALOIS THEORY DOMINIQUE BOURN Abstract. We clarify the relationship between internal profunctors and connectors on pairs (R, S) of equivalence relations which originally appeared in the new profunctorial approach of the Schreier-Mac Lane extension theorem []. This clarification allows us to extend this Schreier-Mac Lane theorem to any exact Mal cev category with centralizers. On the other hand, still in the Mal cev context and in respect to the categorical Galois theory associated with a reflection I, it allows us to produce the faithful action of a certain abelian group on the set of classes (up to isomorphism) of I-normal extensions having a given Galois groupoid. Introduction Any extension between the non-abelian groups K and Y can be canonically indexed by a group homomorphism φ : Y AutK/IntK. The Schreier-Mac Lane extension theorem for groups [23] asserts that the class Ext φ (Y, K) of non-abelian extensions between the groups K and Y indexed by φ is endowed with a simply transitive action of the abelian group Ext φ(y, ZK), where ZK is the center of K and the index φ is induced by φ. A recent extension [] of this theorem to any action representative ([5], [4], [3]) category gave rise to an unexpected interpretation of this theorem in terms of internal profunctors which was closely related with the intrinsic commutator theory associated with the action representative category in question. So that there was a need of clarification about the general nature of the relationship between profunctors and the main tool of the intrinsic commutator theory, namely the notion of connector on a pair (R, S) equivalence relations, see [2], and also [5], [2], [20], [6]. The first point is that many of the observations made in [] in the exact Mal cev and protomodular settings are actually valid in any exact category. The second point is a characterization of those profunctors X Y which give rise to a connector on a pair of equivalence relations: they are exactly those profunctors whose associated discrete bifibration (φ ) : Υ X Y has its two legs φ and γ internally fully faithful. Received by the editors and, in revised form, Transmitted by R.J. Wood. Published on Mathematics Subject Classification: 8G50, 8D35, 8B40, 20J5, 08C05. Key words and phrases: Mal cev categories, centralizers, profunctor, Schreier-Mac Lane extension theorem, internal groupoid, Galois groupoid. c Dominique Bourn, 200. Permission to copy for private use granted. 45

2 452 DOMINIQUE BOURN The third point is the heart of Schreier-Mac Lane extension theorem, and deals with the notion of torsor. It is easy to define a torsor above a groupoid Z as a discrete fibration X Z from the indiscrete equivalence relation on an object X with global support, in a way which generalizes naturally the usual notion of G-torsor, when G is a group. The main point here is that when the groupoid Z is aspherical and abelian ([8]) with direction the abelian group A, there is a simply transitive action of the abelian group T orsa on the set T orsz of classes (up to isomorphism) of Z -torsors. This simply transitive action allows us on the one hand (fourth point) to extend now the Schreier-Mac Lane extension theorem to any exact Mal cev category with centralizers, and on the other hand (fifth point) to produce, still in the exact Mal cev context and in respect to the categorical Galois theory associated with a reflection I, the faithful action of a certain abelian group on the set of classes (up to isomorphism) of I-normal extensions having a fixed Galois groupoid. Incidentally we are also able to extend the notion of connector from a pair of equivalence relations to a pair of internal groupoids. A last word: what is rather amazing here is that the notion of profunctor between groupoids which could rather seem, at first thought, as a vector of indistinction (everything being isomorphic) appears, on the contrary, as a important tool of discrimination. The article is organized along the following lines: Part ) deals with some recall about the internal profunctors between internal categories and their composition, mainly from [8]. Part 2) specifies the notion of profunctors between internal groupoids and characterizes those profunctors X Y whose associated discrete bifibration (φ ) : Υ X Y has its two legs φ and γ internally fully faithful. It contains also our main theorem about the canonical simply transitive action on the Z -torsors, when the groupoid Z is aspherical and abelian. Part 3) deals with some recall about the notion of connector on a pair of equivalence relations, and extends it to a pair of groupoids. Part 4) asserts the Schreier-Mac Lane extension theorem for exact Mal cev categories with centralizers. Part 5) describes the faithful action on the I-normal extensions having a fixed Galois groupoid.. Internal profunctors In this section we shall recall the internal profunctors and their composition... Discrete fibrations and cofibrations. We shall suppose that our ambient category E is a finitely complete category. We denote by CatE the category of internal categories in E, and by () 0 : CatE E the forgetful functor associating with any internal category X its object of objects X 0. This functor is a left exact fibration. Any fibre Cat X E has the discrete equivalence relation X as initial object and the indiscrete equivalence relation X as terminal object.

3 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 453 An internal functor f : X Y is then () 0 -cartesian if and only if the following square is a pullback in E, in other words if and only if it is internally fully faithful: f X Y (d 0,d ) (d 0,d ) X 0 X 0f0 Y f 0 Y 0 0 Accordingly any internal functor f lower quadrangle is a pullback: produces the following decomposition, where the f X Y γ φ (d 0,d ) Z (d 0,d ) X 0 X 0 Y f 0 f 0 Y 0 0 with the fully faithful functor φ and the bijective on objects functor γ. We need to recall the following pieces of definition:.2. Definition. The internal functor f is said to be () 0 -faithful when the previous factorization γ is a monomorphism. It is said to be () 0 -full when this same map γ is a strong epimorphism. It is said to be a discrete cofibration when the following square with d 0 is a pullback: X f Y d 0 d d 0 d X 0 Y f 0 0 It is said to be a discrete fibration when the previous square with d is a pullback. Suppose f is a discrete cofibration; when the codomain Y is a groupoid, the domain X is a groupoid as well and the square with d is a pullback as well; when the codomain Y is an equivalence relation, then the same holds for its domain X. Accordingly, when the codomain Y of a functor f is a groupoid, it is a discrete cofibration if and only if it is a discrete fibration..3. Lemma. Any discrete fibration is () 0 -faithful. A discrete fibration is () 0 -cartesian if and only if it is monomorphic. Proof. Thanks to the Yoneda Lemma, it is sufficient to prove these assertions in Set which is straightforward.

4 454 DOMINIQUE BOURN.4. Profunctors. Let (X, Y ) be a pair of internal categories. Recall from [2] that f 0 g 0 an internal profunctor X Y is given by a pair X 0 U0 Y0 of maps (i.e. a span in E) together with a left action d : U Y U 0 of the category Y and a right action d 0 : U X U 0 of the category X which commute with each others, namely which make commute the left hand side upper dotted square in the following diagram, where all those commutative squares that do not contain dotted arrows are pullbacks: U U X f X p s 0 U Y π 0 π d 0 d s 0 U 0 d 0 x s 0 x 0 X 0 f 0 d g Y g 0 y 0 Y 0 y The middle horizontal reflexive graph is underlying a category U X, namely the category of cartesian maps above X, while the middle vertical reflexive graph is underlying a category U Y, namely the category of cocartesian maps above Y. The pairs (d 0, d ) going out from U Y and U X are respectively coequalized by f 0 and g 0. A morphism of profunctors is a morphism of spans above X 0 Y 0 which commutes with the left and right actions. We define this way the category Prof(X, Y ) of internal profunctors between X and Y. In the set theoretical context, a profunctor X Y is explicitely given by a functor U : X op Y Set. An object of U 0 is then an element ξ U(x, y) for any pair of object in X op Y, in other words U 0 = Σ (x,y) X0 Y 0 U(x, y). Elements of U 0 can be figured as further maps gluing the groupoids X and Y : x ξ y An object of U X is a pair x f ξ x y and we denote d 0 (f, ξ) = U(f, y )(ξ) by ξ.f, while an object of U Y is a pair x ξ y g y and we denote d (ξ, g) = U( x, g)(ξ) by g.ξ. An object of U is thus a triple: f x x y ξ y g We have π 0 (g, ξ, f) = (ξ.f, g) and π (g, ξ, f) = (f, g.ξ). The commutation of the two actions comes from the fact that we have: (g.ξ).f = g.(ξ.f).

5 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 455 In this set theoretical context, the previous diagram can be understood as a map between φ = ξ.f and χ = g.ξ, pictured this way: x φ y ξ f g x χ y Accordingly this defines a reflexive graph given by the vertical central part of the following diagram in Set, with two morphisms of graphs: X f. U g.p Y x 0 X 0 x f 0 d 0. U 0 d.p g 0 y 0 Y 0 y Actually this reflexive graph takes place in the more general scheme of a category we shall denote by U X U Y : its objects are the elements of U 0, a map between ξ and ξ being given by a pair (s, t) of map in X Y such that ξ.s = t.ξ: x ξ y s x ξ t ȳ Clearly the categories U X and U Y are subcategories of U X U Y. Actually, the same considerations hold in any internal context. The object of objects of the internal category U X U Y is U 0, its object of morphisms Υ is given by the pullback in E of the maps underlying the two actions: Υ υ 0 υ U Y d U X d 0 U 0 Moreover there is a pair (φ : Υ X : Υ Y ) of internal functors in E according to the following diagram: X f.υ 0 Υ g.υ Y x 0 X 0 x f 0 d 0.υ 0 U 0 d.υ g 0 y 0 Y 0 y

6 456 DOMINIQUE BOURN Without entering into the details, let us say that this pair of functors is characteristic of the given profunctor under its equivalent definition of a discrete bifibration in the 2- category CatE. Moreover the commutation of the two actions induces a natural morphism of reflexive graph: U d 0. U 0 d.p (π 0,π ) Υ d 0.υ 0 U 0 d.υ The previous observations give rise to the following diagram in CatE: U Y g Y γ U 0 Υ = U X U Y φ U X X f.5. Proposition. Let be given a finitely complete category E and an internal profunctor X Y. Then, in the fibre Cat U0 E, the left hand side quadrangle above is a pullback such that: U X U Y = Υ = U X U Y. Proof. Thanks to the Yoneda embedding, it is enough to check it in Set. A map between ξ and ξ in Υ, as above, is in U Y (resp. in U X ) if and only if s = x (resp. t = y ). Accordingly the intersection of these two subgroupoids is U 0. The second point is a consequence of the fact that any map between ξ and ξ in Υ, as above, has a canonical decomposition through a map in U Y and a map in U X : x ξ y x x ξ.s=t.ξ s y x y ξ Accordingly any subcategory of Υ which contains the two subcategories in question is equal to Υ. What is remarkable is that, in the set theoretical context, the profunctors can be composed on the model of the tensor product of modules, see [2]. The composition can be transposed in E as soon as any internal category admits a π 0 (i.e. a coequalizer of the domain and codomain maps) which is universal (i.e. stable under pullbacks), as it is the case when m E is an elementary topos, see [8]. Let Y 0 n 0 0 U0 Z0 be the span underlying another t y

7 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 457 profunctor Y Z (and let (µ, ν ) : Γ Y Z denote its associated discrete bifibration); then consider the following diagram induced by the dotted pullbacks: U p V V V Z U Y Y V Y θ V Y Z m p θ V0 0 U 0 Y0 V 0 V 0 p U θ n 0 p U0 n 0 W 0 Z 0 U Y U X g Y f 0 g 0 U 0 Y 0 f 0 f X X 0 It produces the following internal category Θ and the following forgetful functor to Y : m 0 n θ Θ : U Y Y V Y g.p U s 0 θ 0 U 0 Y0 V 0 Y : Y y s 0 g 0.p U0 y 0 Y 0 Then take the π 0 of the category Θ (it is the coequalizer θ of the pair (θ 0, θ )) which f0 n produces the dashed span X 0 0 W0 Z0. The π 0 in question being stable under pullbacks, this span is endowed with a left action of Z and a right action of X and gives us the composite (µ, ν ) (φ ). Given an internal category X, the unit profunctor is just given by the Yoneda profunctor, i.e. given by the following diagram, where X is the object of the commutative triangles of the internal category X and X is the object of the triple of composable maps of the internal category X : X X X p s 0 X π 0 π p 2 s X p x s 0 x 0 X 0 x 0 p 2 X 0 p x 0 x X 0 x

8 458 DOMINIQUE BOURN It is easy to check that X X = X 2, namely the domain of the universal internal natural transformation with codomain X. This tensor product : Prof(X, Y ) Prof(Y, Z ) Prof(X, Z ) is associative up to coherent isomorphism. By these units and the tensor composition, we get the bicategory Prof E of profunctors in E [8]..6. X -torsor. From now on, we shall be uniquely interested in the full subcategory GrdE of CatE consisting of the internal groupoids in E. Recall that any fibre Grd X E is a protomodular category. We shall suppose moreover that the category E is at least regular. We say that a groupoid X is connected when it has a global support in its fibre Grd X0 E, namely when the map (d 0, d ) : X X 0 X 0 is a regular epimorphism. We say it is aspherical when, moreover, the object X 0 has a global support, namely when the terminal map X 0 is a regular epimorphism..7. Proposition. Suppose E is a regular category. Let be given a discrete fibration f : X Y with Y an aspherical groupoid. ) If f is a monomorphism, then X is a connected groupoid. 2) If moreover X 0 has global support, then f is an isomorphism. 3) Any discrete fibration f between aspherical groupoids is a levelwise regular epimorphism. Proof. ) Since f is a monomorphic discrete fibration it is () 0 -cartesian according to Lemma.3 and the following commutative square is a pullback: f X Y (d 0,d ) (d 0,d ) X 0 X 0f0 Y f 0 Y 0 0 Accordingly X is connected as soon as such is Y. 2) Then, since f is discrete fibration, the following square that does not contain dotted arrows is still a pullback: X 0 X 0 f 0 f 0 Y 0 Y 0 p p X 0 Y f 0 0 and, by the Barr-Kock Theorem, when X 0 has global support, it is the case also for the following one: f 0 X 0 Y 0

9 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 459 Accordingly f 0 (and thus f ) is an isomorphism. 3) Starting from any discrete fibration f : X Y, take the canonical reg-epi/mono decomposition of f : X q U d 0 d d 0 d X 0 q 0 U 0 m 0 m Y d 0 d Y 0 Since m is a monomorphism, the left hand side squares are a pullback; since q is a regular epimorphism the right hand side squares are still pullbacks. Then m is a monomorphic discrete fibration, and since X 0 has global support, so has U 0. Then, according to 2), the functor m is an isomorphism, and the discrete fibration f is a levelwise regular epimorphism..8. Definition. Let X be an aspherical groupoid in E. A X -torsor is a discrete fibration U X where U is an object with global support: U U τ X U p τ d 0 d X 0 According to the previous proposition, the maps τ and τ are necessarily regular epimorphisms. Moreover it is clear that this determines a profunctor: X. When X is a group G, we get back to the classical notion of G-torsor, if we consider G as a groupoid having the terminal object as object of objects. In order to emphazise clearly this groupoid structure, we shall denote it by K G. 2. Fully faithful profunctors In this section we shall characterize those profunctors X Y between groupoids whose associated discrete bifibration (φ ) : Υ X Y has its two legs φ and γ internally fully faithful, namely those profunctors such that their associated discrete bifibration (φ ): U Y g Y γ U 0 Υ = U X U Y φ U X X f is obtained by the canonical decomposition of the discrete fibrations f and g through the () 0 -cartesian maps.

10 460 DOMINIQUE BOURN When X and Y are internal groupoids, we have substantial simplifications in the presentation of profunctors X Y. First, since any discrete fibration between groupoids becomes a discrete cofibration, any commutative square in the definition diagram, even the dotted ones, becomes a pullback. Accordingly the objects U and Υ defined above coincide, so that the reflexive graph U is actually underlying the groupoid U X U Y On the other hand, the new perfect symmetry of the definition diagram means that the pair (γ, φ ) : U Y X still determines a discrete bifibration, i.e. a profunctor Y X in the opposite direction which we shall denote by (φ ). When E is exact, any internal groupoid admits a π 0 which is stable under pullback; so that, in the context of exact categories, the profunctors between groupoids are composable. We shall be now interested in some special classes of profunctors between groupoids. 2.. Proposition. Suppose E is a finitely complete category. Let be given a profunctor (φ ) : U X Y between groupoids. Then its functorial leg γ : U Y is () 0 -faithful if and only the groupoid U X is an equivalence relation; it is () 0 -cartesian if and only if we have U X = R[g 0 ]. By symmetry, the other leg φ : U X is () 0 -faithful if and only the groupoid U Y is an equivalence relation; it is () 0 -cartesian if and only if we have U Y = R[f 0 ] Proof. Thanks to the Yoneda embedding, it is enough to prove it in Set. Suppose that γ : U Y is faithful. Let be given two parallel arrows in U X U :. x ξ y f f x χ The image of these two arrows of U by the functor γ is y. Accordingly, since this functor is () 0 -faithful, we get f = f, and U X is an equivalence relation. Conversely suppose U X is an equivalence relation. Two maps in U having the same image g by γ determine a diagram: f x f ξ y y y x χ y which itself determines the following diagram: g x g.ξ y f f x χ y y In this way, any profunctor between groupoids can be seen as a double augmented simplicial object in E such that any commutative square is a pullback

11 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 46 But since U X is an equivalence relation, we get f = f. Accordingly the functor γ : U Y is faithful. Suppose we have moreover U X = R[g 0 ]. Any pair (ξ, χ) in U 0 with a map g between their respective codomains y and y, determines a pair (g.ξ, χ) in R[g 0 ]: x g.ξ f y x χ and since we have U X = R[g 0 ], we get a map f such that g.ξ = χ.f which determines an arrow in U whose image by γ is g: x ξ y f x χ y Conversely suppose γ is fully faithful. Since it is faithful we observed that U X R[g 0 ]. Now given any pair (ξ, χ) in R[g 0 ], the fullness property of γ produces a map f which completes the following diagram: g x ξ y f x χ y y and we get R[g 0 ] U X Definition. An internal profunctor between groupoids is said to be faithful when its two legs φ and γ are () 0 -faithful. It is said to be fully faithful when its two legs φ and γ are () 0 -cartesian. It is said to be regularly fully faithful when moreover the maps f 0 and g 0 are regular epimorphisms. Examples. ) Given any internal groupoid X, its associated Yoneda profunctor is a regularly fully faithful profunctor: R 2 [x 0 ] R[x 0 ] X p p 3 s p 2 p 2 s p x s 0 x 0 R[x 0 ] X X 0 x 0 p 2 X 0 p x 0 x x X 0

12 462 DOMINIQUE BOURN It is this precise diagram which makes the internal groupoids monadic above the split epimorphisms, see [6]. 2) Given an abelian group A in E, any A-torsor determines a regularly fully faithful profunctor K A K A. For that, consider the following diagram where the upper right hand side squares are pullbacks by definition of an A-torsor: R( ) R[h ] T T h A R(p ) p p e T T T h p τ T τ T A e and complete it by the horizontal kernel equivalence relations. Then, when A is abelian, the same map h is the quotient of the vertical left hand side equivalence relation. Thanks to the Barr embedding, it is enough to prove it in Set, which is straighforward. Actually this is equivalent to saying that, when A is abelian, a principal left A-object becomes a principal symmetric two-sided object, according to the terminology of [] Proposition. Suppose E is a regular category. A morphism τ : (φ ) (φ, γ ) between profunctors above groupoids having their legs φ and φ () 0-cartesian and regularly epimorphic is necessarily an isomorphism. In particular, a morphism between two regularly fully faithful profunctors is necessarily an isomorphism. Proof. Consider the following diagram which is part of the diagram induced by the morphism τ of profunctors: g R[f 0 ] R(τ) R[f 0] g Y d 0 d d 0 d τ U 0 U 0 g 0 f 0 f 0 X 0 X0 X 0 g 0 y 0 Y 0 y The functor g and g being discrete fibrations, the left hand side upper part of the diagram is a discrete fibration between equivalence relations. Since moreover f 0 is a regular epimorphism, the lower square is a pullback by the Barr-Kock theorem and τ an isomorphism.

13 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY Proposition. Suppose E is an efficiently regular category [9]. Let X Y and Y Z be two profunctors between groupoids whose first leg of their associated discrete bifibrations (φ ) and (µ, ν ) is () 0 -cartesian. Then their profunctor composition does exist in E and has its first leg () 0 -cartesian. When, moreover, their first legs are regularly epimorphic, their profunctor composition has its first leg regularly epimorphic. By symmetry, the same holds concerning the second legs. Accordingly regularly fully faithful profunctors are composable as profunctors, and are stable under this composition. Proof. Let us go back to the diagram defining the composition. When the first leg φ is () 0 -cartesian, we have U Y = R[f 0 ]. By construction the groupoid Θ determines a discrete fibration: U Y Y V Y p U R[f 0 ] θ s 0 U 0 Y0 V θ 0 W 0 θ 0 p s 0 p U0 U 0 f 0 X 0 Since E is an efficiently regular category and the codomain of this discrete fibration is an effective equivalence relation, its domain is an effective equivalence relation as well. Thus, this domain admits a quotient θ, and a factorization f 0 which makes the right hand side square a pullback. Moreover this quotient is stable under pullback, since E is regular. Accordingly the two profunctors can be composed. Let us notice immediately that when m 0 is a regular epimorphism, such is p U0. If moreover f 0 is a regular epimorphism, then f 0 is a regular epimorphism as well, and we shall get the second point, once the first one is checked. Suppose now the first leg φ is () 0-cartesian, i.e. V Z = R[m 0 ]. We have to check that W Z = R[ f 0 ]. Let us consider the diagram where R[p U0 ] is the result of the pulling back along g 0 of V Z = R[m 0 ], and R[p U ] is the result of the result of the pulling back along g of V = R[m ]: R[p U ] p U Y Y V Y p U R[f 0 ] R(θ ) s 0 θ R[p U0 ] W Z R(θ 0 ) p q 0 q θ s 0 U 0 Y0 V θ 0 W 0 θ 0 p U0 f s 0 U 0 X p f Since any of the upper left hand side squares are pullbacks and (θ, θ) is a pair of regular epimorphims, the upper right hand side squares are pullbacks. This implies that the right hand side vertical diagram is a kernel equivalence relation. Accordingly we have W Z = R[ f 0 ]. f 0

14 464 DOMINIQUE BOURN It is clear that when (φ ) : X Y is a fully faithful (resp. regularly fully faithful) profunctor, the profunctor (φ ) : Y X is still fully faihtful (resp. regularly fully faithful) Proposition. Suppose E is an efficiently regular category. Given any regularly fully faithful profunctor (φ ) : X Y, the profunctor (φ ) : Y X is its inverse with respect to the composition of profunctors. Proof. The heart of the composition (φ ) (φ ) is given by the following dotted pullbacks where the unlabelled vertical dotted arrow is g : π 0 U π R[f 0 ] p p R[g 0 ] U 0 f f 0 x X X 0 p g R[f 0 ] Y y x 0 p 0 g 0 U 0 Y 0 f 0 X 0 The commutations of this diagram shows that the maps θ 0 and θ needed in the composition constuction are respectively and π. Accordingly their coequalizer is the regular epimorphism f, and the associated span is just (x 0, x ). So that (φ ) (φ ) is just the Yoneda profunctor associated with X. By symmetry we obtain that (φ ) (φ ) is the Yoneda profunctor associated with Y. Accordingly, when E is a finitely complete efficiently regular category, we get the bigroupoid Rf E of the regularly fully faithful profunctors between internal groupoids in E, and the associated groupoid RfE whose morphisms are the isomorphic classes of regularly fully faithful profunctors. Remark. By the specific (i.e the double choice of the map h as a coequalizer) construction of the example 2 above, we associate with any A-torsor a regularly fully faithful profunctor K A K A. What is very important is that the classical tensor product of A-torsors coincides with the composition of profunctors. Accordingly this construction defines the abelian group T orsa as a subgroup of RfE(K A, K A) The canonical action on the X -torsors when X is abelian. In this section, we shall show that when X is an aspherical abelian groupoid in E, the set T orsx of isomorphic classes of X -torsors is canonically endowed with a simply transitive action of an abelian group of the form T orsa, where A is an internal abelian group in E. Aspherical abelian groupoids Recall from [8] the following: g 0

15 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY Definition. An internal groupoid Z in E is said to be abelian when it is a commutative object in the protomodular fibre Grd Z0 E. We shall see that, in the Mal cev context (see below), this is equivalent to saying that the map (z 0, z ) : Z Z 0 Z 0 is such that [R[(z 0, z )], R[(z 0, z )]] = 0. When moreover the category E is efficiently regular, any aspherical abelian groupoid Z has admits direction [8], namely there exists an abelian group A in E which makes the following upper squares pullbacks squares: R[(z 0, z )] p Z (z 0,z ) Z 0 Z 0 ν Z A e Let us immediately notice that, in the Mal cev context, this makes now central the kernel equivalence relation R[(z 0, z )]. In the set theoretical context, a groupoid Z is abelian when, for any of its object z, the group Aut z of endomaps at z is abelian. The groupoid Z is asherical, when it is non empty and connected. So, when the groupoid Z is aspherical and abelian, all the abelian groups Aut z are isomorphic. Moreover, what is remarkable is that, given any map τ : z z in Z, the induced group homomorphism Aut z Aut z is independent of the map τ. The direction A of Z is then any of these abelian groups. The previous construction provides a direction functor d : AsGrdE AbC from the category of aspherical groupoids in E to the category of abelian groups in E. Suppose you have a () 0 -cartesian functor f : T Z with T aspherical, then the following diagram shows that d(f ) is a group isomorphism, since the lower square is a pullback (and consequently the upper left hand side ones): R[(t 0, t )] R[(z 0, z )] p p f T Z ν Z d(z ) e (t 0,t ) (z 0,z ) T 0 T 0 f 0 f 0 Z 0 Z 0 So, let X Y be any fully faithful profunctor between aspherical groupoids whose associated discrete bifibration is given by the pair (φ ) : Υ X Y. Then necessarily the groupoids X and Y have same direction A, and this fully faithful profunctor determines a group isomorphism d(γ ).d(φ ) : A A. We shall denote by Rf E(X, Y ) the subset of RfE(X, Y ) consisting of those regularly faithful profunctors between aspherical groupoids such that d(γ ).d(φ ) = A (or equivalently d(γ ) = d(φ )).

16 466 DOMINIQUE BOURN 2.8. Proposition. Let E be an efficiently regular category. Suppose the groupoid Z in E is aspherical with direction A and g : X Z is a Z -torsor. Consider the following diagram with the horizontal kernel equivalence relations which make pullbacks the upper left hand side squares: R( ) R[g ] R[g 0 ] q A R(p ) p p X X p X τ X g Z z 0 z g 0 Z 0 Then there is a unique dotted arrow q which completes the previous diagram into a regularly fully faithful profunctor K A Z whose legs of the associated discrete bifibration are such that d(γ ) = d(φ ). Proof. Since the category E is regular, it is sufficient to prove the result in Set, thanks to the Barr embedding []. For sake of simplicity, we denote by ν : R[(z 0, z )] A the mapping which defines the direction of Z. The pullback defining A implies that, given a pair of parallel maps (φ, ψ) : u v in Z, they are equal if and only if ν(φ, ψ) = 0. Let us denote by S the equivalence relation on R[g 0 ] defined by the left hand side vertical diagram. We get (x, x )S(y, y ) if and only if g (x, y) = g (x, y ). On the other hand, we have necessarily: ν( g0 (y), g (y, y ))+ν(g (x, y), g (x, y)) = ν(g (x, y), g (x, y )) = ν(g (x, y), g (x, y )) + ν( g0 (x), g (x, x )), in other words we get: ν( g0 (y), g (y, y )) = ν(g (x, y), g (x, y )) + ν( g0 (x), g (x, x )) So: (x, x )S(y, y ) if and only if ν( g0 (y), g (y, y )) = ν( g0 (x), g (x, x )). Consequently the map q : R[g 0 ] A defined by q(x, x ) = ν( g0 (x), g (x, x )) is such that S = R[q] and, being surjective, it is the quotient map of this equivalence relation S. Conversely suppose you are given a regularly fully faithful profunctor: R( ) R[g ] R[g 0 ] f A R(p ) p p X X p X τ X g Z z 0 z g 0 Z 0 such that d(γ ) = d(φ ). This means that for any pair ((x, y, z), (x, y, z)) such that g 0 (y) = g 0 (z) and g 0 (y ) = g 0 (z), we have ν(g (x, y ), g (x, y)) = f (y, z) f (y, z). Whence: f (y, z) = f (y, z) f (z, z) = ν(g (x, y ), g (x, z)), for any x; in particular we have f (y, z) = ν( g0 (y ), g (y, z)) and thus f (y, z) = q(y, z).

17 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 467 In other words there is a bijection between the sets T orsz and Rf E(K A, Z ). The canonical action Suppose that our aspherical abelian groupoid Z is such that Z 0 =, namely that it is actually an abelian group A. We recalled that the set T orsa is canonically endowed with an abelian group structure which is nothing but a subgroup of the group RfE(K A, K A). Thanks to the previous proposition, we can now precise that this subgroup is Rf E(K A, K A) Theorem. Let E be an efficiently regular category. Let Z be an aspherical abelian groupoid with direction A. Then there is a canonical simply transitive action of the abelian group T orsa on the set T orsz of isomorphic classes of Z -torsors. Proof. The action of the A-torsors on Z -torsors will be naturally given by the composition (= tensor product) of profunctors K A K A Z whose image by the direction functor d will be A since it is the case for both of them. The fact that this action is simply transitive comes from the fact that the regularly fully faithful profunctor K A Z arising from a Z -torsor is an invertible profunctor, see Proposition 2.5. So that, starting from a pair (g, g ) of Z -torsors, the unique A-torsor relating them is necessarily given by the following composition: K A g (g ) Z K A whose image by d is certainly A. Accordingly this group action is nothing else but the simply transitive action of the sub-hom-group Rf E(K A, K A) on the sub-hom-set RfE(K A, Z ) inside the subgroupoid Rf E. So, given a Z -torsor g and an A-torsor h, the tensor product g h will be given by the following diagram: p g0 R[g ] X X g Z (T T ) A R[g 0 ] R[g θ 0 ] θ 0 T X p X X g T T θ p T X ḡ0 h T T T A e Z 0

18 468 DOMINIQUE BOURN 2.0. The additive setting. Let us have a quick look at the translation of the previous theorem in the additive setting. So let A be an efficiently regular additive category, and C an object of A. We are going to make explicit the previous simply transitive action in the slice category A/C (in the pointed category A any abelian group T orsa is trivial). Giving an aspherical internal groupoid A in A/C is equivalent to giving an exact sequence: A α A 0 q C Let us denote by β : B A the kernel of α. The direction of A is then nothing but (p C, ι C ) : B C C. It is well known that a torsor associated with this abelian group in A/C is nothing but an extension: B m H h C and the abelian group of torsors in A/C is nothing but the abelian group Ext A (C, B). On the other hand, giving a A -torsor is equivalent to giving an exact sequence together with a (regular epic) map τ : D A 0 such that q.τ = f and τ.k = α: A k D f C τ A α A 0 q C The action of the abelian group Ext A (C, B) can be described in the following way: starting with the previous B-torsor and A -torsor, take the pullback of f and h, then the result of the action is given by the following 3 3 construction: B B B (,β) ( m,k.β) B A (β, A ) m k H C D w h C f A D C C C k f The kernel of the regular epimorphism h C f is the map m k; so that the 3 3 lemma produces the vertical right hand side exact sequence. The upper right hand side square is then certainly a pushout. Accordingly, the equality α.(β, A ) = α.p A = τ.k.p A = τ.p D.(m k) produces the required factorization τ : D A 0 to have what is equivalent

19 to a A -torsor: INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 469 B A (β, A ) m k w A k A H C D D τ A 0 p D τ D Conversely, starting with what is equivalent to two A -torsors, we get the unique B-torsor relating them by the vertical right hand side exact sequence given by the following 3 3 construction: α B β A (,) (k,k ) B B (,) B D A0 D (k.β) (k.β) τ A0 τ α Kerq A 0 q H C 3. Connector and centralizing double relation We shall describe, here, the strong structural relationship between fully faithful profunctors and connectors between equivalence relations. Consider R and S two equivalence relations on an object X in any finitely complete category E. Let us recall the following definition from [2], see also [24] and [5]: 3.. Definition. A connector for the pair (R, S) is a morphism which satisfies the identities : p : S X R X, (xsyrz) p(x, y, z) ) xrp(x,y,z) ) zsp(x,y,z) 2) p(x,y,y)=x 2 ) p(y,y,z)=z 3) p(x,y,p(y,u,v))=p(x,u,v) 3 ) p(p(x,y,u),u,v)=p(x,y,v) In set theoretical terms, Condition means that with any triple xsyrz we can associate a square: x R p(x, y, z) S y R S z.

20 470 DOMINIQUE BOURN More acutely, any connected pair produces an equivalence relation Σ R in GrdE on the equivalence relation R whose two legs are discrete fibrations, in other words an equivalence relation in DiF E: S X R S p (d 0.,p) (p,d. ) d d 0 It is called the centralizing double relation associated with the connector. It is clear that, conversely, any equivalence relation Σ R in GrdE whose two legs are discrete fibrations determines a connector between R and the image by the functor () 0 : GrdE E of this equivalence relation Σ R. Example ) An emblematical example is produced by a given discrete fibration f : R Z whose domain R is an equivalence relation. For that consider the following diagram: R(d 0 ) R[f ] R[f 0 ] p R(d ) p d 0 R X d 0 d R X d f Z d 0 f 0 Z 0 It is clear that R[f ] is isomorphic to R[f 0 ] X R and that the map p : R[f ] R d X determines a connector for the pair (R, R[f 0 ]). 2) For any groupoid X, we have such a discrete fibration (which we shall denote by ɛ X : Dec X X in GrdE): R[d 0 ] d 2 X p d d 0 X X d 0 which implies the existence of a connector for the pair (R[d 0 ], R[d ]). The converse is true as well, see [5] and [2]; given a reflexive graph : d d s Z 0 Z 0 d 0

21 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 47 any connector for the pair (R[d 0 ], R[d ]) determines a groupoid structure on this graph. 3) For any pair (X, Y ) of objects, the pair (R[p X ], R[p Y ]) of effective equivalence relations is canonically connected. 4) The diagram defining any fully faithful profunctor (φ ) : X Y clearly determines a centralizing double relation, and thus a connector for the pair (R[f 0 ], R[g 0 ]). Remark. The main point, here, is to emphasize that the diagram underlying the double centralizing relation associated with the existence of a connector is the core of the diagram defining a regularly fully faithful profunctor, and that, whenever the category E is exact, this core can be effectively completed into an actual regularly fully faithful profunctor by means of the quotients: Now let us observe that: S X R q R S X p (d 0.,p) (p,d. ) d d 0 x s 0 x 0 d 0 q S R Y d y 0 q S Y 0 X q R X Proposition. Suppose p is a connector for the pair (R, S). Then the following reflexive graph is underlying a groupoid we shall denote by R S: y d.p S X R d 0. X Proof. Thank to the Yoneda embedding, it is enough to prove it in Set. This is straightforward just setting: (zsurv).(xsyrz) = xsp(u, z, y)rv The inverse of the arrow xsyrz is zsp(x, y, z)rx. When S R = X, the groupoid S R is actually an equivalence relation Proposition. Suppose p is a connector for the pair (R, S) and consider the following diagram in GrdE: X R S i S i R S R It is a pullback such that the supremum of i R and i S is R S.

22 472 DOMINIQUE BOURN Proof. By the Yoneda embedding: the first point is straightforward; as for the second one: it is a direct consequence of the fact that any map (xsyrz) in S R is such that: (xsyrz) = (ysyrz).(xsyry) = i R (yrz).i S (xsy). So, when S R = X, the equivalence relation S R is nothing but S R The Mal cev context. Let D be now a Mal cev category, i.e. a category in which any reflexive relation is an equivalence relation [4] [5]. Commutator theory In a Mal cev category, the previous conditions 2) on connectors imply the other ones, and moreover a connector is necessarily unique when it exists, and thus the existence of a connector becomes a property; we then write [R, S] = 0 when this property holds. From [2] recall that: ) R S = X implies [R, S] = 0 2) T S and [R, S] = 0 imply [R, T ] = 0 3) [R, S] = 0 and [R, S ] = 0 imply [R R, S S ] = 0 When D is a regular Mal cev category, the direct image of an equivalence relation along a regular epimorphism is still an equivalence relation. In this case, we get moreover: 4) if f : X Y is a regular epimorphism, [R, S] = 0 implies [f(r), f(s)] = 0 5) [R, S ] = 0 and [R, S 2 ] = 0 imply [R, S S 2 ] = 0 As usual, an equivalence relation R is called abelian when we have [R, R] = 0, and central when we have [R, X ] = 0. An object X in D is called commutative when [ X, X ] = 0. We get also the following precision: 3.5. Proposition. Let D be a Mal cev category. Suppose p is a connector for the pair (R, S). The following diagram in GrdD: X S i S R i R S R is a pushout in GrdD. Proof. Let us consider the following diagram in GrdD: The previously observed decomposition: X S g R f X (xsyrz) = (ysyrz).(xsyry) = i R (yrz).i S (xsy)

23 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 473 in the groupoid S R allows us to construct a unique morphism of reflexive graphs φ : S R X, just setting φ(xsyrz) = f(yrz).g(xsy). Now since D is a Mal cev category, any morphism of reflexive graphs between the underlying graphs of groupoids is necessarily a functor [5]. Finally, recall also the following from [3], see Proposition 3.2: 3.6. Proposition. Let C be a regular Mal cev category. Any decomposition in GrdD of a discrete fibration f : X Y through a regular epic functor: is necessarily made of discrete fibrations. q f X Q Y 3.7. A glance at the notion of centralizing double groupoid. Making a further step, given any pair (U, V ) of internal groupoids in E having the same object of objects U 0, we shall say that they admit a centralizing double groupoid, if there is an internal groupoid in DiF E (objects: internal groupoids; maps: discrete fibrations): W such that its image by the functor () 0 : GrdE E is U ; namely if there is a diagram in E which reproduces the core of an internal profunctor between groupoids, where any commutative square in the following diagram is a pullback: V W d d 0 V d 0 d d 0 d d U U 0 d 0 It is clear that in general this is a further structure. becomes a property: But in a Mal cev category this 3.8. Proposition. Let D be Mal cev category, and (U, V ) a pair of internal groupoids having the same object of objects U 0. A centralizing double groupoid on this pair is unique if it exists. In this case we shall say that the groupoids U and V are connected. Proof. Consider the following pullback of split epimorphisms: W d 0 U d U d d V 0 d d U 0 V d 0 d V U 0

24 474 DOMINIQUE BOURN Since the category D is a Mal cev category, in the pullback above, the pair (s U 0 : U W, s V 0 : V W ) is jointly strongly epic. Accordingly the unicity of the map d V 0 is a consequence of the equations d V 0.s V 0 = V and d V 0.s U 0 = s V 0.d U 0, while the unicity of the map d U is a consequence of the equations d U.s U 0 = U and d U.s V 0 = s U 0.d V. Remark: Actually, in the Mal cev context, a pair (U, V ) of groupoids is connected if and only if there is a pair of maps (d V 0 : W = U U0 V V, d U : W = U U0 V U ) such that the four previous equations are satisfied with moreover the commutation equation d V.d V 0 = d U 0.d U. As previously for the connected equivalence relations, given any pair (U, V ) of groupoids which has a double centralizing groupoid, in a category E, we can observe that the following reflexive graph is underlying a groupoid we shall denote by U V : d.p W U 0 d 0. In order to show this, the quickest argument is to consider the diagram defining the double centralizing groupoid as a double simplicial object where any commutative square is a pullback. The diagonal is then necessarily a simplicial object, while the fact that any of the structural commutative squares of this diagonal is a pullback is a straigthforward consequence of the fact that any of the commutative squares of the original double simplicial object is a pullback Proposition. Let D be a Mal cev category. Suppose the pair (U, V ) of internal groupoids is connected, then the following diagram in GrdD: U 0 V U i V i U U V is a pullback and a pushout. Proof. The proof is the same as for the equivalence relations in the Mal cev context. The first point is straightforward. As for the second one, consider any commutative diagram in GrdD: U 0 V U g X Any map in U V is an object of the pullback W = U U0 V, namely a pair of a map in U and of a map in V. Accordingly it is possible to construct a unique morphism of reflexive graphs φ : W X. Now, since D is a Mal cev category, it is necessarily underlying an internal functor, since U V and X are actual groupoids, again see [5]. f

25 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY Corollary. Let D be a Mal cev category and X Y an internal profunctor. In the following diagram, the left hand side quadrangle is a pushout in GrdD: U Y g Y γ U 0 Υ = U X U Y φ U X X f Example: It is well known that, in the category Gp of groups, an internal groupoid is the same thing as a crossed module. It is easy to check that a pair of crossed modules H h G h H corresponds to a pair of groupoids H and H having a centralizing double groupoid if and only if the restriction of the action of the group G on H to the subgroup h (H ) is trivial (which implies [h(h), h (H )] = 0), and symmetrically the restriction of the action of the group G on H to the subgroup h(h) is trivial. When this is the case, the crossed module corresponding to the groupoid H H is nothing but the factorization φ : H H G induced by the equality [h(h), h (H )] = 0, and the following diagram becomes a pushout inside the category X-M od of crossed modules: H h H φ h H H G Remark. Again, the main point, here, is as above to emphasize that the diagram underlying a double centralizing groupoid is the core of the diagram defining a profunctor; when the category Mal cev D is exact, this core can be effectively completed into an actual profunctor by means of the quotients, thanks to Proposition 3.6: W d 0 U X d s 0 V d 0 d d 0 d s 0 U 0 d 0 x s 0 x 0 X 0 d Y y 0 Y 0 y

26 476 DOMINIQUE BOURN 4. Schreier-Mac Lane extension theorem A first application of the existence of the canonical action on Z -torsors deals with the classification of extensions in D, when the category D is an exact regular Mal cev category which admits centralizers. It was already observed in [7] that, in any finitely complete exact category E, any commutative object X (i.e. such that [ X, X] = 0, with a given commutative connector π) with global support has a direction, namely: there exists an abelian group A in E such that the following squares are pullbacks, where π (x, y, z) = (x, π(x, y, z)): X X X X X p 2 X X ν X A π π p ω p X which is given by the quotient of the upper left hand side horizontal equivalence relation; and consequently any commutative object with global support gave rise to an A-torsor. Accordingly the set of isomorphic classes of commutative objects with global support and direction A has the abelian group structure of T orsa. Now let D be an exact Mal cev category. The previous result applied to the slice category D/Y says that: ) any extension f : X Y which has an abelian kernel equivalence relation produces, as its direction, an abelian group structure E Y in D/Y 2) the set Ext E Y of extensions above Y having an abelian kernel equivalence relation and E Y as direction is nothing but the abelian group T orse. The aim of this section is to show that when, moreover, the Mal cev category D has centralizers of equivalence relations, there is a way of producing an index φ for any extension f : X Y which determines, on the set Ext φ Y of extensions with this given index φ, the simply transitive action of an abelian group of type Ext E Y. This observation is a generalization of the Schreier-Mac Lane extension theorem for groups; it was originated from a first generalization of this Schreier-Mac Lane extension theorem to any action representative category, see []. 4.. Mal cev categories with centralizers. Let now D be a Mal cev category Definition. When R is a equivalence relation on an object X, we define Z(R), and call centralizer of R, the largest equivalence relation on X connected with R, i.e. such that [R, S] = 0. We shall say that the Mal cev category D has centralizers, when any equivalence relation R has a centralizer Z(R). It is clear, for instance, that a naturally Mal cev category in the sense of [9] is nothing but a Mal cev category with centralizers, such that, for any equivalence relation R on

27 INTERNAL PROFUNCTORS AND COMMUTATOR THEORY 477 X, we have Z(R) = X. More generally, recall that, in a Mal cev category, an internal reflexive graph (d 0, d ) : X X 0 is a groupoid if and only if we have [R[d 0 ], R[d ]] = 0, namely R[d ] Z(R[d 0 ]). Of course, there is an extremal situation: 4.3. Definition. Let D be a Mal cev category. A groupoid X in D is said to be eccentral when we have Z(R[d 0 ]) = R[d ]. In other words a groupoid X in D is eccentral if and only if, given any reflexive relation Σ on Dec X (see example 2) of connector) in DiF D, its legs (, p ): Σ p Dec X ɛ X X are coequalized by ɛ X. In the regular context, one important point is that eccentral groupoids are strongly related to centralizers. Before going any further, we have to recall [6] that the functor Dec : GrdD GrdD is underlying a comonad such that the following diagram, in the category GrdD, is a kernel equivalence relation with its quotient: Dec 2 X ɛ Dec X Dec ɛ X Dec X ɛ X X 4.4. Proposition. Suppose the Mal cev category D is regular and X is an eccentral groupoid. Then any regular epimorphic discrete fibration j : R X where R is an equivalence relation on an object X, is such that R[j 0 ] is the centralizer Z(R). Proof. Consider the following diagram in GrdD where Σ is the double centralizing relation associated with a connected pair [R, S] = 0. We shall show that Σ factorizes through R[j ] or, equivalently, that j coequalizes and p. For that take the direct image along the regular epimorphic discrete fibration Dec j of the equivalence relation Dec Σ : Dec Σ Dec j (Σ ) Dec Dec 2 π 0 X Dec p π ɛ Σ Dec R Dec Dec j X Σ p R ɛ R j X The maps π 0 and π are discrete fibrations since they come from a decomposition of a discrete fibration through a regular epic functor. Accordingly Dec j (Σ ) is a reflexive relation in DiF D, which factorizes through Dec 2 X since X is an eccentral groupoid. So ɛ X.Dec j coequalizes Dec and Dec p, and since ɛ Σ is a regular epic functor, the functor j coequalizes and p. ɛ X

28 478 DOMINIQUE BOURN 4.5. Exact Mal cev setting. In this section we shall show that when D is an exact Mal cev category, the existence of centralizers is characterized by the existence of enough eccentral groupoids Proposition. Let D be an exact Mal cev category with centralizers. Given any equivalence relation R, there is a, unique up to isomorphism, regular epic discrete fibration j : R X towards an eccentral groupoid. Proof. Let Σ be the double centralizing relation associated with the connected pair [R, Z(R)] = 0. Since D is exact, we can take a levelwise quotient of this double relation which produces a regular epic discrete fibration: Σ π 0 π R j X We have to show now that the groupoid X is eccentral. Suppose given an equivalence relation Λ on Dec X which is in DiF. Then consider the inverse image of Λ along the regular epic discrete fibration Dec j in the following diagram: Dec j (Λ ) h Λ Dec π 0 π 0 2 X π π Γ Dec R Dec ˇπ 0 Dec j X ɛ R ɛ X p ˇπ 0 Σ R X p j Its direct image Γ along the regular epic discrete fibration ɛ R is an equivalence relation in DiF which is a double centralizing relation associated with R. Accordingly this direct image factorizes through Σ according to Proposition 4.4, and produces the left hand side vertical dotted factorization. Accordingly the pair ( π 0, π ) is coequalized by ɛ X.Dec j. And since h is an epimorphism, the pair (π 0, π ) is coequalized by ɛ X. Accordingly Λ factorizes through Dec 2 X, and X is eccentral. Suppose now there are two regular epic discrete fibrations j and j to eccentral groupoids X and X. Then R[j 0 ] = Z(R) = R[j 0]. Accordingly X 0 is isomorphic to X 0. Since j and j are discrete fibrations, the two equivalence relations R[j ] and R[j ] are part of the double centralizing relation associated with the pair (R, Z(R)). Since D is a Mal cev category, this double centralizing relation is unique (up to isomorphism), and consequently we get R[j ] = R[j ]; so that X is isomorphic to X Theorem. Let D be an exact Mal cev category. Then D has centralizers if and only if D has enough eccentral groupoids with respect to DiF D: namely, from any groupoid T there is a regular epic discrete fibration φ : T X with X an eccentral groupoid. In this case the eccentral groupoid is unique up to isomorphism and we call it the index-groupoid of the groupoid T.