Archimedean Type Conditions in Categories

Arc himedean T yp e Condit ions in Cate gories Elemer E Rosinger Departmen t of Mathematics and Applied Mathematics Univ ersit y of Pretoria Pretoria 0002 South Africa eerosinger@hotmail.com Abstract Tw o concepts of being Archime dean a re defined for arbitrar y cate- gories. 1. The case of usu al semigroups F or con v enience, let us recall tw o v ersions of concepts of being Archime dean in the usual case o f a lgebraic structures. In this regard, a sufficien tly general setup is as follo ws. Let ( E , + , ≤ ) b e a partially ordered semigroup, th us w e hav e satisfied (1.1) x, y ∈ E + = ⇒ x + y ∈ E + where E + = { x ∈ E | x ≥ 0 } . A first in tuitiv e v ersion of the Arc himede an condition, suggested in case ≤ is a line ar or total order on E , is (1.2) ∃ u ∈ E + : ∀ x ∈ E : ∃ n ∈ N : nu ≥ x Here is another formulation used in the literature when ≤ is an arbi- trary partial o r der on E , namely 1 (1.3) ∀ x ∈ E + : x = 0 ⇐ ⇒   ∃ y ∈ E + : ∀ n ∈ N : nx ≤ y   where clearly t he implication ”= ⇒ ” is trivial, and which condition is th us equiv alen t with (1.4) ∀ x ∈ E + : N x is b ounded ab o v e = ⇒ x = 0 Lemma 1. If ( E , + , ≤ ) is a line arly or total ly ordered s emigroup, then (1.3) = ⇒ (1.2). Pro of. Assume indeed that (1.2) do es not hold, then ∀ x ∈ E + : ∃ y ∈ E : ∀ n ∈ N : nx  y and since ≤ is a linear or total order on E , we ha v e ∀ x ∈ E + : ∃ y ∈ E : ∀ n ∈ N : nx ≤ y Ob viously , we can assume that y ∈ E + , t hus (1.4) is con tradicted. 2. The case of categories Let C b e an y category . The issue is to b e able to take a so called ”unit” morphism, like for instance u in (1.2 ), sa y A f − → B and b e able to ”repeat” it, sa y , to the right o f B an y finite n um b er of time. Here the problem is that, in general, we cannot comp ose a morphism in C with it self ev en just t wice. In part icular, w e cannot in general ha v e f ◦ f , let alone f ◦ f ◦ f , and so on. Therefore, when 2 giv en tw o C morphisms whic h can b e comp osed A f − → B g − → C w e hav e to find a w a y to b e able to say that the morphism g is again a ”unit”, that is, mo r e or less t he same with the morphism f from a certain relev an t p oint of view. One simple natural w a y to do that is as follo ws. W e consider the arr o w c ate gory C 2 , associated with C , [H & S, p. 27], namely , the category whose class of ob j e cts is the class of C morphisms , while for any tw o suc h C 2 ob jects A f − → B , A ′ f ′ − → B ′ the corresp onding C 2 morphisms are the pairs ( a, b ), where A a − → A ′ , B b − → B ′ are C morphisms, suc h that the diagram commutes A B f ✲ A ′ a ❄ b ❄ B ′ ✲ f ′ Finally , the C 2 comp osition of morphisms is defined b y ( a ′ , b ′ ) ◦ ( a, b ) = ( a ′ ◦ a, b ′ ◦ b ) in other w ords, by pasting the t w o ab ov e kind of diagrams together, and deleting the middle horizontal arrow. No w, g iv en tw o C morphisms A f − → B , C g − → D 3 w e sa y that they are unitary e quivalent , if and only if, when considered as ob jects in the arrow catego ry C 2 , they are isomorphic, [H & S, p. 36]. With that definition, we can now attempt to define when a category C is Ar chime de an . First, w e consider an extension of the usual v e rsion of the Arc himede an prop ert y in ( 1 .2). N amely , the corresp onding condition in categories is as follows (2.1) ∃ U v − → W C morphism : ∀ A f − → B C morphism : ∃ U 1 v 1 − → W 1 , . . . , U n v n − → W n C morphis ms : 1) v 1 , . . . , v n are in tha t order comp osable in C 2) v 1 , . . . , v n are unita ry equiv a len t with v 3) f is a submorphism of v n ◦ . . . ◦ v 1 Here w e used the follow ing definition. Give n t w o C morphism s A f − → B , C g − → D w e say that f is a submorphism o f g , if and only if there are C mor- phisms C a − → A, B b − → D suc h that (2.2) g = b ◦ f ◦ a 4 As for the extension to categories of the usual condition (1.4), w e can pro ceed as follow s. Give n any C mor phism U v − → W , let us denote by (2.3) N v the class of C morphisms of the form (2.4) v n ◦ . . . ◦ v 1 where U 1 v 1 − → W 1 , . . . , U n v n − → W n are C morphisms whic h a re com- p osable in the giv en order, and are eac h unitar y equiv alent with v . Then w e call C Ar chime de an , if and only if (2.5) ∀ U v − → W C morphism : N v is b ounded in C = ⇒ v = id U Here the follo wing definition was use d. A giv en class N of C mor- phisms is called b ounde d , if and only if there exists a C morphism A f − → B , suc h that ev ery C morphism in N is a submorphism of f . 3. Example As a simple example w e shall illustra t e the t w o general concepts of Arc himedean cat ego ry presen ted in section 2 , in the particular case of categories giv en by quasi or der e d classes, [H & S, p. 19]. W e recall that a category C is a quasi ordered class, if and only if for ev ery t w o of its ob jects A and B , there is at most one single morphism A f − → B . It fo llows that, for simplicit y , suc h a category can b e describ ed b y a r eflexive a nd tr ansitive binary r elatio n ≤ on it s ob jects, whic h is de- fined as follows . F or ev ery t w o ob jects A a nd B in C , we ha v e A ≤ B , if and only if there exists a morphism A f − → B in C . In suc h a case that morphism is, as assumed, unique. In order to illustrate in the ab o v e particular situation the tw o concepts 5 of b eing Arc himedean defined in section 2 for arbitrary categories, w e first ha v e to clarify in the con text of quasi ordered classes the notion of unitary e quivalent , whic h plays a role in b oth men tioned concepts. And for this purp ose, w e ha v e to lo ok at the arrow category C 2 of the quasi ordered class C . It is easy to see that the ob jects in C 2 are precisely the pairs ( A, B ), where A and B are ob jects in C , suc h that A ≤ B . F urther, g iv en tw o ob jects ( A, B ) and ( A ′ , B ′ ) in C 2 , it is immediate tha t , in C 2 , there exist mor phisms b et w een them, if and only if (3.1) A ≤ A ′ , B ≤ B ′ And in suc h a case, the corresp onding morphism is the pair o f pairs (3.2) (( A, A ′ ) , ( B , B ′ )) whic h is the unique suc h morphism in C 2 . Th us C 2 is aga in a quasi ordered class. As for the comp osition of suc h morphisms in C 2 , one readily obta ins that (3.3) (( A, A ′ ) , ( B , B ′ )) ◦ (( A ′ , A ′′ ) , ( B ′ , B ′′ )) = (( A, A ′′ ) , ( B , B ′′ )) Giv en no w t w o morphisms ( A, B ) and ( C , D ) in C , then b y the def- inition in se ction 2, they are unitar y equiv alent, if and o nly if, when considered as ob jects in C 2 , they are isomorphic. In other words, there mus t b e an isomorphism ( A, B ) α − → ( C , D ) in C 2 . This means, [H & S, Prop osition 5.17 , p. 3 6 ], that there exists a unique morphism ( C , D ) β − → ( A, B ) in C 2 , suc h that β ◦ α = id ( A,B ) and α ◦ β = id ( C,D ) . Th us in view of (3.2), it fo llo ws that α = (( A, C ) , ( B , D )) , β = (( C , A ) , ( D , B )) F urthermore, (3 .1 ) applied to α , implies 6 (3.4) A ≤ C , B ≤ D and similarly , applied to β , gives (3.5) C ≤ A, D ≤ B No w as cus tomary , let us consider t he e quivalenc e relation ≈ on the ob jects o f the quasi o rdered class C , defined by (3.6) X ≈ Y ⇔ X ≤ Y , Y ≤ X Then (3.4), (3 .5 ) giv e fo r an y t w o morphisms ( A, B ) and ( C , D ) in C (3.7) ( A, B ) , ( C , D ) are unitary equiv alen t ⇐ ⇒ ⇐ ⇒ A ≈ C ≤ B ≈ D F urther, w e hav e to clarify in the particular case o f quasi ordered classes t he notion of submorphism defined in section 2 for arbitrary categories. Giv en t w o morphisms ( A, B ) and ( C , D ) in C , it is easy to see that ( A, B ) is a submorphism of ( C , D ), if and only if (3.8) C ≤ A ≤ B ≤ D Indeed, since ( A, B ) and ( C , D ) are morphisms in C , w e hav e A ≤ B , C ≤ D . No w, ( A, B ) b eing a submorphism of ( C , D ), it follows in particular tha t ( C , A ) and ( B , D ) are a lso morphisms in C . Th us C ≤ A, B ≤ D . W e are no w in the p osition to reformulate in the par t icular instance of quasi ordered classes conditio n the (2.1) whic h give s the first concept of b eing Arc himedean in the case of arbitrary categories. Namely , in view of (3.7), (3.8), w e obtain 7 (3.9) ∃ U ≤ W ob jects in C : ∀ A ≤ B ob jects in C : ∃ U 1 ≤ U 2 ≤ . . . ≤ U n ≤ U n +1 ob jects in C : 1) U ≈ U m ≤ U m +1 ≈ W, 1 ≤ m ≤ n 2) U 1 ≤ A ≤ B ≤ U n +1 Here w e note that there ar e tw o v ersions of this condition. Na mely , if n = 1 in (3.9), then w e o bta in ∃ U ≤ W ob jects in C : ∀ A ≤ B ob jects in C : ∃ U 1 ≤ U 2 ob jects in C : 1) U ≈ U 1 ≤ U 2 ≈ W 2) U 1 ≤ A ≤ B ≤ U 2 whic h ob viously simplifies to (3.10) ∃ U ≤ W ob jects in C : ∀ A ≤ B ob jects in C : U ≤ A ≤ B ≤ W When how e v er n ≥ 2 in (3.9), then quite s urprisingly , w e obtain the stronger conditio n 8 ∃ U ≤ W ob jects in C : ∀ A ≤ B ob jects in C : ∃ U 1 ≤ U 2 ≤ . . . ≤ U n ≤ U n +1 ob jects in C : 1) U ≈ U 1 ≈ . . . ≈ U m +1 ≈ W 2) U ≈ A ≈ B ≈ W whic h, when simplified, b ecomes ∃ U ≤ W ob jects in C : ∀ A ≤ B ob jects in C : U ≈ A ≈ B ≈ W th us a pa r ticular case o f (3.10). In this w a y , w e obtained Prop osition 1. A quasi ordered category C is Arc himedean in the sense of (2.1) , if and only if it is b ounded. Pro of. In view of (3.8 ), condition (3.1 0) obvious ly me ans that C is b ounded in the sense of t he definition following (2.5). Remark 1. In view of Prop osition 1 ab o v e, in the case of quasi o rdered categories, the concept of b eing Arc himedean giv en in (2.1) do es not reco v er an y of the t w o concepts (1.2) or (1.4 ) whic h are usual in the particular case of partially ordered semigroups. Ho w ev er, quasi ordered categories, let alone, arbitrary categories fo r whic h the concept of b eing Arc himedean in (2.1) w as defined, hav e an ob viously w eak er structure than partially ordered semigroups. Thus 9 one cannot exp ect definition (2.1) to b e able to fully include usual concepts of b eing Arc himedean in suc h a ric her structure lik e partially ordered semigroups. Let us no w turn to the second concept of b eing Arc himedean as given in (2.5 ), and cons ider it in the particular case of quasi ordered cate- gories C . In view of (3.8 ), a class N of morphisms in C is b ounded, if and only if there is a morphism A ≤ B in C , suc h that for ev ery morphism C ≤ D in N , w e ha v e A ≤ C ≤ D ≤ B . No w in order to elucidate condition (2.5) in the case of quasi ordered categories C , w e hav e to clarify the corresp onding particular instances of (2.3). Let therefore U ≤ W b e a morphism in C . Then N ( U, W ) is, according to (2.4) , the class of morphisms in C , of the form (3.11) U 1 ≤ U n +1 where (3.12) U 1 ≤ . . . ≤ U n ≤ U n +1 are ob jects in C , and in view o f (3.7), they satisfy (3.13) U ≈ U m ≤ U m +1 ≈ W, 1 ≤ m ≤ n Clearly , it follo ws that N ( U, W ) is alw ay s bounded, no matter whic h w ould b e the morphism U ≤ W in C . Consequen tly , condition (2.5) b ecomes (3.14) ∀ U ≤ W morphism in C : U = W Therefore, w e obta in Prop osition 2. 10 A quasi ordered category C is Arc himedean in the sense of (2.5) , if and only if all its morphisms are iden tities, that is, it is discrete, [H & S, p. 17]. Remark 2. The commen ts at Remark 1 ab o v e apply again. References [1] [H & S] Herrlic h H, Streck er G E : Category Theory . Allyn & Bacon, New Y ork, 1973 11