RELTIVE GOURST CTEGORIES JULI GOEDECKE ND TMR JNELIDZE bstract. We deine relative Goursat cateories and prove relative versions o the equivalent conditions deinin reular Goursat cateories. These include 3-permutability o equivalence relations, preservation o equivalence relations under direct imaes, a condition on so-called Goursat pushouts, and the denormalised 3 3-Lemma. This extends recent wor by Gran and Rodelo on a new characterisation o Goursat cateories to a relative context. Introduction ccordin to. Carboni, G. M. Kelly and M. C. Pedicchio [2], a Goursat cateory can be deined as a reular cateory satisyin the 3-permutability o equivalence relations, that is, havin RSR = SRS or every two equivalence relations R and S on the same object. However, it is nown that there are several other equivalent deinitions and characterizations, includin the ollowin two, recently obtained by M. Gran and D. Rodelo [4]: 1. reular cateory is Goursat i and only i or every pushout C D o reular epimorphisms in, where and are split epimorphisms, the induced morphism between the ernel pairs o and is a reular epimorphism; such pushouts are called Goursat pushouts. 2. reular cateory is Goursat i and only i it satisies the so-called denormalized 3 3-Lemma. This denormalised 3 3-Lemma was irst introduced in a reular Mal tsev context by D. ourn in [1] and was proved to hold in reular Goursat cateories by S. Lac in [10]. The purpose o the present paper is to extend these two results to characterize what we call relative Goursat cateories by main the ollowin replacements: The reular cateory is replaced with a pair (, E) where E is a class o reular epimorphisms in satisyin suitable conditions; when has all inite limits and coequalizers o ernel pairs and E is the class o all reular epimorphisms in, these conditions mae reular; the Goursat pushouts are required to have all their arrows in E; the 3 3-Lemma is replaced by its E-relative version (see Theorem 3.2 below). In act, we show in detail that our conditions on (, E) allow us to repeat essentially all aruments o [10] and [4]. Our main tool here is the calculus o E-relations developed in [6, 7], which, unlie previously nown versions, does not require E to be part o a actorization system. Its oriinal motivation was to introduce and study relative semi-abelian and relative homoloical cateories in [7, 6, 5], and now we have the implications: relative semi-abelian [7, 6] relative homoloical [5] relative Mal tsev [3] relative Goursat relative reular. Date:?? but this version is dated 26th uust 2010. 2010 Mathematics Subject Classiication. 1820, 18G25, 18G50. Key words and phrases. Goursat; non-abelian relative homoloical alebra; three-permutable equivalence relations; relative reular cateories; relative Goursat cateories; relative denormalised 3x3 lemma; The irst author was supported by the FNRS rant Crédit aux chercheurs 1.5.016.10F. The second author was supported by Claude Leon Foundation Postdoctoral Fellowship. 1
2 JULI GOEDECKE ND TMR JNELIDZE 1. Relations in a relative settin When worin with relations, one usually uses the reular imae actorisation in a reular cateory to obtain composition o relations (see e.. [2]). In a relative settin, reular epimorphisms are replaced with a suitable class E o reular epimorphisms in the round cateory, and a relative actorisation axiom is used instead o reular imae actorisation to compose relations [7] (see also [6]). In this paper, we consider a slihtly more eneral settin or relations than the one in [7], namely, we do not require the existence o all pullbacs as in [7], but only as or pullbacs o morphisms in E to exist. We introduce: Deinition 1.1. relative reular cateory is a pair (, E) where is a cateory with inite products and E is a class o reular epimorphisms in such that the ollowin axioms hold: (E1) E contains all isomorphisms; (E2) pullbacs o morphisms in E exist in and are in E; (E3) E is closed under composition; (E4) i E and E then E; (F) i a morphism in actors as = em with m a monomorphism and e E, then it also actors (essentially uniquely) as = m e with m a monomorphism and e E. Note that i is a cateory with products and all pullbacs and E is a class o reular epimorphisms in containin all isomorphisms, then (, E) is a relative reular cateory (i.e. it satisies axioms (E2) (F)) i and only i (, E) satisies Condition 1.1 o [7]. Note also that this context is not as eneral as the one considered in [8] where products are not required to exist. Thereore, the level o enerality here is between those o [7] and [8]. Remar 1.2 (The absolute case ). s easily ollows rom Deinition 1.1, i is a cateory with inite limits and has coequalizers o ernel pairs and E is the class o all reular epimorphisms in, then (, E) is a relative reular cateory i and only i is a reular cateory. Relative reular cateories provide a convenient settin or the calculus o E-relations in the same way that reular cateories do or the calculus o relations. The ollowin deinitions and properties o E-relations are the relative versions o classical properties o relations. Their proos easily ollow those o the absolute version and appear in [6] (the absolute versions can be ound or example in [2]). Deinition 1.3 (E-relations). Given two objects and in, an E-relation R rom to is a subobject r 1, r 2 : R o such that the morphisms r 1 : R and r 2 : R are in E. We denote such an E-relation by (R, r 1, r 2 ) or just by R, and its opposite (R, r 2, r 1 ) by R ; we will also write R: or an E-relation R rom to. When =, we may also say that R is an E-relation on. We can compose two E-relations (R, r 1, r 2 ) rom to and (S, s 1, s 2 ) rom to C by ormin the pullbac o r 2 and s 1 and then usin the actorisation rom xiom (F) to obtain a monomorphism SR C: P mono P R S R r 1 S E r r 2 s 1 1 s 2 E s 2 SR C mono C xioms (E2), (E3) and (E4) ensure that this composite is aain an E-relation. Moreover, the composition is associative (as we identiy isomorphic relations) and we have: (R ) = R, (SR) = R S, R R R R (orderin as subobjects), i R R and S S then SR S R, or all E-relations R:, R :, S : C, and S : C in. Remar 1.4. Given a morphism : in E, we can use (E1) to view as an E-relation = (, 1, ); its opposite is = (,, 1 ). It is easy to see (c. [2, 7]) that is the ernel pair o, = 1,
RELTIVE GOURST CTEGORIES 3 =, =, or any E-relation (R, r 1, r 2 ) we have R = r 2 r 1. Deinition 1.5. n E-relation (R, r 1, r 2 ) on an object in is said to be relexive i 1 R, symmetric i R R (and thus R = R), transitive i RR R, an equivalence E-relation i it is relexive, symmetric and transitive; an E-eective equivalence E-relation i it is a ernel pair o some morphism in E. s easily ollows rom Deinition 1.5, an E-relation R: which is relexive and transitive satisies RR = R. Note also that the ernel pair o any morphism E is an (E-eective) equivalence E-relation, by pullbac-stability (E2). This allows us to copy the n = 3 version o [2, Theorem 3.5] to a relative version o the same theorem. We ive the proo or convenience. Proposition 1.6. Let (, E) be a relative reular cateory. The ollowin conditions are equivalent: (i) or equivalence E-relations R and S on an object, we have RSR = SRS; (ii) this 3-permutability RSR = SRS holds when R and S are E-eective equivalence E-relations; (iii) every E-relation P rom to satisies P P P P = P P ; (iv) or every relexive E-relation E on an object, the E-relation EE is an equivalence E-relation; (v) or every relexive E-relation E, the E-relation EE is transitive; (vi) or every relexive E-relation E, we have EE = E E. Proo. Clearly (i) (ii). Given an E-relation P rom to we view it as p 1, p 2 : P such that P = p 2 p 1. Then p 1p 1 and p 2p 2 are the ernel pairs o p 1 and p 2 respectively, and thereore E-eective equivalence E-relations. Hence, by (ii) and usin Remar 1.4, we obtain: P P P P = p 2 p 1p 1 p 2p 2 p 1p 1 p 2 = p 2 p 2p 2 p 1p 1 p 2p 2 p 2 = p 2 p 1p 1 p 2 = P P, provin (ii) (iii). Now iven a relexive E-relation E on as in (iv), the relexivity 1 E and the induced 1 E imply 1 EE, ivin relexivity o EE. Symmetry is automatic as (EE ) = EE, and transitivity EE EE = EE ollows rom (iii), thereore (iii) (iv). Clearly (iv) (v), and (v) (vi) since relexivity o E ives E E EE EE EE. It remains to prove (vi) (i). Given two equivalence E-relations R and S on an object, we have R = R, RR = R and the same or S; moreover, their composite E = SR is clearly relexive. Thereore we have SRS = SRR S = R S SR = RSR, which ives (i). The case n = 2 o [2, Theorem 3.5] deines a reular Mal tsev cateory and is stated in its relative version in [3]. In the absolute case o reular cateories, it is possible to orm the direct imae o any endo-relation [2, 4]. In a similar way, we can orm an E-imae o an endo-e-relation in our relative settin. Deinition 1.7 (E-imae). Let (, E) be a relative reular cateory. Given an E-relation (R, r 1, r 2 ) on an object o and a morphism : in E, we deine the E-imae o R alon to be the relation S on which is induced by the (E, mono)-actorisation s 1, s 2 ϕ o the morphism ( ) r 1, r 2 r 1,r 2 R ϕ S s 1,s 2 which exists by axiom (F). We write (R) = S, which is aain an E-relation by axiom (E4). Remar 1.8. When R is a relexive E-relation, the essential uniqueness o (E, mono)-actorisations implies that (R) is also relexive, and when R is a symmetric E-relation, ϕ bein an epimorphism implies that (R) is symmetric. In the next section we will see under which conditions the E-imae (R) o an equivalence E-relation R is aain an equivalence E-relation.
4 JULI GOEDECKE ND TMR JNELIDZE s in the absolute case [2, 4], we have an easy way to orm the E-imae: Lemma 1.9. Let (, E) be a relative reular cateory. Given an E-relation (R, r 1, r 2 ) on an object in and a morphism : in E, the E-imae (R) can be ormed as the composite (R) = R = r 2 r1. Furthermore, usin Remar 1.4 and the deinition o E-imae as well as Lemma 1.9, we easily see the ollowin. Corollary 1.10. Let (, E) be a relative reular cateory. Given a commutative diaram R S r 1 r 2 s 1 s 2 where R and S are E-relations and E, the morphism is in E i and only i S = (R), or equivalently i and only i s 2 s 1 = r 2 r 1. I (R, r 1, r 2 ) and (S, s 1, s 2 ) are ernel pairs with coequalizers r and s in E, then the latter is also equivalent to s s = r r. For the main result in the next section, we need the ollowin lemma rom [3]: Lemma 1.11 ([3, Lemma 3.3]). Let (, E) be a relative reular cateory. Given a morphism o (downward) split epimorphisms h C with in E, the induced morphism between the ernel pairs o h and is also in E. D 2. The relative Goursat axiom We now prove an equivalence o several conditions, which in the absolute case all characterise reular Goursat cateories (see [2] and [4]). Theorem 2.1. Let (, E) be a relative reular cateory. Then the ollowin conditions are equivalent: (i) the E-Goursat axiom: iven a morphism o (downward) split epimorphisms h C in with,, h and in E, the induced morphism between the ernel pairs o and is also in E; (ii) the E-imae o an equivalence E-relation is an equivalence E-relation; (iii) or every relexive E-relation E on an object, the E-relation EE is an equivalence E-relation; (iv) or equivalence E-relations R and S on an object, we have RSR = SRS. Proo. The proo o (i) (ii) is the same as its absolute version iven in [4, Theorem 2.3]. We ive it here or completeness. Let (R, r 1, r 2 ) be an equivalence E-relation on and let : be in E. We want to show that the E-imae (R) = (S, s 1, s 2 ) o R alon is aain an equivalence E-relation. Since S is relexive and symmetric by Remar 1.8, we only have to show that it is transitive, that is, SS S. However, since S is symmetric, it suices to show the existance o a morphism t S : S 1 S, where (S 1, π 1, π 2 ) is the ernel pair o s 1, which maes the ollowin diaram commute: π 1 D t S S 1 S π 2 s 1 S s 2 s 2 (1)
RELTIVE GOURST CTEGORIES 5 Since R is (symmetric and) transitive, there exists a morphism t R : R 1 R, where R 1 is the ernel pair o r 1, main the correspondin diaram or R commute: t R R 1 R r 1 R r2 Usin the morphisms e R and e S which deine the relexivity o R and S, we obtain a diaram r 1 R ϕ S e R s 1 o type (1), where ϕ is the E-part o the (E, mono)-actorisation s 1, s 2 ϕ o ( ) r 1, r 2. Thereore, by (i), the induced morphism ϕ: R 1 S 1 between the ernel pairs o r 1 and s 1 is in E. Since every morphism in E is a reular epimorphism and thereore a stron epimorphism and s 1, s 2 is a monomorphism, we obtain a unique diaonal t S in the square r 2 e S R 1 ϕ S 1 ϕ t R t S S s1,s 2 (s 2 s 2) π 1,π 2 main both trianles commute, which is the required morphism. For (ii) (iii) it is easy to see that or a relexive E-relation (E, e 1, e 2 ) on an object we have EE = e 1 (E 2 ), where E 2 is the ernel pair o e 2. Thereore EE is an equivalence E-relation as the E-imae o the equivalence E-relation E 2. Conditions (iii) and (iv) are equivalent by Proposition 1.6. Finally, or (iv) (i) we aain use the proo rom [4, Theorem 2.3]. For convenience, we copy the proo and add our adapted justiications or the relative settin. Given a diaram such as (1), Lemma 1.11 implies that the induced split epimorphism between the ernel pairs H o h and K o is in E. This means that (H) = K. Now usin Lemma 1.9 and the three-permutability rom (iv) on the ernel pairs H = h h and F =, we see that h(f ) = h h (by Lemma 1.9) = hh h h hh (since hh h = h) = h h h h (by (iv)) = h h (since (H) = K) = hh hh (since = h) = (since hh = 1) = G, where G is the ernel pair o. y Corollary 1.10, this implies that the induced morphism between the ernel pairs F G is in E. s all these conditions characterise Goursat cateories in the absolute case [2, 4], we are now justiied in the ollowin deinition. Deinition 2.2. relative Goursat cateory is a relative reular cateory (, E) in which moreover the ollowin axiom holds: (G) the E-Goursat axiom: iven a morphism o (downward) split epimorphisms h C D
6 JULI GOEDECKE ND TMR JNELIDZE in with,, h and in E, the induced morphism between the ernel pairs o and is also in E. Compare this to the deinition o relative Mal tsev cateories in [3]: the E-Mal tsev axiom (E5) iven there says that or any morphism o split epimorphisms (1) with,, h and in E, the canonical morphism, h to the pullbac D C is also in E. s in the absolute case, this relative Mal tsev axiom implies the E-Goursat axiom (G) (see [3, Lemma 3.4]). 3. The relative 3 3-Lemma In this section we prove the relative version o the so-called denormalized 3 3-Lemma in the context o relative Goursat cateories. Furthermore, ollowin the absolute case layed out in [4], we show that, in a relative reular cateory (, E), the relative 3 3-Lemma is in act equivalent to the E-Goursat axiom. Deinition 3.1. Let (, E) be a relative reular cateory. We will say that the diaram is E-exact when (F, 1, 2 ) is the ernel pair o and is in E. 1 F (2) 2 Note that when (2) is E-exact, the morphisms 1 and 2 are also in E by pullbac-stability (E2). In the proo o the relative 3 3-Lemma, we will need the ollowin Lemma 3.2 ([7, Theorem 2.10]). Let (, E) be a relative reular cateory. Given a diaram h C D with the morphisms, h,, and in E, we have h = i and only i = h and the canonical morphism, h : D C is in E. We irst show that the relative 3 3-Lemma does indeed hold in any relative Goursat cateory. Theorem 3.3 (The relative 3 3-Lemma). Let (, E) be a relative Goursat cateory. Given a commutative diaram F h 1 h 2 F h G 2 H 1 h 1 h 2 1 2 K 2 1 h 2 C D 1 (3) with E-exact columns and middle row, the irst row is E-exact i and only i the third row is E-exact. Proo. The proo is the same as in the absolute case [10]; we repeat it here with the appropriate justiications or the relative case. Suppose the third row is E-exact. Since 1 and 2 are jointly monic, an easy diaram chase proves that (F, h 1, h 2 ) is the ernel pair o h. Thereore, it remains to show that h is in E. Note that since is in E, Corollary 1.10 implies that the E-imae (H) = H is equal to K. We have: = hh hh (since hh = 1) = h h (since h = ) = h h h h (since (H) = K) = hh h h hh (by Theorem 2.1(iv)) = h h (since hh h = h) Thereore G = h(f ) and, by Corollary 1.10, h is in E.
RELTIVE GOURST CTEGORIES 7 Conversely, suppose the irst row is E-exact. y xiom (E4) the morphisms 1, 2 and are in E. We have: = (since = 1) = h h (since h = ) = h h h h (since h(f ) = G) = h h (by Theorem 2.1(iv)) = h h (since = ) = h 2 h 1 (since H = h h = h 2 h 1) = 1 2 (since h i = i ) = 1 2 (since = 1) Thereore, since = 1 2 and the morphisms 1, 2, and are in E, Lemma 3.2 implies that 1e = 1 and 2e = 2 or some e E, where ( 1, 2) is the ernel pair o. It remains to prove that e is an isomorphism. For this, we replace ( 1, 2 ) by ( 1, 2) and by e in the 3 3-diaram (3). Then the three rows, the second column and the third column are E-exact; thereore, by the irst part o the proo, the irst column is also E-exact. Thus and e are both coequalizers o 1 and 2, yieldin that e is an isomorphism. Now we show that the relative 3 3-Lemma is equivalent to the E-Goursat axiom in a relative reular cateory. Theorem 3.4. Let (, E) be a relative reular cateory. The ollowin conditions are equivalent: (i) (, E) is a relative Goursat cateory, that is, the axiom (G) holds in (, E); (ii) the relative 3 3-Lemma holds in (, E); (iii) in a diaram such as (3), i the irst row is E-exact then the third row is also E-exact; (iv) in a diaram such as (3), i the third row is E-exact then the irst row is also E-exact. Proo. The proo is the same as in the absolute case [4]; we aain repeat it or convenience. The part (i) (ii) is Theorem 3.3, and the implications (ii) (iii) and (ii) (iv) are obvious. (iv) (i): Consider the commutative diaram (1) with the assumptions o axiom (G). Let (H, h 1, h 2 ) and (K, 1, 2 ) be the ernel pairs o h and respectively, and let : H K be the induced morphism; since is in E, Lemma 1.11 implies that is also in E. Tain the ernel pairs o, and and the induced morphisms between them, we obtain the diaram (3) with the three columns and the second and the third rows E-exact. Then (iv) implies that the irst row is also E-exact, and thereore the induced morphism between the ernel pairs o and is in E, provin (i). (iii) (i): ain consider the commutative diaram (1). Let (F, 1, 2 ) be the ernel pair o and let (T, t 1, t 2 ) be the E-imae o F alon h. To prove (i) it suices to show that (T, t 1, t 2 ) is the ernel pair o. For this, consider the commutative diaram 2 E H K 1 η 1 η 2 h 1 h 2 1 2 2 F 1 η h T t 2 t 1 C (4) in which (E, η 1, η 2 ), (H, h 1, h 2 ), and (K, 1, 2 ) are the ernel pairs o η, h, and respectively, and 1, 2 : E H are the induced morphisms. Since is in E, Lemma 1.11 implies that is also in E. Moreover, since t 1 and t 2 are jointly monic, it ollows that (E, 1, 2 ) is the ernel pair o, main the irst column E-exact. Then, applyin (iii) to Diaram (4) with the role o columns and rows interchaned, we obtain that the third column is also E-exact. Thereore, (T, t 1, t 2 ) is the ernel pair o, as desired. Remar 3.5. Theorem 3.4 is in act the relative version o Proposition 3.2 o [4]; in particular, our (iii) (iv) corresponds to 3.2(c) 3.2(d) in [4], which itsel can be considered as a non-pointed version o 5.4(b) 5.4(c) in [9]. D
8 JULI GOEDECKE ND TMR JNELIDZE s mentioned in the introduction, we have a chain o implications o various relative cateories: any relative Goursat cateory is by deinition relatively reular, and any relative Mal tsev cateory is relatively Goursat, as proved in [3, Lemma 3.4]. Furthermore, every relative homoloical cateory is relatively Mal tsev by [7, Theorem 2.14] or [3, Proposition 4.5], and any relative semi-abelian cateory is relatively homoloical by deinition, see [7, Deinition 3.2]. This is the same chain o implications as in the absolute case. cnowledements We would lie to than Marino Gran or his suestion to loo at relative Goursat cateories usin the Goursat pushout as a deinition. Reerences [1] D. ourn, The denormalized 3 3 lemma, J. Pure ppl. lebra 177 (2003), 113 129. [2]. Carboni, G. M. Kelly, and M. C. Pedicchio, Some remars on Maltsev and Goursat cateories, ppl. Cate. Structures 1 (1993), 385 421. [3] T. Everaert, J. Goedece, and T. Van der Linden, Resolutions, hiher extensions and the relative Mal tsev axiom, in preparation, 2010. [4] M. Gran and D. Rodelo, new characterisation o Goursat cateories, Pré-Publicações do Departamento de Matemática, Universidade de Coimbra, Preprint Number 10-10, 2010. [5] T. Janelidze, Relative homoloical cateories, J. Homotopy Rel. Struct. 1 (2006), no. 1, 185 194. [6] T. Janelidze, Foundation o relative non-abelian homoloical alebra, Ph.D. thesis, University o Cape Town, 2009. [7] T. Janelidze, Relative semi-abelian cateories, ppl. Cate. Structures 17 (2009), 373 386. [8] T. Janelidze, Incomplete relative semi-abelian cateories, ppl. Cate. Structures, doi:10.1007/s10485-009-9193-4 (2009). [9] Z. Janelidze, The pointed subobject unctor, 3 3 lemmas, and subtractivity o spans, Theory ppl. Cate., 23 (2010), no. 11, 221 242. [10] S. Lac, The 3-by-3 lemma or reular oursat cateories, Homoloy, Homotopy ppl. 6 (2004), no. 1, 1 3. Julia Goedece, Institut de recherche en mathématique et physique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, elium E-mail address: julia.oedece@cantab.net Tamar Janelidze, Department o Mathematical Sciences, Stellenbosch University, Private a X1, Matieland 7602, Stellenbosch, South rica E-mail address: tamar@sun.ac.za