GOURSAT COMPLETIONS DIANA RODELO AND IDRISS TCHOFFO NGUEFEU

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1 Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number GOUSAT COMPLETIONS DIANA ODELO AND IDISS TCHOFFO NGUEFEU Abstract: We characterize categories with weak inite limits whose regular completions give rise to Goursat categories. Keywords: regular category, projective cover, Goursat category, 3-permutable variety. Math. Subject Classiication (2010): 08C05, 18A35, 18B99, 18E10. Introduction The construction o the ree exact category over a category with inite limits was introduced in [3]. It was later improved to the construction o the ree exact category over a category with inite weak limits (weakly lex) in [4]. This ollowed rom the act that the uniqueness o the inite limits o the original category is never used in the construction; only the existence. In [4], the authors also considered the ree regular category over a weakly lex one. An important property o the ree exact (or regular) construction is that such categories always have enough (regular) projectives. In act, an exact category A may be seen as the exact completion o a weakly lex category i and only i it has enough projectives. I so, then A is the exact completion o any o its projective covers. Such a phenomenon is captured by varieties o universal algebras: they are the exact completions o their ull subcategory o ree algebras. Having this link in mind, our main interest in studying this subject is to possibly characterize projective covers o certain algebraic categories through simpler properties involving projectives and to relate those properties to the known varietal characterizations in terms o the existence o operations o eceived February 23, The irst author acknowledges partial inancial assistance by Centro de Matemática da Universidade de Coimbra UID/MAT/00324/2013, unded by the Portuguese Government through FCT/MCTES and co-unded by the European egional Development Fund through the Partnership Agreement PT2020. The second author acknowledges inancial assistance by Fonds de la echerche Scientiique- FNS Crédit Bre Séjour à l étrange018/v 3/5/033 - IB/JN , which supported his stay at the University o Algarve, where this paper was partially written. 1

2 2 DIANA ODELO AND IDISS TCHOFFO NGUEFEU their varietal theories (when it is the case). Such kind o studies have been done or the projective covers o categories which are: Mal tsev [11], protomodular and semi-abelian [5], (strongly) unital and subtractive [6]. The aim o this work is to obtain characterizations o the weakly lex categories whose regular completion is a Goursat (=3-permutable) category (Propositions 3.5 and 3.6). We then relate them to the existence o the quaternary operations which characterize the varieties o universal algebras which are 3-permutable (emark 3.7). 1. Preliminaries In this section, we briely recall some elementary categorical notions needed in the ollowing. A category with inite limits is regular i regular epimorphisms are stable under pullbacks, and kernel pairs have coequalisers. Equivalently, any arrow : A B has a unique actorisation = ir (up to isomorphism), where r is a regular epimorphism and i is a monomorphism and this actorisation is pullback stable. A relation rom to Y is a subobject r 1, : Y. The opposite relation o, denoted o, is the relation rom Y to given by the subobject, r 1 : Y. A relation rom to is called a relation on. We shall identiy a morphism : Y with the relation 1, : Y and write o or its opposite relation. Given two relations Y and S Y Z, we write S Z or their relational composite. With the above notations, any relation r 1, : Y can be seen as the relational composite r o 1. The ollowing properties are well known and easy to prove (see [1] or instance); we collect them in the ollowing lemma: Lemma 1.1. Let : Y be an arrow in a regular category C, and let = ir be its (regular epimorphism,monomorphism) actorisation. Then: (1) o is the kernel pair o, thus 1 o ; moreover, 1 = o i and only i is a monomorphism; (2) o is (i, i), thus o 1 Y ; moreover, o = 1 Y i and only i is a regular epimorphism; (3) o = and o o = o. A relation on is relexive i 1, symmetric i o, and transitive i. As usual, a relation on is an equivalence

3 GOUSAT COMPLETIONS 3 relation when it is relexive, symmetric and transitive. In particular, a kernel pair 1, 2 : Eq() o a morphism : Y is an equivalence relation. By dropping the assumption o uniqueness o the actorization in the deinition o a limit, one obtains the deinition o a weak limit. We call weakly lex a category with weak inite limits. An object P in a category is (regular) projective i, or any arrow : P and or any regular epimorphism g : Y there exists an arrow h : P Y such that gh =. We say that a ull subcategory C o A is a projective cover o A i two conditions are satisied: any object o C is regular projective in A; or any object in A, there exists a (C-)cover o,that is an object C in C and a regular epimorphism C When A admits a projective cover, one says that A has enough projectives. emark 1.2. I C is a projective cover o a weakly lex category A, then C is also weakly lex [4]. For example, let and Y be objects in C and W Y a weak product o and Y in A. Then, or any cover W W o W, W Y is a weak product o and Y in C. Furthermore, i A is a regular category, then the induced morphism W Y is a regular epimorphism. Similar remarks apply to all weak inite limits. 2. Goursat categories In this section we review the notion o Goursat category and the characterizations o Goursat categories through regular images o equivalence relations and through Goursat pushouts. Deinition 2.1. [2, 1] A regular category C is called a Goursat category when the equivalence relations in C are 3-permutable, i.e. S = SS or any pair o equivalence relations and S on the same object. When C is a regular category, (, r 1, ) is an equivalence relation on and : Y is a regular epimorphism, we deine the regular image o along : Y to be the relation () on Y induced by the (regular epimorphism, monomorphism) actorization s 1, s 2 ψ o the composite (

4 4 DIANA ODELO AND IDISS TCHOFFO NGUEFEU ) r 1, : r 1, ψ () s 1,s 2 Y Y. Note that the regular image () can be obtained as the relational composite () = o = r o 1 o. When is an equivalence relation, () is also relexive and symmetric. In a general regular category () is not necessarily an equivalence relation. This is the case in a Goursat category according to the ollowing theorem. Theorem 2.2. [1] A regular category C is a Goursat category i and only i or any regular epimorphism : Y and any equivalence relation on, the regular image () = o o along is an equivalence relation. Goursat categories are well known in universal algebras. In act, by a classical theorem in [10], a variety o universal algebras is a Goursat category precisely when its theory has two quaternary operations p and q such that the identities p(x, y, y, z) = x, q(x, y, y, z) = z and p(x, x, y, y) = q(x, x, y, y) hold. Accordingly, the varieties o groups, Heyting algebras and implication algebras are Goursat categories. The category o topological group, Hausdor groups, right complemented semi-group are also Goursat categories. There are many known characterizations o Goursat categories (see [1, 7, 8, 9] or instance). In particular the ollowing characterization, through Goursat pushouts, will be useul: Theorem 2.3. [7] Let C be a regular category. The ollowing conditions are equivalent: (i) C is a Goursat category; (ii) any commutative diagram o type (I) in C, where α and β are regular epimorphisms and and g are split epimorphisms s Y α (I) β t U W, g gα=β αs=tβ

5 GOUSAT COMPLETIONS 5 (which is necessarily a pushout) is a Goursat pushout: the morphism λ : Eq() Eq(g), induced by the universal property o kernel pair Eq(g) o g, is a regular epimorphism. emark 2.4. Diagram (I) is a Goursat pushout precisely when the regular image o Eq() along α is (isomorphic to) Eq(g). From Theorem 2.3, it then ollows that a regular category C is a Goursat category i and only i or any commutative diagram o type (I) one has α(eq()) = Eq(g). Note that Theorem 2.2 characterizes Goursat categories through the property that regular images o equivalence relations are equivalence relations, while Theorem 2.3 characterizes them through the property that regular images o (certain) kernel pairs are kernel pairs. 3. Projective covers o Goursat categories In this section, we characterize the categories with weak inite limits whose regular completion are Goursat categories. Deinition 3.1. Let C be a weakly lex category: (1) a pseudo-relation on an object o C is a pair o parallel arrows r1 ; a pseudo-relation is a relation i r 1 and are jointly monomorphic; (2) a pseudo-relation r1 on is said to be: relexive when there is an arrow r : such that r 1 r = 1 = r; symmetric when there is an arrow σ : such that = r 1 σ and r 1 = σ; transitive i by considering a weak pullback W p 2 p 1 r 1 r2, there is an arrow t : W such that r 1 t = r 1 p 1 and t = p 2. an pseudo-equivalence relation i it is relexive, symmetric and transitive.

6 6 DIANA ODELO AND IDISS TCHOFFO NGUEFEU emark that the transitivity o a pseudo-relation r1 does not depend on the choice o the weak pullback o r 1 and ; in act, i W p 2 p 1 r 1 r2, is another weak pullback, the actorization W W composed with the transitivity t : W ensures that the pseudo-relation is transitive also with respect to the second weak pullback. The ollowing properties in [12] (Proposition 1.1.9) will be useul in the sequel: Proposition 3.2. [12] Let C be a projective cover o a regular category A. Let r1 be a pseudo-relation in C and consider its (regular epimorphism, monomorphism) actorization in A (r 1, ). p E (e 1,e 2 ) Then, is a pseudo-equivalence relation in C i and only i S is an equivalence relation in A. Deinition 3.3. Let C be a weakly lex category. We call C a weak Goursat category i, or any pseudo-equivalence relation r1 and any reg- ular epimorphism : Y, the composite r1 is also a pseudoequivalence relation. We may also consider weak Goursat categories through a property more similar to the one mentioned in Theorem 2.2:

7 GOUSAT COMPLETIONS 7 Lemma 3.4. Let C be a projective cover o a regular category A. Then C is a weak Goursat category i and only i or any commutative diagram in C r 1 ϕ S s 1 Y s 2 (1) such that and ϕ are regular epimorphism and is a pseudo-equivalence relation, then S is a pseudo-equivalence relation. Proo : (i) (ii) Since r1 is a pseudo-equivalence relation, by assumption r1 is also a pseudo-equivalence relation and then its (regular epimorphism, monomorphism) actorization gives an equivalence relation E e1 e 2 Y in A. We have the ollowing commutative diagram r 1 r 1 e 1 ρ ϕ E σ S Y e 2 s 1 s 2 where σ : S E is induced by the strong epimorphism ϕ ϕ σ ρ E e 1,e 2 S s 1,s 2 Y Y. Then σ is a regular epimorphism and S s1 s 2 Y is a pseudo-equivalence relation (Proposition 3.2).

8 8 DIANA ODELO AND IDISS TCHOFFO NGUEFEU (ii) (i) Let r1 be a pseudo-equivalence relation in C and : Y a regular epimorphism. The ollowing diagram is o the type (1) r 1 r 1 Y. Since r1 is a pseudo-equivalence relation, then by assumption r1 Y is also a pseudo-equivalence relation. We use emark 1.2 repeatedly in the next results. Proposition 3.5. Let C be a projective cover o a regular category A. Then A is a Goursat category i and only i C is a weak Goursat category. Proo : Since C is a projective cover o a regular category A, then C is weakly lex. Suppose that A is a Goursat category. Let r1 be a pseudo-equivalence relation in C and let : Y be a regular epimorphism in C. For the (regular epimorphism, monomorphism) actorizations o r 1, and r 1, we get the ollowing diagram r 1, p E w e 1,e 2 (2) q S s 1,s 2 r 1, Y Y,

9 GOUSAT COMPLETIONS 9 where w : E S is induced by the strong epimorphism p q S w p s 1,s 2 E Y Y. ( ) e 1,e 2 Then w is a regular epimorphism and by the commutativity o the right side o (2), one has S = (E). By Proposition 3.2, we know that E is an equivalence relation in A. Since A is a Goursat category, then S = (E) is also an equivalence relation in A and by Proposition 3.2, we can conclude that r1 is a pseudo-equivalence relation in C. Conversely, suppose that C is a weak Goursat category. Let r1 be an equivalence relation in A and : Y a regular epimorphism. We are going to show that () = S h () = S r 1 s 1 Y s 2 is an equivalence relation; it is obviously relexive and symmetric. In order to conclude that A is a Goursat category, we must prove that S is transitive, i.e that S is an equivalence relation. We begin by covering the regular epimorphism in A with a regular epimorphism in C. For that we take the cover y : Ȳ Y, consider the pullback o y and in A and take its cover α : Y Ȳ x α Y Ȳ Ȳ y y Y.

10 10 DIANA ODELO AND IDISS TCHOFFO NGUEFEU Note that the above outer diagram is a regular pushout, so that o y = x o (3) (Proposition 2.1 in [1]). Next, we take the inverse image x 1 () in A, which in an equivalence relation since is, and cover it to obtain a pseudo-equivalence W in C. By assumption W Ȳ is a pseudo-equivalence relation in C so it actors through an equivalence relation, say V v1 Ȳ, in A. We have v 2 W w x 1 () ρ 1,ρ 2 π x x r 1, h γ W Ȳ Ȳ v v 1,v 2 V y y λ S s 1,s 2 Y Y, where γ and λ are induced by the strong epimorphisms w and v, respectively W v γ w x 1 () ρ 1,ρ 2 and hπ w W λ v V v 1,v 2 Ȳ Ȳ V v 1,v 2 Ȳ Ȳ S s 1,s 2 y y Y Y.

11 GOUSAT COMPLETIONS 11 Since γ is a regular epimorphism, we have V = (x 1 ()). Since λ is a regular epimorphism, we have S = y(v ). One also has V = y 1 (S) because Finally, S is transitive since y 1 (S) = y o Sy = y o ()y = y o o y = x o x o (by (3)) = (x 1 ()) = V. SS = yy o Syy o Syy o (Lemma 1.1(2)) = yy 1 (S)y 1 (S)y o = yv V y o = yv y o (since V is an equivalence relation) = y(v ) = S. Alternatively, weak Goursat categories may be characterized through a property more similar to the one mentioned in emark 2.4: Proposition 3.6. Let C be a projective cover o a regular category A. The ollowing conditions are equivalent: (i) A is a Goursat category; (ii) C is a weak Goursat category; (iii) For any commutative diagram o type (I) in C where β 1 s F Y β 2 λ α (I) β G ρ 1 t U W ρ 2 g F is a weak kernel pair o and λ is a regular epimorphism (in C), then G is a weak kernel pair o g. Proo : (i) (ii) By Proposition 3.5.

12 12 DIANA ODELO AND IDISS TCHOFFO NGUEFEU (i) (iii) I we take the kernel pairs o and g, then the induced morphism ᾱ : Eq() Eq(g) is a regular epimorphism by Theorem 2.3. Moreover, the induced morphism ϕ : F Eq() is also a regular epimorphism. We get F ϕ Eq() 1 s Y 2 λ G ω ᾱ Eq(g) where w : G Eq(g) is induced by the strong epimorphism λ F ᾱ.ϕ α β λ w g 1 t Eq(g) g 1,g 2 U W, g 2 g G U. ρ 1 ρ 1,ρ 2 This implies that ω is a regular epimorphism (ωλ = ᾱϕ) and then G ρ 1 is a weak kernel pair o g. (iii) (ii) Consider the diagram (1) in C where r1 is a pseudo- equivalence relation. We want to prove that S s1 Y is also a pseudoequivalence. Take the (regular epimorphism, monomorphism) actorization s 2 o and S in A and the induced morphism µ making the ollowing diagram commutative r 1 u 1 ρ ϕ U µ u 2 s 1 S Y. s 2 v 1 σ v 2 ρ 2 V ρ 2 U

13 GOUSAT COMPLETIONS 13 Since µ is a regular epimorphism, then V = (U) and consequently, V is relexive and symmetric. Since S is a pseudo-relation associated to V, then S is also a relexive and symmetric pseudo-relation. We just need to prove that V is transitive. To do so, we apply our assumption to the diagram F λ G δ Eq(r 1 ) ϕ(eq(r1 )) G α Eq(r 1 ) χ ϕ(eq(r 1 )) e r 1 ϕ (I) where G is a cover o the regular image ϕ(eq(r 1 )) and F is a cover o the pullback Eq(r 1 ) ϕ(eq(r1 ))G. Since δ is a regular epimorphism, then F is a weak kernel pair o r 1. By assumption G S is a weak kernel pair o s 1, thus ϕ(eq(r 1 )) = Eq(s 1 ). We then have V V = v 2 v1v o 1 v2 o ( since V is symmetric) = v 2 σσ o v1v o 1 σσ o v2 o (Lemma 1.1(2)) = s 2 s o 1s 1 s o 2 = s 2 ϕr1r o 1 ϕ o s o 2 (ϕ(eq(r 1 )) = Eq(s 1 )) = r1r o 1 r2 o o (s i ϕ = r i ) = u 2 ρρ o u o 1u 1 ρρ o u o 2 o = u 2 u o 1u 1 u o 2 o (Lemma 1.1(2)) = UU o (since U is an equivalence relation in A) = U o = V. e S S Y s 1 emark 3.7. When A is a 3-permutable variety and C its subcategory o ree algebras, then the property stated in Proposition 3.6 (iii) is precisely what is needed to obtain the existence o the quaternary operations p and q

14 14 DIANA ODELO AND IDISS TCHOFFO NGUEFEU which characterize 3-permutable varieties. Let denote the ree algebra on one element. Diagram (I) below belongs to C F µ Eq( ) π 1 ι 2 +ι π (I) λ F λµ Eq( 3 ) 3 ι 2. 3 I F is a cover o Eq( )), then F 4 is a weak kernel pair o By assumption F 3 is a weak kernel pair o 3, so that λµ is surjective. We then conclude that λ is surjective and the existence o the quaternary operations p and q ollows rom Theorem 3 in [7]. eerences [1] A. Carboni, G.M. Kelly, M.C. Pedicchio, Some remarks on Mal tsev and Goursat categories, Appl. Categ. Structures 1 (1993), no. 4, [2] A. Carboni, J. Lambek, M.C. Pedicchio, Diagram chasing in Mal cev categories, J. Pure Appl. Algebra 69 (1991) [3] A. Carboni and. Celia Magno, The ree exact category on a let exact one, J. Austr. Math. Soc. Ser., A 33 (1982) [4] A. Carboni and E. Vitale, egular and exact completions, J. Pure Appl. Algebra 125 (1998) [5] M. Gran, Semi-abelian exact completions, Homology, Hom. Appl. 4 (2002) [6] M. Gran and D. odelo, On the characterization o Jónsson-Tarski and o subtractive varieties, Diagrammes, Suppl. Vol (2012) [7] M. Gran and D. odelo, A new characterisation o Goursat categories, Appl. Categ. Structures 20 (2012), no. 3, [8] M. Gran and D. odelo, Beck-Chevalley condition and Goursat categories, J. Pure Appl. Algebra, 221 (2017) [9] M. Gran, D. odelo and I. Tchoo Ngueeu, Some remarks on connecteurs and groupoids in Goursat categories, Logical Methods in Computer Science (2017) Vol. 13(3:14), [10] J. Hagemann and A. Mitschke, On n-permutable congruences, Algebra Universalis 3 (1973), [11] J. osicky and E.M. Vitale, Exact completions and representations in abelian categories, Homology, Hom. Appl. 3 (2001) [12] E. Vitale, Let covering unctors, PhD thesis, Université catholique de Louvain (1994).

15 GOUSAT COMPLETIONS 15 Diana odelo CMUC, Department o Mathematics, University o Coimbra, Coimbra, Portugal and Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, Faro, Portugal address: drodelo@ualg.pt Idriss Tchoo Ngueeu Institut de echerche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium address: idriss.tchoo@uclouvain.be