Intrinsic Schreier special objects
aa r X i v : . [ m a t h . C T ] F e b INTRINSIC SCHREIER SPECIAL OBJECTS
ANDREA MONTOLI, DIANA RODELO, AND TIM VAN DER LINDEN
Abstract.
Motivated by the categorical-algebraic analysis of split epimorph-isms of monoids, we study the concept of a special object induced by the in-trinsic Schreier split epimorphisms in the context of a regular unital categorywith binary coproducts, comonadic covers and a natural imaginary splittingin the sense of our article [20]. In this context, each object comes natur-ally equipped with an imaginary magma structure. We analyse the intrinsicSchreier split epimorphisms in this setting, showing that their properties im-prove when the imaginary magma structures happen to be associative. Wecompare the intrinsic Schreier special objects with the protomodular objects,and characterise them in terms of the imaginary magma structure. We fur-thermore relate them to the Engel property in the case of groups and Liealgebras. Introduction
Recently, two different categorical approaches have been developed which aimto describe the homological properties of monoids, mainly in comparison with theproperties groups have. The first one started with the observation that an im-portant class of split epimorphisms of monoids, called
Schreier split epimorphisms ,satisfies the convenient properties of split epimorphisms of groups [8, 9]. The ideaof considering Schreier split epimorphisms originated from the fact that these splitepimorphisms correspond to monoid actions in the usual sense [21, 18]. Althoughthe category of monoids is not protomodular, Schreier split epimorphisms satisfy theproperties that are typical for split epimorphisms in a protomodular category. Thisled to the notion of an S -protomodular category , with respect to a chosen class S of points—i.e., split epimorphisms with fixed section [10]. In an S -protomodularcategory, it is always possible to identify a full subcategory which is protomodu-lar [2], called in [9] the protomodular core with respect to the class S . The objectsof this subcategory are the S -special objects , namely those objects X for which thesplit epimorphism X ˆ X Ô X, given by the second product projection and thediagonal morphism, belongs to S . The category of monoids is not protomodularbut it is S -protomodular with respect to the class of Schreier split epimorphisms,and its protomodular core is the category of groups.The second approach consists in considering, in a pointed category with finitelimits, a suitable class of objects, called protomodular objects [19]. These are theobjects Y such that every split epimorphism with codomain Y is stably strong . Asplit epimorphism is strongly split if its kernel and section are jointly extremal-epimorphic. It is stably strong if every pullback of it along any morphism is a Date : 12th February 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Imaginary morphism; approximate operation; regular, unital, proto-modular category; monoid; -Engel group, Lie algebra; Jónsson–Tarski variety.The second author acknowledges financial support from the Centre for Mathematics of theUniversity of Coimbra (UID/MAT/00324/2020, funded by the Portuguese Government throughFCT/MCTES).The third author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS. strongly split epimorphism. As proved in [19], in the category of monoids theprotomodular objects are precisely the groups.The notion of protomodular object makes sense in every (pointed) category withfinite limits, while Schreier special objects can apparently be considered only inthe context of a Jónsson–Tarski variety [15], because the notion of Schreier splitepimorphism depends on the existence of a function, which is not a morphism ingeneral, called the Schreier retraction . In order to study this from a categorical per-spective, we introduced in [20] the concept of intrinsic Schreier split epimorphism ,in the context of a regular unital category [3] equipped with a comonadic cover (inthe sense we recall in Subsection 2.3). This approach is inspired by the notion ofimaginary morphism [5]: indeed, the Schreier retraction we need is such an imagin-ary morphism. We showed in [20] that these categories are S -protomodular withrespect to the class of intrinsic Schreier split epimorphisms, and we obtained anintrinsic version of the so-called Schreier special objects. It is shown in [20] thatthe concepts of intrinsic Schreier special object and protomodular object are inde-pendent. Since, however, the two coincide in the category of monoids, the questionof understanding when the two notions are related arises naturally.One of the goals of the present paper is to give an answer to this question.An important ingredient here is the observation that, when considering the Kleislicategory associated with the comonad involved in the definition of an intrinsicSchreier split epimorphism, the definition itself simplifies greatly (Section 5). Also,each object admits a canonical imaginary magma structure whose operation (called imaginary addition in the text) depends on a choice of a natural imaginary splitting,which is part of our initial setting (Section 4). It turns out that an object is in-trinsic Schreier special precisely when its imaginary magma structure is a one-sidedloop structure (Theorem 8.2). Under the assumption that the imaginary additionis associative (Section 6) we are able to extend several stability properties and ho-mological lemmas which hold for Schreier extensions of monoids [8] to our intrinsiccontext (Section 7). Moreover, we prove that every intrinsic Schreier special objectis a protomodular object (Corollary 10.4).It was shown in [20] that there are only two possible choices for the naturalimaginary splitting in the category of monoids, which leads to only two possibleimaginary additions. This is no longer true for the category of groups or Lie algeb-ras, where many options are available. Therefore, we focus on studying intrinsicSchreier special objects with respect to all natural imaginary additions in these cat-egories. We prove that -Engel groups are intrinsic Schreier special with respect toall possible imaginary additions (Proposition 11.10). The similar result also holdsfor Lie algebras (Proposition 12.1).2. Imaginary morphisms
In this section we recall the notions of imaginary morphism needed throughoutthe text. We fix the particular setting for which we consider imaginary morphismsin this work.2.1.
Imaginary morphisms [7] . We take X to be the functor category Set C op ˆ C ,where C is an arbitrary (small) category. Consider functors hom C and A : C op ˆ C Ñ Set and a natural transformation α : hom C ñ A . If all the components α X,Y : hom C p X, Y q Ñ A p X, Y q are injective, then all sets A p X, Y q contain (anisomorphic copy of) hom C p X, Y q . So, we may think of A p X, Y q as an extension of hom C p X, Y q , and indeed in [7] the triple p C , A, α q was called an extended cat-egory . The elements of A p X, Y qz hom C p X, Y q will be called imaginary morph-isms . Sometimes it will be convenient to call a morphism in hom C p X, Y q a realmorphism to emphasise that it is an actual morphism in C . NTRINSIC SCHREIER SPECIAL OBJECTS 3
We use arrows of the type X , ❴❴❴ Y to represent an element of A p X, Y q , which could be an imaginary morphism or not.To distinguish those which are not, i.e., the elements of A p X, Y q corresponding toa real morphism, say f : X Ñ Y , we tag the dashed arrow with the name of thatreal morphism overlined (instead of α X,Y p f q ): X f , ❴❴❴ Y. It is possible to define an extended composition, denoted by ˝ , between real andimaginary morphisms as follows: X a , ❴❴❴ v ˝ a ; P ❯ ❩ ❴ ❞ ✐ ♥ Y v , V, where v ˝ a “ A p X , v qp a q and U u , a ˝ u ; P ❯ ❩ ❴ ❞ ✐ ♥ X a , ❴❴❴ Y, where a ˝ u “ A p u, Y qp a q . If a corresponds to a real morphism, i.e., a “ f “ α X,Y p f : X Ñ Y q , then thesame is true for v ˝ a and a ˝ u . Indeed, by the naturality of α we have A p X , v qp α X,Y p f qq “ α X,V p vf q , so that v ˝ f “ vf p“ α X,V p vf qq corresponds to the real morphism vf . Similarly, f ˝ u “ f u p“ α U,Y p f u qq corresponds to the real morphism f u . In particular, weobtain identity axioms v ˝ Y “ v and X ˝ u “ u . There is also an associativityaxiom, which follows from the fact that A is a functor p v ˝ a q ˝ u “ A p u, V qp A p X , v qp a qq “ A p u, v qp a q“ A p U , v qp A p u, Y qp a qq “ v ˝ p a ˝ u q . Definition 2.2.
We say that a real morphism f : X Ñ Y admits an imagin-ary splitting when there exists an imaginary morphism s such that the followingdiagram commutes Y s , ❴❴❴ f ˝ s “ Y ; P ❯ ❩ ❴ ❞ ✐ ♥ X f , Y. Comonadic covers.
We assume that C is a regular category equipped with acomonad p P, δ, ε q whose counit ε is a regular epimorphism. We let ε X : P p X q ։ X denote the chosen cover of some object X in C : ε X is a regular epimorphism. Notethat for any morphism f : X Ñ Y in C f ε X “ ε Y P p f q (1)and P p f q δ X “ δ Y P p f q , (2)where P “ P P . Also ε P p X q δ X “ P p X q “ P p ε X q δ X (3)and P p δ X q δ X “ δ P p X q δ X , (4)for all objects X in C . ANDREA MONTOLI, DIANA RODELO, AND TIM VAN DER LINDEN
Example . If V is a variety of universal algebras, then we may consider the freealgebra comonad p P, δ, ε q . For any algebra X , we have ε X : P p X q ։ X and δ X : P p X q ã Ñ P p X q , r x s ÞÑ x r x s ÞÑ rr x ss where r x s denotes the one letter word x ; such words are the generators of P p X q .In this case, any function f : X Ñ Y between algebras X and Y extends uniquelyto a morphism P p X q Ñ Y r x s ÞÑ f p x q in V .2.5. Imaginary morphisms induced from comonadic covers.
The idea be-hind functions extending to real morphisms in Example 2.4 can be captured throughthe notion of imaginary morphism: it is like a function (not a morphism) X Y of algebras X and Y that extends to an actual morphism of algebras P p X q Ñ Y .More precisely, given a regular category C with comonadic covers we define thefunctor A : C op ˆ C ÝÑ Set , p X, Y q ÞÝÑ hom C p P p X q , Y q u Ò Ó v Ó A p u, v qp U, V q ÞÝÑ hom C p P p U q , V q where A p u, v q “ hom C p P p u q , v q . So, A is just the functor hom C p P op ˆ C q .The components of α : hom C ñ A are defined, for all objects X , Y by α X,Y : hom C p X, Y q ÝÑ hom C p P p X q , Y q .X f Ñ Y ÞÝÑ P p X q ε X ։ X f Ñ Y Note that α is indeed a natural transformation because ε is (see (1)). Also, thecomponents α X,Y are injective, for all objects
X, Y , since ε X is a regular epimorph-ism. Since the elements of A p X, Y q “ hom C p P p X q , Y q are actual morphisms in C ,an arrow of the type X Y corresponds to a morphism P p X q Ñ Y . Accordingto Subsection 2.1:— if X f Ñ Y is a real morphism, then X f Y corresponds to the morphism P p X q fε X ÝÑ Y , so f “ f ε X ;— an imaginary morphism X a Y is a (real) morphism P p X q a Ñ Y which is not of the type a “ f ε X , for some real morphism X f Ñ Y ;— the composition of a real morphism with an imaginary one is defined by $’’’’&’’’’% X a , ❴❴❴ v ˝ a ; P ❯ ❩ ❴ ❞ ✐ ♥ Y v , V , where v ˝ a “ va : P p X q ÝÑ V , U u , a ˝ u ; P ❯ ❩ ❴ ❞ ✐ ♥ X a , ❴❴❴ Y , where a ˝ u “ aP p u q : P p U q ÝÑ Y . Convention 2.6.
From now on, we only consider imaginary morphisms that areinduced from comonadic covers.
Remark . It is clear that in this setting the existence of an imaginary splitting(Definition 2.2) for a morphism f implies that f is a regular epimorphism ( f ˝ s “ Y implies that f s “ ε Y , which is a regular epimorphism). The converse holds whenthe values of P are projective objects in C . If f : X ։ Y is a regular epimorphism, NTRINSIC SCHREIER SPECIAL OBJECTS 5 composition ˝ composition in C X a , ❴❴❴ v ˝ a ; P ❯ ❩ ❴ ❞ ✐ ♥ Y v , V P p X q a , Y v , VX a , ❴❴❴ v ˝ a ; P ❯ ❩ ❴ ❞ ✐ ♥ Y v , ❴❴❴ V P p X q δ X , (3) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ P p X q P p a q , ε P p X q (cid:30) ( ❍❍❍❍❍❍❍❍❍ P p Y q ε Y , (1) v , VP p X q a D ⑥⑥⑥⑥⑥⑥⑥⑥⑥ U u , a ˝ u ; P ❯ ❩ ❴ ❞ ✐ ♥ X a , ❴❴❴ Y P p U q P p u q , P p X q a , YU u , ❴❴❴ a ˝ u ; P ❯ ❩ ❴ ❞ ✐ ♥ X a , ❴❴❴ Y P p U q δ U , (3) P p U q P p ε U q , P p U q P p u q , P p X q a , Y Table 1.
Composition in the Kleisli categorythen f admits an imaginary splitting because P p Y q is projective P p Y q D s x (cid:2) ③③③③③③③③ ε Y (cid:12) (cid:18) (cid:12) (cid:18) X f , , Y ; thus f s “ ε Y . So the existence of imaginary splittings characterises regular epi-morphisms in this setting. Moreover, P p f q is a split epimorphism since P p f q P p s q δ Y “ P p ε Y q δ Y (3) “ P p Y q . The Kleisli category
Let C be a regular category with comonadic covers. We denote by K theKleisli category associated to the comonad p P, δ, ε q . Its objects are those of C and hom K p X, Y q “ hom C p P p X q , Y q . The morphisms of K are the imaginary morphismstogether with those of the type f : X Y , for some real morphism f : X Ñ Y (Subsection 2.5).The composition in K will also be denoted by ˝ (as in Subsections 2.5 and 2.1) A a , ❴❴❴ b ˝ a ; P ❯ ❩ ❴ ❞ ✐ ♥ B b , ❴❴❴ C, where b ˝ a corresponds to the morphism in C P p A q δ A , P p A q P p a q , P p B q b , C. Remark . If any of the morphisms in a composite in K corresponds to a realmorphism, then this composite coincides with the one defined in Subsections 2.5and 2.1—See Table 1.The comonad p P, δ, ε q gives rise to an adjunction K Õ C , where the right adjointis the embedding I : C Ñ K : X f , Y ÞÑ X f , ❴❴❴ Y ANDREA MONTOLI, DIANA RODELO, AND TIM VAN DER LINDEN
Consequently, K has a limit for every finite diagram in C , which is just the limit ofthat diagram in C , embedded into K .4. Imaginary addition in unital categories
In this section we define an imaginary addition on each object X of a unitalcategory with comonadic covers, i.e., an imaginary morphism µ X : X ˆ X X such that µ X ˝ x X , y “ X and µ X ˝ x , X y “ X . Such an imaginary additionprovides one of the tools needed to define intrinsic Schreier split epimorphisms inSection 5.4.1. Unital categories [3] . A pointed and finitely complete category is called unital when, for all objects
A, B , A x A , y , A ˆ B B x , B y l r is a jointly extremal-epimorphic pair. Example . As shown in [1], a variety of universal algebras V is unital if and onlyif it is a Jónsson-Tarski variety [15]. Recall that a Jónsson–Tarski variety is suchthat its theory contains a unique constant and a binary operation ` satisfying theidentities x ` “ x “ ` x . So an algebra is a unitary magma, possibly equippedwith additional operations.A pointed finitely complete category C is unital if and only if for any punctualspan in C A s , C g , , f l r l r B, t l r f s “ A , gt “ B , f t “ , gs “ the factorisation x f, g y : C ։ A ˆ B is a strong epimorphism (Theorem 1.2.12 in [1]).Consequently, a pointed regular category with binary coproducts is unital if andonly if for all objects A, B , the comparison morphism r A,B “ v A
00 1 B w : A ` B ։ A ˆ B is a regular epimorphism.4.3. Natural imaginary splittings [20] . If C is a regular unital category withbinary coproducts and comonadic covers, then for all objects A, B , the comparisonmorphism r A,B “ v A
00 1 B w : A ` B ։ A ˆ B is a regular epimorphism. When P p A ˆ B q is a projective object, as in the varietal case, there exists a (not necessarilyunique) morphism t A,B : P p A ˆ B q Ñ A ` B such that r A,B t A,B “ ε A ˆ B (5)(see Remark 2.7). Example . Let V be a Jónsson–Tarski variety. For any pair of algebras p A, B q in V , we can make the following choices of an imaginary splitting for r A,B : the directimaginary splitting t d rp a, b qs ÞÑ a ` b which sends a generator rp a, b qs P P p A ˆ B q to the sum of a “ ι A p a q with b “ ι B p b q in A ` B (where ι A and ι B are the coproduct inclusions); and the twisted imaginarysplitting t w rp a, b qs ÞÑ b ` a which does the same, but in the opposite order. Note that each of those choicesdetermines a natural transformation t : P pp¨q ˆ p¨qq ñ p¨q ` p¨q NTRINSIC SCHREIER SPECIAL OBJECTS 7 such that rt “ ε p¨qˆp¨q , where r : p¨q ` p¨q ñ p¨q ˆ p¨q and ε p¨qˆp¨q : P pp¨q ˆ p¨qq ñ p¨q ˆ p¨q . It was shown in [20] that when V is the category Mon of monoids, then the abovechoices of natural imaginary splittings (direct and twisted) are the only options.As we shall see below, this is far from being true in general.We make the existence of a natural t into an axiom. Let C be a pointed regular(unital) category C with binary coproducts and comonadic covers. Suppose alsothat there exist t A,B such that (5) holds and that they are the components of anatural transformation t : P pp¨q ˆ p¨qq ñ p¨q ` p¨q where rt “ ε p¨qˆp¨q . Then all r A,B are necessarily regular epimorphisms (becausethe ε A ˆ B are) and, consequently, C is a unital category. In [20] such a naturaltransformation t was called a natural imaginary splitting . Remark . Any natural imaginary splitting t : P pp¨q ˆ p¨qq ñ p¨q ` p¨q has thefollowing properties:1. t A, is isomorphic to ε A P p A q ε A , , – (cid:12) (cid:18) A A – (cid:12) (cid:18) A – (cid:12) (cid:18) P p A ˆ q t A, , ε A ˆ A ` r A, , , A ˆ , for all objects A in C ;2. the naturality of t gives the commutative diagram P p A ˆ B q t A,B , P p u ˆ v q (cid:12) (cid:18) A ` B u ` v (cid:12) (cid:18) P p C ˆ D q t C,D , C ` D (6)for all u : A Ñ C , v : B Ñ D in C ;3. from (5), we deduce p A q t A,B “ π A ε A ˆ B (1) “ ε A P p π A q (7)and p B q t A,B “ π B ε A ˆ B (1) “ ε B P p π B q (8)for all objects A and B in C ;4. using properties 1. and 2. above, we obtain the (regular epimorphism, mono-morphism) factorisations P p A q P px A , yq , ε A ' . ' . ❱❱❱❱❱❱❱❱❱❱❱❱❱❱ P p A ˆ B q t A,B , A ` BA ι A ❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (9)and P p B q P px , B yq , ε B ' . ' . ❱❱❱❱❱❱❱❱❱❱❱❱❱❱ P p A ˆ B q t A,B , A ` B,B ι B ❣❣❣❣❣❣❣❣❣❣❣❣❣ (10)for all objects A and B in C . ANDREA MONTOLI, DIANA RODELO, AND TIM VAN DER LINDEN
Imaginary addition.
Let C be a regular unital category with binary cop-roducts, comonadic covers and a natural imaginary splitting t . For every object X ,we consider the imaginary morphism µ X : X ˆ X X given by P p X ˆ X q t X,X , µ X ❨ ❩ ❬ ❪ ❫ ❴ ❵ ❛ ❝ ❞ ❡ ❢ ❣ X ` X p X X q , X. (11)We call µ X an imaginary addition on X since: X x X , y , µ X ˝x X , y“ X ❱ ❲ ❳ ❩ ❭ ❴ ❜ ❞ ❢ ❣ ✐ X ˆ X µ X , ❴❴❴❴❴ X (12)and X x , X y , µ X ˝x , X y“ X ❱ ❲ ❳ ❩ ❭ ❴ ❜ ❞ ❢ ❣ ❤ X ˆ X µ X , ❴❴❴❴❴ X. (13)Indeed, µ X ˝ x X , y “ p X X q t X,X P px X , yq (9) “ p X X q ι ε X “ ε X “ X and µ X ˝ x , X y “ p X X q t X,X P px , X yq (10) “ p X X q ι ε X “ ε X “ X . We adapt Definition 3.15 in [7] to the unital context and call the family p µ X : X ˆ X X q X P C a natural addition . Here natural means that for any morphism f : X Ñ Y the diagram X ˆ X µ X , ❴❴❴❴ f ˆ f (cid:12) (cid:18) X f (cid:12) (cid:18) Y ˆ Y µ Y , ❴❴❴❴ Y (14)commutes. In fact, f ˝ µ X “ f p X X q t X,X “ p Y Y q p f ` f q t X,X (6) “ p Y Y q t Y,Y P p f ˆ f q “ µ Y ˝ p f ˆ f q . Intrinsic Schreier split extensions
In this section we recall the notion of a Schreier split epimorphism of monoidsand its extended categorical version, the notion of an intrinsic Schreier split epi-morphism. We actually give a simplified version of the intrinsic definition by usingthe direct composition of imaginary morphisms, which is simply the compositionin the Kleisli category associated with the comonad of the comonadic covers.5.1.
Schreier split extensions of monoids [8, 9] . We recall the definition andthe main properties concerning Schreier split epimorphisms.A split epimorphism of monoids f with chosen section s and kernel KK ✤ , k , p X, ¨ , q f , , Y s l r (15)is called a Schreier split epimorphism if, for every x P X , there exists a uniqueelement a P K such that x “ k p a q ¨ sf p x q . Equivalently, if there exists a uniquefunction q : X K such that x “ kq p x q ¨ sf p x q for all x P X . We emphasise the NTRINSIC SCHREIER SPECIAL OBJECTS 9 fact that q is just a function (not necessarily a morphism of monoids) by using anarrow of type .The uniqueness property may be replaced [9, Proposition 2.4] by an extra con-dition on q : the couple p f, s q is a Schreier split epimorphism if and only if (S1) x “ kq p x q ¨ sf p x q , for all x P X ; (S2) q p k p a q ¨ s p y qq “ a , for all a P K , y P Y . Remark . Recall from [8] that Schreier split epimorphisms are also called righthomogenous split epimorphisms. A split epimorphism as in (15) is called lefthomogenous if, for every x P X , there exists a unique element a P K such that x “ sf p x q ¨ k p a q . Proposition 5.3. [8, Proposition 2.1.5]
Given a Schreier split epimorphism asin (15) , the following hold: (S3) qk “ K ; (S4) qs “ ; (S5) q p q “ ; (S6) kq p s p y q ¨ k p a qq ¨ s p y q “ s p y q ¨ k p a q , for all a P K, y P Y ; (S7) q p x ¨ x q “ q p x q ¨ q p sf p x q ¨ kq p x qq , for all x, x P X . A split epimorphism as in (15) is said to be strong when p k, s q is a jointlyextremal-epimorphic pair. It is stably strong if every pullback of it along anymorphism is strong. Any Schreier split epimorphism is (stably) strong (see [8],Lemma 2.1.6 and Proposition 2.3.4), thus f is the cokernel of its kernel k . So, sucha split epimorphism is in fact a Schreier split extension .As shown in [17], the definition of a Schreier split epimorphism makes sense alsoin the wider context of Jónsson-Tarski varieties.5.4.
Intrinsic Schreier split extensions [20] . We recall our approach towardsSchreier extensions. Here C will denote a regular unital category with binary cop-roducts, comonadic covers and a natural imaginary splitting t . Definition 5.5.
A split epimorphism f with chosen section s and kernel kK ✤ , k , X f , , Y, s l r (16)is called an intrinsic Schreier split epimorphism (with respect to t ) if thereexists an imaginary morphism q : X K (i.e., a morphism q : P p X q Ñ K ), calledthe imaginary (Schreier) retraction , such that (iS1) µ X ˝ x k ˝ q, sf y “ X , i.e., the diagram X x k ˝ q,sf y , ❴❴❴❴ X (cid:29) ' ❋❋❋❋❋❋ X ˆ X µ X (cid:12) (cid:18) ✤✤✤ X commutes; (iS2) q ˝ µ X ˝ p k ˆ s q “ π K , i.e., the diagram K ˆ Y k ˆ s , π K & - ❚❚❚❚❚❚❚❚❚❚❚ X ˆ X µ X , ❴❴❴❴ X q (cid:12) (cid:18) ✤✤✤✤ K commutes. The original definition in [20] expressed the above axioms through their corres-ponding morphisms and equalities in C . However, using the composition in theKleisli category K , as above, gives a better understanding of the link with (S1) and (S2) .The imaginary retraction of an intrinsic Schreier split epimorphism is necessarilyunique (see [20, Proposition 5.3]) and we also have (by [20, Proposition 5.4]): (iS3) K k , q ˝ k “ K : ❘ ❲ ❬ ❴ ❝ ❣ ❧ X q , ❴❴❴❴ K, i.e., qP p k q “ ε K ; (iS4) Y s , q ˝ s “ : ❘ ❲ ❬ ❴ ❝ ❣ ❧ X q , ❴❴❴❴ K, i.e., qP p s q “ ; (iS5) X , q ˝ X “ K : ❘ ❱ ❬ ❴ ❝ ❤ ❧ X q , ❴❴❴❴ K, i.e., qP p X qp“ q P p X q q “ K ; (iS6) Y ˆ K s ˆ k (cid:12) (cid:18) x k ˝ q ˝ µ X ˝p s ˆ k q ,sπ Y y , ❴❴❴❴❴❴❴❴❴❴❴ X ˆ X µ X (cid:12) (cid:18) ✤✤✤✤ X ˆ X µ X , ❴❴❴❴❴❴❴❴❴❴❴ X, i.e., µ X ˝ x k ˝ q ˝ µ X ˝ p s ˆ k q , sπ Y y “ µ X ˝ p s ˆ k q .In order to get the intrinsic version of (S7) , we will need a further assumption,that will be discussed in the next section.If we apply this intrinsic definition to the category Mon of monoids, we regainthe original definition of a Schreier split epimorphism (= right homogeneous splitepimorphism). Also, left homogeneous split epimorphisms (see Remark 5.2) fit thepicture. Indeed:
Theorem 5.6. [20, Theorem 5.10]
In the case of monoids, the intrinsic Schreiersplit epimorphisms with respect to the direct imaginary splitting t d are precisely theSchreier split epimorphisms. Similarly, the intrinsic Schreier split epimorphismswith respect to the twisted imaginary splitting t w are the left homogeneous splitepimorphisms. This result extends to Jónsson–Tarski varieties [20].5.7. S -protomodular categories [8] . We recall now the definition of an S -protomodular category, with respect to a class S of points (i.e., of split epimorph-isms with a fixed section) in a pointed category C with finite limits.We denote by Pt p C q the category of points in C , whose morphisms are pairs ofmorphisms which form commutative squares with both the split epimorphisms andtheir sections. The functor cod : Pt p C q Ñ C associates with every split epimorphismits codomain. It is a fibration, usually called the fibration of points . For eachobject Y of C , we denote by Pt Y p C q the fibre of this fibration, whose objects arethe points with codomain Y .Let S be a class of points in C which is stable under pullbacks along any morph-ism. If we look at it as a full subcategory S - Pt p C q of Pt p C q , then it gives rise to asubfibration S - cod of the fibration of points. Definition 5.8. [8, Definition 8.1.1] Let C be a pointed finitely complete category,and S a pullback-stable class of points. We say that C is S -protomodular when:1. every point in S - Pt p C q is a strong point;2. S - Pt p C q is closed under finite limits in Pt p C q . NTRINSIC SCHREIER SPECIAL OBJECTS 11
As shown in [10], S -protomodular categories satisfy, relatively to the class S ,many of the properties of protomodular categories [2]. In particular, a relative ver-sion of the Split Short Five Lemma holds: given a morphism of S -split extensions,i.e., a diagram K ✤ , k , γ (cid:12) (cid:18) X f , , g (cid:12) (cid:18) Y s l r h (cid:12) (cid:18) K ✤ , k , X f , , Y, s l r such that the two rows are S -split extensions (points in S with their kernel)and the three squares involving, respectively, the split epimorphisms, the kernels,and the sections commute, g is an isomorphism if and only if both γ and h areisomorphisms. In Section 7 we will show that, when S is the class of intrinsicSchreier split extensions, a stronger version of this lemma holds. Moreover, we willdiscuss the validity of other homological lemmas. Example . [8] The category Mon of monoids is S -protomodular with respect tothe class S of Schreier split epimorphisms. Example . [17] Every Jónsson–Tarski variety is an S -protomodular categorywith respect to the class of Schreier split epimorphisms. Example . [20] Every regular unital category with binary coproducts, equippedwith comonadic covers and a natural imaginary splitting, is S -protomodular withrespect to the class S of intrinsic Schreier split epimorphisms. Consequently, anysuch split epimorphism is an intrinsic Schreier split extension .The reader may find several other examples in [11].6. The associativity axiom
In order to improve the behaviour of the intrinsic Schreier split extensions, itis useful to consider an additional assumption, concerning the associativity of theimaginary addition µ X .Let C be a regular unital category with binary coproducts, comonadic covers anda natural imaginary splitting t . Suppose that, for every object X , the imaginaryaddition µ X satisfies the associativity axiom µ X ˝ p µ X ˆ X q “ µ X ˝ p X ˆ µ X q ,i.e., the diagram X ˆ X ˆ X µ X ˆ X , ❴❴❴❴❴ X ˆ µ X (cid:12) (cid:18) ✤✤✤✤ X ˆ X µ X (cid:12) (cid:18) ✤✤✤✤ X ˆ X µ X , ❴❴❴❴❴❴❴ X commutes. In particular, for arbitrary imaginary morphisms a, b, c : A X , weget µ X ˝ x µ X ˝ x a, b y , c y “ µ X ˝ x a, µ X ˝ x b, c yy . (17) Example . In Gp and in Mon , the direct and twisted imaginary splittings induceassociative imaginary additions.Among the properties, listed in the previous section, of Schreier split epimorph-isms of monoids, there is one, namely the property given by (S7) , which uses theassociativity of the monoid operation (for X ). Hence it is not so surprising that wemay prove its intrinsic version when we assume the associativity axiom. Proposition 6.2.
Suppose that the natural addition p µ X : X ˆ X X q X P C isassociative. Given an intrinsic Schreier split extension (16) with imaginary retrac-tion q , the following diagram commutes (iS7) X ˆ X µ X (cid:12) (cid:18) ✤✤✤ x q ˝ π ,sfπ ,k ˝ q ˝ π y , ❴❴❴❴❴❴❴❴❴ K ˆ X ˆ X K ˆp q ˝ µ X q , ❴❴❴❴❴❴❴❴❴ K ˆ K µ K (cid:12) (cid:18) ✤✤✤ X q , ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ K i.e., µ K ˝ p K ˆ p q ˝ µ X qq ˝ x q ˝ π , sf π , k ˝ q ˝ π y “ q ˝ µ X .Proof. Using Lemma 6.3 below, it suffices to prove that µ X ˝ x k ˝ µ K ˝ p K ˆ p q ˝ µ X qq ˝ x q ˝ π , sf π , k ˝ q ˝ π y , sf ˝ µ X y ““ µ X ˝ x k ˝ q ˝ µ X , sf ˝ µ X y which, by p iS1 q , is the same as µ X ˝ x k ˝ µ K ˝ p K ˆ p q ˝ µ X qq ˝ x q ˝ π , sf π , k ˝ q ˝ π y , sf ˝ µ X y “ µ X . We have µ X ˝ x k ˝ µ K ˝ p K ˆ p q ˝ µ X qq ˝ x q ˝ π , sf π , k ˝ q ˝ π y , sf ˝ µ X y (14) “ µ X ˝ x µ X ˝ p k ˆ k q ˝ x q ˝ π , q ˝ µ X ˝ x sf π , k ˝ q ˝ π yy , µ X ˝ p sf ˆ sf qy“ µ X ˝ x µ X ˝ x k ˝ q ˝ π , k ˝ q ˝ µ X ˝ x sf π , k ˝ q ˝ π yy , µ X ˝ p sf ˆ sf qy (17) “ µ X ˝ x k ˝ q ˝ π , µ X ˝ x k ˝ q ˝ µ X ˝ x sf π , k ˝ q ˝ π y , µ X ˝ p sf ˆ sf qyy“ µ X ˝ x k ˝ q ˝ π , µ X ˝ x k ˝ q ˝ µ X ˝ x sf π , k ˝ q ˝ π y , µ X ˝ x sf π , sf π yyy (17) “ µ X ˝ x k ˝ q ˝ π , µ X ˝ x µ X ˝ x k ˝ q ˝ µ X ˝ x sf π , k ˝ q ˝ π y , sf π y , sf π yy“ µ X ˝ x k ˝ q ˝ π , µ X ˝ x µ X ˝ x k ˝ q ˝ µ X ˝ p s ˆ k q ˝ x f π , q ˝ π y ,sπ Y ˝ x f π , q ˝ π y , sf π yy“ µ X ˝ x k ˝ q ˝ π , µ X ˝ x µ X ˝ x k ˝ q ˝ µ X ˝ p s ˆ k q , sπ Y y˝x f π , q ˝ π y , sf π yy p iS6 q “ µ X ˝ x k ˝ q ˝ π , µ X ˝ x µ X ˝ p s ˆ k q ˝ x f π , q ˝ π y , sf π yy“ µ X ˝ x k ˝ q ˝ π , µ X ˝ x µ X ˝ x sf π , k ˝ q ˝ π y , sf π yy (17) “ µ X ˝ x k ˝ q ˝ π , µ X ˝ x sf π , µ X ˝ x k ˝ q ˝ π , sf π yyy p iS1 q “ µ X ˝ x k ˝ q ˝ π , µ X ˝ x sf π , π yy (17) “ µ X ˝ x µ X ˝ x k ˝ q ˝ π , sf π y , π y p iS1 q “ µ X ˝ x π , π y“ µ X . (cid:3) Lemma 6.3.
Let (16) be a split epimorphism with an imaginary morphism q suchthat (iS2) holds. If µ X ˝ x k ˝ a, s ˝ b y “ µ X ˝ x k ˝ c, s ˝ d y , where a, c : A K and b, d : A Y are imaginary morphisms, then a “ c . NTRINSIC SCHREIER SPECIAL OBJECTS 13
Proof. µ X ˝ x k ˝ a, s ˝ b y “ µ X ˝ x k ˝ c, s ˝ d yñ q ˝ µ X ˝ x k ˝ a, s ˝ b y “ q ˝ µ X ˝ x k ˝ c, s ˝ d yñ q ˝ µ X ˝ p k ˆ s q ˝ x a, b y “ q ˝ µ X ˝ p k ˆ s q ˝ x c, d y p iS2 q ñ π K ˝ x a, b y “ π K ˝ x c, d yñ a “ c. (cid:3) Remark . As an immediate consequence of Lemma 6.3, we obtain the uniquenessof the imaginary retraction for any intrinsic Schreier split extension (16) (which wasalready known from [20, Proposition 5.3]). Given two possible imaginary retractions q, q : X K , (iS1) gives µ X ˝ x k ˝ q, sf y “ X “ µ X ˝ x k ˝ q , sf y ; consequently, q “ q .7. Stability properties and homological lemmas
In this section we prove that certain stability properties for Schreier extensions ofmonoids shown in [8] still hold for intrinsic Schreier extensions in our context: C willdenote a regular unital category with binary coproducts, comonadic covers and anatural imaginary splitting t . Moreover, we will observe that some of these stabilityproperties allow to extend the validity of some classical homological lemmas to ourintrinsic context.In order to extend those proofs in [8] which use the associativity of the monoidoperations, we assume the associativity axiom holds. This is the case, in particular,of the first stability property we consider: Proposition 7.1. (See [8, Proposition 2.3.2])
Suppose that the natural addition p µ X : X ˆ X X q X P C is associative. Then the intrinsic Schreier split extensionsare stable under composition.Proof. Suppose that K ✤ , k , X f , , q f l r ❴ ❴ ❴ ❴ Y s l r and L ✤ , l , Y g , , q g l r ❴ ❴ ❴ ❴ Z t l r are intrinsic Schreier extensions. We want to prove that M ✤ , m , X gf , , Z st l r is an intrinsic Schreier extension, where m is the kernel of gfM ✤ , m , f (cid:12) (cid:18) X f (cid:12) (cid:18) L s L R ✤ , l , (cid:12) (cid:18) Y s L R g (cid:12) (cid:18) , Z. t L R We must define the imaginary retraction q : X M . The imaginary morphism µ X ˝ x k ˝ q f , sl ˝ q g ˝ f y : X X is such that the composition with gf gives thefollowing equalities in C : gf p X X q t X,X P px kq f , slq g P p f qyq δ X “ gf p X X q t X,X P pp kq f q ˆ p slq g P p f qqq P px P p X q , P p X q yq δ X (6) “ gf p X X q pp kq f q ` p slq g P p f qq t P p X q ,P p X q P px P p X q , P p X q yq δ X “ p gf kq f gf slq g P p f q q t P p X q ,P p X q P px P p X q , P p X q yq δ X “ . This gives a unique morphism q : P p X q Ñ M in C , i.e., an imaginary morphism q : X M , such that m ˝ q “ µ X ˝ x k ˝ q f , sl ˝ q g ˝ f y . Next we prove that q isthe imaginary retraction for the split epimorphism gf : (iS1) µ X ˝ x m ˝ q, stgf y“ µ X ˝ x µ X ˝ x k ˝ q f , sl ˝ q g ˝ f y , stgf y (17) “ µ X ˝ x k ˝ q f , µ X ˝ x sl ˝ q g ˝ f, stgf yy“ µ X ˝ x k ˝ q f , µ X ˝ p s ˆ s q ˝ x l ˝ q g , tg y ˝ f y (14) “ µ X ˝ x k ˝ q f , s ˝ µ Y ˝ x l ˝ q g , tg y ˝ f y“ X , where in the last step we use (iS1) for g and then (iS1) for f .Next we prove that m ˝ q ˝ µ X ˝ p m ˆ p st qq “ m ˝ π M and use the fact that m isa monomorphism, to conclude (iS2) : m ˝ q ˝ µ X ˝ p m ˆ p st qq“ µ X ˝ x k ˝ q f , sl ˝ q g ˝ f y ˝ µ X ˝ p m ˆ p st qq“ µ X ˝ x k ˝ q f ˝ µ X ˝ p m ˆ p st qq , sl ˝ q g ˝ f ˝ µ X ˝ p m ˆ p st qqy (14) “ µ X ˝ x k ˝ q f ˝ µ X ˝ p m ˆ p st qq , sl ˝ q g ˝ µ Y ˝ p f ˆ f q ˝ p m ˆ p st qqy“ µ X ˝ x k ˝ q f ˝ µ X ˝ p m ˆ p st qq , sl ˝ q g ˝ µ Y ˝ pp f m q ˆ p f st qqy“ µ X ˝ x k ˝ q f ˝ µ X ˝ p m ˆ p st qq , sl ˝ q g ˝ µ Y ˝ pp lf q ˆ t qy“ µ X ˝ x k ˝ q f ˝ µ X ˝ p m ˆ p st qq , sl ˝ q g ˝ µ Y ˝ p l ˆ t q ˝ p f ˆ Z qy“ µ X ˝ x k ˝ q f ˝ µ X ˝ p m ˆ p st qq , sl ˝ π L ˝ p f ˆ Z qy , where in the last equality we use (iS2) for g . Now we use sl ˝ π L ˝p f ˆ Z q “ sf mπ M and (iS7) applied to f . This gives m ˝ q ˝ µ X ˝ p m ˆ p st qq “ ¨ ¨ ¨“ µ X ˝ x k ˝ µ K ˝ p K ˆ p q f ˝ µ X qq ˝ x q f ˝ π , sf π , k ˝ q f ˝ π y˝p m ˆ p st qq , sf mπ M y“ µ X ˝ x k ˝ µ K ˝ x q f ˝ π ˝ p m ˆ p st qq , q f ˝ µ X ˝ x sf π , k ˝ q f ˝ π y˝p m ˆ p st qqy , sf mπ M y . Note that, part of the composite above is q f ˝ µ X ˝ x sf π , k ˝ q f ˝ π y ˝ p m ˆ p st qq (14) “ µ K ˝ p q f ˆ q f q ˝ x sf π , k ˝ q f ˝ π y ˝ p m ˆ p st qq“ µ K ˝ x q f ˝ sf π ˝ p m ˆ p st qq , q f ˝ k ˝ q f ˝ π p m ˆ p st qqy p iS3 q “ µ K ˝ x q f ˝ sf π ˝ p m ˆ p st qq , q f ˝ stπ y p iS4 q “ . NTRINSIC SCHREIER SPECIAL OBJECTS 15
Thus, m ˝ q ˝ µ X ˝ p m ˆ p st qq “ ¨ ¨ ¨“ µ X ˝ x k ˝ µ K ˝ x K , y ˝ q f ˝ mπ M , sf mπ M y (12) “ µ X ˝ x k ˝ q f , sf y ˝ mπ M p iS1 q “ X ˝ mπ M “ m ˝ π M . (cid:3) Proposition 7.2. (See [8, Proposition 2.3.2])
Consider split epimorphisms X f , Y g , s l r Z t l r in C . If p gf, st q is an intrinsic Schreier split extension, then so is p g, t q .Proof. We use the same notation as in Proposition 7.1. We define the imaginaryretraction for g as q g “ f ˝ q ˝ s : Y L . From (iS1) for gf , we have µ X ˝ x m ˝ q, stgf y “ X ñ f ˝ µ X ˝ x m ˝ q, stgf y “ f (14) ñ µ Y ˝ x f m ˝ q, tgf y “ f . Then µ Y ˝ x l ˝ q g , tg y “ µ Y ˝ x lf ˝ q ˝ s, tg y“ µ Y ˝ x f m ˝ q ˝ s, tgf s y“ µ Y ˝ x f m ˝ q, tgf y ˝ s “ f ˝ s “ Y , which proves (iS1) for g . As for (iS2) for g , we have q g ˝ µ Y ˝ p l ˆ t q “ f ˝ q ˝ s ˝ µ Y ˝ p l ˆ t q (14) “ f ˝ q ˝ µ X ˝ pp sl q ˆ p st qq“ f ˝ q ˝ µ X ˝ pp ms q ˆ p st qq“ f ˝ q ˝ µ X ˝ p m ˆ p st qq ˝ p s ˆ Z q“ f ˝ π M ˝ p s ˆ Z q“ f s ˝ π L “ π L , where we use (iS2) for gf in the fifth equality. (cid:3) Lemma 7.3.
Suppose that the values of P are projective objects in C . Let a , b : A X be imaginary morphisms and z : Z ։ A be a regular epimorphism. If a ˝ z “ b ˝ z , then a “ b .Proof. a ˝ z “ b ˝ z corresponds to the equality aP p z q “ bP p z q in C . Then a “ b ,since P p z q is a split epimorphism (see Remark 2.7). (cid:3) In the following Eq p f q denotes the kernel pair of a morphism f . Proposition 7.4. (See [8, Proposition 2.3.5] and [11, Proposition 4.8])
Supposethat the values of P are projective objects in C . Consider the following commutative diagram Eq p γ q ✤ , κ , γ (cid:12) (cid:18) γ (cid:12) (cid:18) Eq p g q ρ l r ❴ ❴ ❴ ❴ ϕ , , g (cid:12) (cid:18) g (cid:12) (cid:18) Eq p h q σ l r h (cid:12) (cid:18) h (cid:12) (cid:18) K ✤ , k , γ (cid:12) (cid:18) L R X q l r ❴ ❴ ❴ ❴ ❴ ❴ f , , g (cid:12) (cid:18) (cid:12) (cid:18) L R Y s l r h (cid:12) (cid:18) (cid:12) (cid:18) L R K ✤ , k , X f , , Y. s l r Note that, by the commutativity of limits, κ is the kernel of ϕ . If the top two rowsare intrinsic Schreier split extensions and g and h are regular epimorphisms, thenthe bottom row is also an intrinsic Schreier split extension.Proof. C is an S -protomodular category (Example 5.11), thus it is an S -Mal’tsevcategory [11, Theorem 5.4]. By Proposition 3.2 in [11], γ is a regular epimorphism.Since P p X q is a projective object and g is a regular epimorphism, then it admitsan imaginary splitting t : X X , with g ˝ t “ X (see Remark 2.7). We definethe imaginary retraction for the bottom row as q “ γ ˝ q ˝ t : X K . We mustprove (iS1) µ X ˝ x k ˝ q, sf y “ µ X ˝ x kγ ˝ q ˝ t, sf y“ µ X ˝ x gk ˝ q ˝ t, sf ˝ g ˝ t y“ µ X ˝ x gk ˝ q ˝ t, g ˝ s f ˝ t y“ µ X ˝ p g ˆ g q ˝ x k ˝ q , s f y ˝ t (14) “ g ˝ µ X ˝ x k ˝ q , s f y ˝ t “ g ˝ X ˝ t “ X , where we use (iS1) applied to the second row in the next to last equality.For (iS2) , we precompose the equality we wish to prove with the regular epi-morphism γ ˆ hq ˝ µ X ˝ p k ˆ s qp γ ˆ h q “ γ ˝ q ˝ t ˝ µ X ˝ pp kγ q ˆ p sh qq“ γ ˝ q ˝ t ˝ µ X ˝ pp gk q ˆ p gs qq“ γ ˝ q ˝ t ˝ µ X ˝ p g ˆ g qp k ˆ s q (14) “ γ ˝ q ˝ t ˝ g ˝ µ X ˝ p k ˆ s q p˚q “ γ ˝ q ˝ µ X ˝ p k ˆ s q“ γ ˝ π K “ π K ˝ p γ ˆ h q , where we use (iS2) for the second row in the next to last equality. Then (iS2) follows from Lemma 7.3.To finish, we just need to prove the equality γ ˝ q ˝ t ˝ g p˚q “ γ ˝ q . Actually,we prove this equality in C (not in K ) and to do so, we use the compatibility ofthe first two rows with respect to the imaginary retractions [20, Proposition 5.7]: γ i ρ “ q P p g i q , for i P t , u . We have γq P p t q δ X P p g q (2) “ γq P p t q P p g q δ X “ γq P p tP p g qq δ X “ γq P p g x tP p g q , ε X yq δ X , NTRINSIC SCHREIER SPECIAL OBJECTS 17 where x tP p g q , ε X y is the unique morphism making the following diagram commut-ative P p X q x tP p g q ,ε X y (cid:29) ' ε X (cid:18) (cid:25) tP p g q ' . Eq p g q g (cid:12) (cid:18) g , X g (cid:12) (cid:18) (cid:12) (cid:18) X g , , X. Using the compatibility mentioned earlier, γq P p t q δ X P p g q “ γq P p g q P px tP p g q , ε X yq δ X “ γγ ρP px tP p g q , ε X yq δ X “ γγ ρP px tP p g q , ε X yq δ X “ γq P p g q P px tP p g q , ε X yq δ X “ γq P p ε X q δ X (3) “ γq . (cid:3) Corollary 7.5. (See [8, Corollary 2.3.6])
Suppose that the values of P are projectiveobjects in C . Consider the diagram K ✤ , k , X f , , g (cid:12) (cid:18) (cid:12) (cid:18) Y s l r h (cid:12) (cid:18) (cid:12) (cid:18) K ✤ , k , X f , , Y, s l r where the three squares involving, respectively, the split epimorphism, the kernels,and the sections commute. If the top row is an intrinsic Schreier split extension,then so is the bottom row.Proof. Take the kernel pairs of the regular epimorphisms K , g and h . This givesa ˆ diagram whose top row is an intrinsic Schreier split extension, since theseextensions are closed under arbitrary pullbacks (see [20, Proposition 6.1]). ApplyingProposition 7.4 to this ˆ diagram, we conclude that p f, s q is an intrinsic Schreiersplit extension. (cid:3) In order to get the validity, in our context, of one of the classical homologicallemmas, namely the ˆ -Lemma, we need another stability property of the class ofintrinsic Schreier split epimorphisms. This property was called equi-consistency in [11]: Definition 7.6. [11, Definition . ] Let S be a pullback-stable class of points.Consider any commutative diagram K ✤ , k , w (cid:12) (cid:18) X f , , ❴ (cid:12) (cid:18) u (cid:12) (cid:18) Y s l r ❴ (cid:12) (cid:18) v (cid:12) (cid:18) K ✤ , k , t (cid:12) (cid:18) t (cid:12) (cid:18) R f , , r (cid:12) (cid:18) r (cid:12) (cid:18) S s l r s (cid:12) (cid:18) s (cid:12) (cid:18) K ✤ , k , L R X f , , e R L R Y, s l r e S L R (18)where x r , r y : R X ˆ X and x s , s y : S Y ˆ Y are equivalence relations, p f, s q and p f , s q are split epimorphisms, p f , s q is the induced split epimorphismbetween the kernels of r and s , and the diagram is completed by taking kernelsand the induced dotted morphisms. S is equi-consistent if, whenever the points p f, s q , p r , e R q and p f , s q belong to S , p f , s q is in S , too. Proposition 7.7. (See [11, Proposition 6.4])
Suppose that the natural addition p µ X : X ˆ X X q X P C is associative. Then the class of intrinsic Schreier splitextensions is equi-consistent.Proof. Consider the commutative diagram (18). Suppose that p f, s q , p r , e R q and p f , s q are intrinsic Schreier split epimorphisms, and denote by q , q and χ theimaginary retractions for p f, s q , p f , s q and p r , e R q , respectively. In particular, wehave t i w ˝ q “ q ˝ r i u, i P t , u . (19)First, we consider the imaginary morphism α “ µ R ˝x s f u ˝ χ, e R k ˝ q ˝ r y : R R . We have r ˝ µ R ˝ x s f u ˝ χ, e R k ˝ q ˝ r y (14) “ µ X ˝ p r ˆ r q ˝ x s f u ˝ χ, e R k ˝ q ˝ r y (18) “ µ X ˝ x , k ˝ q ˝ r y“ µ X ˝ x , X y ˝ k ˝ q ˝ r (13) “ k ˝ q ˝ r and r ˝ µ R ˝ x s f u ˝ χ, e R k ˝ q ˝ r y (14) “ µ X ˝ p r ˆ r q ˝ x s f u ˝ χ, e R k ˝ q ˝ r y (18) “ µ X ˝ x sf r u ˝ χ, k ˝ q ˝ r y ; thus XR α , ❴❴❴❴❴❴❴ k ˝ q ˝ r ; ♦♦♦♦♦♦♦ µ X ˝x sfr u ˝ χ,k ˝ q ˝ r y + ❖❖❖❖❖❖❖ R r L R r (cid:12) (cid:18) X. (20) NTRINSIC SCHREIER SPECIAL OBJECTS 19
Second, we consider the imaginary morphism β “ µ R ˝ x e R k ˝ q ˝ r ˝ α, s f u ˝ χ y : R R . We have r ˝ µ R ˝ x e R k ˝ q ˝ r ˝ α, s f u ˝ χ y (14) “ µ X ˝ p r ˆ r q ˝ x e R k ˝ q ˝ r ˝ α, s f u ˝ χ y (18) “ µ X ˝ x k ˝ q ˝ r ˝ α, y“ µ X ˝ x X , y ˝ k ˝ q ˝ r ˝ α (12) “ k ˝ q ˝ r ˝ α and r ˝ µ R ˝ x e R k ˝ q ˝ r ˝ α, s f u ˝ χ y (14) “ µ X ˝ p r ˆ r q ˝ x e R k ˝ q ˝ r ˝ α, s f u ˝ χ y (18) , (20) “ µ X ˝ x k ˝ q ˝ µ X ˝ x sf r u ˝ χ, k ˝ q ˝ r y , sf r u ˝ χ y“ µ X ˝ x k ˝ q ˝ µ X ˝ p s ˆ k q ˝ x f r u ˝ χ, q ˝ r y , sf r u ˝ χ y“ µ X ˝ x k ˝ q ˝ µ X ˝ p s ˆ k q ˝ x f r u ˝ χ, q ˝ r y , sπ Y ˝ x f r u ˝ χ, q ˝ r yy“ µ X ˝ x k ˝ q ˝ µ X ˝ p s ˆ k q , sπ Y y ˝ x f r u ˝ χ, q ˝ r y p iS6 q “ µ X ˝ p s ˆ k q ˝ x f r u ˝ χ, q ˝ r y“ µ X ˝ x sf r u ˝ χ, k ˝ q ˝ r y ; thus XR β ❴❴❴❴❴❴ , ❴❴❴❴❴❴ µ X ˝x sfr u ˝ χ,k ˝ q ˝ r y / ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ k ˝ q ˝ r ˝ α ) / ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ R r = qqqqqqqqqqqq r ! * ▼▼▼▼▼▼▼▼▼▼▼▼ i R , R r L R r (cid:12) (cid:18) X. (21)Third, we consider the imaginary morphism γ “ µ R ˝ x uk ˝ q ˝ χ, e R k ˝ q ˝ r ˝ α y : R R . We have r ˝ µ R ˝ x uk ˝ q ˝ χ, e R k ˝ q ˝ r ˝ α y (14) “ µ X ˝ p r ˆ r q ˝ x uk ˝ q ˝ χ, e R k ˝ q ˝ r ˝ α y (18) “ µ X ˝ x , k ˝ q ˝ r ˝ α y“ µ X ˝ x , X y ˝ k ˝ q ˝ r ˝ α (13) “ k ˝ q ˝ r ˝ α and r ˝ µ R ˝ x uk ˝ q ˝ χ, e R k ˝ q ˝ r ˝ α y (14) “ µ X ˝ p r ˆ r q ˝ x uk ˝ q ˝ χ, e R k ˝ q ˝ r ˝ α y (18) , (20) “ µ X ˝ x kt w ˝ q ˝ χ, k ˝ q ˝ µ X ˝ x sf r u ˝ χ, k ˝ q ˝ r yy“ µ X ˝ p k ˆ k q ˝ x t w ˝ q ˝ χ, q ˝ µ X ˝ x sf r u ˝ χ, k ˝ q ˝ r yy (19) “ µ X ˝ p k ˆ k q ˝ x q ˝ r u ˝ χ, q ˝ µ X ˝ x sf r u ˝ χ, k ˝ q ˝ r yy (14) “ k ˝ µ K ˝ p K ˆ p q ˝ µ X qq ˝ x q ˝ r u ˝ χ, x sf r u ˝ χ, k ˝ q ˝ r yy“ k ˝ µ K ˝ p K ˆ p q ˝ µ X qq˝x q ˝ π ˝ x r u ˝ χ, r y , x sf π ˝ x r u ˝ χ, r y , k ˝ q ˝ π ˝ x r u ˝ χ, r yyy“ k ˝ µ K ˝ p K ˆ p q ˝ µ X qq ˝ x q ˝ π , x sf π , k ˝ q ˝ π yy ˝ x r u ˝ χ, r y p iS7 q “ k ˝ q ˝ µ X ˝ x r u ˝ χ, r y“ k ˝ q ˝ µ X ˝ x r u ˝ χ, r ˝ e R r y“ k ˝ q ˝ µ X ˝ p r ˆ r q ˝ x u ˝ χ, e R r y (14) “ k ˝ q ˝ r ˝ µ R ˝ x u ˝ χ, e R r y p iS1 q “ k ˝ q ˝ r , where the last (iS1) is with respect to the intrinsic Schreier extension p r , e R q ; thus XR γ , ❴❴❴❴❴❴❴ k ˝ q ˝ r ˝ α ; ♦♦♦♦♦♦♦ k ˝ q ˝ r + ❖❖❖❖❖❖❖ R r L R r (cid:12) (cid:18) X. (22)Next we use the fact that R is transitive together with (20), (21) and (22) todeduce the existence of an imaginary morphism δ : R R such that the followingdiagram commutes XR δ , ❴❴❴❴❴❴❴ k ˝ q ˝ r ; ♦♦♦♦♦♦♦ k ˝ q ˝ r + ❖❖❖❖❖❖❖ R r L R r (cid:12) (cid:18) X. We are now able to define the imaginary retraction q for p f , s q : R q ❆❆ (cid:26) % ❆❆ δ (cid:16) (cid:23) ❪ ❬ ❳ ❚ ▼ ❅ ✲ ! R ) ✦ ✩ ✬ ✱ ✹ ❈ ❙ K ! K (cid:12) (cid:18) k , R f (cid:12) (cid:18) (cid:12) (cid:18) , , S. Note that s i f ˝ δ “ f r i ˝ δ “ f k ˝ q ˝ r i “ , i P t , u , from which we get f ˝ δ “ . NTRINSIC SCHREIER SPECIAL OBJECTS 21
To finish we must prove (iS1) and (iS2) for p f , s q . To obtain the equality for (iS1) R x k ˝ q ,s f y , ❴❴❴❴ R (cid:31) ( ■■■■■■ R ˆ R µ R (cid:12) (cid:18) ✤✤✤✤ R, we prove that r i ˝ µ R ˝ x k ˝ q , s f y “ r i ˝ R “ r i , i P t , u , by using (iS1) for p f, s q .To obtain the equality for (iS2) K ˆ S k ˆ s , π K % , ❙❙❙❙❙❙❙❙❙❙❙ R ˆ R µ R , ❴❴❴❴ R q (cid:12) (cid:18) ✤✤✤✤ K , we prove that r i ˝ k ˝ q ˝ µ R ˝ p k ˆ s q “ r i ˝ k ˝ π K , i P t , u , which uses (iS2) for p f, s q . (cid:3) We say that a morphism f : X Ñ Y is an intrinsic Schreier special morph-ism if the split epimorphism p f , e f q (or, equivalently, the split epimorphism p f , e f q )is intrinsic Schreier, where Eq p f q f , f , X e f l r is the kernel pair of f . If this intrinsicSchreier special morphism is a regular epimorphism, then it is automatically thecokernel of its kernel, thus it gives rise to an extension (Proposition 5.6 in [11]).Thanks to the stability properties we proved in this section, we can apply Proposi-tion . and Theorem . in [11] and get the following version of the ˆ -Lemma: Theorem 7.8.
Consider the commutative diagram K k , ❴ (cid:12) (cid:18) w (cid:12) (cid:18) X f , ❴ (cid:12) (cid:18) u (cid:12) (cid:18) Y ❴ (cid:12) (cid:18) v (cid:12) (cid:18) K ✤ , k , w (cid:12) (cid:18) (cid:12) (cid:18) X f , , u (cid:12) (cid:18) (cid:12) (cid:18) Y v (cid:12) (cid:18) (cid:12) (cid:18) K k , L R X f , Y, where the three columns and the middle row are intrinsic Schreier special extensions.The upper row is an intrinsic Schreier special extension if and only if the lower oneis. We conclude this section by proving the stronger version of the Split Short FiveLemma we mentioned in Section 5.
Proposition 7.9. (See [8, Proposition 2.3.10])
Suppose that the values of P areprojective objects in C . Consider the diagram K ✤ , k , γ (cid:12) (cid:18) X f , , g (cid:12) (cid:18) q l r ❴ ❴ ❴ ❴ Y s l r h (cid:12) (cid:18) K ✤ , k , X f , , q l r ❴ ❴ ❴ ❴ ❴ Y, s l r where both rows are intrinsic Schreier split extensions and the three squares in-volving, respectively, the split epimorphism, the kernels, and the sections commute.Then (a) g is a regular epimorphism if and only if γ and h are regular epimorphisms; (b) g is a monomorphism if and only if γ and h are monomorphisms.Proof. (a) If g is a regular epimorphism, then so is h , from the commutativityof the diagram. Moreover, the compatibility for the imaginary retractions gives γq “ qP p g q . Then γ is a regular epimorphism since so are q and P p g q (by (iS2) and Remark 2.7).Conversely, suppose that γ and h are regular epimorphisms. We take the (reg-ular epimorphisms, monomorphism) factorisation g “ me , and prove that m isan isomorphism. Since the bottom row is an intrinsic Schreier split extension, weknow that p k, s q is an jointly extremal-epimorphic pair (see Subsection 5.4). Since q and hε Y are regular epimorphisms, then p kq, shε Y q is also a jointly extremal-epimorphic pair. In C , it is easy to check the commutativity of M (cid:12) (cid:18) m (cid:12) (cid:18) P p X q kq , ε M P p ek q σP p q q δ X B ②②②②②②②②②② X P p Y q , shε Y l r ε M P p es q ] g ❋❋❋❋❋❋❋❋❋❋ where σ is a splitting of the split epimorphism P p γ q (Remark 2.7). Thus, m is anisomorphism and g is a regular epimorphism.(b) If g is a monomorphism, then so are γ and h , from the commutativity of thediagram. For the converse, suppose that a, b : U Ñ X are morphisms such that ga “ gb . Then, f ga “ f gb , from which we get f a “ f b (since f g “ hf and h is a monomorphism). On the other hand, we deduce kqP p g q P p a q “ kqP p g q P p b q and kγq P p a q “ kγq P p b q , from the compatibility for imaginary retractions ([20,Proposition 5.7]). This gives q P p a q “ q P p b q , since kγ is a monomorphism. Thus q ˝ a “ q ˝ b , as imaginary morphisms. Then a “ X ˝ a p iS1 q “ µ X ˝ x k ˝ q , s f y ˝ a “ µ X ˝ x k ˝ q ˝ a, s f a y“ µ X ˝ x k ˝ q ˝ b, s f b y“ µ X ˝ x k ˝ q , s f y ˝ b p iS1 q “ X ˝ b “ b. (cid:3) NTRINSIC SCHREIER SPECIAL OBJECTS 23 Intrinsic Schreier special objects
Let C be a pointed and finitely complete category and S a class of points in C which is stable under pullbacks along arbitrary morphisms. Recall from [10] thatan object Y is called an S -special object when the split epimorphism Y ✤ , x Y , y , Y ˆ Y π , , Y x Y , Y y l r (23)(or, equivalently, the split epimorphism p π , x Y , Y yq ) belongs to the class S . If C is an S -protomodular category, then the full subcategory formed by the S -specialobjects is protomodular ([10], Proposition . ), and it is called the protomodularcore of C with respect to the class S . When C is the category of monoids, and S is either the class of Schreier split epimorphisms or the one of left homogeneous splitepimorphisms, the protomodular core is the category of groups. More generally,when V is a Jónsson–Tarski variety, an algebra in V is a Schreier special object ifand only if it has a right loop structure [20, Proposition 7.5] (see Subsection 8.1for the right loop axioms). Similarly, an algebra in V is special with respect to theclass of left homogeneous split epimorphisms (see Remark 5.2) if and only if it hasa left loop structure (see Subsection 8.1 for the left loop axioms).Now we want to study what happens in the intrinsic Schreier setting. So, let C bea regular unital category with binary coproducts, comonadic covers and a naturalimaginary splitting t . An object is an intrinsic Schreier special object whenthe split epimorphism (23) is an intrinsic Schreier split epimorphism. This meansthat there exists an imaginary morphism q : Y ˆ Y Y such that: (iSs1) the diagram Y ˆ Y xx Y , y˝ q, x Y , Y y π y , ❴❴❴❴❴❴❴❴❴❴❴❴ Y ˆ Y ' . ❲❲❲❲❲❲❲❲❲❲❲❲❲❲ Y ˆ Y ˆ Y ˆ Y µ Y ˆ Y (cid:12) (cid:18) ✤✤✤✤ Y ˆ Y commutes; (iSs2) the diagram Y ˆ Y x Y , yˆx Y , Y y , π ) ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ Y ˆ Y ˆ Y ˆ Y µ Y ˆ Y , ❴❴❴❴❴❴❴❴❴ Y ˆ Y q (cid:12) (cid:18) ✤✤✤ Y commutes.In this context, we also have: (iSs3) Y x Y , y , q ˝x Y , y“ Y ❚ ❳ ❬ ❴ ❝ ❢ ❥ Y ˆ Y q , ❴❴❴❴ Y, i.e., qP px Y , yq “ ε Y ; (iSs4) Y x Y , Y y , q ˝x Y , Y y“ ❚ ❳ ❬ ❴ ❝ ❢ ❥ Y ˆ Y q , ❴❴❴❴ Y, i.e., qP px Y , Y yq “ .So, if Y is an intrinsic Schreier special object, then the identities (iSs3) and (iSs4) make q : Y ˆ Y Y an imaginary subtraction . Indeed, the identities (iSs3) and (iSs4) correspond to the varietal axioms for a subtraction, i.e., q p x, q “ x, q p x, x q “ . right loop axiom “imaginary” commutative diagram p x ´ y q ` y “ x (iL1) Y ˆ Y x q,π y , ❴❴❴❴❴❴ π + ❖❖❖❖❖❖❖❖ Y ˆ Y µ Y (cid:12) (cid:18) ✤✤✤ Y p x ` y q ´ y “ x (iL2) Y ˆ Y x µ Y ,π y , ❴❴❴❴❴❴ π + ❖❖❖❖❖❖❖❖ Y ˆ Y q (cid:12) (cid:18) ✤✤✤ Y Table 2.
The right loop axioms and their corresponding diagrams8.1.
Imaginary (one-sided) loops.
Consider an intrinsic Schreier special object Y . We now show that the imaginary addition given in (11) and the imaginarysubtraction q : Y ˆ Y Y satisfy the axioms of a (one-sided) loop (like those of aright loop or a left loop). We say then that Y has the structure of an imaginaryone-sided loop .We must prove the following right loop or left loop axioms " p x ´ y q ` y “ x p x ` y q ´ y “ x or " x ` p´ x ` y q “ y ´ x ` p x ` y q “ y in the imaginary context; we consider the left-hand side axioms. Table 2 gives theright loop axioms and their corresponding “imaginary” commutative diagrams. Theonly difference in the diagrams is that µ Y and q are swapped, just as “ ` ” and “ ´ ”are swapped in the right loop axioms.The commutativity of (iL1) follows from composing (iSs1) with π . From (14)we know that π ˝ µ Y ˆ Y “ µ Y ˝ p π ˆ π q . Then, we just have to prove that p π ˆ π q ˝ xx Y , y ˝ q, x Y , Y y π y “ x q, π y . In fact, p π ˆ π q ˝ xx Y , y ˝ q, x Y , Y y π y corresponds to the real morphism p π ˆ π qxx Y , y q, x Y , Y y π ε Y ˆ Y y “ x q, π ε Y ˆ Y y “ x q, π y . The commutativity of (iL2) follows from (iSs2) . In this case we must show that µ Y ˆ Y ˝ px Y , y ˆ x Y , Y yq “ x µ Y , π y . The imaginary morphism µ Y ˆ Y ˝ px Y , y ˆx Y , Y yq corresponds to the real morphism p Y ˆ Y Y ˆ Y q t Y ˆ Y,Y ˆ Y P px Y , y ˆ x Y , Y yq (6) “ p x Y , y x Y , Y y q t Y,Y “ x p Y Y q , p Y q y t Y,Y (11) , (8) “ x µ Y , π ε Y ˆ Y y“ x µ Y , π y . The converse is also true. Indeed, suppose the object Y has the structure ofan imaginary one-sided loop, in the sense that it is equipped with an imaginarymorphism q : Y ˆ Y Y which satisfies, together with the imaginary addition µ Y , (iL1) and (iL2) . Then q is the imaginary Schreier retraction for the splitepimorphism (23). To show this, we need to show that (iSs1) and (iSs2) hold. (iSs2) follows immediately from (iL2) , because, as we already observed, µ Y ˆ Y ˝px Y , y ˆ x Y , Y yq “ x µ Y , π y . In order to prove (iSs1) , we use the previous NTRINSIC SCHREIER SPECIAL OBJECTS 25 equality p π ˆ π q ˝ xx Y , y ˝ q, x Y , Y y π y “ x q, π y to get π ˝ µ Y ˆ Y ˝ xx Y , y ˝ q, x Y , Y y π y “ µ Y ˝ x q, π y p iL1 q “ π “ π ˝ Y ˆ Y ; also π ˝ µ Y ˆ Y ˝ xx Y , y ˝ q, x Y , Y y π y (14) “ µ Y ˝ p π ˆ π q ˝ xx Y , y ˝ q, x Y , Y y π y“ µ Y ˝ x π x Y , y ˝ q, π x Y , Y y π y“ µ Y ˝ x , π y“ µ Y ˝ x , Y y ˝ π (13) “ π “ π ˝ Y ˆ Y . Combining these two equalities we get (iSs1) . Hence
Theorem 8.2.
In a regular unital category with binary coproducts, comonadiccovers and a natural imaginary splitting, an object is an intrinsic Schreier specialobject if and only if its canonical imaginary magma structure is a one-sided loopstructure. A non-varietal example
In this section we give an example of a non-varietal category for which thereexists a natural imaginary splitting, and we analyse what are the intrinsic Schreiersplit epimorphisms and the intrinsic Schreier special objects in that context.Take C “ Set op ˚ , which is a semi-abelian category [14, 4, 1], so it is a regularunital category with binary coproducts. We consider the powerset monad p P, δ, ε q in Set ˚ , where: P p X, x q “ p P p X q “ t A Ď X | x P A u , t x uq ,ε p X,x q p x q “ t x, x u ,δ p X,x q pt A i u i P I q “ ď i P I A i , where each A i P P p X q .The monad p P, δ, ε q may be seen as a comonad in Set op ˚ . Moreover, it is easyto check that each P p X, x q is projective in Set op ˚ , so that Set op ˚ is equipped withcomonadic projective covers; we are in the conditions of Subsection 4.3. A naturalimaginary splitting in Set op ˚ corresponds to a natural transformation t : p¨q ˆ p¨q ñ P pp¨q ` p¨qq in Set ˚ . We define, for any pair of pointed sets p A, ˚q and p B, ˚q , t A,B : p A ˆ B, p˚ , ˚qq Ñ p P p A ` B q , t˚uq : p a, b q ÞÑ t a, b, ˚u (24)It is easy to check that t is a natural transformation and that it satisfies the oppositeof equality (5), for all pointed sets p A, ˚q and p B, ˚qp P p A ` B q , t˚uqp A ˆ B, p˚ , ˚qq t A,B : ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ p A ` B, ˚q . v A
00 1 B w l r ε A ` B L R An intrinsic Schreier split epimorphism in
Set op ˚ corresponds to a split mono-morphism in Set ˚ . The following diagram represents a split monomorphism, given by an injection f , and its cokernel in Set ˚ p K, ˚q p X, ˚q k l r s , , p Y, ˚q , ? _ f l r where K “ X z Y Y t˚u , k p y q “ ˚ , for all y P Y and k p x q “ x , for all x P X z Y . Itis an intrinsic Schreier split monomorphism if there exists a morphism of pointedsets q : p K, ˚q Ñ p P p X q , t˚uq such that the opposite of equalities (iS1) and (iS2) hold. Note that q p x q P P p X q , i.e., ˚ P q p x q Ď X , for all x P X z Y .The opposite of (iS1) is given by the commutativity of the following diagram(we omit the fixed points to make it easier to read) P p X q δ X (cid:12) (cid:18) P p P p X q ` P p X qq P p p q q l r P p X q ˆ P p X q t P p X q ,P p X q l r P p X q X. ε X l r x qk,ε X fs y L R The commutativity of the diagram above always holds for any element y P Y . Forany element x P X z Y , we get t q p x q , t f s p x q , ˚u , t˚uu δ X (cid:12) (cid:18) t q p x q , t f s p x q , ˚u , t˚uu P p p q q l r p q p x q , t f s p x q , ˚uq t P p X q ,P p X q l r q p x q Y t f s p x q , ˚u “ t x, ˚u x. ε X l r x qk,ε X fs y L R From the equality q p x q Y t f s p x q , ˚u “ t x, ˚u , and the fact that s p x q P Y and x P X z Y , we deduce that s p x q “ ˚ and q p x q “ t x, ˚u . As a consequence the splitmonomorphism is isomorphic to the binary coproduct p X z Y Y t˚u , ˚q p X, ˚q s , k l r – (cid:12) (cid:18) p Y, ˚q _? f l r p X z Y Y t˚u , ˚q pp X z Y Y t˚uq ` Y, ˚q p Y q , p K q l r p Y, ˚q _? ι l r It is easy to see that the opposite of equality (iS2) always holds.We have just proved that in
Set op ˚ , with respect to the natural imaginary splitting(24), the only intrinsic Schreier split epimorphisms correspond to binary productprojections. Moreover, a pointed set p Y, ˚q is an intrinsic Schreier special object ifand only if (23) is a product projection, i.e., if and only if it is the zero object.Note that we could also apply the same approach to the finite powerset monadin Set ˚ .10. Intrinsic Schreier special objects vs. protomodular objects
Recall from [19] that an object Y in a finitely complete category is called a protomodular object when all points over it p f : X Ñ Y, s : Y Ñ X q are stablystrong. More precisely, for any pullback Z ˆ Y X , π Z (cid:12) (cid:18) X f (cid:12) (cid:18) Z g , x Z ,sg y L R Y, s L R NTRINSIC SCHREIER SPECIAL OBJECTS 27 the pair px , k y , x Z , sg yq , where k is the kernel of f , is a jointly extremal-epimorphicpair. If the point p f, s q is stably strong, then it is strong, i.e., p k, s q is a jointlyextremal-epimorphic pair.In the category Mon of monoids the notion of a Schreier special object and thenotion of a protomodular object both coincide with that of a group: a monoidis a Schreier special object if and only if it is a group [10] if and only if it is aprotomodular object [19].The question of understanding under which conditions these two notions coincidearises naturally. In general, neither of these notions implies the other, as we showedin [20]. Indeed, the variety
HSLat of Heyting semilattices provides an example ofa category where all objects are protomodular, but not every object is Schreierspecial ([20], Example 7.7).On the other hand, the cyclic group C “ pt , u , `q gives an example of aSchreier special object in the Jónsson-Tarski variety of unitary magmas Mag , be-cause it is a right loop. However, we gave an example of a point X Ô C whichis not strong. Consequently, C is not a protomodular object ([20], Example 7.4).Of key importance here is that the unitary magma p X, `q is non-associative . Aswe shall prove next in Corollary 10.4, the presence of the associativity axiom (Sec-tion 6) gives a link between intrinsic Schreier special objects and protomodularobjects: then Every intrinsic Schreier special object is a protomodular object.
The proof of this statement follows the same proof for monoids, i.e., that aSchreier special monoid Y is necessarily a group; the inverse of an element y P Y isgiven by q p , y q , where q is the imaginary retraction for (23) (see Proposition 3.1.6of [8]). Also, that all points over a group are necessarily Schreier split epimorphisms(see Corollary 3.1.7 of [8]). Lemma 10.1. If (23) satisfies (iSs1) , then µ Y ˝ x q ˝ x , Y y , Y y “ : Y x q ˝x , Y y , Y y , ❴❴❴❴❴❴ + ❖❖❖❖❖❖❖❖ Y ˆ Y. µ Y (cid:12) (cid:18) ✤✤✤ Y (25) Proof.
In Subsection 8.1 we saw that (iL1) follows from (iSs1) . If we precompose (iL1) with x , Y y : Y Ñ Y ˆ Y , we get µ Y ˝ x q, π y ˝ x , Y y “ π ˝ x , Y yô µ Y ˝ x q ˝ x , Y y , Y y “ (cid:3) Lemma 10.2.
Suppose that the natural addition p µ X : X ˆ X X q X P C is asso-ciative. If Y is an intrinsic Schreier special object, then Y x Y ,q ˝x , Y yy , ❴❴❴❴❴❴ + ❖❖❖❖❖❖❖❖ Y ˆ Y µ Y (cid:12) (cid:18) ✤✤✤ Y (26) µ Y ˝ x Y , q ˝ x , Y yy “ . Proof.
In Subsection 8.1 we saw that (iL2) follows from (iSs2) . If we precompose (iL2) with x µ Y ˝ x Y , q ˝ x , Y yy , Y y : Y Y ˆ Y , we obtain q ˝ x µ Y , π y ˝ x µ Y ˝ x Y , q ˝ x , Y yy , Y y “ π ˝ x µ Y ˝ x Y , q ˝ x , Y yy , Y y , which is equivalent to q ˝ x µ Y ˝ x µ Y ˝ x Y , q ˝ x , Y yy , Y y , Y y “ µ Y ˝ x Y , q ˝ x , Y yy (17) ô q ˝ x µ Y ˝ x Y , µ Y ˝ x q ˝ x , Y y , Y yy , Y y “ µ Y ˝ x Y , q ˝ x , Y yy (25) ô q ˝ x µ Y ˝ x Y , y , Y y “ µ Y ˝ x Y , q ˝ x , Y yyô q ˝ x µ Y ˝ x Y , y , Y y “ µ Y ˝ x Y , q ˝ x , Y yy (12) ô q ˝ x Y , Y y “ µ Y ˝ x Y , q ˝ x , Y yyô q ˝ x Y , Y y “ µ Y ˝ x Y , q ˝ x , Y yy p iSs4 q ô “ µ Y ˝ x Y , q ˝ x , Y yy . (cid:3) Proposition 10.3.
Suppose that the natural addition p µ X : X ˆ X X q X P C isassociative. If Y is an intrinsic Schreier special object, then any split epimorphism (16) satisfies (iS1) .Proof. We define an imaginary morphism ρ : X K through the universal prop-erty of the kernel K ✤ , k , X f , , Y. s l r X µ X ˝p X ˆ s q˝p X ˆp q ˝x , Y yqq˝x X ,f y L R ✤✤✤✤ ρ ] g Indeed, f ˝ µ X ˝ p X ˆ s q ˝ p X ˆ p q ˝ x , Y yqq ˝ x X , f y (14) “ µ Y ˝ p f ˆ f q ˝ x X , s ˝ q ˝ x , Y y f y“ µ Y ˝ x f , q ˝ x , Y y f y“ µ Y ˝ x Y , q ˝ x , Y yy ˝ f (26) “ . Now we must check (iS1) for (16): µ X ˝ x k ˝ ρ, sf y “ µ X ˝ x µ X ˝ x X , s ˝ q ˝ x , Y y f y , sf y (17) “ µ X ˝ x X , µ X ˝ x s ˝ q ˝ x , Y y f, sf yy“ µ X ˝ x X , µ X ˝ p s ˆ s q ˝ x q ˝ x , Y y , Y y ˝ f y (14) “ µ X ˝ x X , s ˝ µ Y ˝ x q ˝ x , Y y , Y y ˝ f y (25) “ µ X ˝ x X , y“ µ X ˝ x X , y (12) “ X . (cid:3) Corollary 10.4.
Suppose that the natural addition p µ X : X ˆ X X q X P C isassociative. Then every intrinsic Schreier special object is a protomodular object.Proof. This follows from Proposition 10.3 and Proposition 5.8 in [20], which statesthat any split epimorphism satisfying (iS1) is stably strong. (cid:3)
NTRINSIC SCHREIER SPECIAL OBJECTS 29
Remark . Even if the natural addition p µ X : X ˆ X X q X P C is associative,the converse of Corollary 10.4 may be false. As mentioned above, the variety HSLat of Heyting semilattices provides an example of a category where all objects areprotomodular, but not all are Schreier special objects. The natural addition for
HSLat , given by the meet, is associative.
Remark . The variety
Loop of (left and right) loops gives an example where thenatural addition is non-associative. All loops are intrinsic Schreier special objects(see Section 8) and they are also all protomodular objects (because
Loop is a semi-abelian category, thus a protomodular category). So, the fact that all intrinsicSchreier special objects are protomodular objects does not imply that the naturaladdition is associative.From the remark above, in Gp all objects are intrinsic Schreier special withrespect to its usual group operation. Also, all objects are protomodular since Gp isa protomodular category. So the two notions coincide in Gp , just as in the case of Mon . However, in
Mon there are only two possible choices of imaginary splittings(see Subsection 4.3). In Gp there are countably many possibilities. Given groups X and Y , a natural imaginary splitting t : P p X ˆ Y q Ñ X ` Y may be defined bymaking t prp x, y qsq equal to any combination of alternating products of x or x ´ ,and of y or y ´ , for which the products of the x s gives x and the products of the y s gives y . For example x ´ yx or xyx ´ y ´ xy .Although these notions are independent in general, as we have already observed,there are special properties of the category of groups that make the notions coin-cide. From Corollary 10.4, we know that the associativity of the group operationimplies that all intrinsic Schreier special objects are protomodular objects. Thisassociativity is not enough to guarantee that every protomodular object is intrinsicSchreier special (Remark 10.6). This leads us to the following question: What property of Gp guarantees that all protomodular objects are in-trinsic Schreier special ones? We cannot answer this question now, but we can see that groups lack a certainhomogeneity, in the sense that the concept of an intrinsic Schreier special objectstrongly depends on the chosen natural imaginary splitting. We can eliminate thisdiscrepancy by considering groups which satisfy the property with respect to all natural imaginary splittings. Then we find:11.
The variety of -Engel groups Recall that a -Engel group is a group E that satisfies the commutator identity rr x, y s , y s “ for all x , y P E . The aim of this section is to show that -Engelgroups are intrinsic Schreier special objects with respect to all natural imaginarysplittings: Proposition 11.10.We begin by recalling the definition and main properties of -Engel groups neededin the sequel, which can be found in [12, 13, 16].Here we denote the conjugation of an element x by an element y as y x “ yxy ´ and we write r x, y s for xyx ´ y ´ , so that r xy, z s “ x r y, z sr x, z s and r x, yz s “r x, y s y r x, z s . Note also that r x, y s ´ “ r y, x s . Definition 11.1.
A group E is called a -Engel group if it satisfies any of thefollowing equivalent conditions, for all elements x , y P E ,1. rr x, y s , x s “ ;2. rr x, y s , y s “ ;3. r x, y ´ s “ r x, y s ´ ;4. r x ´ , y s “ r x, y s ´ ; r x, y k s “ r x, y s k , for all k P Z ;6. r x k , y s “ r x, y s k , for all k P Z . Example .
1. Any abelian group is obviously a -Engel group.2. The group of quaternions Q is a -Engel group which is not abelian.3. The smallest non -Engel (thus non-abelian) group is the symmetric group S (which is isomorphic to the dihedral group D ).4. The dihedral group D is -Engel, but the Dihedral group D is not (seeExample 11.6). Lemma 11.3.
Let E be a -Engel group. Then: r xy, z s “ x pr y, z sr x, z sq , for all x , y , z P E ; r x, yz s “ y pr x, y sr x, z sq , for all x , y , z P E .Proof. r xy, z s “ x r y, z sr x, z s “ x r y, z s x ´ r x, z s . . “ x r y, z sr x, z s x ´ .2. r x, yz s “ r x, y s y r x, z s “ r x, y s y r x, z s y ´ . . “ y r x, y sr x, z s y ´ . (cid:3) It is known (and in fact not hard to see) that the free object on two generatorsin the variety
Eng p Gp q of -Engel groups is -nilpotent. This allows us to provethe following result. Lemma 11.4.
In a -Engel group E , let a , b and c be products of given elements x , y of E , or their inverses. Then: rr a, b s , c s “ ; r ab, c s “ r b, c sr a, c s and r a, bc s “ r a, b sr a, c s ; r a ´ , b s “ r a, b s ´ “ r a, b ´ s ; r a k , b s “ r a, b s k “ r a, b k s , for all k P Z .Proof.
1. Follows from the fact that the free -Engel group on two generators x and y is necessarily -nilpotent.2. r ab, c s “ a r b, c sr a, c s “ a r b, c s a ´ r a, c s . “ r b, c sr a, c s . The proof of the secondclaim is similar.3. r a ´ , b sr a, b s . “ r aa ´ , b s “ r , b s “ . The proof of the second claim is similar.4. Is just a particular case of . . and . . . (cid:3) We now look at a specific natural imaginary splitting in Gp : the one defined bythe function t X,Y : X ˆ Y X ` Y : p x, y q ÞÑ x ´ yx , (27)for any pair of groups X and Y . It is easy to see that this t is indeed a naturalimaginary splitting. When X “ Y , we write x ˚ y “ µ Y p x, y q “ ∇ Y p t Y,Y p x, y qq “ x ´ yx . It is easy to check that x ˚ “ x “ ˚ x and that x ˚ x ´ “ “ x ´ ˚ x ; however ˚ is not associative.A group Y is an intrinsic Schreier special object with respect to (27) if thereexists an imaginary retraction q : Y ˆ Y Y such that (iSs1) and (iSs2) hold.In this case (iSs1) means that q p x, y q ˚ y “ x , for all x , y P Y , and (iSs2) means that q p x ˚ y, y q “ x , for all x , y P Y —see Section 8. Proposition 11.5. If Y is a -Engel group, then Y is an intrinsic Schreier specialobject with respect to the natural imaginary splitting (27) . NTRINSIC SCHREIER SPECIAL OBJECTS 31
Proof.
We define the imaginary retraction by q p x, y q “ x ˚ y ´ . Then, for all x , y P Y , (iSs1) holds: q p x, y q ˚ y “ p x ˚ y ´ q ˚ y “ p x ´ y ´ x q ˚ y “ x ´ yxyx ´ y ´ x x ´ y ´ x “ x ´ y r x, y s xy ´ x . . “ x ´ y r y ´ , x s xy ´ x “ x ´ yy ´ xyx ´ xy ´ x “ x. As for (iSs2) , the equality q p x ˚ y, y q “ p x ˚ y q ˚ y ´ “ x holds by swapping y and y ´ in (iSs1) . (cid:3) Example . The Dihedral group D is generated by elements a and b such that a “ , b “ and abab “ . We have D “ t , a, a , a , a , b, ab, a b, a b, a b u , where the elements b , . . . , a b are all inverses to themselves. We have— D is not a -Engel group: r a, ab s ab “ a , while ab r a, ab s “ a b .— D is an intrinsic Schreier special object with respect to the natural ima-ginary splitting (27). It suffices to build the Cayley table for the product ˚ and observe that it gives a Latin square. The fact that it is a Latin squareguarantees the existence of a unique element, which is equal to q p x, y q , sat-isfying the equality (iSs1) q p x, y q ˚ y “ x . The equality (iSs2) follows fromthe uniqueness of each q p x, y q .— D is not an intrinsic Schreier special object with respect to the naturalimaginary splitting which gives rise to x ˚ y “ r x, y s xy . For example, q p , b q should be the unique element of D such that q p , b q ˚ b “ . However, allof the elements b , . . . , a b satisfy this equality.This example shows that the converse of Proposition 11.5 is false. However, wemay claim the following: Proposition 11.7.
If a group Y is an intrinsic Schreier special object with respectto the natural imaginary splitting (27) and such that q p x, y q “ x ˚ y ´ , then Y is a -Engel group.Proof. It suffices we use (iSs1) p x ˚ y ´ q ˚ y “ x to see that Y is in Eng p Gp q . Indeed, this is equivalent to x “ p x ´ y ´ x q ˚ y “ x ´ yxyx ´ y ´ x x ´ y ´ x “ x ´ yxyx ´ y ´ xy ´ x , which may be rewritten as “ x ´ yxyx ´ y ´ xy ´ x, so “ x ´ yxyx ´ y ´ xy ´ . This gives y ´ xyx ´ “ xyx ´ y ´ , or, equivalently, r y ´ , x s “ r x, y s “ r y, x s ´ . (cid:3) Next we aim to prove the that a -Engel group Y is an intrinsic Schreier specialobject with respect to all natural imaginary splittings t . So, we need to extendProposition 11.5 to all t . Lemma 11.8. If t is a natural imaginary splitting in Gp , then for each pair ofgroups X , Y and all x P X , y P Y we have that t X,Y p x, y q P X ` Y may be writtenas a product x k y l ¨ ¨ ¨ x k m y l m , for some m P N and k , . . . , k m , l , . . . , l m P Z such that ÿ ď i ď m k i “ “ ÿ ď i ď m l i . Proof. If X “ Y “ Z , then t Z , Z p k, l q P Z ` Z must be of the form k l ¨ ¨ ¨ k m l m ,for some m P N and k , . . . , k m , l , . . . , l m P Z such that ř ď i ď m k i “ k and ř ď i ď m l i “ l , for all p k, l q P Z ˆ Z (see Subsection 4.3). Consider the grouphomomorphisms f : Z Ñ X : 1 ÞÑ x and g : Z Ñ Y : 1 ÞÑ y . The naturality of t gives the commutative diagram (see (6)) Z ˆ Z t Z , Z , ❴❴❴ f ˆ g (cid:12) (cid:18) Z ` Z f ` g (cid:12) (cid:18) X ˆ Y t X,Y , ❴❴❴ X ` Y, from which we conclude that t X,Y p x, y q “ t X,Y p f ˆ g qp , q “ p f ` g q t Z , Z p , q .Suppose that t Z , Z p , q “ k l ¨ ¨ ¨ k m l m , where ř ď i ď m k i “ “ ř ď i ď m l i . We get t X,Y p x, y q “ x k y l ¨ ¨ ¨ x k m y l m , as desired. (cid:3) Proposition 11.9. If Y is a -Engel group and t is a natural imaginary splittingin Gp , then the induced operation x ˚ y “ µ Y p x, y q “ ∇ Y p t Y,Y p x, y qq may be writtenas x ˚ y “ r x, y s k xy for some k P Z .Proof. Lemma 11.8 tells us that x ˚ y “ x k y l ¨ ¨ ¨ x k m y l m , for some m P N and k , . . . , k m , l , . . . , l m P Z such that ř ď i ď m k i “ “ ř ď i ď m l i . We rewrite the expression above as x ˚ y “ p x k y l ¨ ¨ ¨ x k m y p l m ´ q x ´ y q xy, where the product in brackets is such that the sums of the exponents of the x ’s and y ’s are zero. Hence this expression is a commutator word in x and y : it is a productof (nested) commutators. By Lemma 11.4, all higher-order commutators in thisproduct vanish; furthermore, the expression is equal to a product of commutatorsof the form r x, y s or r y, x s “ r x, y s ´ . Hence it is of the form r x, y s k for some integer k . (cid:3) Proposition 11.10. If Y is a -Engel group, then Y is intrinsic Schreier specialwith respect to all natural imaginary splittings in Gp .Proof. The proof is similar to that of Proposition 11.5. We define the imaginaryretraction by q p x, y q “ x ˚ y ´ “ r x, y ´ s k xy ´ (Proposition 11.9). We use the NTRINSIC SCHREIER SPECIAL OBJECTS 33 properties in Definition 11.1 and Lemma 11.4 to prove that (iSs1) holds: p x ˚ y ´ q ˚ y “ pr x, y ´ s k xy ´ q ˚ y “ “ r x, y ´ s k xy ´ , y ‰ k r x, y ´ s k xy ´ y “ “ r x, y s ´ k xy ´ , y ‰ k r x, y s ´ k x “ ´ r x,y s ´ k r x, y srr x, y s ´ k , y s ¯ k r x, y s ´ k x “ ´´ r x,y s ´ k r x, y s ¯ ´ r x,y s ´ k rr x, y s ´ k , y s ¯¯ k r x, y s ´ k x “ r x, y s k r x, y s ´ k x “ x. As for (iSs2) , the equality q p x ˚ y, y q “ p x ˚ y q ˚ y ´ “ x follows from (iSs1) byreplacing y with y ´ . (cid:3) It remains an open question whether or not the converse of Proposition 11.10holds; we are currently working on this question. Essentially the same result holdsfor Lie algebras, as we shall explain now.12.
Lie algebras
Let K be a field, and consider the variety Lie K of K -Lie algebras. Recall thata -Engel Lie algebra is a Lie algebra e that satisfies the commutator identity rr x, y s , y s “ for all x , y P e . The aim of this section is to relate the variety Eng p Lie K q of -Engel Lie algebras over K to the Schreier special objects with respectto all natural imaginary splittings: Theorem 12.1. We can actually just follow thepattern of the previous section; since furthermore things are somewhat simpler here,we will only sketch the basic idea.We may proceed as in Proposition 11.5, now taking the natural imaginary split-ting in Lie K defined by t x , y : x ˆ y x ` y : p x, y q ÞÑ x ` y ` r x, y s . Recall that the free Lie algebra over K on a single generator is K itself, equippedwith the trivial bracket. Mimicking the proof of Lemma 11.8, we see that for anypair of K -Lie algebras x and y and any x P x , y P y , necessarily t x , y p x, y q “ x ` y ` φ p x, y q , where φ p x, y q is an expression in terms of Lie brackets of x ’s and y ’s. Now usingessentially the same proof as in groups, we see that if y is -Engel, all higher-orderbrackets vanish, and we deduce that t x , y p x, y q “ x ` y ` k r x, y s for some k P K . As in Proposition 11.9, it follows that x ˚ y “ x ` y ` k r x, y s . It isthen again easy to check that (iSs1) and (iSs2) hold. Proposition 12.1.
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Andrea Montoli, Dipartimento di Matematica “Federigo Enriques”, Università de-gli Studi di Milano, Via Saldini 50, 20133 Milano, Italy
Email address : [email protected] Diana Rodelo, Departamento de Matemática, Faculdade de Ciências e Tecnologia,Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal and CMUC,Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
Email address : [email protected] Tim Van der Linden, Institut de Recherche en Mathématique et Physique, Uni-versité catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, 1348 Louvain-la-Neuve, Belgium
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