aa r X i v : . [ m a t h . C T ] F e b Complete objects in categories
James Richard Andrew GrayFebruary 22, 2021
Abstract
We introduce the notions of proto-complete, complete, complete ˚ andstrong-complete objects in pointed categories. We show under mild condi-tions on a pointed exact protomodular category that every proto-complete(respectively complete) object is the product of an abelian proto-complete(respectively complete) object and a strong-complete object. This to-gether with the observation that the trivial group is the only abeliancomplete group recovers a theorem of Baer classifying complete groups.In addition we generalize several theorems about groups (subgroups) withtrivial center (respectively, centralizer), and provide a categorical explana-tion behind why the derivation algebra of a perfect Lie algebra with trivialcenter and the automorphism group of a non-abelian (characteristically)simple group are strong-complete. Recall that Carmichael [19] called a group G complete if it has trivial cen-ter and each automorphism is inner . For each group G there is a canonicalhomomorphism c G from G to Aut p G q , the automorphism group of G . This ho-momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg ´ . It can be readily seen that a group G is complete if and onlyif c G is an isomorphism. Baer [1] showed that a group G is complete if andonly if every normal monomorphism with domain G is a split monomorphism.We call an object in a pointed category complete if it satisfies this latter condi-tion. Completeness, which corresponds to being injective in abelian categories,has been studied in other contexts (although not always under that name) andas explained by B. J. Gardner in [23] the following is known. Completenesscorresponds to:1. having trivial center (annihilator) and each derivation being inner in eachcategory of Lie algebras;2. having multiplicative identity in each category of associative algebras;and follows from: 1. the existence of a multiplicative identity in each category of alternativealgebras and each category of autodistributive algebras.Note that complete objects need not be injective nor absolute retracts, in factBaer showed in [1] that there are no non-trivial absolute retracts of groups.One of the main purposes of this paper is to introduce and study four (threemain) notions of completeness, and to give a categorical explanation of Baer’sresult mentioned above. In addition to completeness which we have alreadydefined above we call an object X proto-complete if every normal monomor-phism with domain X which is the kernel of a split epimorphism is a splitmonomorphism, and strong-complete if it satisfies the same condition exceptwith the additional requirement that the normal monomorphisms in questionare required to have a unique splitting. The forth notion is obtain by replacingnormal by Bourn-normal in the definition of completeness. Let us immedi-ately mention that pointed protomodular categories [10] in which every objectis strong-complete turn out to be precisely what F. Borceux and D. Bourn in[4] called coarsely action representable. In the pointed protomodular contextwe show that strong-completeness implies completeness (Proposition 4.7) andthat every proto-complete (respectively complete) object, satisfying certain ad-ditional conditions which automatically hold in every such variety of universalalgebras, is the product of an abelian proto-complete (respectively complete)object and a strong-complete object (Theorem 4.16). We show that a partialconverse to the previous fact holds (Proposition 4.17). We give classificationsof proto-completeness and strong-completeness (see Theorems 4.19 and 4.25)relating to the existence of generic split extensions in the sense of [7], which areclosely related to Problem 6 of the open problems of [8]. For a group G thesetheorems imply: (a) G is proto-complete if and only if c G is a split epimorphism;(b) G is strong-complete (= complete) if and only if c G is an isomorphism.Other aims include a brief study of objects with trivial center and of subob-jects with trivial centralizer, in Section 3, and to study characteristic monomor-phism and their interaction with completeness in Section 5. The main results ofSection 3, applied to the category of groups, recover the following known factsabout groups:(i) If G has trivial center, then c G : G Ñ Aut p G q is a normal monomorphismwith trivial centralizer (Proposition 3.6);(ii) If n : N Ñ G is a normal monomorphism with trivial centralizer, theneach automorphism of N admits at most one extension to G (Proposition3.7).In Section 5 we provide a common categorical explanation behind why thederivation algebra of a perfect Lie algebra with trivial center, and the automor-phism group of a (characteristically) simple group are (strong) complete. Thisexplanation depends on several facts including: (a) two new characterizationsof characteristic monomorphisms with domain satisfying certain conditions (seeTheorems 5.2 and 5.3); (b) Theorem 5.5, which generalizes the following fact for2 group G : the homomorphism c G : G Ñ Aut p G q is a characteristic monomor-phism if and only if G has trivial center and Aut p G q is (strong) complete. In this section we recall preliminary definitions and introduce some notation.Let C be a pointed category with finite limits. We will write 0 for boththe zero object and for a zero morphism between objects. For objects A and B we will write A ˆ B for the product of A and B , and write π and π forthe first and second product projections, respectively. For a pair of morphisms f : W Ñ A and g : W Ñ B we will write x f, g y : W Ñ A ˆ B for the uniquemorphism with π x f, g y “ f and π x f, g y “ g . For objects A and B we write A ` B for the coproduct (when it exists), and write ι and ι for the first andsecond coproduct inclusion. For a pair of morphisms u : A Ñ Z an v : B Ñ Z we write r u, v s : A ` B Ñ Z for the unique morphism with r u, v s ι “ u and r u, v s ι “ v .The category C is called unital [3] if for each pair of objects A and B themorphisms x , y : A Ñ A ˆ B and x , y : B Ñ A ˆ B are jointly strongly epi-morphic. When these morphism are only jointly epimorphic C is called weaklyunital [32]. A pair of morphisms f : A Ñ X and g : B Ñ X in a (weakly)unital category are said to (Huq)-commute if there exists a (unique) morphism ϕ : A ˆ B Ñ X making the diagram A x , y / / f . . A ˆ B ϕ (cid:15) (cid:15) B x , y o o g p p X commute. Let us recall some well-known facts and definitions related to thecommutes relation (see e.g. [3], and the references there for the unital context,and [24, 26] for the weakly unital context.) Lemma 2.1.
Let C be a weakly unital category and let e : S Ñ A , f : A Ñ X , g : B Ñ X , f : A Ñ X , g : B Ñ X and h : X Ñ Y be morphisms in C ,then(i) f and g commute if and only if g and f commute;(ii) if f and g commute, then f e and g commute;(iii) if f and g commute, then hf and hg commute;(iv) f ˆ f and g ˆ g commute if and only if f and g , and f and g commute.Moreover, the converse of (ii) holds when e is a pullback stable regular epimor-phism, while the converse of (iii) holds when C is unital and h is a monomor-phism. efinition 2.2. Let C be a weakly unital category. The centralizer of a mor-phism f : A Ñ B is the terminal object in the category of morphisms commutingwith f . We will write z f : Z X p A, f q Ñ X for the centralizer of f when it exits. Notethat z f is always a monomorphism. When f “ X the centralizer of f is calledthe center of X and will be denoted z X : Z p X q Ñ X .A split extension in C is a diagram X κ / / A α / / B β o o (1)in C where κ is the kernel of α and αβ “ B . A morphism of split extensionsin C is a diagram X κ / / u (cid:15) (cid:15) A v (cid:15) (cid:15) α / / B w (cid:15) (cid:15) β o o X κ / / A α / / B β o o (2)in C where the top and bottom rows are split extensions (the domain andcodomain, respectively), such that κ u “ vκ , β w “ vβ and α v “ wα . Let usdenote by SplExt p C q the category of split extensions, and by K the functorsending (1) and (2) to X and u , respectively. Let us write KGpd p C q for thecategory with objects 8-tuples p X, G , G , d, c, e, m, k q consisting of objects andmorphisms such that the diagram on the left G ˆ x d,c y G m / / G d / / c / / G e o o X k / / G d / / G e o o (3)is a groupoid and the diagram on the right is a split extension. A morphism p X, G , G , d, c, e, m, k q Ñ p X , G , G , d , c , e , m , k q is a triple p u, v, w q where u : X Ñ X , v : G Ñ G and w : G Ñ G are morphisms in C such that thediagram on the left is a functor G ˆ x d,c y G m / / v ˆ v (cid:15) (cid:15) G v (cid:15) (cid:15) d / / c / / G w (cid:15) (cid:15) e o o G ˆ x d ,c y G m / / G d / / c / / G e o o X k / / u (cid:15) (cid:15) G v (cid:15) (cid:15) d / / G w (cid:15) (cid:15) e o o X k / / G d / / G e o o and the diagram on the right is a morphism of split extensions. Note that theforgetful functor U : KGpd p C q Ñ SplExt p C q is monadic since it is essentiallythe same as the forgetful functor from the category of internal groupoids in C to the category of split epimorphisms in C shown to be monadic by D. Bournin [9] (see also the discussion above Theorem 4.2 of [28]).4 pointed category C can be equivalently defined to be (Bourn)-protomodular[10] if the split short five lemma holds, that is, for each morphism of split exten-sions (2) if u and w are isomorphisms, then v is an isomorphism. A category C is semi-abelian in the sense of G. Janelidze, L. Marki and W. Tholen [31] if it ispointed, (Barr)-exact [2], protomodular and has binary coproducts. Following,F. Borceux, G. Janelidze, G. M. Kelly, in [7] we define a generic split extensionwith kernel X to be a terminal object in a fiber K ´ p X q of K : SplExt p C q Ñ C .We denote such a generic split extension as follows: X k / / r X s ˙ X p / / r X s . i o o (4)A semi-abelian category is called action representable if each object admints ageneric split extension with kernel X . Examples of action representable cate-gories such include the category of groups where r X s “ Aut p X q is the automor-phism group of X , and the category of Lie algebras over a commutative ring R where r X s “ Der p X q is the Lie algebra of derivations of X (see [7]). Otherexamples can be found in [8], [5], [6], [25], [27]. For a pointed protomodularcategory C and for an object X in C such that the generic split extension withkernel X exists, the object r X s is called the split extension classifier for X andhas the following universal property: For each split extension (1) there exist aunique morphism v : B Ñ X such that there exist u : A Ñ r X s ˙ X making thediagram a morphism of split extensions X κ / / A α / / u (cid:15) (cid:15) B β o o v (cid:15) (cid:15) X k / / r X s ˙ X p / / r X s i o o (5)(see the last theorem of Section 6 of [7].)D. Bourn and G. Janelidze call a split extension with kernel X faithful [16]when there is at most one morphism to it from any split extension in K ´ p X q .A semi-abelian category is called action accessible [16] if for each X in C there isa morphism from each split extension to a faithful one in K ´ p X q . Examples ofaction accessible categories include the category of not-nessesarily unital rings,associative algebras and more generally categories of interest in the sense of G.Orzech [35] (see [16] and [34].)Throughout the paper we denote by the category with objects 0 and 1, andwith one non-identity morphism 0 Ñ
1. We will identify the functor category C with the category of morphism of C , and its objects will be written astriples p X, Z, f q where X and Z are objects and f : X Ñ Z is a morphism in C . A morphism p X, Z, f q Ñ p X , Z , f q with be written as a pair p u, v q where u : X Ñ X and v : Z Ñ Z are morphisms in C with f u “ vf .Recalling from [27], in the semi-abelian context , or from [28] for the generalpointed context, that each generic split extension in C has codomain a generic5plit extension in C , we will denote a generic split extensions of a morphism f : X Ñ Z in C as follows: X f (cid:15) (cid:15) k / / r X, Z, f s ˙ X p / / q ˙ f (cid:15) (cid:15) r X, Z, f s i o o q (cid:15) (cid:15) Z k / / r Z s ˙ Z p / / r Z s . i o o (6)Note that according to the universal property of generic split extensionsthere is also a unique morphism X k / / r X, Z, f s ˙ X p / / q ˙ (cid:15) (cid:15) r X, Z, f s i o o q (cid:15) (cid:15) X k / / r X s ˙ X p / / r X s . i o o When X admits a generic split extension, writing p R, r , r q for the kernelpair of p , it turns out that the unique morphism X x ,k y / / R r / / s (cid:15) (cid:15) r X s ˙ X x , y o o p (cid:15) (cid:15) X k / / r X s ˙ X p / / r X s i o o is an algebra structure for the monad induced by the adjunction where U : KGpd p C q Ñ SplExt p C q is a right adjoint, and hence determines a groupoidstructure on (4). Putting all the structure together we obtain an object p X, r X s˙ X, r X s , p , p , i, m, k q in KGpd p C q which turns out to be terminal in the cat-egory of X -groupoids, that is, the fiber p KU q ´ p X q (see Proposition 5.1 of[4] where this is proved directly in the pointed protomodular context, or com-bine Proposition 2.26 and Theorem 4.1 via the remarks before Theorem 4.2 of[28], for the pointed finitely complete context). Writing ∇ p X q for the objectin KGpd p C q with underlying groupoid the indiscrete groupoid on X and with x , y : X Ñ X ˆ X as kernel of its domain morphism π : X ˆ X Ñ X , we callthe morphism c X : X Ñ r X s which forms part of the unique morphism of X -groupoids ∇ p X q Ñ p X, r X s ˙ X, r X s , p , p , i, m, k q the conjugation morphismof X (see Theorem 3.1 of [4]). In particular this means it forms part of theunique morphism X x , y / / X ˆ X ϕ (cid:15) (cid:15) π / / X x , y o o c X (cid:15) (cid:15) X k / / r X s ˙ X p / / r X s , i o o (7)6nd that p k “ c X . The kernel of c X turns out to be z X : Z p X q Ñ X the centerof X (see Proposition 5.2 of [16] and Proposition 4.1 of [4] which is closelyrelated).For a semi-abelian category C let us briefly recall the equivalence between SplExt p C q and Act p C q the category of internal object actions from [7]. Theobjects of Act p C q are triples p B, X, ζ q where B and X and are objects in C and ζ : B X Ñ X is a morphism making p X, ζ q an algebra over a certain monadwhose object functor sends X to B X where κ B,X : B X Ñ B ` X is the kernelof r , s : B ` X Ñ B . Given morphisms g : B Ñ B and f : X Ñ X theunique morphism g f : B X Ñ B X making the diagram B X g f (cid:15) (cid:15) κ B,X / / B ` X g ` f (cid:15) (cid:15) r , s / / B g (cid:15) (cid:15) ι o o B X s κ B ,X / / B ` X , s / / B ι o o a morphism of split extensions, makes ´5˚ : C ˆ C Ñ C a functor. A morphism p B, X, ζ q Ñ p B , X , ζ q in Act p C q is a pair p g, f q , where g : B Ñ B and f : X Ñ X are morphisms in C such that f ζ “ ζ p g f q .Given a split extension (1) the unique morphism ζ : B X Ñ X making thediagram B X κ B,X / / ζ (cid:15) (cid:15) ✤✤✤ B ` X r , s / / r β,κ s (cid:15) (cid:15) B ι o o X κ / / A α / / B β o o a morphism of split extensions, makes p B, X, ζ q an object in Act p C q and thisassignment determines the object map of a functor SplExt p C q Ñ Act p C q which is an equivalence of categories. Using this equivalence of categories G.Janelidze produced an equivalence of categories between internal categories in C and internal crossed modules in C [30]. Let us write γ X : X X Ñ X for the action corresponding to the domain of morphism of split extensions(7). When star multiplicative graphs coincide with multiplicative graphs (whichN. Martins-Ferreira and T. Van der Linden showed, in [33], happens exactlywhen the Huq [29] and Smith-Pedicchio [36] commutators coincide) an inter-nal crossed module can be equivalently defined as a triple p B, X, ζ, f q where p B, X, ζ q is an object in Act p C q and f : X Ñ B is a morphism in C suchthat p f, X q : p X, X, γ X q Ñ p B, X, ζ q and p B , f q : p B, X, ζ q Ñ p
B, B, γ B q are morphisms in Act p C q (see [30] noting that γ X “ r , s κ X,X and that r , f s κ B,X “ γ B p B f q ). When C is a semi-abelian category and τ is theaction corresponding to split extension (4) then, via the equivalence betweeninternal categories and internal crossed modules, it follows that the quadruple pr X s , X, τ, c X q is the terminal object in the category internal crossed moduleswith domain of the underlying morphism the object X .7 monomorphism m : S Ñ X in a finitely complete category C is Bourn-normal [12] to an equivalence relation r , r : R Ñ X if there exists a morphism˜ m : S ˆ S Ñ R such that (either and hence both of) the squares of the diagramon left, or equivalently the diagram on the right S m (cid:15) (cid:15) S ˆ S ˜ m (cid:15) (cid:15) π / / π o o S m (cid:15) (cid:15) X R r / / r o o X S ˆ S ˜ m / / ˆ m (cid:15) (cid:15) R x r ,r y (cid:15) (cid:15) S ˆ X m ˆ / / X ˆ X are pullbacks. When C is pointed, such a morphism ˜ m exists as soon as thereis a morphism k : S Ñ R making the lower left hand square in the diagram S x , y / / ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ m { { S ˆ S ˜ m (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ π / / m ˆ m | | S m (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ x , y o o m { { S k / / m (cid:15) (cid:15) R x r ,r y (cid:15) (cid:15) r / / X s o o X x , y / / X ˆ X π / / X x , y o o (8)a pullback. Note that, writing s for the unique morphism such that r s “ X “ r s , this means that the entire diagram consists of morphisms of splitextensions. Note also that this means that for an equivalence relation p R, r , r q if k is the kernel of r , then the composite r k is Bourn-normal to p R, r , r q .We call a morphism m a Bourn-normal monomorphism as soon as there existsan equivalence relation to which it is Bourn-normal. We will say that:(i) An object X in a weakly unital category has trivial center if Z p X q “ f : A Ñ X in a weakly unital category has trivial centralizerif Z X p A, f q “
Lemma 3.1.
Let C be a (weakly) unital category. An object X in C has trivialcenter if and only if for each object Y the morphism x , y : X Ñ X ˆ Y has aunique section. roof. The claim follows by noting that for each commutative diagram X x , y / / X ●●●●●●●●● X ˆ Y ϕ (cid:15) (cid:15) Y x , y o o f { { ①①①①①①①①① Xf is a zero morphism if and only if ϕ “ π .Since for any morphisms e : S Ñ A , f : A Ñ X and g : B Ñ X in a weaklyunital category, if f and g commute, then so do f e and g , we obtain: Proposition 3.2.
Let e : S Ñ A and f : A Ñ X be morphisms in a weaklyunital category C . If f e has trivial centralizer, then so does f . In particularwhen A “ X and f “ X this means that if e has trivial centralizer, then X hastrivial center. Proposition 3.3.
Let C be a weakly unital category and let p X, p π i : X Ñ X i q i P I q be the product of a family of objects p X i q i P I in C .(i) A morphism f : A Ñ X commutes with X if and only if π i f : A Ñ X i commutes with X i (ii) If the center z X i : Z p X i q Ñ X i of each X i exists and the product of family p Z p X i q i P I q exists, then the center of X exists and is the product of thefamily of morphisms p z X i : Z p X i q Ñ X i q i P I ;(iii) if the center of X exists, then the center of each X i exist and their productexists and is the center of X .Proof. Since (ii) is a straight forward consequence of (i) we prove (i) and (iii).To prove (i) let f : A Ñ X be a morphism in C . Since each π i being a splitepimorphism is a pullback stable regular epimorphism it follows by Proposi-tion 3.14 in [26] that π i f commutes with π i if and only if π i f commutes with1 X i . Therefore we need only show that f commutes with 1 X if and only if π i f commutes with π i . However, this follows from the fact that the existence of amorphism ϕ : A ˆ X Ñ X such that ϕ x , y “ f and ϕ x , y “ X is equivalentto the existence of a family of morphisms p ϕ i : A ˆ X Ñ X i q i P I such that foreach i in I , ϕ i x , y “ π i f and ϕ i x , y “ π i . To prove (iii) let z : Z Ñ X bethe center of X . For each i in I let λ i : X i Ñ X be the unique morphism with π j λ i identity if i “ j and zero otherwise, and let z i : Z Ñ X be the morphismobtained by pullback as displayed in the square of the diagram Z π i z (cid:31) (cid:31) ρ i ❆❆❆❆ Z i η i / / z i (cid:15) (cid:15) Z z (cid:15) (cid:15) X i λ i / / X.
9e will show that z i is the center of X i . Suppose f : A Ñ X i a centralmorphism. Since as easily follows from (i) the morphism λ i f : A Ñ X iscentral, it follows that there exists a unique morphism ˜ f : A Ñ Z such that z ˜ f “ λ i f and hence a unique morphism ¯ f : A Ñ Z i such that z i ¯ f “ f and η i ¯ f “ ˜ f . Since z i is a monomorphism this is sufficient to show that it is thecenter of X i . Since z commutes with 1 X it follows that π i z commutes with π i and hence, by Proposition 3.14 in [26], that π i z is central. This means thatthere is a unique morphism ρ i : Z Ñ Z i such that z i ρ i “ π i z . We will showthat the family p Z, p ρ i : Z Ñ Z i q i P I q is a product. Suppose p α i : A Ñ Z i q i P I is family of morphisms. By composition we obtain a family p z i α i : A Ñ X i q i P I and hence a unique morphism α : A Ñ X such π i α “ z i α i for each i in I . Sinceby applying (i) we see that α is central it follows that there exists a uniquemorphism ¯ α : A Ñ Z such that z ¯ α “ α . However, since for each i in I themorphism z i is a monomorphism and z i ρ i ¯ α “ π i z ¯ α “ π i α “ z i α i it follows that ρ i ¯ α “ α i . The proof of claim is completed by noting that the family p ρ i q i P I is jointly monomorphic. To see why note that p π i z q i P I is jointly monomorphic(since z is a monomorphism and p π i q i P I is jointly monomorphic) and for each i in I , z i is a monomorphism and z i ρ i “ π i z .As a corollary we obtain: Corollary 3.4.
Let C be a weakly unital category and let p X, p π i : X Ñ X i q i P I q be the product of a family of objects p X i q i P I in C . The object X has trivialcenter if and only if for each i in I the object X i has trivial center. Recall that an object in the category of X -groupoids is called faithful [16] if itadmits at most one morphism from each object in the category of X -groupoids.Recall also that a faithful X -groupoid has underlying split extension faithful(see Lemma 3.2 [16]), and that an internal reflexive graph in a protomodular(more generally Mal’tsev) category admits at most one groupoid structure. Wewill therefore, in the protomodular (more generally Mal’tsev) context, considergroupoids as special kinds of reflexive graphs. Lemma 3.5.
Let C be a pointed protomodular category. If X has trivial centerand X x , y / / X ˆ X ϕ (cid:15) (cid:15) π / / π / / X c (cid:15) (cid:15) x , y o o X κ / / A α / / α / / B β o o (9) is a morphism of X -groupoids, then p A, α , α q is an equivalence relation and c is a Bourn-normal monomorphism normal to p A, α , α q . When, in addition,the X -groupoid at the bottom of (9) is faithful, then c has trivial centralizer.Proof. Since according to Lemma 2.4 of [20] (see also Proposition 5.2 of [16])the morphism ker p c q : Ker p c q Ñ X is a subobject of X such that x , y and10 , y ker p c q commute, it easily follows that ker p c q and 1 X commute (see Ob-servation 5.3 of [16] or Corollary 2.6 of [20]). This means that ker p c q “ c is a monomorphism [11]. Since c “ cπ x , y “ α ϕ x , y “ α κ it follows that the diagram X κ / / c (cid:15) (cid:15) A α / / x α ,α y (cid:15) (cid:15) B β o o B x , y / / B ˆ B π / / B x , y o o (10)commutes. Therefore, since ker px α , α yq “ ker p c q “ x α , α y is a monomorphism and c is Bourn-normal to the equivalencerelation p A, α , α q . For a morphism u : S Ñ B it follows from Lemma 2.1 thatthe conditions:(i) u and c commute;(ii) x α , α y κ “ x , y c “ p c ˆ c qx , y and x α , α y βu “ x , y u “ p u ˆ u qx , y commute;(iii) κ and βu commuteare all equivalent. Therefore, if the upper split extension in (10) is faithful itfollows that u and c commute if and only if u “ Proposition 3.6.
Let C be a pointed protomodular category and let X be anobject with trivial center, such that the generic split extension with kernel X exists.(i) The morphism c X is a Bourn-normal monomorphism with trivial central-izer, and the object r X s has trivial center.(ii) Every X -groupoid is an equivalence relation.(iii) The morphism c X is terminal in the category of Bourn-normal monomor-phisms with kernel X .(iv) When C is in addition semi-abelian, for each internal crossed module p B, X, ζ, f q the morphism f is a normal monomorphism.Proof. The claims (i) and (ii) are direct corollaries of the previous lemmas,while the (iv) is obtained from (ii) via the equivalence of categories betweeninternal groupoids and internal crossed modules. The final claim follows bynoting that there is an equivalence of categories between X -groupoids whichare equivalence relations and Bourn-normal monomorphisms with domain X ,and under this equivalence the terminal X -groupoid is sent to c X .11ecalling that for a group homomorphism f : X Ñ Y the group r X, Y, f s isthe subgroup of Aut p X q ˆ Aut p Y q consisting of those pairs of automorphisms p θ, φ q such that f θ “ φf , and q and q are the first and second projections, onesees that the following proposition applied to the category of groups explainswhy for a normal subgroup S of a group X if the centralizer of S in X is trivial,then each automorphism of S has at most one extension to X . Proposition 3.7.
Let C be a pointed protomodular category, and let m : S Ñ X be a normal monomorphism in C such that the generic split extensions with ker-nel S and p S, X, m q exist in C and C , respectively. If m has trivial centralizer,then q : r S, X, m s Ñ r S s is a monomorphism.Proof. By considering the diagram (8), via the universal property of the splitextension classifiers of p S, X, m q and S , there is a (unique) morphism u : X Ñr S, X, m s making the diagram S c S (cid:15) (cid:15) m / / X u (cid:15) (cid:15) X c X (cid:15) (cid:15) r S s r S, X, m s q / / q o o r X s commute. Let κ : K Ñ r
S, X, m s be the kernel of q . We will show that K “ I / / (cid:15) (cid:15) J / / λ (cid:15) (cid:15) K / / κ (cid:15) (cid:15) (cid:15) (cid:15) S m / / X u / / r S, X, m s q / / r S s in which all squares are pullbacks. Since S has trivial center and c S “ q um it follows that I “ Ker p c S q “ Z p S q “
0. This means that m and λ commute(see e.g. Proposition 3.3.2 of [3]) and hence J “
0. However, since X hastrivial center it follows, by Proposition 3.6, that c X is a normal monomorphism,and hence since q is a monomorphism (see Proposition 4.5 of [27]) it followsthat u is a normal monomorphism too. This means that u and κ commute andtherefore so do c X and q κ . But, according to Proposition 3.6, c X has trivialcentralizer, and hence K “ θ of a group X with trivial center admits a unique extension ϕ to the automorphism group Aut p X q in such a way that c X θ “ ϕc X . Proposition 3.8.
Let C be a pointed protomodular category and let X be anobject in C such that the generic split extension with kernel X exists. If thegeneric split extension with kernel p X, r X s , c X q exists in C , and X has trivial enter, then there is a unique morphism ϕ : r X s ˙ X Ñ rr X ss ˙ r X s making thediagram X c X (cid:15) (cid:15) k / / r X s ˙ X ϕ (cid:15) (cid:15) p / / r X s i o o c r X s (cid:15) (cid:15) r X s k / / rr X ss ˙ r X s p / / rr X ss i o o a generic split extension in C .Proof. Let X be an object with trivial center. Consider the diagram X k / / c X (cid:15) (cid:15) r X s ˙ X p / / x p ,p y (cid:15) (cid:15) r X s i o o r X s ( ( θ / / r X, r X s , c X s q (cid:15) (cid:15) q / / r X sr X s x , y / / r X s ˆ r X s π / / r X s x , y o o c r X s / / rr X ss where the morphism in C displayed on the right is the unique morphism ob-tained from the split extension on the left via the universal property of the splitextension classifier of p X, r X s , c X q and which has lower morphism c r X s accordingto Lemma 4.2 in [27]. The universal property of the split extension classifier r X s then shows that q θ “ r X s . However, since by Proposition 3.6 the morphism c X is a Bourn-normal monomorphism with trivial centralizer, it follows, by theprevious proposition, that q : r X, r X s , c X s Ñ r X s is a monomorphism andhence an isomorphism. This means that θ is an isomorphism which completesthe claim. In this section we study four notions of completeness and explain how Baer’sresult can be recovered categorically. For a pointed category C we call a mor-phism a protosplit monomorphism [8] if it is the kernel of a split epimorphism.As mentioned above we define: Definition 4.1.
Let C be a pointed category. An object X is called(i) proto-complete if every protosplit monomorphism with domain X is a splitmonomorphism;(ii) complete if every normal monomorphism with domain X is a split monomor-phism;(iii) complete ˚ if every Bourn-normal monomorphism with domain X is a splitmonomorphism; iv) strong-complete if every protosplit monomorphism with domain X is asplit monomorphism with a unique section. Remark 4.2.
Since in a pointed Barr-exact category every Bourn-normal monomor-phism is a normal monomorphism it follows that completeness and completeness ˚ coincide in the pointed Barr-exact context. Most of the content of the following lemma follows from Corollary 3.3.3 of[3].
Lemma 4.3.
Let X be a proto-complete object in a pointed protomodular cat-egory C with finite limits. For each split extension X κ / / A α / / B β o o there exists a morphism θ : B Ñ X and an isomorphism ψ : A Ñ B ˆ X makingthe diagram X κ / / A α / / ψ (cid:15) (cid:15) B β o o X x , y / / B ˆ X π / / B x ,θ y o o an isomorphism of split extensions.Proof. Suppose that X is a proto-complete object in a pointed protomodularcategory C and let X κ / / A α / / B β o o be a split extension. By assumption there exists a morphism λ : A Ñ X suchthat λκ “ X . It easily follows that the diagram X κ / / A α / / x α,λ y (cid:15) (cid:15) B β o o X x , y / / B ˆ X π / / B x ,λβ y o o is a morphism of split extensions and hence by protomodularity x α, λ y is anisomorphism.We will also need the following easy lemma (see e.g. Corollary 3.3.3 of [3]). Lemma 4.4.
Let C be a pointed protomodular category and let m : S Ñ X be a Bourn normal monomorphism. If m is a split monomorphism, then m is a product inclusion , that is, there exists an object T and an isomorphism ϕ : S ˆ T Ñ X such that ϕ x , y “ m . roposition 4.5. Let C be a pointed protomodular category. An object X in C is strong-complete if and only if X is proto-complete and has trivial center.Proof. Let X be a proto-complete object in C . Since every protosplit normalmonomorphism with domain X is, by Lemma 4.3 up to isomorphism, of theform x , y : X Ñ X ˆ Y for some Y , the claim follows from Lemma 3.1 Remark 4.6.
As mentioned above categories in which every object is strong-complete are called coarsely action representable. These include the oppositecategory of pointed sets.
Proposition 4.7.
In a pointed protomodular category the implications hold:strong-completeness ñ completeness ˚ ñ completeness ñ proto-completeness.Proof. Trivially completeness ˚ ñ completeness ñ proto-completeness. Suppose X is strong-complete, we need to show that it is complete ˚ . Let n : X Ñ B bea Bourn-normal monomorphism and let r , r : R Ñ B be the (projections ofthe) relation of which it is the zero class. This means that there are morphisms k : X Ñ R and ˜ n : X ˆ X Ñ R such that the diagram X x , y / / ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ n | | X ˆ X ˜ n (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧ π / / n ˆ n } } X n (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ x , y o o n | | X k / / n (cid:15) (cid:15) R x r ,r y (cid:15) (cid:15) r / / B s o o B x , y / / B, ˆ B π / / B x , y o o (11)in which s is the unique morphism r s “ r s “ B , is a morphism of splitextensions. According to Lemma 4.3 the middle split extension is isomorphicto a split extension of the form X x , y / / B ˆ X π / / B x ,θ y o o and hence composing the upper morphism of diagram (11) with this isomor-phism we obtain a morphism of split extensions of the form X x , y / / X ˆ X π / / x nπ ,φ y (cid:15) (cid:15) X x , y o o n (cid:15) (cid:15) X x , y / / B ˆ X π / / B, x ,θ y o o for some morphism φ : X ˆ X Ñ X . This implies that φ is a splitting of x , y and hence must be π (since X is strong-complete). It now follows that θn “ π x , θ y n “ π x nπ , π yx , y “ X as desired.15ince in an abelian category every monomorphism is a normal monomor-phism and every protosplit monomorphism is a split monomorphism we seethat: Proposition 4.8.
Let C be an abelian category.(i) Every object X satisfies the condition: if κ : X Ñ A is the kernel of asplit epimorphism which is also a normal epimorphism, then κ is a splitmonomorphism;(ii) Every object is proto-complete;(iii) An object is complete if and only if it is (regular) injective;(iv) An object is strong-complete if and only if it is a zero object.More generally (i), (iii) and (iv) hold if C op is semi-abelian.Proof. Noting that (i) and (ii) are equivalent for an abelian category. We provethe dual of (i), (iii) and (iv) for C a semi-abelian category. To prove the dualof (i) suppose that α : A Ñ B is the cokernel of a normal split monomorphism κ : X Ñ A . If λ is a splitting of κ , then the diagram X κ / / A α / / x α,λ y (cid:15) (cid:15) BX x , y / / B ˆ X π / / B is a morphism of short exact sequences and hence by the short five lemma x α, λ y is an isomorphism and α a split epimorphism. The dual of (iii) is a standard factabout regular projective objects in regular categories. The dual of (iv) followseasily from the fact that if X is an object in C , then x , y and x , y are bothsections of the normal epimorphism π , and x , y “ x , y if and only if 1 X “ X is a zero object.On the other hand in the category Rng of not necessarily unitary rings,proto-completeness, completeness and strong-completeness coincide and are equiv-alent to being unitary. Recall that as mentioned in the introduction it is knownthat completeness is equivalent to being unitary.
Proposition 4.9.
Let X be an object in Rng . The object X is a unitary if andonly if it satisfies any of Definition 4.1 (i)-(iv).Proof. Let X be a ring. If X is proto-complete, then we see that the morphism k , which forms part of the split extension X k / / Z ˙ X p / / Z i o o X ”, splits. Since a surjectivering homomorphism sends a multiplicative identity to a multiplicative identityit follows that X is unitary. Conversely if X is unitary and X is an ideal of Y , then the map l : Y Ñ X defined y ÞÑ ey (where e is the multiplicativeidentity in X ) is a morphism which splits the inclusion. Indeed, using that e isa multiplicative identity it immediately follows that it splits the inclusion, andsince multiplication by an element is always an abelian group homomorphismone only needs to check that l preserves multiplication. However since for any y P Y , ey is in X and hence eye “ ey it follows that l p y y q “ ey y “ ey ey “ l p y q l p y q . Finally note that X has trivial center since ex “ x for all x P X . Remark 4.10.
The previous proof can be easily adapted to prove that proto-completeness, completeness and strong-completeness coincide and are equivalentto being unitary for the category of algebras over an arbitrary ring. It is alsoeasy to show that a morphism f : X Ñ Y in Rng is a unitary ring morphism ifand only if it is strong-complete in the category of morphisms of
Rng . In thisway one can recover the category of unitary rings from the category of rings.
Remark 4.11.
The Propositions 4.7 and 4.8 show that in general for SC , C and PC the classes of strong-complete, complete and proto-complete objects ina semi-abelian category the inclusions SC Ă C Ă PC are strict. However wewill recall the above mentioned fact that a group is strong-complete if and onlyif it is what we called complete, and we will show that a Lie algebra over acommutative ring is strong-complete if and only if it is complete if and only ifit is proto-complete. We will also show that there are groups, in particular thegroup Z { Z is such an example, that are proto-complete but not complete. Proposition 4.12.
Let C be a pointed protomodular category, let S and T beobjects in C , and let X “ S ˆ T . If X is proto-complete, complete, complete ˚ or strong-complete, then so are both S and T .Proof. For the cases where X is proto-complete, complete or complete ˚ notethat if n : S Ñ Y is a protosplit/normal/Bourn-normal monomorphism, then n ˆ T is too. This means that in each case n ˆ T is a split monomorphism.Since the diagram S x , y (cid:15) (cid:15) n / / Y x , y (cid:15) (cid:15) S ˆ T n ˆ T / / Y ˆ T commutes and the composite of two split monomorphisms is a split monomor-phism it follows that x , y n “ p n ˆ T qx , y is a split monomorphism and hence n is too. The claim now follows from Corollary 3.4 and Proposition 4.5 .The converse of the implications of the previous proposition don’t hold ingeneral, but do hold in certain categories.17 xample 4.13. The product of strong-complete objects need not be proto-complete.If X is a non-zero strong-complete group, then X ˆ X has trivial center andhence is strong-complete if and only if it is proto-complete. However the auto-morphism X ˆ X Ñ X ˆ X defined by p x, y q ÞÑ p y, x q is clearly not inner (whichis sufficient via Baer’s theorem). On the other hand since the product of unitaryrings is unitary we see that product of strong-complete objects in the category ofrings is always strong-complete. It is also easy to check that the converse of theimplications of the previous proposition do hold in each abelian category. Recall that every unital variety of universal algebras has centers of objects.Recall also, that in a unital category C , central subobjects are normal whenever C satisfies the condition that for each composite f “ αn where α is a splitepimorphism and n is a normal monomorphism f is a normal monomorphismas soon as it is a monomorphism. In particular every semi-abelian variety ofuniversal algebras and every semi-abelian algebraically cartesian closed categorysatisfies these properties. Furthermore, we have (the probably known fact whichwe couldn’t find a reference for): Proposition 4.14.
Let C be a pointed protomodular category. If m : S Ñ X isa central monomorphism in C , then m is Bourn-normal.Proof. Suppose m : S Ñ X is a central monomorphism with cooperator ϕ : S ˆ X Ñ X . It follows that the diagram S m (cid:15) (cid:15) x , y / / S ˆ X π / / x ϕ,π y (cid:15) (cid:15) X x , y o o X x , y / / X ˆ X π / / X x , y o o is a morphism of split extensions and hence by protomodularity x ϕ, π y is amonomorphism and m a Bourn-normal monomorphism.For an object X in a pointed weakly unital category consider the condition: Condition 4.15.
The center z X : Z p X q Ñ X exists, x z X , z X y : Z p X q Ñ X ˆ X is a normal monomorphism, and the cokernels of z X and x z X , z X y exist. Note that the above condition holds for each object in a semi-abelian cat-egory admitting centers and hence in each semi-abelian variety of universalalgebras. On the other hand it holds for each abelian object in a unital cate-gory.
Theorem 4.16.
Let C be a pointed protomodular category and let X be anobject in C . If X is proto-complete (respectively complete or complete ˚ ) andsatisfies Condition 4.15, then X is the product of its center Z p X q which isan abelian proto-complete (respectively complete or complete ˚ ) object and thequotient object X { Z p X q which is a strong-complete object. roof. Let X be a proto-complete object in C and let p Z, z q be the center of X .Consider the diagram Z x z,z y (cid:15) (cid:15) Z z (cid:15) (cid:15) X x , y / / X ˆ X π / / ϕ (cid:15) (cid:15) X x , y o o c (cid:15) (cid:15) X x , y / / B ˆ X π / / B x ,θ y o o in which c and ϕ are cokernels of z and x z, z y , respectively, and the lower inducedsplit extension is of the form presented by Lemma 4.3 . This means that thediagram X x , y / / ●●●●●●●●● X ˆ X π ϕ (cid:15) (cid:15) X x , y o o X { { ✇✇✇✇✇✇✇✇✇ X commutes and so the morphism π ϕ x , y is central. It follows there exists¯ f : X Ñ Z such that z ¯ f “ π ϕ x , y and hence cπ ϕ x , y “
0. Therefore, sincethe morphisms x , y and x , y are jointly epimorphic it follows that cπ ϕ “ cπ and hence that cθc “ cπ x , θ y c “ cπ ϕ x , y “ cπ x , y “ c . This means that c is a split epimorphism which means that its kernel being central is a productinclusion . Indeed, if γ : B ˆ Z Ñ X is the cooperator of z and θ , then since x , y and x , y are jointly epimorphic it follows that the diagram Z x , y / / B ˆ Z π / / γ (cid:15) (cid:15) B x , y o o Z z / / X c / / B θ o o is a morphism of split extensions and γ is an isomorphism. Since x , y : Z Ñ B ˆ Z is the center of B ˆ Z it follows by Proposition 3.3 that B has trivialcenter. The claim now follows from Propositions 4.5, 4.7 and 4.12 .The previous theorem raises the question of whether the product of a proto-complete abelian object and a strong-complete object is necessarily proto-complete.This turns out to be false in general. Let S be the symmetric group on a threeelement set. It is well-known (and easy to check) that S is complete=strong-complete. If the group X “ Z { Z ˆ S is proto-complete, then by the pre-vious theorem since x , y : Z { Z Ñ X is the center of X we must have theAut p X q – S – Aut p S q and so every automorphism of X must be of the form1 ˆ φ where φ P Aut p S q . However, recalling that the group S is isomor-phic to the semi-direct product Z { Z ˙ θ Z { Z where θ : Z { Z Ñ Aut p Z { Z q is19the isomorphism) defined by 1 ÞÑ p x ÞÑ ¨ x q and hence forms part of a splitextension Z { Z k / / S p / / Z { Z i o o and writing a : Z { Z ˆ Z { Z Ñ Z { Z for the addition morphism of Z { Z , wesee that the morphism x a p Z { Z ˆ p q , π y : X Ñ X is an automorphism which isnot of this form. On the other hand we do have: Proposition 4.17.
Let C be a pointed protomodular category in which central-izers of Bourn-normal monomorphisms exists and are Bourn-normal. Suppose A is an abelian and complete ˚ object of C , and B is a strong-complete object of C . If hom p B, A q “ t u , then B ˆ A is complete ˚ .Proof. Suppose n : B ˆ A Ñ Y is a Bourn-normal monomorphism. Since thepullback of the centralizer of a monomorphism f along f is the center of itsdomain it follows that there is a unique morphism m making the diagram A m / / x , y (cid:15) (cid:15) Z Y p B ˆ A, n q z n (cid:15) (cid:15) B ˆ A n / / Y a pullback. This means that n x , y (being the binary meet of Bourn-normalmonomorphisms) is Bourn-normal and hence by Lemma 4.4 there is an iso-morphism θ : Y Ñ C ˆ A such that θn x , y “ x , y . Therefore, sincehom p B, A q “ t u it follows that π θn x , y “ θn x , y “ x ˜ n, y where ˜ n “ π θn x , y . Since x , y and x , y are jointly epimorphic, it followsthat θn “ ˜ n ˆ
1. Noting that the diagram B ˜ n / / x , y (cid:15) (cid:15) C x , y (cid:15) (cid:15) B ˆ A ˜ n ˆ / / C ˆ A is a pullback, we see that ˜ n is a Bourn-normal monomorphism and hence it and n are split monomorphisms. Lemma 4.18.
Let C be a pointed protomodular category and let X be an objectadmitting a generic split extension with kernel X . The following are equivalent:(a) the generic split extension with kernel X is of the form X x , y / / r X s ˆ X π / / r X s x ,θ y o o for some morphism θ ; b) c X is a split epimorphism.Proof. (a) ñ (b): Consider the diagram X x , y / / r X s ˆ X θ ˆ (cid:15) (cid:15) π / / r X s θ (cid:15) (cid:15) x ,θ y o o X x , y / / X ˆ X ϕ (cid:15) (cid:15) π / / X x , y o o c X (cid:15) (cid:15) X x , y / / r X s ˆ X π / / r X s . x ,θ y o o where the lower part is the unique morphism into the generic split extensionwith kernel X which defines c X . The universal property of the generic splitextension implies that the composite c X θ “ r X s .(b) ñ (a): Suppose that c X is a split epimorphism with splitting θ : r X s Ñ X and consider the diagram X x , y / / r X s ˆ X θ ˆ (cid:15) (cid:15) π / / r X s θ (cid:15) (cid:15) x ,θ y o o X x , y / / X ˆ X ϕ (cid:15) (cid:15) π / / X x , y o o c X (cid:15) (cid:15) X k / / r X s ˙ X p / / r X s i o o in which the lower part is the unique morphism into the generic split extensionwith kernel X which defines c X . The claim now follows from the fact that since c X θ “ r X s , the split short five lemma implies that ϕ p θ ˆ q is an isomorphism. Theorem 4.19.
Let C be a pointed protomodular category. For an object X in C the following are equivalent:(a) X is proto-complete and satisfies Condition 4.15,(b) the generic split extension with kernel X exists and is of the form X x , y / / r X s ˆ X π / / r X s x ,θ y o o for some morphism θ .(c) the generic split extension with kernel X exists and c X is a split epimor-phism. roof. The equivalence of (b) and (c) follow from the previous lemma. Supposethe assumptions in (a) hold. According to Theorem 4.16 there exists objects A and r X s in C such that A is abelian and proto-complete, r X s is strong-complete,and X “ r X s ˆ A . We will show that X x , y / / r X s ˆ X π / / r X s x , x , yy o o (12)is a generic split extension with kernel X . For any split extension with kernel X , which by Lemma 4.3 is (up to isomorphism) of the form displayed at the topof the following diagram, the composite of the morphisms X x , y / / B ˆ X π / / x π , x π π ,π π ´ gπ yy (cid:15) (cid:15) B x , x f,g yy o o X x , y / / B ˆ X π / / f ˆ (cid:15) (cid:15) B x , x f, yy o o f (cid:15) (cid:15) X x , y / / r X s ˆ X π / / r X s , x , x , yy o o gives a morphism to (12). It remains only to show that this is the unique suchmorphism. By protomodularity it is sufficient to show that for a morphism ofsplit extensions X x , y / / B ˆ X π / / u (cid:15) (cid:15) B x , x f,g yy o o v (cid:15) (cid:15) X x , y / / r X s ˆ X π / / r X s , x , x , yy o o we must have v “ f . To do so note that the diagram r X s x , y / / r X s (cid:15) (cid:15) x , x , yy & & X x , y ●●●●●●●●● x , y / / B ˆ X u (cid:15) (cid:15) r X s X π o o r X s ˆ X π o o commutes, and so since r X s is strong-complete, and x , x , yy is normal withsplitting π π it follows that π π u “ π π . This means that v “ π π x , x , y v “ π π u x , x f, g yy “ π π x , x f, g yy “ f as desired. The claim now follows by noting that: (b) implies that every pro-tosplit monomorphism with domain X factors through x , y : X Ñ r X s ˆ X c X and ϕ in (7), whose kernels are z X and x z X , z X y respectively, are normalepimorphisms. Corollary 4.20.
Let C be a pointed Barr-exact protomodular category admittingcenters. If X is complete (respectively proto-complete), then, r X s , the splitextension classifier for X exists and is strong-complete, Z p X q is an abeliancomplete (respectively proto-complete) object, and X – Z p X q ˆ r X s . Corollary 4.21.
Let C be a pointed Barr-exact protomodular category admittingcenters. An object X in C is complete and abelian, if and only if the splitextension classifier for X is the zero object.Proof. Let X be an object. Since every complete object has a split extensionclassifier, we may assume that the split extension classifier for X exists, andhence so does c X . Recall that z X is the kernel of c X , X is abelian if and onlyif z X is an isomorphism, and according to Theorem 4.19, X is proto-completeif and only if c X splits. Now, if r X s – c X “ z X is an isomorphism. Conversely, if z X is anisomorphism and c X is a split epimorphism, then c X being the cokernel of z X is zero and r X s – Proposition 4.22. If X is a group such that Aut p X q – , then X is isomorphicto or Z { Z . The only (up to isomorphism) non-zero abelian proto-completegroup is Z { Z .Proof. Since X is abelian it follows that the map sending x to ´ x is an auto-morphism and hence must be equal to 1 X . This means that for each x in X ,0 “ x ` ´ x “ x ` x “ x and so X is a vector space over Z { Z . Since anyvector space has a non-trivial automorphism corresponding to any non-trivialpermutation of elements in its basis it follows that X has dimension at mostone and hence is isomorphic to Z { Z or 0. Proposition 4.23. If X is a Lie algebra over a commutative ring R such that Der p X q – , then X – . There are no non-zero abelian proto-complete Liealgebras.Proof. Since X is abelian it follows that any R -linear map is a derivation sothat in particular the identity map is a derivation and hence must be equal to0. Proposition 4.24.
Let C be an anti-additive action representable semi-abeliancategory. If r X s – , then X – .Proof. The claim is immediate since X is necessarily abelian and hence isomor-phic to 0. 23ecall that for an object X , in a pointed protomodular category admittinga generic split extension with kernel X , the object Out p X q is codomain of thecokernel q : r X s Ñ Out p X q of c X . The following theorem should be comparedto Theorems 9.1 and 9.2 of [4]. Theorem 4.25.
Let C be a pointed protomodular category. For an object X the following are equivalent:(a) X is strong-complete;(b) the generic split extension with kernel X exists and the conjugation mor-phism c X : X Ñ r X s is an isomorphism;(c) the split extension X x , y / / X ˆ X π / / X x , y o o (13) is a generic split extension;(d) the generic split extension with kernel X exists, X has trivial center, and Out p X q is trivial.If in addition every abelian complete object in C is a zero object, then theseconditions are further equivalent to:(e) X is complete and satisfies Conditon 4.15.Proof. Considering the definition of c X and recalling that the kernel of c X iscenter of X and cokernel of c X is Out p X q it easily follows that (b), (c) and (d)are equivalent. Using again the fact that the center of X is kernel of c X theequivalence of (a) and (b) follows from Theorem 4.19 . The final claim followsfrom Theorem 4.16 since it implies that Z p X q is abelian and complete and henceby assumption a zero object. Remark 4.26.
Note that Conditions 4.25 (a) and (c) are equivalent for anobject X in a pointed finitely complete category. The content of the following proposition is well-known, we include a proofto keep the paper more self contained.
Proposition 4.27.
There are no (non-zero) abelian complete groups. Thegroup Z { Z is proto-complete but not complete.Proof. Since every complete group is proto-complete and the only non-zeroabelian proto-complete group is Z { Z it suffices to prove the final claim. How-ever, this follows trivially from fact that the canonical monomorphism from Z { Z into Z { Z is a normal monomorphism which isn’t a split monomorphism. Remark 4.28.
Note that Baer’s theorem which can stated using our terminol-ogy, as: every complete group is strong-complete, is now a corollary of Theorem4.25 via the previous proposition. Furthermore, we have that if a group G isproto-complete, then G is complete or G – Z { Z ˆ H where H is complete. Characteristic monomorphisms
The main purpose of this section is show that there is a common categoricalexplanation behind why the derivation algebra of a perfect Lie algebra withtrivial center and the automorphism group of a (characteristically) simple non-abelian group are complete.Recall that a subgroup S of a group X is called characteristic if every auto-morphism of X restricts to an automorphism of S . Recall also that a subgroup S of X is characteristic if and only if whenever X is normal in Y , then S isnormal in Y . Generalizing this latter condition A. S. Cigoli and A. Montoliintroduced and studied the notion of a characteristic subobjects in semi-abeliancategories [22]. Later D. Bourn gave a different definition in a more general con-text [14], which coincides with the previously mentioned one in the semi-abeliancontext. Here we say that a morphism u : S Ñ X in a category C is character-istic monomorphism if for each Bourn-normal monomorphism n : X Ñ Y thecomposite un is a Bourn-normal monomorphism. In [21] it was shown that for asemi-abelian category, a morphism u : S Ñ X is a characteristic monomorphismif and only if for each protosplit monomorphism κ : X Ñ A the composite κu is a normal monomorphism. This fact essentially remains true in any pointed Mal’tsev category using the above definition of characteristic monomorphism.Recall that a category is Mal’tsev if it is finitely complete and each (internal)reflexive relation is an (internal) equivalence relation. Note that Mal’tsev cat-egories were first introduced and studied in [17] with exactness as part of thedefinition, exactness was removed in [18].
Proposition 5.1.
Let C be a pointed finitely complete Mal’tsev category andlet u : S Ñ X be morphism in C . The following are equivalent:(a) u is a characteristic monomorphism;(b) for each protosplit normal monomorphism κ : X Ñ A the composite κu isa Bourn-normal monomorphism;(c) for each reflexive relation p R, r , r q on an object Y with k : X Ñ R as kernelof r , there exists a monomorphism of reflexive relations v : p T, t , t q Ñp R, r , r q with l : S Ñ T as kernel of t such that vl “ ku ;(d) the same as (c) but replace “reflexive relation” by “equivalence relation”.Proof. Trivially (a) implies (b) and (c) is equivalent to (d). Now suppose that(b) holds and let p R, r , r q be a reflexive relation on Y with k : X Ñ R as kernel of r . Since r is a split epimorphism we know that k is a protosplitmonomorphism and hence ku is a Bourn-normal monomorphism by assumption.Let us write p ¯ T , ¯ t , ¯ t q for the equivalence relation on R that k is zero-class ofand ¯ l : S Ñ ¯ T for the kernel of ¯ t so that ¯ t ¯ l “ ku . Now, suppose that e : Y Ñ R and ¯ f : R Ñ ¯ T are the unique morphisms such that r i e “ Y and ¯ t i ¯ f “ R and25onsider the pullback of split extensions S S (cid:31) (cid:31) ❄❄❄❄❄ l / / u (cid:15) (cid:15) T t / / v (cid:15) (cid:15) w (cid:31) (cid:31) ❄❄❄❄❄ Y f o o e (cid:31) (cid:31) ❄❄❄❄❄ S ¯ l / / ku (cid:15) (cid:15) ¯ T ¯ t / / x ¯ t , ¯ t y (cid:15) (cid:15) R ¯ f o o X k / / k (cid:31) (cid:31) ❄❄❄❄❄ R r / / x er , y (cid:31) (cid:31) ❄❄❄❄❄ Y e o o e (cid:31) (cid:31) ❄❄❄❄❄ R x , y / / R ˆ R π / / R. x , y o o Setting t “ r v we obtain the desired monomorphism of reflexive relations v : p T, t , t q Ñ p R, r , r q proving (b) implies (c). To complete the claim wewill show that (d) implies (a). Suppose that (d) holds and n : X Ñ Y is aBourn-normal monomorphism. By assumption there is an equivalence relation p R, r , r q on Y with k : X Ñ R as kernel of r such that n “ r k . According to(d) there is a monomorphism of equivalence relations v : p T, t , t q Ñ p R, r , r q with l : S Ñ T as kernel of t such that vl “ ku . This means that t l “ r vl “ r ku “ nu proving that nu is Bourn-normal.Recall that a pointed category with finite limits can be equivalently definedto be strongly protomodular in the sense of D. Bourn [11] if it is protomodularand for each morphism of split extensions X f (cid:15) (cid:15) κ / / A α / / g (cid:15) (cid:15) B β o o Z σ / / C γ / / B δ o o the morphism f is Bourn-normal if and only if σf is Bourn-normal. Note thatpart (i) of the following theorem is essentially known, see Proposition 3.1 of [14]. Theorem 5.2.
Let C be a pointed protomodular category, let X be an object in C such that the generic split extension with kernel X exists, and let u : S Ñ X be a morphism.(i) The morphism u : S Ñ X is a characteristic monomorphism if and only ifthe composite ku of u and k : X Ñ r X s ˙ X is a Bourn-normal monomor-phism;(ii) If u : S Ñ X is a characteristic monomorphism, then the generic splitextension with kernel p S, X, u q exists in C and q : r S, X, u s Ñ r X s is anisomorphism.Furthermore, when C is strongly protomodular the converse of (ii) holds. roof. Suppose u : S Ñ X is a morphism in C . By definition, if u is a charac-teristic monomorphism, then ku : S Ñ r X s ˙ X is Bourn-normal. Conversely,suppose ku : S Ñ r X s ˙ X is Bourn-normal and κ : X Ñ A is the kernel of asplit epimorphism α : A Ñ B with splitting β : B Ñ A . Accordingly there is aunique morphism of split extensions X κ / / A f (cid:15) (cid:15) α / / B β o o g (cid:15) (cid:15) X k / / r X s ˙ X p / / r X s . i o o (14)By protomodularity the right and downward directed arrows of the right handsquare of (14) form a pullback and hence the morphism x f, α y : A Ñ pr X s˙ X qˆ B is a monomorphism. Therefore, since x f, α y κu “ x ku, y which is a Bourn-normal monomorphism (being the intersection of the Bourn-normal monomor-phisms 1 ˆ ku and x , y - see e.g. Proposition 3.2.6 of [4]) it follows that κu isa Bourn-normal monomorphism as desired. Now suppose u is a characteristicmonomorphism, H “ r X s˙ X and p ¯ T , ¯ t , ¯ t q is the equivalence relation that ku isthe zero class of. Since C is protomodular there is a unique (up to isomorphism)equivalence relation to which ku is normal. This means that p ku, p ¯ T , ¯ t , ¯ t qq isthe normalizer of ku (in the sense of [15]) and hence by Proposition 2.4 of [15]the front face of the pullback of split extensions S S (cid:31) (cid:31) ❄❄❄❄❄ l / / u (cid:15) (cid:15) T t / / v (cid:15) (cid:15) w (cid:31) (cid:31) ❄❄❄❄❄ r X s f o o i (cid:31) (cid:31) ❄❄❄❄ S ¯ l / / ku (cid:15) (cid:15) ¯ T ¯ t / / x ¯ t , ¯ t y (cid:15) (cid:15) H ¯ f o o X k / / k (cid:31) (cid:31) ❄❄❄❄❄ H p / / x ip , y (cid:31) (cid:31) ❄❄❄❄❄ r X s i o o i (cid:31) (cid:31) ❄❄❄❄ H x , y / / H ˆ H π / / H x , y o o is a K -precartesian. However, by Lemma 2.7 of [15] this means that the backface is also K -precartesian and hence by Lemma 2.6 of [28] is the generic splitextension with kernel p S, X, u q in C .To see that the final claim follows from strong protomodularity. Just notethat in the diagram (6) with f “ u and q an isomorphism, strong protomodu-larity implies that the composite kf is Bourn-normal. Theorem 5.3.
Let C be a pointed protomodular category, and let X be an objectin C with trivial center such that the generic split extension with kernel X exists.A morphism u : S Ñ X is a characteristic monomorphism if and only if thecomposite c X u : S Ñ r X s is Bourn-normal. roof. Since X has trivial center, the morphism c X is monomorphism and hence(by protomodularity) so is the middle morphism x p , p y in the morphism X k / / c X (cid:15) (cid:15) r X s ˙ X p / / x p ,p y (cid:15) (cid:15) r X s i o o r X s x , y / / r X s ˆ r X s π / / r X s x , y o o of split extensions. Since x , c X u y “ x p , p y ku is a Bourn-normal monomor-phism and x p , p y is a monomorphism, it follows that ku is a Bourn-normalmonomorphism. The claim now follows by the previous proposition. Proposition 5.4.
Let C be a pointed protomodular category. Every normalsubobject of a proto-complete object is characteristic.Proof. Let X be a proto-complete object and let n : S Ñ X be a Bourn-normalmonomorphism. If κ : X Ñ A is a protosplit monomorphism, then accordingto Lemma 4.3 there is an isomorphism ψ : B ˆ X Ñ A , where B is the (objectpart of the) cokernel of κ , such that κ “ ψ x , y . Since x , n y is Bourn-normalit follows that κn “ ψ x , y n is Bourn-normal. Theorem 5.5.
Let C be a pointed protomodular category and let X be an objectin C such that the generic split extension with kernel X exists. There is a genericsplit extension with kernel r X s and the conjugation morphism c X : X Ñ r X s isa characteristic monomorphism if and only if X has trivial center and r X s isstrong-complete.Proof. Since (i) the morphism c X is a Bourn-normal monomorphism if andonly if X has trivial center (for the “if” part use Proposition 3.6 , and for theconverse just recall that the center is the kernel of c X ); (ii) by Theorem 4.25 if r X s is strong-complete, then there is a generic split extension with kernel r X s ,it follows that in addition to the assumptions above we may assume that X hastrivial center and that there is a generic split extension with kernel r X s , andprove that c X is a characteristic monomorphism if and only if r X s is strong-complete. Suppose that c X is a characteristic monomorphism. Then accordingto Theorem 5.2 the generic split extension with kernel p X, r X s , c X q exists in C .By Proposition 3.8 this means that the split extension classifier of p X, r X s , c X q is pr X s , rr X ss , c r X s q and so by Theorem 5.2 (again) the morphism c rr X ss is anisomorphism. Therefore, r X s is strong-complete by Theorem 4.25. The conversefollows from Proposition 5.4.We obtain the following known fact as a corollary: Proposition 5.6.
If a group X is characteristically simple (i.e. it has no propercharacteristic subobjects) and non-abelian, then Aut p X q is complete (=strong-complete). roof. Suppose that X is a characteristically simple non-abelian group andsuppose that θ is an automorphism of Aut p X q . Since the center is alwayscharacteristic it follows that Z p X q “ c X is a normal monomorphism.Forming the pullback K u / / v (cid:15) (cid:15) X c X (cid:15) (cid:15) X θc X / / Aut p X q we see that c X u is normal and hence by Theorem 5.3 that u is characteristic.By assumption this means that either K “ u is an isomorphism. However,since by Proposition 3.6 c X has trivial centralizer it follows that K is not trivialand hence u must be an isomorphism. Essentially the same argument impliesthat v is an isomorphism which proves that c X is characteristic and hence bythe previous theorem implies that Aut p X q is complete. Proposition 5.7.
Let C be a category of interest in the sense of G. Orzech [35]such that the group operation (required to exists) is commutative, and let X bean object in C .(i) If X is perfect, then every normal monomorphism with domain X is acharacteristic monomorphism;(ii) if X is perfect, has trivial center, and the generic split extension withkernel X exists, then r X s is strong-complete.Proof. Recall that for such a variety of universal algebras a subobject S ď Y is normal if and only if for each s in S , each y in Y and each binary operation ˚ (excluding addition), s ˚ y and y ˚ s are in S . Recall also that r X, X s is thesubalgebra of X generated by elements of the form x ˚ x where x and x areelements of X and ˚ is a binary operation (exlcuding addition). Now suppose X is perfect (i.e. r X, X s “ X ), X is a normal subobject of Y , and Y is a normalsubobject of Z . Since C is category of interest for binary operations ˚ and ‚ weknown that for some postive integer n there are binary operations ˚ , ..., ˚ n and ‚ , ..., ‚ n and a term w such that (in particular) for all x , x in X and z in Z p x ˚ x q ‚ z “ w p x ˚ p x ‚ z q , ..., x ˚ m p x ‚ m z q , x ˚ m ` p x ‚ m ` z q , ..., x ˚ n p x ‚ n z qq , where m is an integer between 0 and n . Therefore, since each x j ‚ i z isin Y it follows that x k ˚ l p x j ‚ z q is in X and hence p x ˚ x q ‚ z is in X . A similarcalculation shows that z ‚ p x ˚ x q is in X . Since X is perfect we know that X is generated by products and hence via the previous calculations is normalin Z . This proves (i). Combining (i) with Proposition 3.6 and Theorem 5.5 weobtain (ii). Remark 5.8.
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