Relative Commutator Theory in Semi-Abelian Categories

RELA TIVE COMMUT A TOR THEO R Y IN SEM I-ABELIAN CA TEGORIES TOMAS EVERAER T AND TIM V AN DER LINDEN Abstract. Basing ourselves on t he concept of double cen tral extension f rom categorical Galois theory , we study a notion of commutato r which is defined relative to a Birkhoff sub category B of a semi-ab elian category A . This com- mut ator c haracterises Janelidze and Kel l y’s B -central extensions; when the subcategory B is determined by the ab elian ob jects in A , i t coincides with Huq’s comm utator; and when the category A is a v ariet y of Ω- groups, it coin- cides with the r elativ e commutator int ro duced b y the first author. 1. Introduction The aim of this article is to fill in the q ue s tion mar k in the diagra m ? o o o o o o o O O O O O O O O O Janelidze & K elly Huq Evera ert o o o o o O O O O O F ro ehlich Higgins which relates several no n-equiv alent concepts of c ommuting normal sub obje cts , here named after the authors who intro duced them. This diag r am is mea nt to b e read in the following manner. The bo ttom tr iangle restr icts itself to theories which make sense for v arieties of Ω- groups, while the top tr ia ngle extends those theories to a categor ic al context. In the left hand side column we hav e theories which a re one-dimensional and r elative ; the theories in the right ha nd side column, how ever, ar e two-dimensional and absolute , while the ones in the middle column are two-dimensional and r elative . So w e are lo o king for a c ate goric al c ommutator t he ory which is b oth r elative and t wo- dimensional . Le t us explain in more detail what this means for us. 1.1. The b ottom triangle. Recall that a v ariety of Ω -groups [26] is a v ar ie ty in the sense of universal a lgebra which is p ointed (i.e., it ha s exactly one constant) and has among st its op erations a nd identities those of the v ariety of groups. Apart from groups, the examples include the v a rieties of ab elian groups, of non-unital ring s, o f commutativ e algebr as, of mo dules a nd of Lie algebr a s, and also the categ ories of 2010 Mathematics Subje ct Classific ation. 18E10, 17D10, 20J. Key wor ds and phr ases. categorical Galois theory , semi- abeli an category , commu tator, double cen tral extension. The first author was supported b y F onds v oor W etensc happelijk Onderzo ek (FWO- Vlaanderen). The second author works as char g´ e de r ech er ches for F onds de la Recherc he Scien tifique–FNRS; his r esearc h wa s supported by Centro de Matem´ atica da U ni v ersidade de Coimbra and by F unda¸ c˜ ao para a Ciˆ enc ia e a T ecnologia (grant num b er SFRH/BPD/38797/2007). 1 2 TOMAS E VERAER T AND TIM V AN DER LINDEN crossed mo dules and of precross ed mo dules a re known (essentially from [36]) to b e equiv alent to v arie ties of Ω-gro ups. In this context there a re tw o clas sical a ppr oaches to commutator theor y . On the one hand, there is the Higgins commut ator of norma l s ubo b jects [26] which has a s particular cases the ordinar y commutators of gro ups, ring s, etc. It is t w o- dimensi onal in the sense that any tw o nor mal sub ob jects (i.e., idea ls or kernels) N and M of an ob ject A in a v ariety of Ω-gr oups A have a c o mmut ator [ N , M ] Ω , namely , the no rmal sub ob ject of the join M ∨ N = M · N of M a nd N gener ated by the set { w ( m n ) w ( n ) − 1 w ( m ) − 1 | w is a term, m ∈ M and n ∈ N } , where the nota tion “ m ∈ M ” mea ns that m is a finite sequence ( m 1 , . . . , m r ) of element s in M . C a ll an ob ject A o f A ab elian when it can b e endow ed with the structur e of an internal ab elian gr oup (necessarily in a unique wa y). The sub c ategory of A deter mined by the ab elian ob jects is denoted by Ab A . It is well known (and ea sily verified) that whe n A is a v ariety of Ω-g roups, an alg ebra A is in Ab A pr ecisely w hen the pro duct map A × A → A (sending a pair of elements ( a, a ′ ) to its pro duct aa ′ ) is a ho momorphism in the v a riety . F rom this it follows immediately that the Higgins commutator characterises the abe lia n ob jects: A is ab elian if and o nly if [ A, A ] Ω = 0 . On the other hand there is the relativ e notion of central extensio n due to F r¨ ohlich [23] (see a lso Lue [37] and F urta do-Co elho [24]). This notion o f central extension corres po nds to a one-di mensional commutator. Here o ne sta rts from a v ariety of Ω-groups A to g ether with a chosen subv ariety B o f A . The subv ariety B is c o mpletely determined by a set of iden tities of terms o f the form w ( x ) = 1; the set of a ll corres po nding ter ms w ( x ) is deno ted by W B = { w ( x ) | w ( b ) = 1 , ∀ B ∈ B , ∀ b ∈ B } , and an ob ject A o f A b elong s to B if and only if w ( a ) = 1 for all w ∈ W B and all a ∈ A . An extension f : A → B in A is a reg ular epimorphism, i.e., a surjective ho- momorphism. Let K denote the kernel of f . The nor mal sub ob ject [ K, A ] Ω B of A generated by the s e t { w ( k a ) w ( a ) − 1 | w ∈ W B , k ∈ K and a ∈ A } is called the rel ativ e commutator (with resp ect to B ) o f K and A . (Note that F r¨ ohlich us es the no tation V 1 for the relative commutator.) The extensio n f is cen tral (with resp ect to B ) when [ K, A ] Ω B is zero . It is easily seen that this relative commutator characterises ob jects of B as follows: A b elongs to B if and only if [ A, A ] Ω B is zer o. In the absolute case when the subv ariety B cons is ts of a ll ab elian ob jects in A , it was s hown in [24] that the tw o commutators coincide, [ K, A ] Ω Ab A = [ K , A ] Ω . (Note here that K ∨ A = A .) The main adv an tage o f the relative approa ch is that one may consider many situations whic h are not cov er ed by the Higgins comm utator. F or instance, the notio n of central extensio n of pr ecrosse d mo dules rela tive to the subv ar iet y of crossed mo dules is of this type. The main adv an tage of the Higgins commutator is tha t it is t w o-dimensional. So the Higg ins commutator is tw o- dimensional and absolute, the F r¨ ohlich commutator is one - dimensional and r elative, and in the one- dimensional abs olute ca se the tw o commutators coincide. What ab out the tw o-dimensional rela tive ca se? RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 3 In his article [15] the fir st author of the present a rticle aims at a nswering prec is ely this question. He intro duces a tw o -dimensional relative commutator for v arieties of Ω- groups which restricts to the Higgins commutator in the absolute case and which characterises F r ¨ ohlich’s rela tive cent ral extensions. Given any pair of nor mal sub o b jects M and N o f an ob ject A o f A , the commutator [ M , N ] B is the no rmal sub o b ject of M ∨ N gene r ated by the set { w ( m n ) w ( n ) − 1 w ( m ) − 1 w ( p ) | w ∈ W B , m ∈ M , n ∈ N , p ∈ M ∧ N } . The examples give an indication of how go od his definition is. F or instance, when considering the v a riety of pr ecross e d mo dules together w ith the s ubv arie t y of crossed mo dules, the r elative commutator o btained is the so -called Peiffer c ommutator , which is exa ctly wha t one would exp ect. 1.2. The left hand s ide column. Ba sing themselves on ideas from categ orical Galois theory [28, 4], in the ar ticle [32] Janelidze and K elly introduce a g eneral notion of central extensio n, rela tive with resp ect to a B ir khoff sub ca tegory B of a (Barr) exact category A . This notion of re la tive central extension is a g eneralisa tio n of F r ¨ ohlic h’s definition. In what follows, we shall restr ic t ourse lves to the case of semi -ab elian cat- egories [3 4]: p o inted, exa c t and pro tomo dular w ith binary sums. So let A be a semi-ab elian categ ory a nd B a Birkhoff sub category of A —full, reflective a nd closed under sub ob jects and regular quotients; a Birkhoff sub categor y of a v ariety is nothing but a subv ariety . Let I : A → B denote the reflector , and η : 1 A ⇒ I the unit o f the adjunction. Rec a ll from [32] that the clo sure of B under sub o b jects and regular quotients is equiv a lent to the co nditio n tha t the commutativ e squar e A f , 2 η A   B η B   I A I f , 2 I B ( A ) is a pusho ut of reg ular epimor phisms, for any reg ula r epi f : A → B . An extension in A is a regular epimorphis m. Such an extensio n f : A → B is called trivial (with resp ect to B ) when the induced commutativ e square ( A ) is a pullbac k. f is cen tral (with resp ect to B ) when it is lo c al ly trivial in the sense that there exists a reg ular epimorphism p : E → B such that the pullback p ∗ ( f ) : E × B A → E of f along p is a trivia l extension. Since, in the pr esent context, this implies that f ∗ ( f ) is trivial, w e hav e that f is central if and only if it is normal : either o ne of the pro jections in the kernel pair ( R [ f ] , f 0 , f 1 ) of f is a trivial extension. It is ex pla ined in the ar ticles [32, 11] why these c e nt ral extensio ns reduce to F r¨ ohlich’s when the catego ry A is a v ariety o f Ω- g roups. This notion of relative central extens io n induces a one-dim e nsional relative comm utator as follows [20, 19]. Let [ − ] B : A → A de no te the r adi c al induced by B : the functor which maps a n ob ject A of A to the o b ject [ A ] B defined throug h the s hort e x act se q uence 0 , 2 [ A ] B µ A , 2 A η A , 2 I A , 2 0 , and a morphism a : A ′ → A to its (co)restr iction [ a ] B : [ A ′ ] B → [ A ] B . Let aga in f : A → B be an extension and let K be its kernel. By pro tomo dularity , f is B - central if a nd only if for the kernel pair ( R [ f ] , f 0 , f 1 ) o f f , the (co)restr ictions [ f 0 ] B , [ f 1 ] B : [ R [ f ]] B → [ A ] B 4 TOMAS E VERAER T AND TIM V AN DER LINDEN of the t wo pro jections a r e isomorphisms (see [11]). Hence the kernel [ K, A ] B of [ f 0 ] B measures ho w far f is from being central: f is B -cen tral if and o nly if [ K, A ] B is zero. The ob ject [ K , A ] B may b e considere d as a normal sub ob ject o f A via the com- po site µ A ◦ [ f 1 ] B ◦ ker [ f 0 ] B : [ K, A ] B → A ; the induced extens io n A/ [ K, A ] B → B is the B -centralisation of f . W e int er- pret [ K , A ] B as a commutator of K with A , relative to the Bir k hoff sub categ ory B of A . When A is a v ariety of Ω-gr o ups, [ K, A ] B coincides with the relative com- m utator [ K , A ] Ω B , b ecause they induce the sa me cent ral e x tensions. And as in the v arietal c ase, an o b ject A of A belo ngs to B if a nd only if [ A, A ] B = 0 , b ecause the extension A → 0 is a split epimorphism, and therefore central if and only if it is trivial [32]. 1.3. The right hand s i de column. In his article [27], Huq introduces a catego- rical no tion of commutator of co terminal morphisms which makes se nse in quite diverse algebra ic settings. Using “o ld- style” a xioms, he formulates his results for those c a tegories we would now adays call semi-ab elia n [34]. Reca st in more mo dern terminology by Bo urn, his definition takes the following shap e [9]. In a s e mi- ab elian category , co nsider tw o cotermina l morphisms, m : M → A and n : N → A , and the resulting square o f s olid a r rows M h 1 M , 0 i z           m  $ ? ? ? ? ? ? ? M × N , 2 Q A. q l r N h 0 , 1 N i Z d ? ? ? ? ? ? ? n : D        L R The colimit o f this square co nsists of an o b ject Q together with four morphisms with co domain Q as indicated in the diagr am. The mo rphism q turns o ut to b e a normal epimorphism; its kernel is deno ted [ m, n ] H : [ M , N ] H → A and called the Huq commutator of m and n . It is conv enien t for us to r estrict its us e to the s ituation when M a nd N are norma l s ubo b jects of A , i.e., m a nd n are kernels. The commutator [ M , N ] H bec omes the o r dinary commutator of nor mal subgroups M and N in the ca se of groups, the ideal generated b y M N + N M in the case of rings, the Lie bra ck et in the case of Lie a lgebras, and so on. More generally , when c omputed in the join M ∨ N , we know fro m [27] that in a ny v ariety of Ω-groups the Huq commutator [ M , N ] H coincides with the Higg ins co mm utator [ M , N ] Ω . Just as the Higg ins commutator, the Huq c o mmut ator characterises the Birkhoff sub catego ry Ab A o f A of abelia n ob jects in A . This is a consequence o f the fact that, in a semi-ab elian categ ory A , an ob ject A admits at mo st one internal ab elian group s tructure, and such a structure is entirely determined by a morphism m : A × A → A which satisfies m ◦ h 1 A , 0 i = 1 A = m ◦ h 0 , 1 A i [2 7, 8]. 1.4. The questio n mark. By now it is clear, we hop e, tha t the purp ose of the present a rticle is to introduce a ca tegorica l version o f the relative commutator for v arieties of Ω-groups, in s uch a way that (1) it characterises the B -cen tral ex tensions o f A , (2) it coincides with the Huq commutator when B is Ab A . RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 5 In [22] the pr e sent authors a lready introduced a r elative concept o f commu ting normal sub ob jects, based on categor ical Galois theory and v a lid in the context of semi-ab elian ca tegories. This notio n was shown to b e compatible with the re la tive commutator for v a rieties of Ω-groups . What we still hav e to do now is · expla in how this induces a tw o- dimensional commutator; · prove that this commutator satisfies (1) a nd (2) ab ove; · explo re the commutator’s basic prop erties. One may ask whether it is worth the effort at all to le ave the context of Ω-g roups and study a relative commutator from a categor ical p ersp ective. W e claim tha t the categoric al a pproach not o nly provides us with a conceptua l explana tion o f the definitions (in terms of Galois theor y) but also with interesting new examples. F or instance, in the case of lo ops vs. gr oups considered in [22], the commutator b eco mes an asso ciator , and it effectiv ely mea sures how well tw o no rmal s ublo ops of a lo op asso ciate with e ach other . 1.5. Defini ti on of the comm u tator. Let us now briefly sketch how the relative commutator [ − , − ] B is defined. Let A again be a semi-ab elian catego ry a nd B a Birkhoff sub ca tegory o f A . M and N will b e norma l sub ob jects of an ob ject A of A . R M and R N are the equiv alence r elations on the join M ∨ N (taken in the lattice of nor mal s ub o b jects of A ) corres po nding to M and N , a nd R M  R N r 1 , 2 r 0 , 2 p 1   p 0   R N     R M , 2 , 2 M ∨ N is the largest double equiv a lence r elation on R M and R N : the ob ject R M  R N “consists of ” all quadruples ( x, y , z , t ) ∈ M ∨ N where ( x, z ), ( y , t ) ∈ R M and ( x, y ), ( z , t ) ∈ R N . The commutator o f M a nd N is the meet [ M , N ] B = K [[ p 0 ] B ] ∧ K [[ r 0 ] B ] of the kernels of the morphisms [ p 0 ] B and [ r 0 ] B in the following diagr am, obtained by apply ing the functor [ − ] B to the diagram a b ove. [ R M  R N ] B [ r 1 ] B , 2 [ r 0 ] B , 2 [ p 1 ] B   [ p 0 ] B   [ R N ] B     [ R M ] B , 2 , 2 [ M ∨ N ] B ( B ) It may b e co ns idered as a normal sub ob ject o f M ∨ N . 1.6. Interpr etation i n term s of doubl e central extensions. W e hav e to ex- plain why [ M , N ] B is defined the way it is. The rea son co mes from catego rical Galois theory , in particular the theory of higher c entr al extensions . Just like the concept o f ce ntral ex tension which is defined with resp ect to the adjunction A I , 2 ⊥ B , ⊃ l r ( C ) one may consider do uble central ex tensions which a re defined with r esp ect to the reflection of extensio ns to central extensions —the a djunction Ext A I 1 , 2 ⊥ CExt B A ⊃ l r ( D ) 6 TOMAS E VERAER T AND TIM V AN DER LINDEN where E xt A is the ca tegory of extensions and co mm utative sq uares b etw een them, and CExt B A its full subc a tegory determined by those extensions which a re central. The reflector I 1 takes a n extension f : A → B with kernel K a nd maps it to the central extension I 1 f : A/ [ K , A ] B → B . This may b e rep eated ad infinitum, so that notions of n - fold c entr al extension are obtained, but for the pres ent purp os es the second step is sufficient. Do uble central extensions, fir st introduced b y Janelidze for groups [29], are an impor ta nt to ol in semi-ab elian (co)homolo gy [19, 30, 4 1], and turn o ut to b e precisely what is needed to understa nd how the re lative commutator works. W e refer the r eader to the articles [19, 16] for more details on higher central extensions. As we explain b e low, the commutator [ M , N ] B is zer o if and o nly if any (hence, all) of the four comm utativ e sq uares in the diagram ( B ) is a pullback. Galois theor y shows that this condition is equiv a lent to the squa re M ∨ N q M , 2 q N   M ∨ N M   M ∨ N N , 2 0 ( E ) being a double central extensio n. (Here q M denotes the cokernel of the normal monomorphism M → M ∨ N .) When this happ ens, we say that M and N commute (with resp ect to B ) . Accordingly , given a ny tw o no rmal sub ob jects M a nd N of an o b ject A , the commutator [ M , N ] B is the smallest nor mal sub ob ject J o f M ∨ N such that M /J and N /J c ommut e; it is the nor mal s ub o b ject which must b e divided o ut of M ∨ N to turn the double ex tens io n ( E ) into a double central ex tension. 1.7. Structure of the text. In the following sections we shall expla in why the commutator has the prop erties (1) and (2 ) men tioned in 1.4. With this purp ose in mind, the text is structured as follows. In Sectio n 2 we provide the necessa r y background for understanding the definition of the commutator: semi-ab elia n cat- egories, normal sub ob jects, double extensio ns and double cent ral extensions. Its basic technical pro p e rties a nd the pro of of (1) a re given in Section 3. In Section 4 we prov e (2): the commutator [ − , − ] B coincides with the Huq co mmut ator in case B is Ab A . F inally , Section 5 bring s up s o me further rema rks and unansw ered ques - tions. Contents 1. Int ro duction 1 2. Preliminarie s 7 3. Definition a nd basic pr op erties 11 4. The absolute case: ab elia nisation 15 5. F urther re ma rks 18 References 21 RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 7 2. Preliminaries W e r ecall some basic definitions a nd res ults which we shall need in the following sections. 2.1. Sem i-ab eli an categories. A c ategory is regul ar when it is finitely complete with co equaliser s of kernel pa irs and with pullback-stable re gular epimorphisms [1]. In a reg ular catego ry , a ny mor phism f may b e factor ed as a regular e pimorphism follow ed b y a monomorphism (called the i mage of f ), and this im age factor- isation is unique up to iso morphism. Giv en a monomor phism m : M → A and a regular e pimorphism f : A → B , the di rect image f ( m ) : f M → B of m a long f is the image of the comp osite f ◦ m . When a catego ry is p ointed and regular , protomo dul arit y ca n b e defined via the following pr o p erty , which is equiv alen t to the Short Five Lemma [5 , 7]: given any c o mmut ative dia gram K [ f ′ ] ker f ′ , 2 k   A ′ f ′ , 2 a   B ′ b   K [ f ] ker f , 2 A f , 2 B ( F ) such that f and f ′ are regular epimorphisms, k is an isomor phism if and o nly if the right hand squa re b ◦ f ′ = f ◦ a is a pullback. (Here, we use the nota tion ker f : K [ f ] → A for the kernel of f .) A hom ologi cal ca tegory is p o inted, reg ular and pr otomo dular [3]. In s uch a category , a reg ular epimor phism is alwa ys the cokernel of its kernel, and there is the following no tio n of short exac t s e q uence. A short e xact se quence is any sequence K k , 2 A f , 2 B with k = ker f a nd f a reg ular epimor phism. W e deno te this situation by 0 , 2 K k , 2 A f , 2 B , 2 0 . The following pro p erty ho lds. Lemma 2.2. [7] Consider a morphism of short ex act se quenc es su ch as ( F ) ab ove. The left hand side squar e ker f ◦ k = a ◦ ker f ′ is a pul lb ack if and only if b is a mono.  A (Barr) e xact category is regular a nd such that every in ternal equiv alence relation is a kernel pair [1]. A homolo gical catego r y is exact if and o nly if the dir ect image o f a normal monomorphism a lo ng a regular e pimo rphism is aga in a normal monomorphism. A sem i-ab el ian catego ry is homolog ical a nd exact with binary copro ducts [34]. A regular pushout square is a commutativ e squar e X c , 2 d   C g   D f , 2 Z ( G ) such that all its maps and the compar ison map h d, c i : X → D × Z C to the pullback of f with g are regular epimor phisms. In a s e mi-ab elian categor y , every pusho ut of a regular epimor phism along a reg ula r epimorphism is a reg ular pushout [14], and the fo llowing dual to Lemma 2.2 holds: 8 TOMAS E VERAER T AND TIM V AN DER LINDEN 0   0   0   0 , 2 M ∧ N (i) , 2   N n   , 2 N M ∧ N   , 2 0 0 , 2 M m , 2   A , 2   (ii) A M   , 2 0 0 , 2 M M ∧ N , 2   A N , 2   A M ∨ N   , 2 0 0 0 0 Figure 1. The 3 × 3 diag ram induced by M , N norma l in A Lemma 2. 3. [11] Given a morphism of short ex act se quenc es such as ( F ) ab ove with a and b r e gular epi, the right hand side s quar e f ◦ a = b ◦ f ′ is a (r e gular) pushout if and only if k is a r e gular epimorphism.  2.4. Norm al sub ob jects. A normal sub ob ject N of an ob ject A of a semi- ab elian categor y is a sub ob ject represented b y a normal monomorphism n : N → A . Let M and N b e tw o no r mal sub ob jects of A with representing no rmal mo no morph- isms m and n . T a king into acco unt Lemma 2.2 and the stability of no rmal mono- morphisms under regular images , we may alwa ys form the 3 × 3 diagram in Figure 1 (in which all rows and columns a re shor t exact sequences). The meet M ∧ N a nd the join M ∨ N of the sub ob jects M and N a r e taken in the la ttice of no r mal sub o b jects of A . W e see that M ∧ N is co mputed as the pullback (i) and M ∨ N is obtained throug h the pushout (ii ) , as the kernel of the comp osite morphism A → A/ ( M ∨ N ). O f course , M ∧ N coincides with the meet M ∩ N in the lattice of (all) sub ob jects of A . O ne could also compute the join of M and N as (or dinary) sub o b jects o f A by ta king the image M ∪ N of the morphis m h m n i : M + N → A . It is known [2, 27] that b oth c o nstructions yield the same res ult. W e s hall give an alternative pr o of o f this fac t b elow, but first we prove a weaker prop er ty . Let us fix some notatio n: w e write j for the normal mono morphism repres ent- ing M ∨ N , and m ′ : M → M ∨ N a nd n ′ : N → M ∨ N for the induced factoris a - tions. Since m ′ and n ′ are norma l mono mo rphisms, we may a lso consider the join of M and N as no r mal sub ob jects of M ∨ N . W e denote it by M g N and wr ite j ′ : M g N → M ∨ N for the representing normal monomor phism. Lemma 2.5 . The t wo joins M ∨ N and M g N c oincide : j ′ is an isomorphism. Pr o of. First o f all note that the commutativ e square M ∨ N , 2 j   M ∨ N M   A , 2 A M is a pullback by pr otomo dularity , so that the right hand vertical morphism is a monomorphism becaus e, in a protomo dular category , pullbac ks r eflect monos [5 ]. (One could, alternatively , use Le mma 2.2 to prove that this morphism is a mono- morphism.) N ow, the normal monomorphisms m ′ and n ′ induce a 3 × 3 diagram similar to Figur e 1, a nd j induces a morphism b etw een the tw o 3 × 3 diagr ams, of RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 9 which we co nsider only the last row: 0 , 2 N M ∧ N , 2 M ∨ N M , 2   M ∨ N M g N   , 2 0 0 , 2 N M ∧ N , 2 A M , 2 A M ∨ N , 2 0 W e hav e just explained why the middle vertical morphism is a monomorphism. Hence, using the same a rguments as ab ov e, we find that also the right hand vertical morphisms is a mono. Since the comp osite M ∨ N → ( M ∨ N ) / ( M g N ) → A/ ( M ∨ N ) is zer o, we find that ( M ∨ N ) / ( M g N ) = 0, i.e ., the factorisa tion j ′ is a n iso morph- ism.  Now, taking this lemma into ac c ount, w he n A = M ∨ N in the 3 × 3 diagram ab ov e, the o b ject A/ ( M ∨ N ) is zero , and we rega in the No ether isomorphism s [3] N M ∧ N ∼ = M ∨ N M and M M ∧ N ∼ = M ∨ N N . ( H ) W e are ready to prov e the identit y M ∨ N = M ∪ N . Notation 2.6. Giv en a normal s ubo b ject J of an ob ject A , the induced q uotient of A is denoted q A J : A → A/J ; we wr ite R A J for the kernel pa ir A × A/J A o f q A J . Most of the time A will b e a join M ∨ N , in which ca se we dro p the A from the notation a nd simply wr ite q J : M ∨ N → ( M ∨ N ) /J for the quo tient and R J for the kernel pa ir o f q J . Prop ositi o n 2.7. [2, 27] If M and N ar e normal in A , t hen their join as normal sub obje cts M ∨ N c oincide s with their join as sub obje ct s M ∪ N . H enc e the morphism h coker n, coker m i : A → ( A/ N ) × ( A/ M ) is a r e gular epimorphism if and only if s u ch is the morphism h m n i : M + N → A. Pr o of. If J is a sub ob ject of M ∨ N containing M and N , then by Lemma 2.2 it induces a factoris ation of the first o f the isomorphisms ( H ) as a mor phism N / ( M ∧ N ) → J / M follow ed by a monomo rphism j : J / M → ( M ∨ N ) / M . This j is also a s plit epimorphism; hence it is an iso morphism, and J is equa l to M ∨ N by the Sho rt Five Lemma. Now M ∪ N is a subo b ject of M ∨ N co ntaining M and N , and the tw o joins coincide. As to the latter statement, the first condition holds if and o nly if the squar e A , 2   A M   A N , 2 0 is a reg ular pushout. Since, in a semi-ab elian category , a pushout o f regular epi- morphisms is necessa rily regular , this happ ens when A = M ∨ N . But then A 10 TOMAS E VERAER T AND TIM V AN DER LINDEN is M ∪ N by the fo r mer par t of the pro of, and the seco nd co ndition holds only when this is the cas e .  Given a monomorphism m : M → A , the normal closure M A of M in A always exists, and is computed a s the kernel of the cokernel of m . It is the smallest norma l sub o b ject of A that contains M . 2.8. Doubl e (cen tral) extensions . A double extens ion is a regula r pushout square ( G ). F or instance, given a ny tw o norma l sub o b jects M and N of an ob ject A of A , the induced pusho ut square ( E ) is a double extensio n. Recall from [19] that pullbacks of double ex tensions exist in Ext A and are degree-wis e pullba cks in A . Moreov er, double extensions are pullback-stable. The categor y of double e x tensions in A a nd commutativ e cub es b etw een them is denoted Ext 2 A . Double central extensio ns are defined with r e sp ect to the adjunction ( D ) in the same w ay as cent ral extensions are defined with resp ect to the adjunction ( C ) . More pr ecisely , a double extension ( G ), considered as a map ( c, f ) : d → g in the category Ext A , is trivial when the left ha nd comm utative squar e below, induced by the unit of the a djunction ( D ), is a pullbac k in Ext A ; this means that the right hand commutativ e sq uare, in which the vertical morphisms a re the canonical quotient maps, is a pullback in A . d ( c,f ) , 2   g   I 1 d , 2 I 1 g X c , 2   C   X [ K [ d ] ,X ] B , 2 C [ K [ g ] ,C ] B The s q uare ( G ) is a double cen tral extensi on (wi th resp e ct to B ) when its pullback a long some double extensio n is a triv ial do uble extension. It is a doubl e normal e xtension (with res p ect to B ) when the fir st pro jection of its kernel pair R [ c ] c 0 , 2 r   X d   R [ f ] f 0 , 2 D is a trivial double extension. (Alternatively , one could use the square o f seco nd pro jections.) By proto mo dularit y , this amounts to the (one-dimensiona l, r elative) commutators [ K [ r ] , R [ c ]] B and [ K [ d ] , X ] B being iso morphic. Similar to the one- dimensional case, do uble central extensions and double no rmal extensions coincide. 2.9. Hi gher extensi ons. In what follows we s ha ll also need three-fold extensions, so let us r ecall the definition of n -fold extens ion for arbitra ry n . Given n ≥ 0 , denote by Arr n A the c a tegory of n -dimensional a rrows in A . (Zero-dimensio nal arrows—as well as zer o-dimensional extens io ns—are just ob jects of A .) A (one- fold) extension is a regular epimorphism in A . F or n ≥ 1, an ( n + 1 ) -fold extension is a co mm utative square ( G ) in Arr n − 1 A (an ar row in Arr n A ) such that all its maps and the compar ison map h d, c i : X → D × Z C to the pullback of f with g are n -fold extensio ns. Thus for n = 2 we regain the notion of double extension. RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 11 A three-fold ex tension is a commutativ e cub e X , 2   C   X ′ , 2   : D        C ′   : D        D , 2 Z D ′ , 2 : D Z ′ : D        X ′ , 2   D ′ × Z ′ C ′   X , 2 D × Z C of which a ll faces as well as the induced right-hand square are double extensions. Since, in a semi-ab elian ca teg ory , reg ula r epimo rphisms are norma l, the three-fold extension a bove is co mpletely determined by the ob ject X ′ and the three norma l sub o b jects giv en by the kernels of its “initial r ibs ” X ′ → X , X ′ → C ′ and X ′ → D ′ . Conv er sely , given an ob ject X ′ and three normal subob jects J , M and N of X ′ , the fo llowing lemma determines when the induced cub e is a three-fold extension. Lemma 2.10 . Given normal su b obje cts J , M and N of an obje ct X ′ in a semi- ab elian c ate gory, the cu b e obtaine d by pushing out the induc e d quotients is a t hr e e- fold ext ension if and only if q X ′ J ( M ∧ N ) = q X ′ J M ∧ q X ′ J N . Pr o of. Since, in a se mi-ab elian categor y , pushouts of r egular epimorphisms a re regular , the induced cub e is a three-fold extensio n as so on as the sq uare X ′ , 2   X ′ M × X ′ M ∨ N X ′ N   X ′ J , 2 X ′ J ∨ M × X ′ J ∨ M ∨ N X ′ J ∨ N is a do uble extension. W e a lready know that all mor phisms in this square ar e regular e pimorphisms, so by Lemma 2.3 it is a double extension if a nd only if q X ′ J ( M ∧ N ) = q X ′ J M ∧ q X ′ J N .  F urther results on higher- dimensional extensio ns and ce nt ral extensions may be found in [1 9] and [16]. Let us just reca ll here that, for any n ≥ 0, a split epimorphism of n -fold extensions is always a n ( n + 1)-fold extension, and it is an ( n + 1)-fold central extension if a nd only if it is a tr ivial ( n + 1 )-fold ex tens ion. Higher-dimensiona l central ex tensions are impo rtant in homolog y whe r e they app ear in the higher Hopf formulae, a nd in cohomology where (in the absolute case, and in low dimensio ns ) they ar e cla s sified by the cohomolog y gro ups [25, 41]. 3. Definition and basic pr oper ties In this section we reca ll the categorica l definition of the relative commutator fro m the introduction and we explo re its basic prop er ties: co mpatibility with the cent- ral e x tensions introduced by Janelidze and K elly (Pr op osition 3.2), ba s ic stability prop erties (Theorem 3.9) and the cas e of Ω-gro ups (Prop o s ition 3.10). In what follows, A will b e a semi-a b e lia n catego r y and B a B irkhoff sub categ ory of A . 12 TOMAS E VERAER T AND TIM V AN DER LINDEN Definition 3.1 . Let M and N b e normal s ub o b jects of an ob ject A o f A . W e say that M and N com mute (with resp ect to B ) when the double ex tens ion M ∨ N q M , 2 q N   M ∨ N M   M ∨ N N , 2 0 ( I ) is central (with res p ect to B ). Is is immediately clear that this notion of comm uting sub ob jects characterises the B -central extensions of A and the o b jects of B : Prop ositi o n 3. 2. An extension f : A → B in A is B -c entr al if and only if the obje ct A and t he kernel K of f c ommute. An obje ct A of A lies in B if and only if A c ommutes with itself. Pr o of. The fir s t result holds b ecause the double extension A q A , 2 f = q K   0 B , 2 0 , being a s plit epimo rphism o f extensions, is central if and only if it is trivial, which happ ens precis ely when f is a central e x tension. The s econd result fo llows from the fir st, since A is in B if and only if the split epimorphism A → 0 is a B -cen tral extension.  Lemma 3.3. [22, Prop os ition 2.9] L et M and N b e normal sub obje cts of an ob- je ct A . M and N c ommute if and only if any of the four c ommut ative squar es in the diagr am [ R M  R N ] B [ r 1 ] B , 2 [ r 0 ] B , 2 [ p 1 ] B   [ p 0 ] B   [ R N ] B [ π 1 ] B   [ π 0 ] B   [ R M ] B [ ρ 1 ] B , 2 [ ρ 0 ] B , 2 [ M ∨ N ] B ( J ) is a pul lb ack.  Definition 3.4. Let M and N b e no rmal sub ob jects of an ob ject A . L et [ R M ] B × [ M ∨ N ] B [ R N ] B denote the pullback o f the morphisms [ π 0 ] B and [ ρ 0 ] B from Diag r am ( J ). The comm utator [ M , N ] B is the kernel of the mor phism h [ p 0 ] B , [ r 0 ] B i : [ R M  R N ] B → [ R M ] B × [ M ∨ N ] B [ R N ] B , considered a s a no r mal sub ob ject of M ∨ N . Remark 3.5. Two normal sub ob jects M and N o f an ob ject A commute if and only if [ M , N ] B is zero. Indeed, the morphism h [ p 0 ] B , [ r 0 ] B i is a reg ular (hence, normal) epimorphism b ecaus e the square [ π 0 ] B ◦ [ r 0 ] B = [ ρ 0 ] B ◦ [ p 0 ] B is a double extension a s a split epimorphism of split epimorphisms. Hence its kernel is zero if and only if it is an isomorphism—which, by Lemma 3.3, means tha t M and N commute. RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 13 Remark 3.6. The kernel of h [ p 0 ] B , [ r 0 ] B i ma y indeed be considered as a normal sub o b ject of M ∨ N , na mely , through the comp osition of ker h [ p 0 ] B , [ r 0 ] B i : K [ h [ p 0 ] B , [ r 0 ] B i ] → [ R M  R N ] B with ρ 1 ◦ p 1 ◦ µ R M  R N : [ R M  R N ] B → M ∨ N . First o f a ll, this comp osite is a monomor phis m, b ecause µ R M  R N ◦ ker h [ p 0 ] B , [ r 0 ] B i = ker h p 0 , r 0 i ◦ µ K [ h [ p 0 ] B , [ r 0 ] B i ] and b oth µ K [ h [ p 0 ] B , [ r 0 ] B i ] and ρ 1 ◦ p 1 ◦ ker h p 0 , r 0 i a re mo nomorphisms. Now µ R M  R N ◦ ker h [ p 0 ] B , [ r 0 ] B i is a nor ma l monomor phism as a meet o f tw o normal monomorphisms. This follows fro m Lemma 2.2, since the induced mor phism [ R M ] B × [ M ∨ N ] B [ R N ] B → R M × M ∨ N R N is a mo nomorphism. Hence ρ 1 ◦ p 1 ◦ µ R M  R N ◦ ker h [ p 0 ] B , [ r 0 ] B i is normal, being the direct imag e o f this latter no rmal monomorphism along the regular epimorphism ρ 1 ◦ p 1 . Remark 3. 7. O n the o ther hand, there is no r eason why [ M , N ] B should b e a normal sub ob ject of A . A co un terexample is given in [38]. Remark 3.8. T he co mmutator [ M , N ] B is no thing but L 2 of the double exten- sion ( I ) as consider ed in the article [19]. Theorem 3.9. L et M , N , L (r esp. M ′ , N ′ ) b e normal sub obje cts of an obje ct A (r esp. A ′ ). L et J b e a normal sub obje ct of M ∨ N . The fol lowing hold: (1) [0 , N ] B = 0 ; (2) [ M , N ] B = [ N , M ] B ; (3) [ M , N ] B ≤ M ∧ N ; (4) if N ≤ L then [ M , N ] B ≤ [ M , L ] B as su b obje cts of A ; (5) q J [ M , N ] B ≤ [ q J M , q J N ] B ; (6) [ M × M ′ , N × N ′ ] B = [ M , N ] B × [ M ′ , N ′ ] B ; (7) q J [ M , N ] B = [ q J M , q J N ] B as so on as q J ( M ∧ N ) = q J M ∧ q J N , which happ en s, for instanc e, when either M ≤ N or J ≤ M ∧ N ; (8) [ M , N ] B is the smal lest normal sub obje ct J of M ∨ N such t hat q J M and q J N c ommute. Pr o of. The fir s t prop er ty holds b ecause, for any ob ject N , the squar e N q N   q 0 N   0 0 is a do uble central extension with resp ect to B . Pr op erty (2) follows from the symmetry of Diagram ( J ); se e [16] for a detailed explanatio n. (3) follows fro m the 14 TOMAS E VERAER T AND TIM V AN DER LINDEN definition of [ M , N ] B . T o see this, consider the diagram K [[ r 0 ] B ] k 1   k 0   ker [ r 0 ] B , 2 [ R M  R N ] B [ r 1 ] B , 2 [ r 0 ] B , 2 [ p 1 ] B   [ p 0 ] B   [ R N ] B [ π 1 ] B   [ π 0 ] B   K [[ ρ 0 ] B ] ker [ ρ 0 ] B , 2 l   [ R M ] B [ ρ 1 ] B , 2 [ ρ 0 ] B , 2 µ R M   [ M ∨ N ] B µ M ∨ N   M ker ρ 0 , 2 R M ρ 1 , 2 ρ 0 , 2 M ∨ N . Since [ M , N ] B , b eing the kernel of h [ p 0 ] B , [ r 0 ] B i , may be computed a s the meet of the kernels of [ p 0 ] B and [ r 0 ] B , it is a lso the kernel of k 0 . Hence, cons idered as a sub o b ject of M ∨ N via Rema rk 3.6, it is a s ubo b ject of M thro ugh the morphism l ◦ k 1 . Likewise, [ M , N ] B is contained in N . The four th pro per ty follo ws from the functor iality of the constr uc tio n of [ − , − ] B . So do es the fifth. T o s e e tha t the re la tive co mm utator pr eserves bina ry pr o ducts, it s uffices to no te that the zero- dimensional commutator [ − ] B preserves them, and that joins commut e with pro ducts. The fo r mer pro pe r ty is well known. It is a consequence of the fact that the reflector I : A → B preser ves pullbacks of s plit epimorphisms along split epimor phisms (be c ause the comp onents of the unit are extensions) to gether with the fact tha t a s plit epimorphism of split e pimorphisms in Ext A is alwa ys a thr ee-fold extension. The latter prop erty holds be c ause the pro duct of tw o regula r pushouts is a reg ular pushout: pr o ducts of pullbacks ar e pullbacks, pro ducts o f regular epis ar e r egular epis. T o prov e (7), first of all recall that the squar e ( A ) induced by the unit η is a pushout o f regular epimor phisms for a ny regular epimo rphism f , by the Bir khoff condition. Hence, by Lemma 2.3, the zero-dimensio nal co mmu tator [ − ] B : A → A preserves extens io ns. Now assume that q J ( M ∧ N ) = q J M ∧ q J N . Then by Lemma 2 .10 the left hand side co mmutative cub e M ∨ N J , 2   M ∨ N M ∨ J   M ∨ N , 2   : D       M ∨ N M   β : D      M ∨ N N ∨ J , 2 0 M ∨ N N , 2 α : D 0               R M ∨ J J  R N ∨ J J , 2 , 2     R N ∨ J J     R M  R N , 2 , 2     : D      R N     : D      R M ∨ J J , 2 , 2 M ∨ N J R M , 2 , 2 : D M ∨ N : D       is a three - fold extension. As a consequence, so ar e all the c o mmut ative cube s in the right hand side dia gram, b eing pullbacks o f three-fo ld extensio ns . This is still true if we apply the functor [ − ] B to the right hand side diag ram, since [ − ] B preserves extensions and b ecause a split epimorphism of ex tensions is a do uble extension, and a split epimorphism of double extensions a three - fold extensio n. The identit y in (7 ) now follows. If M ≤ N then q J ( M ∧ N ) = q J M = q J M ∧ q J N . If, on the other hand, we assume that J ≤ M ∧ N , then the morphism α a nd, b y symmetry , also β , are isomor phisms. This implies that the left hand s ide cub e above is a three- fold extension, so that q J ( M ∧ N ) = q J M ∧ q J N by Lemma 2.10. RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 15 Prop erties (3) and (7 ) tog ether imply that q [ M ,N ] B M and q [ M ,N ] B N co mm ute. Using (5) it is now easily seen that [ M , N ] B is minimal a mongst all J such tha t [ q J M , q J N ] B = 0.  It was shown in [22] that t w o normal sub ob jects of an Ω-g roup commut e in the sense of [15] if and only if they commute in the s ense of our Definition 3 .1. Since b oth notions of re lative commutator satisfy the same universal prop erty (see Theorem 3.9 (8)), we find: Prop ositi o n 3. 10. L et A b e a variety of Ω -gr oups and B a subvariety of A . Given any two normal sub obje cts M and N of an obje ct A of A , we have [ M , N ] Ω B ∼ = [ M , N ] B . In p articular, the c ommutator [ M , N ] Ω B is zer o if and only if the double extension ( I ) is c entr al.  Remark 3 .11. This alr eady gives us the examples work ed out in [15]: precross ed mo dules vs. crosse d mo dules, where the relative commutator is the Peiffer commu- tator, for instance. An example which is not a consequence of this theorem—lo ops vs. groups , where the relative commutator is a n asso ciato r—was co nsidered in the article [2 2]. Another example which falls outside the scop e of [15] is the case of compact Hausdor ff top olog ical groups vs . profinite groups. Here, the relative com- m utator [ M , N ] B is the co nnected comp one nt of the intersection M ∩ N , as follows from r esults in [17]. Mo re g enerally , in any situation where the refle c tor I : A → B is pr ot o additiv e [18, 17] (for instance, when A is a be lia n), one ha s the iden tit y [ M , N ] B = [ M ∩ N ] B for a ny ob ject A of A and any pair o f normal s ubo b jects M and N of A . The “a bsolute” cas e of ab elianisatio n is trea ted in the following section. Remark 3.1 2 . It suffices to consider the case B = 0 (wher e 0 is the ca tegory with one ob ject and one arrow) to see that the equality in Statement (5) of The o rem 3.9 do es not hold in gener a l. The case B = 0 shows, furthermore, that unlike the Smith/Pedicc hio comm utator—cf. Lemma 4.2—the c o mmu tator [ − , − ] B need not preserve binary joins. 4. The absolute case: abelianisa tion In the case o f Ω-gro ups, the re la tive commutator [ − , − ] Ω B in A reduces to the Higgins commutator when B is the Birkhoff s ub ca tegory Ab A of all ab elian ob jects of A . L ikewise, when A is an arbitrary semi- a b elian category and B is Ab A , the relative [ − , − ] B is no thing but the Huq c ommut ator. T o show this we take a detour via the Smith/Pedicc hio commutator of equiv alence re la tions. Firs t, in Lemma 4.4, we pr ov e that the equiv ale nce rela tion corr esp onding to the c ommut ator o f tw o normal sub o b jects is e x actly the commutator o f the equiv alence relatio ns corr e- sp onding to those no r mal s ubo b jects. Then we prove Pro p osition 4.6 which states that the Huq commutator of a pair of no r mal sub ob jects M a nd N of a n ob ject A is the nor malisation o f the Smith/Pedicchio comm utator of the corres p o nding equiv a- lence r elations, when M ∨ N = A . Combining b oth results, we obtain Theorem 4.7: given any tw o nor mal sub ob jects M a nd N o f A , their Huq co mmutator [ M , N ] H , computed in M ∨ N , coincides with [ M , N ] Ab A . 4.1. The comm utator of equiv alence rel atio ns. In his bo ok [43], Smith intro- duced a commutator of equiv alence re lations in the co ntext of Mal’tsev v arie ties. It was extended to a purely categor ical se tting by Pedicc hio [39] and may be pres ented in a manner which is similar to the definition of the Huq commut ator of norma l sub o b jects [3, 13]. 16 TOMAS E VERAER T AND TIM V AN DER LINDEN Let A b e an ob ject o f a se mi- ab elian category A . The largest equiv alence relation on A is deno ted by ∇ A = ( A × A, π 0 , π 1 ) and the smallest one by ∆ A = ( A, 1 A , 1 A ). Two equiv alence relatio ns R = ( R, r 0 , r 1 ) and S = ( S, s 0 , s 1 ) on A are said to cen trali se eac h other when they admit a centra lising double rel ation C , 2 , 2     S     R , 2 , 2 A, ( K ) i.e., a (unique) do uble equiv alence rela tion C on R a nd S such that any of the four commutativ e squares in ( K ) is a pullback. (Then a ll of the commutativ e s quares in ( K ) are pullbacks.) R and S cent ralise each o ther if and o nly if there exists a partial Mal’ts e v op eratio n on R and S , a morphism p : R × A S → A which s atisfies p ( α, α, γ ) = γ and p ( α, γ , γ ) = α . The commuta tor [ R, S ] S of R and S is the universal eq uiv alence relation on A which, when divided out, ma kes them centralise ea ch other. Co nsider the pullback R × A S p R   p S , 2 S s 0   i S l r R r 1 , 2 i R L R A l r L R of r 1 and s 0 ; then [ R, S ] S is the kernel pair R [ q ] of the morphism q in the diagra m R i R z           r 0  $ ? ? ? ? ? ? ? R × A S , 2 Q A q l r S i S Z d ? ? ? ? ? ? ? s 0 : D        L R where the do tted arrows denote the colimit of the outer square. The direct im- ages q R and q S of R and S alo ng the r egular epimorphism q centralise each other ; hence R and S do so if and o nly if [ R , S ] S = ∆ A . The following pro p erties of this commutator will b e useful for us. Lemma 4.2. [3, 12, 3 9] L et R , S , S ′ b e e quivalenc e r elations on an obje ct A and f : A → B a r e gular epimorphism. The fol lowing hold: (1) [∆ A , S ] S = ∆ A ; (2) [ R , S ] S = [ S , R ] S ; (3) [ R , S ] S ≤ R ∧ S ; (4) if S ≤ S ′ then [ R, S ] S ≤ [ R , S ′ ] S ; (5) [ R , S ∨ S ′ ] S = [ R , S ] S ∨ [ R, S ′ ] S ; (6) if [ R, S ] S = ∆ A then [ f R, f S ] S = ∆ B .  The double central extensio ns with resp ect to the Bir khoff sub categor y Ab A of ab elian ob jects in a semi-ab elian categor y A hav e b een characterised in terms of this commutator o f equiv alence rela tions as follows. Lemma 4.3. [4 1, 21] A double ext ension ( G ) in a semi-ab elian c ate gory A satisfies [ R [ d ] , R [ c ]] S = ∆ A = [ R [ d ] ∧ R [ c ] , ∇ A ] S if and only if it is c entr al with r esp e ct to Ab A .  This immediately implies that [ − , − ] Ab A corres p o nds to [ − , − ] S in the following sense: RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 17 Lemma 4.4 . Given any t wo normal sub obje cts M and N of A , [ R M , R N ] S = R [ M ,N ] Ab A . Pr o of. By definition, M and N commute when the squar e ( I ) is a double central extension with resp ect to Ab A . According to Lemma 4.3, this ha pp ens if a nd only if [ R M , R N ] S = ∆ M ∨ N = [ R M ∧ R N , ∇ M ∨ N ] S . ( L ) Using ∇ M ∨ N = R M ∨ R N we see tha t [ R M ∧ R N , ∇ M ∨ N ] S = [ R M ∧ R N , R M ] S ∨ [ R M ∧ R N , R N ] S ≤ [ R M , R N ] S and the second equality in ( L ) follows from the fir st. Hence [ M , N ] Ab A is zero if a nd o nly if [ R M , R N ] S = ∆ M ∨ N . The co mm utator [ R M , R N ] S now coincides with R [ M ,N ] Ab A bec ause these t w o equiv alence relations satisfy the same universal prop erty .  4.5. The Huq commutator. It is well known that in general, the Huq comm u- tator do es not corr esp ond to the commutator of equiv ale nce relations: the rela- tion R A [ M ,N ] H need not b e isomo rphic to [ R A M , R A N ] S for arbitrar y norma l sub ob jects M and N of an ob ject A —a counterexample is given in [10] for digr oups, a v ar iety of Ω-gr oups. There are essentially tw o ways to remedy this situatio n. On the one hand, the context may be strengthened to that o f Mo or e c ate gories by impo sing the str ong pr otomo dularity axiom [3, 40]; but then the theo r y no lo nger applies to all v arieties of Ω- g roups. O n the o ther hand, it is known that the induced notions of centralit y coincide in any s emi-ab elian ca tegory (see [2 5, P rop osition 2.2 ]). That is to sa y , R A [ M ,N ] H is isomorphic to [ R A M , R A N ] S when N is equal to A . In fact, acco rding to an unpublished r esult by M. Gra n and the first author (pr esented here a s Pro- po sition 4.6 b e low) this assumption is to o str ong: a s we shall see, the commutators coincide a s so on as A = M ∨ N . Two coter minal morphisms m : M → A and n : N → A comm ute when there exists a (necessa rily unique) morphism ϕ m,n : M × N → A such that m = ϕ m,n ◦ h 1 M , 0 i and n = ϕ m,n ◦ h 0 , 1 N i . It is clear that m and n commut e if a nd only if their Huq commutator [ m, n ] H : [ M , N ] H → A is zer o, se e Subsection 1 .3. Prop ositi o n 4.6 . Given any two n ormal sub obje cts M and N of A such t hat M ∨ N = A we have R [ M ,N ] H = [ R M , R N ] S . Pr o of. W e s how that the r epresenting normal monomorphisms m and n of M a nd N commute if and only if the equiv alenc e rela tions R M and R N centralise ea ch other; the result then follows, bec ause the commutators [ − , − ] H and [ − , − ] S satisfy the same universal prop er t y . One implication is Prop os itio n 3.2 in [13] which s tates that m and n commute whenever [ R M , R N ] S is ∆ A . Indeed, if p : R M × A R N → A is a partial Mal’tsev op era tion o n R M and R N , then its r estriction to M × N is the needed ϕ m,n . T o prov e the other implication, supp ose tha t ϕ m,n : M × N → A exists. By assumption, the mor phism h m n i : M + N → A , a nd hence also ϕ m,n , is a reg ula r epimorphism. This implies that R M = ϕ m,n ( ϕ − 1 m,n R M ) and R N = ϕ m,n ( ϕ − 1 m,n R N ). Since the imag e s of tw o equiv alence re la tions which centralise each other still cent- ralise ea ch other (by (6) in Lemma 4.2), it suffices to show that so do ϕ − 1 m,n R M and ϕ − 1 m,n R N . Now these re la tions turn o ut to b e pa rticularly simple. Via Lemma 2.2, 18 TOMAS E VERAER T AND TIM V AN DER LINDEN the No ether isomorphism N / ( M ∧ N ) ∼ = ( M ∨ N ) / M implies that the left hand side square in the diag ram with exact rows 0 , 2 M × ( M ∧ N )   , 2 M × N ϕ m,n   , 2 N M ∧ N ∼ =   , 2 0 0 , 2 M , 2 M ∨ N , 2 M ∨ N M , 2 0 is a pullback, so ϕ − 1 m,n R M = ∇ M × R N M ∧ N . Similar ly , ϕ − 1 m,n R N = R M M ∧ N × ∇ N . Since [ M ∧ N , M ] H = 0 = [ M ∧ N , N ] H , Prop ositio n 2 .2 in [25] may b e used to see that b oth [ ∇ M , R M M ∧ N ] S = ∆ M and [ R N M ∧ N , ∇ N ] S = ∆ N , so tha t [ ϕ − 1 m,n R M , ϕ − 1 m,n R N ] S = ∆ M × N —which finishes the pro of.  Combining L e mma 4.4 with Pro p o sition 4.6, we obtain Theorem 4.7. Give n any two normal sub obje cts M and N of A , their Hu q c om- mutator [ M , N ] H , c ompute d in M ∨ N , c oincid es with [ M , N ] Ab A .  Remark 4.8 . Given a ny monomorphism i : A → B , t w o coter minal morphisms m : M → A a nd n : N → A co mmute if and only if i ◦ m and i ◦ n commute— bo th in Huq’s s ense and r elative with resp ect to any B . This implies that the concept of “ commuting subob jects” is independent of the surr ounding o b ject A . As a co nsequence, [ M , N ] Ab A A = [ M , N ] H . 5. Fur ther remarks 5.1. Findi ng the right con text. W e hav e defined the relative commutator in the framework of s emi-ab elian catego ries. How ev er, lo o king a t the diag ram in the int ro duction, this is not entirely satis fa ctory , bec a use: · Central extensions were defined in [32] in the context of exact ca tegor- ies A , relative to a c hoice of admissible Birkhoff subca tegory; and it was shown that if A is Mal’tsev (every r eflexive r elation in ternal in A is an equiv alence r elation) then any Bir khoff sub catego ry is admissible. Mor e recently , V. Ro ssi proved in [4 2] the admissibility o f B irkhoff sub catego ries in a context which includes every regula r Mal’tsev categ ory tha t is “ a lmost exact” in the sens e tha t every r egular epimo rphism is an effective des cent morphism. · The Huq commutator can b e co nsidered in a context, as gener al as that of finitely co complete unital categor ies; in particular, in any finitely co com- plete po int ed Mal’tsev c a tegory [9]. Thu s o ne may as k if it is p ossible to consider the r e lative co mm utator in a more general context than that o f semi-a b e lian categories , say , in finitely co complete, po inted, reg ular, “ a lmost e xact” Mal’tsev categ ories? W e do not know the ans wer, but let us mention here tw o appa r ent obs tacles and comment on either of these . (1) Double central extensio ns, o n which c oncept the no tion of rela tive commut a- tor dep ends, w ere defined in [19] in the semi-ab elia n c ontext. One re ason for this was that the cons truction of the left adjoint to the inclusion functor CE xt B A → Ext A given in [1 9] is only v alid if A is semi- ab elian (and B is a Birk hoff sub catego ry o f A ). In this ca se, the s ame construc tio n ca n b e applied to higher dimensio ns, g iving us, in par ticular, a left adjoint to the inclus ion functor CExt 2 B A → Ext 2 A o f double central ex tens ions into double extensions. The existence of the latter adjoint or, RELA TIVE COMMUT A TOR THEOR Y IN SEM I-ABELIAN CA TEGORIES 19 more precise ly , of the reflection into CExt 2 B A o f double e x tensions o f the for m ( E ) is what a llows us to define the relative co mmut ator. There is no a prio ri reas o n, though, why the left adjoints Ext A → CExt B A a nd Ext 2 A → CExt 2 B A could not exist when the catego ry A is not s emi-ab elian. In fact, the for mer adjoint is known to exist in a wide v ariety of cases (see [33, 31]). F or instance, it e x ists if A is a finitely co co mplete exact Mal’tsev ca tegory and B the Birkhoff sub categ ory o f ab elian ob jects, and in this cas e the characterisation of Lemma 4.3 ab ove remains v alid (se e [21]). (2) In an ex act Mal’tsev catego ry any pusho ut of regular epimor phisms is a regular pusho ut [14], and w e hav e used this prop e r ty to conclude the crucial fact that the squar e ( E ) is a lwa y s a double extension. F urthermore, we know from [14] that in every regula r, but no t exact, Mal’tsev categ ory there exist pusho ut squares of regula r epimorphisms that a re not double extensions. This seems to indicate that exactnes s is unav oidable in defining a rela tive commutator. Howev er , we can say the following. First of all we r e c all from [6] that a finitely complete catego ry A is Mal’ts e v if and only if for any squa re of split e pimorphisms X d   c , 2 C g   l r D f , 2 L R Z l r L R which “reas o nably” commutes (in the sense that it r epresents a split epimorphism in the category of split epimorphisms, with given s plitting, in A ), the facto r isation h d, c i : X → D × Z C to the pullback of f with g is a strong epimor phism. A finitely complete po int ed category A is called unital if the same prop erty holds, but only in the case where Z is the zero ob ject. Equiv a lent ly , A is unital if for a n y tw o ob jects C and D the “pro duct injections” h 0 , 1 C i : C → D × C and h 1 D , 0 i : D → D × C are jointly strongly epimorphic [6 , 8]. A third characterisation of unital ca tegories is given by the following prop os ition. Prop ositi o n 5.2. If A is a finitely c omplete p ointe d c ate gory, then the fi rst c on- dition implies the se c ond: (1) A is unital; (2) for any p air of st r ong epimorph isms c and d D X d l r c , 2 C such that the kernels ker d and ker c ar e jointly str ongly epimorphic, t he in- duc e d m orphism t o the pr o duct h d, c i : X → D × C is a str ong epimorphi sm. If, mor e over, A has finite c opr o ducts, then the two c onditions ar e e quivalent. Pr o of. Assume that A is unital and that d a nd c are as in (2). Fir st of all note that a morphism is a strong epimorphis m if it is jo int ly s trongly epimorphic with a zer o morphism. Since ker d and ker c are jointly str o ngly epimorphic, a nd d is a strong epimorphism, this implies that the comp os ite d ◦ ker c is stro ngly epimorphic. Similarly , c ◦ k er d is a stro ng epimorphism. Since A is unital, the pr o duct injections h 0 , 1 C i and h 1 D , 0 i ar e jointly stro ngly epimorphic, hence, by the ab ov e, so ar e h 0 , 1 C i ◦ c ◦ ker d = h d, c i ◦ ker d and h 1 D , 0 i ◦ c ◦ k er c = h d, c i ◦ ker c . Hence h d, c i is a str ong epimorphis m. Conv er sely , for any t w o o b jects D and C of A , applying co ndition (2 ) to the “copro duct pr o jections” D D + C h 1 D 0 i l r h 0 1 C i , 2 C 20 TOMAS E VERAER T AND TIM V AN DER LINDEN gives us that the pro duct injections h 1 D , 0 i and h 0 , 1 C i ar e jointly stro ngly epi- morphic. Hence A is unital.  Now suppo se tha t A is finitely co complete, reg ular a nd unital. Then, in particu- lar, any tw o normal subob jects M and N of an ob ject A in A admit a unio n M ∪ N , and the ab ove pr op osition implies that the squa r e M ∪ N q M , 2 q N   M ∪ N M   M ∪ N N , 2 0 ( M ) is a double extension (here q M and q N are the co kernels of the inclusions in M ∪ N of M and N , r esp ectively). This indicates tha t it mig ht b e p oss ible, after all, to consider the relative co mm utator in a non-exact context, but we would need to hav e an appr opriate no tio n of double central e xtension. In that case, we could s ay that M and N co mmu te if and only if the double extension ( M ) is central. 5.3. Stabili t y unde r regul ar im ages. W e proved in Theorem 3.9 that p [ M , N ] B = [ p M , pN ] B ( N ) for any regular epimorphism p : A → B a nd normal sub ob jects M and N o f A such that either K [ p ] ≤ M ∧ N o r M ≤ N . As no ted in Remark 3.12, this identit y need not hold for arbitrar y p , M and N . How ever, w e know from [27] tha t ( N ) do es hold for a rbitrary p , M a nd N if B = Ab A , and the same is true, for instance, for the Peiffer commut ator of precros sed modules or the asso c ia tor of lo o ps (considered in [2 2]). This suggests to lo ok for necessa ry a nd s ufficient co nditions on the Bir khoff sub c ategory B for [ − , − ] B to be stable under r e gular images , i.e., for the identit y ( N ) to hold for any reg ular epimor phism p : A → B and a ny normal sub ob jects M and N of A . W e do not hav e a satisfacto ry answer to this question, although a characterisation o f s uch B in the case of Ω-gr oups was given in [15], in terms of the ident ities that define the s ub v ariety B . Let us just reca ll here the following necessa ry condition, again ta ken from the article [15]: we need the s ubca tegory Ab A of a belia n ob jects of A to b e contained in B . Indeed, if we assume that the r elative co mmut ator [ − , − ] B is stable under r eg- ular images, and that A is a n ab elian ob ject with “multiplication” π : A × A → A , then [ A, A ] B = h π  A × 0  , π  0 × A  i B = π   A × 0 , 0 × A  B  ⊆ π   A × 0  ∧  0 × A   = 0 . How ever, the conv erse is no t true. The condition B ⊇ Ab A do es not imply the stability under regular images of [ − , − ] B ; a co un terexample was given in [15]. A simila r question may b e asked with resp ect to preser v ation of joins, see Re- mark 3.12. 5.4. Hi gher dim ensions . In this ar ticle, we considered what we hav e called zer o- dimensional, one- dimensional and tw o-dimensional r e la tive comm utators, but what ab out higher dimensions? Keeping in mind ex amples such as the asso c iator of loops, this do es not seem to b e an unreas o nable question to ask . Let us wr ite [ L, M , N ] B for a three-dimens ional relative comm utator defined on triples o f normal sub ob jects L , M , N of a n o b ject A of A , with res pe ct to a Birkho ff sub categ ory B of A . 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