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1 The copyright o this thesis rests with the. No quotation rom it or inormation derived rom it is to be published without ull acknowledgement o the source. The thesis is to be used or private study or non-commercial research purposes only.

2 FOUNDTION OF RELTIVE NON-BELIN HOMOLOGICL LGEBR Tamar Janelidze Thesis Presented or the Degree o DOCTOR OF PHILOSOPHY in the Department o Mathematics and pplied Mathematics UNIVERSITY OF CPE TOWN ugust 2009 Supervisors: Pro. H.-P. Künzi, Pro. W. P. Tholen, Pro. G. Janelidze

3 bstract We consider pairs (C, E), where C is a pointed category and E a class o regular/normal epimorphisms in C, satisying various exactness properties. The purpose o this thesis is: 1. To introduce and study suitable notions o a relative homological and a relative semiabelian category. In the absolute case, where E is the class o all regular epimorphisms in C, the pair (C, E) is relative homological/semi-abelian i and only i C is homological/semiabelian; that is, we obtain known concepts. ccordingly we extend known analysis o the axiom systems, and in particular show that suitable lists o old style and new style axioms are equivalent; this requires developing a relative version o what is usually called the calculus o relations. We then present various non-absolute examples, where these results can be applied. 2. To ormulate and prove relative versions o classical homological lemmas; this includes Five Lemma, Nine Lemma, and Snake Lemma. i

4 Contents Introduction 1 1 Preliminaries Regular and normal epimorphisms Regular and Barr exact categories Protomodular categories Homological categories Semi-abelian categories Calculus o E-relations Category o E-relations Properties o the E-relations Equivalence E-relations Relative homological categories xioms or incomplete relative homological categories Relative homological categories Examples Homological lemmas in incomplete relative homological categories E-exact sequences The Five Lemma The Nine Lemma The Snake Lemma ii

5 5 Relative semi-abelian categories xioms or incomplete relative semi-abelian categories Relative semi-abelian categories Examples Bibliography 101 iii

6 Introduction The title o the thesis ( Foundation o relative non-abelian homological algebra ) is suggested by classical work o S. Eilenberg and J. C. Moore [14]. Relative homological algebra in abelian categories also appears in the irst two books in homological algebra, namely in [12] and [30], and in a number o papers o many students and ollowers o Samuel Eilenberg and Saunders Mac Lane. The term non-abelian has several meanings; here it means suitable or non-abelian groups, or rings, or algebras. ccordingly, the non-abelian categories o our interest include semi-abelian categories in the sense o G. Janelidze, L. Márki, and W. Tholen [23], and, more generally, homological categories in the sense o F. Borceux and D. Bourn [3] and protomodular categories in the sense o D. Bourn [6]. On the other hand, the term relative reers, just as in the abelian case, to a distinguished class E o regular/normal epimorphisms in the ground category C - in contrast to the absolute case, where the role o E is played by the class o all regular epimorphisms in C. nd in act various axioms we impose on (C, E) make the ground category C semi-abelian or homological only in the absolute case. In particular, we do not exclude the trivial case o C being an arbitrary pointed category and E the class o isomorphisms in C. In the abelian case this approach goes back to N. Yoneda [34], whose quasi-abelian categories can in act be deined as pairs (C, E) where C is an additive category in which the short exact sequences K B with B in E have the same properties as all short exact sequences in an abelian category. The purpose o the thesis is two-old: 1. Detailed study o axiom systems or relative semi-abelian and relative homological categories; the aim was to obtain the relative versions o the results o [23] or semi-abelian categories and o homological categories. For, we develop the calculus o E-relations which 1

7 easily ollows its well-known absolute version, in which E is the class o all regular epimorphisms in C (see e.g. [10]). Those results o [23], symbolically expressed as OLD=NEW, actually have a long history behind them, which begins with Mac Lane s amous Duality or Groups [29]: ter observing that several basic concepts in group theory can be described in abstract categorical terms, making some o them dual to each other, Mac Lane proposes a number o undamental categorical constructions to be used in developing categorical group theory. He then says: urther development can be made by introducing additional careully chosen dual axioms. This will be done below only in the more symmetrical abelian case The careul choice took more than iteen years o many researchers, who arrived to a very non-dual list o non-abelian axioms, which produced categorical versions o many known results, especially in homological algebra and Kurosh-mitsur radical theory; at the same time it seemed to be too technical, and eventually was nearly ignored and/or orgotten. The development o topos theory in the sixties/seventies strongly supports a new approach in categorical algebra that arrives to Barr exact categories [1]. Barr exact category is abelian whenever it is additive; yet, every variety o universal algebras is Barr exact, making the old and new approaches seemingly incomparable. The new concept o a protomodular category, due to Bourn (see [6]), which turned out to be the missing link, is introduced only in 1990, i.e. ater twenty years. nd the main conclusion o [23] is that a pointed category satisies the old orgotten axioms i and only i is Barr exact and Bourn protomodular, and has inite coproducts. Such categories were called semi-abelian or two reasons: (a) a category C is abelian i and only i both C and its dual category C op are semi-abelian; (b) a Barr exact category is semi-abelian i and only i it admits semidirect products in the sense o D. Bourn and G. Janelidze [8], and then it is abelian i and only i its semidirect products are (direct) products. ccordingly, the present work in act deals with a number o categorical axioms and exactness properties studied by various authors or many years, and it involves relativisation o both old and new axioms systems. 2. The study o relative versions o the so-called classical homological lemmas, especially the Five Lemma, Nine Lemma, and Snake Lemma. The absolute versions are well-known in the abelian case, and in the non-abelian case given in [3]. The proo o the Snake Lemma that we give involves partial composition o internal relations in C and goes back at least to S. Mac Lane [28] (see also e.g. [10] or the so-called calculus o relations in regular categories). 2

8 The thesis consists o the ollowing chapters: Chapter 1: We begin with recalling the relevant properties o regular and normal epimorphisms, and then give a brie overview o regular, Barr exact [1], Bourn protomodular [6], homological [3], and semi-abelian [23] categories (see also [2]). Chapter 2: We develop what we call a relative calculus o relations. That is: or a pair (C, E), in which C is a pointed category and E is a class o regular epimorphisms in C satisying certain conditions, we study the relations (R, r 1, r 2 ) : B in C having the morphisms r 1 and r 2 in E. Our calculus o E-relations extends its well-known absolute version, in which E is the class o all regular epimorphisms in C (see e.g. [10]). Chapter 3: In the irst section we introduce a notion o incomplete relative homological category in such a way that we have: (a) Trivial Case : (C, Isomorphisms in C) is an incomplete relative homological category or every pointed category C; (b) bsolute Case : (C, Regular epimorphisms in C) is an incomplete relative homological category i and only i C is a homological category in the sense o [3]. In the second section we consider the special case o C being initely complete and cocomplete, and deine a relative homologcial category accordingly. In the third section we consider various examples. Chapter 4: We extend Five Lemma, Nine Lemma, and Snake Lemma to the context o incomplete relative homological categories. The proos ollow the proos given in [3], although the proo o Snake Lemma (as already mentioned) substantially uses the calculus o E-relations described in Chapter 2. Chapter 5: In the irst section we introduce a notion o incomplete relative semi-abelian category in such a way that we have: (a) Trivial Case : (C, Isomorphisms in C) is an incomplete relative semi-abelian category or every pointed category C; 3

9 (b) bsolute Case : (C, Regular epimorphisms in C) is an incomplete relative semiabelian category i and only i C is a semi-abelian category in the sense o [23]; (c) relative semi-abelian category (C, E) is an incomplete relative homological category in which: (i) every equivalence E-relation is E-eective (i.e. every equivalence E- relation is the kernel pair o some morphism in E); (ii) i a morphism : B is in E then the coproduct Ker() + B exists in C. nd, we prove the (incomplete) relative version o the main result o [23], which asserts that the old-style axioms and the new-style axioms or the semi-abelian categories are equivalent. For, we again use the calculus o E-relations described in Chapter 2. In the second section we consider the special case o C being initely complete and cocomplete, and deine a relative semi-abelian category accordingly. In the third section we consider various examples. The results obtained appear as the three papers [24], [25], [26], and the ourth paper [27] is submitted or the publication. 4

10 Chapter 1 Preliminaries 1.1 Regular and normal epimorphisms Deinition morphism : B in a category C is said to be a regular epimorphism, i it is the coequalizer o some pair o parallel morphisms in C. Proposition In a category C with kernel pairs, every regular epimorphism is the coequalizer o its kernel pair. Proposition Let C be a category with pullbacks. The composite g o regular epimorphisms : B and g : B C in C is a regular epimorphism whenever is a pullback stable epimorphism. In particular, the class o pullback stable regular epimorphisms in C is closed under composition. Proo. Let : B and g : B C be regular epimorphisms in C, and let ( 1, 2 ), (g 1, g 2 ), and (h 1, h 2 ) be the kernel pairs o, g, and g respectively. To prove that g is a regular epimorphism it suices to prove that g is the coequalizer o h 1 and h 2. For, consider the commutative diagram C B C B h h 2 g 1 g 2 B 1 2 h 1 B g C h b c C 5

11 in which: - = h1, h 2 is the canonical morphism, i.e. since (g 1, g 2 ) is the kernel pair o g and gh 1 = gh 2, there exists a unique morphism : C B C B with g 1 = h1 and g 2 = h2 ; - h : C is any morphism with h h 1 = h h 2 ; - h = 1, 2 is the canonical morphism, i.e. since (h 1, h 2 ) is the kernel pair o g and g 1 = g 2, there exists a unique morphism h : B C with h 1 h = 1 and h 2 h = 2, yielding h 1 = h h 1 h = h h 2 h = h 2 ; - Since is the coequalizer o 1 and 2, and h 1 = h 2, there exists a unique morphism b : B C with b = h. We have to show that there exists a unique morphism c : C C with cg = h (since g is an epimorphism we do not need to prove the uniqueness o c), but since g is the coequalizer o g 1 and g 2, it is suicient to show that bg 1 = bg 2. It is easy to see that the morphism : C B C B is the composite o the canonical morphisms h 1, h 2 : C B C and π 1, π 2 : B C B C B, where (B C, π 1, π 2 ) is the pullback o g and g. Then, since is a pullback stable epimorphism, we obtain that the morphisms h 1, h 2 and π 1, π 2 are epimorphisms, and thereore their composite also is an epimorphism. We have: bg 1 = bh1 = h h 1 = h h 2 = bh 2 = bg 2, and since is an epimorphism we conclude that bg 1 = bg 2, as desired. Proposition Let C be a category with pullbacks. I the composite g o : B and g : B C in C is a regular epimorphism, then so is the morphism g i any one o the ollowing conditions hold: (i) is an epimorphism; (ii) g is a pullback stable epimorphism. 6

12 Proo. Under the condition (i), consider the ollowing diagram C B C B h 1 h 2 g 1 g 2 B g C g where (g 1, g 2 ) is the kernel pair o g, (h 1, h 2 ) is the kernel pair o g, is the canonical morphism, and g : B C is any morphism with g g 1 = g g. To prove that g is a regular epimorphism it suices to prove that g is the coequalizer o g 1 and g 2 ; hence, we need prove the existance o a unique morphism c : C C such that cg = g. Since g is the coequalizer o h 1 and h 2, the equalities c C (g )h 1 = g g 1 = g g 2 = (g )h 2 imply the existance o a unique morphism c : C C with c(g) = g. Since is an epimorphism, the last equality implies cg = g ; and, since g is an epimorphism, such c is unique. Next, suppose condition (ii) holds instead o condition (i). Consider the pullback diagram we have: B C π 2 π 1 B g C g - Since π 2 can be obtained as a pullback o the pullback o g along g, it is a split epimorphism, and hence a pullback stable regular epimorphism; - Since g is a regular epimorphism and π 2 is a pullback stable regular epimorphism, by Proposition 1.1.3, gπ 1 = gπ 2 is a regular epimorphism; - Since g is a pullback stable epimorphism, π 1 also is an epimorphism. 7

13 Since π 1 is an epimorphism and gπ 1 is a regular epimorphism, the irst part o the proo implies that g is an epimorphism, as desired. Deinition morphism : B in a category C is said to be a strong epimorphism, i or every commutative diagram o the orm B g h C m D (1.1) where m is a monomorphism, there exists a unique morphism β : B C with β = g and mβ = h. s easily ollows rom Deinition 1.1.5, i a category C has equalizers then every regular epimorphism is strong. Indeed, i : B is a regular epimorphism in C, then it is the coequalizer o some pair o parallel morphisms ( 1, 2 ). For any commutative diagram (1.1) with a monomorphism m, we have g 1 = g 2, thereore there exists a unique morphism β : B C with β = g; and mβ = h since mβ = h and is an epimorphism. Deinition morphism : B in a category C is said to be a normal epimorphism, i it the cokernel o some morphism in C. Proposition In a category C with kernels, every normal epimorphism is the cokernel o its kernel. Using the same arguments as in the proos o Proposition and Proposition we can prove the ollowing: Proposition Let C be a category with pullbacks. The composite g o normal epimorphisms : B and g : B C in C is a normal epimorphism whenever is a pullback stable epimorphism. In particular, the class o pullback stable normal epimorphisms in C is closed under composition. Proposition Let C be a category with pullbacks. I the composite g o : B and g : B C in C is a normal epimorphism, then so is the morphism g i any one o the ollowing conditions hold: (i) is an epimorphism; (ii) g is a pullback stable epimorphism. 8

14 1.2 Regular and Barr exact categories Deinition category C is said to be regular (see e.g. [2]), i: (a) C has inite limits; (b) C has a pullback stable (regular epi, mono)-actorization system. I a morphism in any category C actors as = me in which e is a regular epimorphism and m is a monomorphism, then e is the coequalizer o the kernelpair o, provided that the latter exists. nd, conversely, i e is the coequalizer o the kernelpair o and m is the morphism with = me, then m is a monomorphism i the regular epimorphisms in C are pullback stable. Thereore, we have: Proposition Let C be a category with inite limits. The ollowing conditions are equivalent: (i) C has a pullback stable (regular epi, mono)-actorization system. (ii) (a) C has coequlizers o kernel pairs; (b) Regular epimorphisms in C are pullback stable. Regular categories admit a good calculus o relations (see e.g. [10]): Recall, that a relation R rom an object to an object B in C, written as R : B, is a subobject r 1, r 2 : R B, where r 1, r 2 is the canonical morphism rom R to the product B induced by r 1 : R and r 2 : R B; as subobjects, the relations rom to B orm an ordered set with inite meets. Equivalently, we can deine a relation R : B as a triple (R, r 1, r 2 ) in which R is an object in C and r 1 and r 2 are jointly monic morphisms. For the given relations (R, r 1, r 2 ) : B and (S, s 1, s 2 ) : B C the composite SR is the relation rom to C deined as the mono part o the (regular epi, mono)-actorization o the morphism r 1 p 1, s 2 p 2 : R B S C, where (R B S, p 1, p 2 ) is the pullback o r 2 9

15 and s 1 : R B S p 1 p 2 e R SR r 1 r 2 S s 2 s 1 t 1 m t 2 B C π 1 π 2 C That is, the composite o the relations R : B and S : B C is the subobject SR C, i.e. it is the triple (SR, t 1, t 2 ) where t 1 = π 1 m and t 2 = π 2 m and π 1 and π 2 are the irst and the second product projections o C respectively; regularity o C implies that such a composition is associative. Recall, that a relation R : in a regular category C is said to be an equivalence relation i it is relexive, symmetric, and transitive, i.e. 1 R, R R, and RR R. It is easy to see that or a given morphism : B, the kernelpair o is an equivalence relation rom to (or more details about the relations in a regular category see also [11]). Theorem For a regular category the ollowing conditions are equivalent, and deine a Mal tsev category: (a) For equivalence relations R and S on an object, the relation SR is an equivalence relation. (b) For such equivalence relations we have SR = RS. (c) Every relation R : B is diunctional; that is, RR R = R. (d) Every relexive relation is an equivalence relation. (e) Every relexive relation is transitive. (e) Every relexive relation is symmetric. For the proo, see Theorem 3.6 o [10] (see also Theorem 1 o [15]). Deinition category C is said to be Barr-exact [1], i: 10

16 (a) C is a regular category; (b) Every equivalence relation in C is eective, i.e. it is the kernel pair o some morphism in C. Theorem (Theorem 5.7 o [10]). regular category C is an exact Mal tsev category i and only i, given regular epimorphisms r : B and s : C with a common domain, their pushout s C r v B u D exists in C, and moreover, the canonical morphism r, s : B D C is a regular epimorphism. 1.3 Protomodular categories Let C be a category and B an object in C. Recall that Pt C (B) = Pt(C B) is a category, whose objects are triples (,, g), in which is an object in C, and : B and g : B are morphisms in C satisying g = 1 B. morphism α : (,, g) (,, g ) in Pt C (B) is deined as a morphism α : in C such that α = and αg = g. Note that i C has pullbacks, then every morphism v : B B in C induces the pullback unctor v : Pt C (B ) Pt C (B) which pulls back o (,, g ) along v. Deinition category C is said to be protomodular (in the sense o D. Bourn [6]), i the ollowing conditions hold: (a) C has pullbacks; (b) For every morphism v : B B in C, the pullback unctor v : Pt C (B ) Pt C (B) relects isomorphisms. It is easy to see that i C has a zero object 0, then in Deinition 1.3.1(b) it suices to consider the morphism 0 B : 0 B instead o an arbitrary morphism v : B B. Indeed: in the presence o a zero object C, the category Pt C (0) is isomorphic to the category C, and since 0 B = v0 B, the relection o isomorphisms 0 B = 0 B v implies the same or v. Since pulling back : B along 0 B is taking the kernel o, we obtain the ollowing 11

17 Corollary I C is a category with pullbacks and a zero object, then C is protomodular i and only i or every object B in C, the kernel unctor ker B : Pt C (B) C relects isomorphisms. This proves the ollowing Proposition Let C be a category with pullbacks and a zero object. The ollowing conditions are equivalent: (i) C is protomodular. (ii) The Split Short Five Lemma holds true in C, that is: or every commutative diagram u K k w B K k B v (3.1) with k = ker(), k = ker( ), and and split epimorphisms, w is an isomorphisms i u and v are isomorphisms. Remark I a protomodular category is also regular, then the Split Short Five Lemma is equivalent to the Regular Short Five Lemma, which states: given the commutative diagram (3.1) with k = ker() and k = ker( ), i and are regular epimorphisms and u and v are isomorphisms, then w is an isomorphism [6]. Proposition I C is a pointed protomodular category with pullbacks, then: (i) Every regular epimorphism in C is a normal epimorphism. (ii) Every split epimorphism in C is a normal epimorphism. Proo. (i): Let : B be a regular epimorphism in C, q = coker(ker()), and let ( 1, 2 ) and (q 1, q 2 ) be the kernel pairs o and q respectively. It is a well known act, that in this situation the morphisms 1 and q 1 (and also 2 and q 2 ) are split epimorphisms, and Ker(q 1 ) Ker( 1 ). This gives us a commutative diagram Ker(q 1 ) k q 1 Ker( 1 ) k h B 1 12

18 in which: = Coker(Ker()), k = ker( 1 ), k = ker(q 1 ), h is the canonical morphism between the pullbacks, and 1 and q 1 are split epimorphisms. Hence, by protomodularity we obtain that h is an isomorphism. Since and q are regular epimorphisms and they have isomorphic kernel pairs, we conclude that is a normal epimorphism, as desired. Since every split epimorphism is a regular epimorphism, (ii) ollows directly rom (i). Theorem (Proposition o [3]). ny initely complete protomodular category C is a Mal tsev category. This statement was in act irst proved in [7]. 1.4 Homological categories Homological categories, according to [3], provide the most convenient setting or proving non-abelian versions o various standard homological lemmas, such as the Five Lemma, the 3 3-Lemma, and the Snake Lemma. We recall: Deinition (Deinition o [3]). category C is homological when (a) C is pointed; (b) C is regular; (c) C is protomodular. Or, equivalently, a category C is homological i and only i the ollowing conditions hold: (a) C has inite limits; (b) C has a zero object; (c) C has coequalizers o kernel pairs; (d) Regular epimorphisms in C are pullback stable; (e) The (Split) Short Five Lemma holds in C. s usually, a sequence o morphisms... i 1 i 1 i i i+1... (4.1) 13

19 in a homological category C is said to be exact at i, i the mono part o the (regular epi, mono)-actorization o i 1 is the kernel o i. nd, (4.1) is said to be an exact sequence, i it is exact at i or each i (unless the sequence either begins or ends with i ). Proposition (Lemma o [3]). In a pointed protomodular category C, in particular in a homological category, the sequence 0 B g C 0 is exact i and only i = ker(g) and g is a regular epimorphism. Proposition (Proposition o [3]). Let C be a homological category. (i) The sequence 0 B is exact i and only i is a monomorphism. (ii) The sequence 0 B g C is exact i and only i = ker(g). (iii) The sequence B 0 is exact i and only i is a regular epimorphism. (iv) The sequence B g C 0 is exact i and only i g = coker(). We now recall the above mentioned homological lemmas involving exact sequences (see again [3]): Theorem (The Five Lemma). Let C be a homological category. I in a commutative diagram B α β γ g C h D δ k E ɛ B g C h D k E 14

20 the two rows are exact sequences, and the morphisms α, β, δ, and ɛ are isomorphisms, then γ is also an isomorphism. Theorem (The Nine Lemma). Let C be a homological category. I in a commutative diagram X Y Z 0 0 X Y 0 X Y Z Z the three columns and the middle row are exact sequences, then the irst row is an exact sequence i and only i the last row is an exact sequence. Theorem (The Snake Lemma). Let C be a homological category. I in a commutative diagram X Y u v g Z 0 X Y g Z the two rows are exact sequences, then there exists a morphism d : Ker(w) Coker(u), such that the sequence w Ker(u) Ker(v) Ker(w) d Coker(u) Coker(v) Coker(w) where the unlabeled arrows are the canonical morphisms, is exact. 1.5 Semi-abelian categories The notion o an abelian category was introduced by S. Mac Lane in 1950 in his paper Duality or groups [29]; it was however more restrictive than the one used today, which 15

21 was given by D.. Buchsbaum in Exact Categories and Duality [9] in 1955 (under the name exact category ). Let us recall that a category C is said to be abelian ([9], and, see also [16], [18]) i the ollowing conditions hold: (a) C has a zero object; (b) C has binary products and binary coproducts; (c) Every morphism has a kernel and a cokernel; (d) Every monomorphism is a kernel, every epimorphism is a cokernel. Categories o abelian groups and o modules are abelian categories, which is certainly not the case or the categories o groups, rings or algebras over rings; the easiest way to see this is just to note that not all o their monomorphisms are normal. The semi-abelian categories, introduced by G. Janelidze, L. Marki, and W. Tholen in 1999 (published in 2002; see [23]), play, however, the same role or groups, rings, and algebras, as the abelian categories do or the abelian groups and modules. We recall: Deinition category C is said to be semi-abelian, i: (a) C has a zero object and coproducts; (b) C is Barr-exact; (c) C is protomodular. That is, a category C is semi-abelian, i it satisies the ollowing Condition ( New-style axioms ). (a) C has a zero object, inite limits, and coproducts; (b) C has coequalizers o kernel pairs; (c) The regular epimorphisms in C are pullback stable; (d) The Split Short Five Lemma holds in C; (e) ll equivalence relations r 1, r 2 : R in C are eective equivalence relations. 16

22 Conditions 1.5.2(a)-1.5.2(e) are regarded as the new-style axioms or a semi-abelian category. s proved in [23], these conditions are equivalent to the old-style axioms involving normal monomorphisms and normal epimorphisms: Condition ( Old-style axioms ). (a) C has a zero object, inite limits, and coproducts; (b) C has cokernels o kernels, and every morphism with a zero kernel is a monomorphism; (c) The normal epimorphisms in C are pullback stable; (d) (Homann s axiom) I in a commutative diagram w B B and are normal epimorphisms, w is a monomorphism, v is a normal monomorphism, and ker( ) w, then w is a normal monomorphism; (e) For every commutative diagram m B B with and normal epimorphisms and m and m monomorphisms, i m is a normal monomorphism then m also is a normal monomorphism. Protomodularity in terms o the old-style axioms is the Homann s axiom [19], while the Barr s exactness condition is Condition 1.5.3(e). Remark Using the notion o a homological category, a semi-abelian category can be deined as a homological category with coproducts in which every equivalence relation is eective (see Proposition o [3]). Thereore, one can obtain the old-style axioms also or the homological categories, i.e. deine the homological categories using normal epimorphisms and the Homann s axiom. This will be done in a more general setting in Chapter 4. v m 17

23 Chapter 2 Calculus o E-relations 2.1 Category o E-relations Throughout this chapter we assume that (C, E) is a pair in which C is a category and E is a class o regular epimorphisms in C containing all isomorphisms and satisying the ollowing Condition (a) The class E is closed under composition; (b) I E and g E then g E; (c) diagram o the orm B g g has a limit in C provided, g,, and g are in E, and either (i) = g and = g, or (ii) (, g) and (, g ) are relexive pairs (that is, h = 1 B = gh and h = 1 B = g h or some h and h ), and and g are jointly monic. B (d) I B π 2 π 1 B 18

24 is a pullback and and are in E, then π 1 and π 2 are also in E; (e) I h 1 : H and h 2 : H B are jointly monic morphisms in C and i α : C and β : B D are morphisms in E, then there exists a morphism h : H X in E and jointly monic morphisms x 1 : X C and x 2 : X D in C making the diagram commutative. α C x 1 h 1 H X h 2 h B x 2 β Remark I the morphisms : B and : B are in E, then the pullback ( B, π 1, π 2 ) o and exists in C by Condition 2.1.1(c), and π 1 and π 2 are in E by Condition 2.1.1(d). Thereore, the kernel pair o (and ) also exists in C. The two basic examples o a pair (C, E) satisying Condition are: 1. Trivial case : C is a category and E is the class o all isomorphisms in C. 2. bsolute case : C is a regular category and E is the class o all regular epimorphisms in C. Proposition The actorization in Condition 2.1.1(e) is unctorial. That is, i h 1 H h X B α β x 1 x q 2 C D k x l H h 1 m h h n 2 X B α β x 1 x C 2 D 19 h 2 D (1.1)

25 is a commutative diagram in C, in which: - (h 1, h 2 ), (x 1, x 2 ), (h 1, h 2 ), and (x 1, x 2 ) are jointly monic pairs; - α, β, h, α, β, and h are morphisms in E; - m, k, q, l, and n are any morphisms making the diagram (1.1) commutative; then there exists a morphism x : X X or which the diagram (1.1) is still commutative. Proo. Since h is in E, the kernel pair (u 1, u 2 ) o h exists by Remark Since x 1 and x 2 are jointly monic and the equalities x 1h qu 1 = mαh 1 u 1 = mx 1 hu 1 = mx 1 hu 2 = mαh 1 u 2 = x 1h qu 2, x 2h qu 1 = nβh 2 u 1 = nx 2 hu 1 = nx 2 hu 2 = nβh 2 u 2 = x 2h qu 2 hold, we conclude that h qu 1 = h qu 2. Since h is the coequalizer o u 1 and u 2, the last equality implies the existence o a unique morphism x : X X with h q = xh. It remains to prove that x 1 x = mx 1 and x 2 x = nx 2; however, since h is an epimorphism, the latter ollows rom ollowing equalities: x 1xh = x 1h q = mx 1 h, x 2xh = x 2h q = nx 2 h. Proposition Let B B g h g h C D (1.2) be a commutative diagram in C. I and are in E and (g, h) and (g, h ) are jointly monic pairs, then there exists a unique isomorphism β : B B with g β = g, β =, and h β = h. 20

26 Proo. Since and are in E, the kernel pairs o and exist by Remark 2.1.2; moreover, they coincide since (g, h) and (g, h ) are jointly monic pairs and the diagram (1.2) is commutative. Since every regular epimorphism is the coequalizer o its kernel pair, we conclude that there exists a unique isomorphism β : B B with β =, and since and are epimorphisms we obtain g β = g and h β = h. Remark s ollows rom Proposition 2.1.4, the actorization in Condition 2.1.1(e) is unique up to an isomorphism. Proposition I a morphism in C actors as = em in which e is in E and m is a monomorphism, then it also actors (essentially uniquely) as = m e in which m is a monomorphism and e is in E. Proo. Under the assumptions o Condition 2.1.1(e), take h 1 = h 2 = m and α = β = e, then we obtain the desired actorization o. Deinition n E-relation R rom an object to an object B in C, written as R : B, is a triple R = (R, r 1, r 2 ) in which r 1 : R and r 2 : R B are jointly monic morphisms in E. Let (R, r 1, r 2 ) = R : B and (S, s 1, s 2 ) = S : B C be the E-relations in C and let (P, p 1, p 2 ) be the pullback o s 1 and r 2 ; by Remark this pullback does exist and p 1 and p 2 are in E. Since p 1 and p 2 are jointly monic and r 1 and s 2 are in E, using Condition 2.1.1(e) we obtain the commutative diagram P p 1 p 2 e R T r S 1 s 2 r t 2 s 1 1 t 2 B C (1.3) in which e is in E, t 1 and t 2 are jointly monic, and such actorization (t 1 e = r 1 p 1 and t 2 e = s 2 p 2 ) is unique up to an isomorphism by Remark Moreover, since r 1, p 1, s 2, and p 2 are in E, the morphisms t 1 and t 2 are also in E by Conditions 2.1.1(a) and 2.1.1(b). ccordingly, we introduce: 21

27 Deinition I R : B and S : B C are the E-relations in C, then their composite SR : C is the E-relation (T, t 1, t 2 ) in which T, t 1, and t 2 are deined as in the diagram (1.3). Proposition The composition o E-relations in C is associative (i we identiy isomorphic relations). Proo. Let R : B, S : B C, and T : C D be the E-relations in C. Consider the commutative diagram in which: X x 1 x 2 x P e p 1 p 2 q 2 X e 1 x Z 1 x 2 q 1 z 2 z R 1 e S 2 T r 2 s 1 s 2 t 1 r 1 SR T (SR) t 2 p 1 p z z B C D (1.4) - (P, p 1, p 2 ) is the pullback o s 1 and r 2, (, q 1, q 2 ) is the pullback o t 1 and s 2, and (X, x 1, x 2 ) is the pullback o q 1 and p 2 ; these pullbacks do exist and the morphisms p 1, p 2, q 1, q 2, x 1, and x 2 are in E by Remark (SR, p 1, p 2 ) is the composite o the E-relations R : B and S : B C, and e 1 : P SR is the canonical morphism, i.e. e 1 is the morphism in E with p 1 e 1 = r 1 p 1 and p 2 e 1 = s 2 p 2. - (Z, z 1, z 2 ) is the pullback o t 1 and p 2, this pullback does exist and the morphisms z 1 and z 2 are in E by Remark 2.1.2; since p 2 e 1x 1 = t 1 q 2 x 2, there exists a unique morphism x : X Z with z 2 x = q 2 x 2 and z 1 x = e 1 x 1. 22

28 - (T (SR), z 1, z 2 ) is the composite o the E-relations SR : C and T : C D, and e 2 : Z T (SR) is the canonical morphism, i.e. e 2 is the morphism in E with z 1 e 2 = p 1 z 1 and z 2 e 2 = t 2 z 2. - Since x 1 and x 2 are jointly monic and r 1 p 1 and t 2 q 2 are in E, by Condition 2.1.1(e) there exists a morphism e : X X in E and jointly monic morphisms x 1 : X and x 2 : X D or which r 1 p 1 x 1 = x 1 e and t 2q 2 x 2 = x 2 e. We irst prove that the square e 1 x 1 = z 1 x in the diagram (1.4) is the pullback o e 1 and z 1. For, consider the commutative diagram e 1 y 1 X x 2 x 1 P x p 2 Z q 2 q 1 z 2 S T z 1 SR s 2 t 1 p 2 C which is a part o the diagram (1.4) with the new arrows y 1, y 2, y, and ȳ deined as ollows: ȳ - y 1 : Y P and y 2 : Y Z are any two morphisms with e 1 y 1 = z 1 y 2. - Since (, q 1, q 2 ) is the pullback o s 2 and t 1, and t 1 z 2 y 2 = p 2 z 1y 2 = p 2 e 1y 1 = s 2 p 2 y 1, there exists a unique morphism y : Y with z 2 y 2 = q 2 y and q 1 y = p 2 y 1. - Since (X, x 1, x 2 ) is the pullback o q 1 and p 2 and q 1 y = p 2 y 1, there exists a unique morphism ȳ : Y X with x 2 ȳ = y and x 1 ȳ = y 1. Since z 1 and z 2 are jointly monic and the equalities y Y y 2 z 1 xȳ = e 1 x 1 ȳ = e 1 y 1 = z 1 y 2, 23

29 z 2 xȳ = q 2 x 2 ȳ = q 2 y = z 2 y 2 hold, we conclude that xȳ = y 2. That is, there exists a morphism ȳ : Y X with x 1 ȳ = y 1 and xȳ = y 2, and since x 1 and x 2 are jointly monic, such ȳ is unique, proving that (X, x, x 1 ) is the pullback o e 1 and z 1. ter this, since e 1 is in E, the morphism x : X Z is in E by Remark 2.1.2, thereore, the composite e 2 x is also in E by Condition 2.1.1(a). We obtain: r 1 p 1 x 1 = x 1 e and t 2 q 2 x 2 = x 2 e in which e E and x 1 and x 2 are jointly monic morphisms; and, z 1 e 2x = r 1 p 1 x 1 and z 2 e 2x = t 2 q 2 x 2 in which e 2 x E and z 1 and z 2 are jointly monic morphisms. Thereore, by Proposition we have X (T S)R. Similarly we can prove that X T (SR). Hence, T (RS) (T R)S, as desired. Remark s ollows rom the proo o Proposition 2.1.9, to construct the composite o the E-relations (R, r 1, r 2 ) : B, (S, s 1, s 2 ) : B C, and (T, t 1, t 2 ) : C D, we simply take the pullbacks (P, p 1, p 2 ), (, q 1, q 2 ), and then the composite (X, x 1, x 2 ) : D will be the E-relation obtained rom the ollowing actorization: X x 1 x 2 P e x x 1 X 2 p 1 p 2 q 1 q 2 R S T r 2 s 1 s 2 t r 1 1 t 2 B C D (1.5) Using the induction principle, we can compose any inite number o the E-relations accordingly. For each E-relation R : B in C there is an opposite E-relation R : B given by the triple (R, r 2, r 1 ), and we have: Proposition I (R, r 1, r 2 ) : B and (S, s 1, s 2 ) : B C are the E-relations in C, then: (i) (R ) = R. 24

30 (ii) (SR) = R S. The objects o C and the E-relations between them orm a category Rel(C, E), in which the identity E-relation on an object is the E-relation (, 1, 1 ) :. It is in act an order-enriched category with (R, r 1, r 2 ) (R, r 1, r 2 ) i and only i there exists a morphism r : R R with r 1 r = r 1 and r 2 r = r 2 (the relevant properties will be given in the next section). 2.2 Properties o the E-relations Proposition Let (R, r 1, r 2 ) : B, (R, r 1, r 2 ) : B, (S, s 1, s 2 ) : B C, and (S, s 1, s 2 ) : B C be the E-relations in C. We have: (i) I R R then R R. (ii) I R R then SR SR. (iii) I R R and S S then SR S R. Proo. (i) is obvious. (ii): I R R then there exists a morphism r : R R with r 1 r = r 1 and r 2 r = r 2. Let (P, p 1, p 2 ) be the pullback o r 2 and s 1 and let (P, p 1, p 2 ) be the pullback o r 2 and s 1. Consider the commutative diagram: P p 1 p 2 R SR r 1 S s 2 r 2 s 1 r B p C r s 1 r 1 2 s 2 R SR S p 1 p 2 P 25

31 s ollows rom Proposition 2.1.3, since the pairs o morphisms (p 1, p 2 ) and (p 1, p 2 ) are jointly monic, and the morphisms r 1, s 2, r 1, and s 2 are in E, in order to prove that SR SR it suices to prove that there exists a morphism p : P P or which p 1 p = rp 1 and p 2 p = p 2. However, since r 2 rp 1 = r 2 p 1 = s 1 p 2, the latter ollows rom the act that the square s 1 p 2 = r 2 p 1 is the pullback o r 2 and s 1. (iii): I R R and S S then by (ii) we have SR SR and SR S R ; thereore, SR S R. Remark ny morphism : B in E can be considered as an E-relation (, 1, ) rom to B. The opposite E-relation rom B to will then be the triple (,, 1 ). Proposition Let (R, r 1, r 2 ) : B be an E-relation in C. I RR 1 B then r 1 : R is an isomorphism. Proo. Let (R, r 1, r 2 ) : B be an E-relation in C and let RR 1 B. Consider the commutative diagram u u,v X P p 1 p 2 e R RR R r 2 p 1 p 2 r 1 r r 2 B 1 1 B 1 B B B in which: - (P, p 1, p 2 ) is the kernel pair o r 1 : R. - (RR, p 1, p 2 ) is the composite o the E-relations R and R, and e : P R R is the canonical morphism; since RR 1 B, there exists a morphism : RR B with 1 B = p 1 and 1 B = p 2. v 26

32 - u, v : X R are any two morphisms with r 1 u = r 1 v, and u, v : X P is the unique morphism with p 1 u, v = u and p 2 u, v = v. We have: e u, v = p 1e u, v = r 2 p 1 u, v = r 2 u, e u, v = p 2e u, v = r 2 p 2 u, v = r 2 v, yielding r 2 u = r 2 v. Since u and v are any two morphisms with r 1 u = r 1 v and since r 1 and r 2 are jointly monic, we obtain u = v. Thereore, r 1 is a monomorphism, and since r 1 is in E, we conclude that r 1 is an isomorphism, as desired. Similarly we can prove that i R R 1 then r 2 is an isomorphism. Proposition I (R, r 1, r 2 ) : B is an E-relation in C then R = r 2 r 1. Proo. Let (R, r 1, r 2 ) be an E-relation rom to B. s ollows rom Remark 2.2.2, r 1 is the E-relation rom to R and r 2 is the E-relation rom R to B. Since the pullback o an identity morphism is again an identity, and since E contains all isomorphisms, the composite r 2 r 1 : B is the E-relation obtained rom the ollowing actorization: R 1 R 1 R 1 R R R r R 1 r 2 r 1 1 R 1 R r 2 R B That is, r 2 r 1 is the E-relation (R, r 1, r 2 ) rom to B, proving the desired. Proposition I : B and g : C B are the morphisms in E, then the E-relation g rom to C in C is given by the pullback ( B C, p 1, p 2 ) o along g. Proo. Let : B and g : C B be the morphisms in E, and let (P, p 1, p 2 ) be the pullback o along g; by Remark 2.1.2, the morphisms p 1 and p 2 are in E. s ollows rom Remark 2.2.2, is the E-relation rom to B and g is the E-relation rom B to C. Since E contains all isomorphisms, the composite g : C is the E-relation obtained rom 27

33 the ollowing actorization: P p 1 p 2 1 P P 1 C 1 C p 1 g p 2 B C That is, (P, p 1, p 2 ) is the E-relation g rom to C, proving the desired. Remark s ollows rom Proposition 2.2.5, i : B is a morphism in E, then the E-relation : is given by the pullback ( B, 1, 2 ) o with itsel. That is, = ( B, 1, 2 ) is the kernel pair o, and thereore 1. Proposition I a morphism : B is in E, then = 1 B. Proo. Let : B be a morphism in E. s ollows rom Remark 2.2.2, is the E-relation rom to B and is the E-relation rom B to. Since E contains all isomorphisms and is in E, the composite is the E-relation obtained rom the ollowing actorization: 1 1 B 1 B B B B That is, is the identity E-relation (B, 1 B, 1 B ) rom B to B, as desired. Remark It ollows rom Proposition that or every morphism : B in E the ollowing equalities hold. =, = 28

34 Theorem Let D k C h B g (2.1) be a diagram in C. I the morphisms, g, h, and k are in E, then: (i) kh g i and only i the diagram (2.1) commutes. (ii) kh = g i and only i the diagram (2.1) commutes and the canonical morphism h, k : D B C is in E. Proo. Consider the diagram (2.1) in which the morphisms, g, h, and k are in E. By Proposition 2.2.5, we have g = ( B C, p 1, p 2 ); and, the composite kh is the E-relation (X, x 1, x 2 ) rom to C obtained rom the ollowing actorization: D 1 D 1 D e D X D h k x 1 1 D 1 D x 2 D C (i): Let kh g ; that is, there exists a morphism x : X B C with p 1 x = x 1 and p 2 x = x 2. To prove that the diagram (2.1) is commutative, it suices to prove that there exists a morphism d : D B C with p 1 d = h and p 2 d = k. For, consider the commutative diagram h x 1 p 1 D X e x B C x 2 p 2 g B 29 k C (2.2)

35 and take d = xe; then p 1 d = p 1 xe = x 1 e = h and p 2 d = p 2 xe = x 2 e = k, as desired. Conversely, suppose the diagram (2.1) is commutative, i.e. h = gk. To prove kh g, we need to show that there exists a morphism x : X B C with p 1 x = x 1 and p 2 x = x 2. For, consider the diagram (2.2); since e is in E and x 1 e = h = gk = gx 2 e we conclude that x 1 = gx 2. Thereore, since ( B C, p 1, p 2 ) is the pullback o and g, there exists a unique morphism x : X B C with p 1 x = x 1 and p 2 x = x 2, as desired. (ii): Let kh = g. s ollows rom (i), the diagram (2.1) is commutative; thereore the diagram (2.2) is also commutative and since g = ( B C, p 1, p 2 ) and kh = (X, x 1, x 2 ), we conclude that x : X B C is an isomorphism. Since h, k = xe and e is in E, by Condition 2.1.1(a), the morphism h, k is also in E. Conversely, suppose the diagram (2.1) is commutative and the canonical morphism h, k : D B C is in E. s ollows rom (i), kh g and thereore, there exists a morphism x : X B C with p 1 x = x 1 and p 2 x = x 2. To prove that kh = g it suices to prove that x is an isomorphism. For, consider the commutative diagram (2.2). Since p 1 xe = x 1 e and p 2 xe = x 2 e we conclude that h, k = xe, and since h, k and e are in E, the morphism x is also in E by Condition 2.1.1(b). Moreover, since p 1 and p 2 are jointly monic and p 1 x = x 1 and p 2 x = x 2, x is a monomorphism. Thereore, since every morphism in E is a normal epimorphism, we conclude that x is an isomorphism, as desired. 2.3 Equivalence E-relations Deinition n E-relation R : in C is said to be (a) a relexive E-relation i 1 R; (b) a symmetric E-relation i R R (so that R = R); (c) a transitive E-relation i RR R; (d) an equivalence E-relation i it is relexive, symmetric, and transitive. s ollows rom Deinition 2.3.1, i R is a relexive and a transitive E-relation then RR = R; indeed, since R is relexive we have R RR, which together with transitivity gives RR = R. Proposition The composite o relexive E-relations in C is a relexive E-relation. 30

36 Proo. I R : and S : are relexive E-relations in C, then 1 R and 1 S. Thereore, 1 SR by Proposition 2.2.1(iii), proving that SR : is a relexive E-relation. Proposition Let R : and S : be equivalence E-relations in C. I the composite SR is an equivalence E-relation, then SR = S R (i.e. SR is the smallest equivalence E-relation containing both S and R). Proo. Let T : be an equivalence E-relation with R T and S T. Since T is an equivalence E-relation, we have T T T. Thereore, SR T by Proposition 2.2.1(iii), proving the desired. Proposition I a morphism : B is in E, then the kernel pair ( B, 1, 2 ) o is an equivalence E-relation in C. Proo. I : B is a morphism in E, then by Remark the kernel pair o is the E-relation : and we have 1, thereore, is a relexive E-relation. Moreover, it is symmetric since ( ) = by Proposition , and it is transitive since = by Remark Deinition n E-relation R : B in C is said to be diunctional i RR R = R. Theorem I (R, r 1, r 2 ) : and (S, s 1, s 2 ) : are equivalence E-relations in C then the ollowing conditions are equivalent: (a) SR : is an equivalence E-relation. (b) SR = RS. (c) Every E-relation is diunctional. (d) Every relexive E-relation is an equivalence E-relation. (e) Every relexive E-relation is symmetric. () Every relexive E-relation is transitive. Proo. 31

37 (a) (b): Let SR : be an equivalence E-relation in C. Since R, S, and SR are symmetric E-relations, we have: R = R, S = S, and SR = (SR). Thereore, SR = R S = RS, as desired. (b) (a): Let SR = RS. We have: - 1 R and 1 S since R and S are relexive. - R R and S S since R are S are symmetric. - RR R and SS S since R and S are transitive. Using Proposition 2.2.1(iii), we obtain: - 1 SR, thereore SR is relexive. - S R SR; since (SR) = (RS) = S R, we conclude that (SR) SR, thereore, SR is symmetric. - SSRR SR; since SRSR = SSRR, we conclude that SRSR SR, thereore, SR is transitive. That is, SR is a relexive, symmetric, and a transitive E-relation, proving that SR is an equivalence E-relation. (b) (c): Let (U, u 1, u 2 ) : X Y be an arbitrary E-relation in C. By Proposition 2.2.4, U = u 2 u 1 ; thereore, to prove that the E-relation U : X Y is diunctional, i.e. UU U = U, it suices to prove u 2 u 1 u 1 u 2 u 2 u 1 = u 2 u 1. Since u 1 and u 2 are in E, by Remark 2.2.6, the E-relations u 1 u 1 : U U and u 2 u 2 : U U are the kernel pairs o u 1 and u 2 respectively, thereore, they are the equivalence E-relations by Proposition Hence, by (b), u 1 u 1 u 2 u 2 = u 2 u 2 u 1 u 1, and multiplying the last equality on the let by u 2 and on the right by u 1, using Proposition we obtain u 2 u 1 u 1 u 2 u 2 u 1 = u 2 u 1, as desired. (c) (d): Let (U, u 1, u 2 ) : X X be a relexive E-relation in C. U is symmetric since U = 1 X U 1 X UU U = U, and U is transitive since UU = U1 X U UU U = U. Thereore, U is an equivalence E-relation in C. (d) (e) is obvious. (e) (c): Let (U, u 1, u 2 ) : X Y be an arbitrary E-relation in C. The proo is essentially the same as the proo o (b) (c): here u 1 u 1 u 2 u 2 = u 2 u 2 u 1 u 1 since the composite o 32

38 the equivalence E-relations u 1 u 1 : U U and u 2 u 2 : U U is relexive by Proposition 2.3.2, thereore, by (e) the composite u 1 u 1 u 2 u 2 is also symmetric. (c) (a): I R : and S : are equivalence E relations in C, then their composite SR : is a relexive E-relation by Proposition Since (c) implies (d), we conclude that SR is an equivalence E-relation. (c) (): Since (c) implies (d), and (d) implies (), we conclude that (c) implies (). () (c): Let (U, u 1, u 2 ) : X Y be an arbitrary E-relation in C. s stated in the proo o (b) (c), to prove that the E-relation U : X Y is diunctional, it suices to prove u 2 u 1 u 1 u 2 u 2 u 1 = u 2 u 1. Since the kernel pairs o u 1 and u 2 are the equivalence E-relations u 1 u 1 : U U and u 2 u 2 : U U respectively, by Proposition their composite u 2 u 2 u 1 u 1 : U U is a relexive E-relation; thereore, u 2 u 2 u 1 u 1 is transitive by (), and we have u 2 u 2 u 1 u 1 u 2 u 2 u 1 u 1 = u 2 u 2 u 1 u 1. Multiplying the last equality on the let by u 2 and on the right by u 1, using Proposition we obtain u 2 u 1 u 1 u 2 u 2 u 1 = u 2 u 1, as desired. Remark Theorem is the relative version o Theorem Consider the ollowing Condition (a) C is pointed; (b) I : B is in E then the kernel o exists in C; (c) I in a commutative diagram K k K k w B B k = ker(), k = ker( ), and and are in E, then w is an isomorphism. Remark Condition 2.3.8(c) is the relative version o the Short Five Lemma. ccordingly, we will say that Condition 2.3.8(c) is the E-Short Five Lemma. Theorem I (C, E) satisies Condition 2.3.8, then every relexive E-relation in C is transitive. 33

39 Proo. Let (R, r 1, r 2 ) : be a relexive E-relation in C. We have: - The pullback (P, p 1, p 2 ) o r 2 and r 1 exists in C and the morphisms p 1 and p 2 are in E by Remark Since p 2 is in E, the kernel k : K P o p 2 exists by Condition 2.3.8(b). - Since r 1, r 2, p 1, and p 2 are in E, the composites r 1 p 1 and r 2 p 2 are also in E by Condition 2.1.1(a), and thereore the limit o the diagram P R r1 r 1 p 1 r 2 p 2 r 2 (3.1) exists by Condition 2.1.1(c). Since R is a relexive E-relation there exists a morphism α : R with r 1 α = r 2 α = 1. Consider the commutative diagram: R s P αr 1 1 R p 2 R p 1 r 1 R r2 Since the square r 2 p 1 = r 1 p 2 is a pullback and r 2 αr 1 = r 1, there exists a unique morphism s : R P with p 1 s = αr 1 and p 2 s = 1 R, yielding that p 2 is a split epimorphism. Next, let 34

40 (, q 1, q 2 ) be the limit o the diagram (3.1) and consider the commutative diagram: R 1 R t q 1 s q 2 P R r1 r 1 p 1 r 2 p 2 Since r 1 p 1 s = r 1 αr 1 = r 1 and r 2 p 2 s = r 2, there exists a unique morphism t : R with q 1 t = 1 R and q 2 t = s. We have p 2 q 2 t = p 2 s = 1 R, thereore the composite p 2 q 2 is a split epimorphism. Furthermore, since r 2 p 1 k = r 1 p 2 k = 0 = r 2 p 2 k and (, q 1, q 2 ) is the limit o the diagram (3.1), there exists a unique morphism : K making the diagram K p 1 k q 1 k q 2 r 2 P R r1 r 1 p 1 r 2 p 2 commutative. Since r 1 and r 2 are jointly monic and (, q 1, q 2 ) is the limit o the diagram (3.1), we conclude that q 2 is a monomorphism. Thereore, since p 2 is in E by Proposition we obtain the actorization p 2 q 2 = m 1 e 1 in which e 1 is in E and m 1 is a monomorphism. Since p 2 q 2 is a split epimorphism, it is a strong epimorphism and thereore m 1 is an isomorphism. Hence, since E contains all isomorphism, p 2 q 2 is in E by Condition 2.1.1(a). We have r 2 35

41 p 2 q 2 = p 2 k = 0, let us prove that = ker(p 2 q 2 ). For, consider the commutative diagram X x x K q R x q 2 K k P p 2 R in which q = p 2 q 2 and x : X is any morphism with qx = 0. Since p 2 q 2 x = qx = 0 and k = ker(p 2 ), there exists a unique morphism x : X K with k x = q 2. Since q 2 is a monomorphism and q 2 x = q 2 x, we conclude x = x, and since k is a monomorphism such x is unique, proving = ker(p 2 q 2 ). Since p 2 and p 2 q 2 are in E, by the E-Short Five Lemma we conclude that q 2 is an isomorphism. Finally, consider the commutative diagram E e 2 e 1 e P RR q 1 q 1 t 2 r 1 R r1 r 2 t 2 where e : P RR is the canonical morphism, i.e. it is the morphism in E or which t 1 e = r 1 p 1 and t 2 e = r 2 p 2, and (E, e 1, e 2 ) is the kernel pair o e which does exist by Remark Since r 1 and r 2 are jointly monic we conclude that q 1 q2 1 e 1 = q 1 q2 1 e 2, thereore, since e is in E (i.e. it is is a regular epimorphism and thereore it is the coequalizer o its kernel pair), there exists a morphism r : RR R with re = q 1 q2 1. Moreover, since e is an epimorphism, r 1 re = r 1 q 1 q2 1 = t 1 e and r 2 re = r 2 q 1 q2 1 = t 2 e we obtain r 1 r = t 1 and r 2 r = t 2. That is, there exists a morphism r : RR R with r 1 r = t 1 and r 2 r = t 2, proving that R is a transitive E-relation. Theorem together with Theorem gives Corollary I (C, E) satisies Condition 2.3.8, then every relexive E-relation in C is an equivalence E-relation. 36

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