A Logic of Orthogonality

Transcription

1 A Logic of Orhogonaliy J. Adámek, M. Héber and L. Soua Sepember 3, 2006 Thi paper wa inpired by he hard-o-believe fac ha Jiří Roický i geing ixy. We are happy o dedicae our paper o Jirka on he occaion of hi birhday. Abrac A logic of orhogonaliy characerize all orhogonaliy conequence of a given cla Σ of morphim, i.e. hoe morphim uch ha every objec orhogonal o Σ i alo orhogonal o. A imple four-rule deducion yem i formulaed which i ound in every cocomplee caegory. In locally preenable caegorie we prove ha he deducion yem i alo complee (a) for all clae Σ of morphim uch ha all member excep a e are regular epimorphim and (b) for all clae Σ, wihou rericion, under he e-heoreical aumpion ha Vopěnka Principle hold. For finiary morphim, i.e. morphim wih finiely preenable domain and codomain, an appropriae finiary logic i preened, and proved o be ound and complee; here he proof follow immediaely from previou join reul of Jiří Roický and he fir wo auhor. 1 Inroducion The famou orhogonal ubcaegory problem ak wheher given a cla Σ of morphim of a caegory he full ubcaegory Σ of all objec orhogonal o Σ i reflecive. Recall ha an objec i orhogonal o Σ iff i hom-funcor ake member of Σ o iomorphim. In he realm of locally preenable caegorie for he orhogonal ubcaegory problem (a) he anwer i affirmaive whenever Σ i mall more generally, a proved by Peer Freyd and Max Kelly [7], i i affirmaive whenever Σ = Σ 0 Σ 1 where Σ 0 i mall and Σ 1 i a cla of epimorphim, and Suppored by he Czech Gran Agency, Projec 201/06/0664 Financial uppor by he Cener of Mahemaic of he Univeriy of Coimbra and he School of Technology of Vieu 1

2 (b) auming he large-cardinal Vopěnka Principle, he anwer remain affirmaive for all clae Σ, a proved by he fir auhor and Jiří Roický in [3]. The problem o which he preen paper i devoed i dual : we udy he orhogonaliy conequence of clae Σ of morphim by which we mean morphim uch ha every objec of Σ i alo orhogonal o. Example: if Σ i reflecive, hen all he reflecion map are orhogonaliy conequence of Σ. Anoher imporan example: given a Gabriel- Ziman caegory of fracion C Σ : A A[Σ 1 ], hen every morphim which C Σ ake o an iomorphim i an orhogonaliy conequence of Σ. In Secion 2 we recall he precie relaionhip beween Σ and A[Σ 1 ]. We formulae a very imple logic for orhogonaliy conequence (inpired by he calculu of fracion and by he work of Grigore Roçu [12]) and prove ha i i ound in every cocomplee caegory. Tha i, whenever a morphim ha a formal proof from a cla Σ, hen i an orhogonaliy conequence of Σ. In he realm of locally preenable caegorie we alo prove ha our logic i complee, ha i, every orhogonaliy conequence of Σ ha a formal proof, provided ha or (a) Σ i mall more generally, compleene hold whenever Σ = Σ 0 Σ 1 where Σ 0 i mall and Σ 1 i a cla of regular epimorphim (b) Vopěnka Principle i aumed. (We recall Vopěnka Principle in Secion 4.) In fac he compleene of our logic for all clae of morphim will be proved o be equivalen o Vopeňka Principle. Thi i very imilar o reul of Jiří Roický and he fir auhor concerning he orhogonal ubcaegory problem, ee 6.24 and 6.25 in [3]. Our logic i quie analogou o he Injeciviy Logic of [4] and [1], ee alo [12]. There a morphim i called an (injeciviy) conequence of Σ provided ha every objec injecive w.r.. member of Σ i alo injecive w.r... Recall ha an objec i injecive w.r.. a morphim iff i hom-funcor ake o an epimorphim. Recall furher from [1] ha he deducion yem for Injeciviy Logic ha ju hree deducion rule: ranfinie compoiion puhou cancellaion i (i < α) u if i an α-compoie of he i if i a puhou 2

3 We recall he concep of α-compoie in 3.2 below. In locally preenable caegorie he correponding logic i complee and ound for e Σ of morphim, a proved in [1]. Bu he Injeciviy Logic i no complee for clae Σ of morphim in a harp conra o he cae of Orhogonaliy Logic. We give an (abolue) couner-example a he end of our preen paper. Now boh ranfinie compoiion and puhou are ound rule for orhogonaliy, oo. In conra, cancellaion i no ound and ha o be ubiued by he following weaker form: weak cancellaion u v u Furher we have o add a fourh rule in cae of orhogonaliy: coequalizer if f g i a coequalizer uch ha f = g We obain a 4-rule deducion yem for which he above compleene reul (a) and (b) will be proved. The above logic are infiniary, in fac, ranfinie compoiion i a cheme of deducion rule, one for every ordinal α. We alo udy he correponding finiary logic by rericing ourelve o e Σ of finiary morphim, meaning morphim wih finiely preenable domain and codomain. Boh in he injeciviy cae and in he orhogonaliy cae one imply replace ranfinie compoiion by wo rule: ideniy id A and compoiion 1 2 if = 2 1 Thi finiary logic i proved o be ound and complee for e of finiary morphim. In fac, in [10] a decripion of he caegory of fracion A ω [Σ 1 ] (ee 2.1) a a dual o he heory of he ubcaegory Σ i preened; our proof of compleene of he finiary logic i an eay conequence. The reul of Peer Freyd and Max Kelly menioned a he beginning goe beyond locally preenable caegorie, and alo our preceding paper [1] i no rericed o hi conex. Nonehele, he preen paper udie he orhogonaliy conequence and i logic in locally preenable caegorie only. 3

4 2 Finiary Logic and he Calculu of Fracion 2.1. Aumpion Throughou he paper A denoe a locally preenable caegory in he ene of Gabriel and Ulmer; he reader may conul he monograph [3]. A locally preenable caegory i a cocomplee caegory A uch ha, for ome infinie cardinal λ, here exi a e A λ of objec repreening all λ-preenable objec up-o an iomorphim and uch ha a compleion of A λ under λ-filered colimi i all of A. The caegory A i hen aid o be locally λ-preenable. Recall ha a heory of a locally λ-preenable caegory A i a mall caegory T wih λ-mall limi 1 uch ha A i equivalen o he caegory Con λ (T ) of all e-valued funcor on T preerving λ-mall limi. For every locally λ-preenable caegory i follow ha he dual A op λ of he above full ubcaegory i a heory of A: A = Con λ (A op λ ) Morphim wih λ-preenable domain and codomain are called λ-ary morphim Noaion For every cla Σ of morphim of A we denoe by Σ he full ubcaegory of all objec orhogonal o Σ. If Σ i mall, hi ubcaegory i reflecive, ee e.g. [7]. We denoe, whenever Σ i reflecive, by R Σ : A Σ a reflecor funcor and by η A : A R Σ A he reflecion map; wihou lo of generaliy we will aume R Σ η A = id RΣ A = η RΣ A Obervaion If Σ i a reflecive ubcaegory, hen orhogonaliy conequence of Σ are preciely he morphim uch ha R Σ i an iomorphim. In fac, if : A B i an orhogonaliy conequence of Σ, hen R Σ A i orhogonal o, which yield a commuaive riangle A η A u R Σ A The unique morphim ū : R Σ B R Σ A wih ū η B = u i invere o R Σ : hi follow from he diagram A B u η B η A ū R Σ A R Σ B R Σ 1 Limi of diagram of le han λ morphim are called λ-mall limi. Analogouly λ-wide puhou are puhou of le han λ morphim. 4 B

5 Converely, if : A B i urned by R Σ o an iomorphim, hen every objec X orhogonal o Σ i orhogonal o : given f : A X we have a unique f : R Σ A X wih f = f η A, and we ue f (R Σ ) 1 η B : B X. I i eay o check ha hi i he unique facorizaion of f hrough Remark The above obervaion how a connecion of he orhogonaliy logic wih he calculu of fracion of Peer Gabriel and Michel Ziman [8], ee alo Secion 5.2 in [5]. Given a cla Σ of morphim in A, i caegory of fracion i a caegory A[Σ 1 ] ogeher wih a funcor C Σ : A A[Σ 1 ] univeral w.r.. he propery ha C Σ ake member of Σ o iomorphim. (Tha i, if a funcor F : A B ake member of Σ o iomorphim, hen here exi a unique funcor F : A[Σ 1 ] B wih F = F C Σ.) The caegory of fracion i unique up-o iomorphim of caegorie whenever i exi, and i doe exi if Σ i mall, ee [5], Example (ee [5], 5.3.1) For every reflecive ubcaegory B of A, R : A B he reflecor, pu Σ = { R i an iomorphim}. Then B = Σ A[Σ 1 ]. More preciely, here exi an equivalence E : A[Σ 1 ] Σ uch ha E C Σ = R = R Σ Example (ee [6]) In he caegory Ab of abelian group conider he ingle morphim Σ = {Z 0} where Z i he group of ineger. Then clearly Oberve ha Σ = {0}. Ab[Σ 1 ] = {0} becaue he coreflecor F : Ab Ab of he full ubcaegory Ab of all orion group ake Z 0 o an iomorphim, bu F i he ideniy funcor on Ab. Thi of coure implie ha C Σ : Ab Ab[Σ 1 ] i monic on Ab Definiion (ee [8]) A cla Σ of morphim i aid o admi a lef calculu of fracion provided ha (i) Σ conain all ideniy morphim, (ii) Σ i cloed under compoiion, (iii) for every pan here exi a commuaive quare f wih Σ f f wih Σ 5

6 and (iv) for every parallel pair f, g equalized by a member of Σ here exi a member of Σ coequalizing he pair: f g 2.8. Theorem (ee [10] IV.2) Le Σ be a e of finiary morphim of a locally finiely preenable caegory A. If Σ admi a lef calculu of fracion in he ubcaegory A ω, hen Σ i a locally finiely preenable caegory whoe heory i dual o A ω [Σ 1 ]. More preciely: Le C Σ : A ω A ω [Σ 1 ] be he canonical funcor from A ω ino he caegory of fracion of Σ in A ω, ee 2.4. Then here exi an equivalence funcor I : Con ω (A ω [Σ 1 ] op ) Σ uch ha for he incluion funcor E : A ω A and he Yoneda embedding Y : A ω [Σ 1 ] Con ω (A ω [Σ 1 ] op ) he following diagram A ω C Σ E A ω [Σ 1 ] Y Con ω (A ω [Σ 1 ] op ) (2.1) A R Σ I Σ commue Corollary Le Σ admi a lef calculu of fracion in A ω. Then he orhogonaliy conequence of Σ in A ω are preciely he finiary morphim uch ha C Σ i an iomorphim. In fac, ince I Y i a full embedding, we know ha C Σ i an iomorphim iff (I Y C Σ ) i one, hu, hi follow from Obervaion Example (refer o 2.6) For Σ = {Z 0}, he malle cla Σ 0 in Ab (rep., in Ab ω ) conaining Σ and admiing a lef calculu of fracion i he cla of all (rep., all finiary) morphim which are ideniie or have codomain 0. One ee eaily ha Ab[Σ 1 0 ] = {0} = Ab ω [Σ 1 0 ] = Σ 0 = Σ Remark In a finiely cocomplee caegory A for every e Σ of finiary morphim here i a canonical exenion of Σ o a e Σ admiing a lef calculu of fracion in A ω : le Σ be he cloure in A ω of Σ {id A } A Aω 6

7 under (a) compoiion (b) puhou and (c) weak coequalizer in he ene ha Σ conain for every pair f, g : A B, a coequalizer of f, g whenever f = g for ome member of Σ. We will ee in Obervaion 2.16 below ha Σ and Σ have he ame orhogonaliy conequence Theorem (ee [5], 5.9.3) If a e Σ admi a lef calculu of fracion, hen he cla of all morphim aken by C Σ o iomorphim i he malle cla Σ conaining Σ and uch ha given hree compoable morphim u v wih u and v u boh in Σ, hen lie in Σ Remark Apply he above heorem o Σ of Remark 2.11: if Σ denoe he cloure of Σ under weak cancellaion in he ene ha from u Σ and v u Σ we derive Σ, hen Σ i preciely he cla aken by C Σ o iomorphim. Thi lead u o he following Definiion The Finiary Orhogonaliy Deducion Syem coni of he following deducion rule: ideniy id A compoiion puhou if 2 1 i defined if i a puhou coequalizer if f g i a coequalizer and f = g weak cancellaion u v u We ay ha a morphim can be proved from a e Σ of morphim uing he Finiary Orhogonaliy Logic, in ymbol Σ provided ha here exi a formal proof of from Σ uing he above five deducion rule (in A ω ). 7

8 2.15. Remark A formal proof of i a finie li 1, 2,..., k of finiary morphim uch ha = k and for every i = 1,..., k eiher i Σ, or i i he concluion of one of he deducion rule whoe aumpion lie in he e { 1,..., i 1 }. For a locally preenable caegory he Finiary Orhogonaliy Logic i he applicaion of he relaion and = o finiary morphim of A Obervaion In every finiely cocomplee caegory he Finiary Orhogonaliy Logic i ound: if a finiary morphim ha a proof from a e Σ of finiary morphim hen i an orhogonaliy conequence of Σ. Shorly: Σ implie Σ =. I i ufficien o check individually he oundne of he five deducion rule. Every objec X i clearly orhogonal o id A ; and i i orhogonal o 2 1 whenever X i orhogonal o 1 and 2. The oundne of he puhou rule i alo elemenary:! Suppoe i a coequalizer of f, g : A B and le f = g. Whenever X i orhogonal o, i i orhogonal o. In fac, given a morphim p : B X, X f A A B g p X hen from p f = p g i follow ha p f = p g (due o X ) and hu p uniquely facor hrough = coeq(f, g). Finally, le X be orhogonal o u and v u, B A u B v C D r p r q w w X hen we how X. Given p : A X here exi q : C X wih p = q (u ). Then r = q u fulfil p = r. Suppoe r fulfil p = r. We have, ince X v u, a unique w : D X wih r = w v u and a unique w wih r = w v u. The equaliy w v u = w v u implie w v = w v, hu, r = w v u = w v u = r. 8

9 2.17. Theorem In locally finiely preenable caegorie he Finiary Orhogonaliy Logic i complee for e Σ of finiary morphim: Σ = implie Σ. Proof Le be an orhogonaliy conequence of Σ and le Σ be he e of all finiary morphim ha can be proved from Σ; we have o verify ha Σ. Due o he fir four deducion rule, Σ clearly admi a lef calculu of fracion in A ω. Hence C Σ i, by Corollary 2.9, an iomorphim. Theorem 2.12 implie (due o weak cancellaion) ha Σ Example demonraing ha we canno, for he finiary orhogonaliy logic, work enirely wihin he full ubcaegory A ω : le u denoe by Σ = ω he aemen ha every finiely preenable objec X Σ i orhogonal o. Then i i in general no rue ha, given a e of finiary morphim Σ, hen Σ = ω implie Σ. Le A = Rel(2, 2) be he caegory of relaional rucure on wo binary relaion α and β. We denoe by he iniial (empy) objec, 1 a erminal objec (a ingle node which i a loop of α and β), T a one-elemen objec wih α = and β a loop and, for every prime p 3, by A p he objec on {0, 1,..., p 1} whoe relaion β i a clique (ha i, wo elemen are relaed by β iff hey are diinc) and he relaion α i a cycle of lengh p wih one addiional edge from 1 o 0: p p Conider he e Σ of finiary morphim given by Σ = {, u} { A p ; p 3 a prime} where : T 1 and u : are he unique morphim. Orhogonaliy of a relaional rucure X o Σ implie ha every loop of he relaion β i a join loop of boh relaion (due o ) and uch a loop i unique (due o u). Moreover, he given objec X ha a unique morphim from each A p. If X i finiely preenable (i.e., in hi cae, finie), hen one of hee morphim f : A p X i no monic; given i j wih f(i) = x = f(j), hen x i a loop of β in X (recall ha β i a clique in A p ), hu, X ha a 9

10 unique join loop of α and β, in oher word, a unique morphim 1 X. Conequenly, X i orhogonal o 1. Thi prove Σ = ω ( 1). However 1 canno be deduced from Σ in he Finiary Deducion Syem becaue he objec Y = p 3 p prime i orhogonal o Σ bu no o 1. In fac, Y ha no loop of β, hu, Y i orhogonal o and u. Furhermore for every prime p 3 he coproduc injecion i p : A p Y i he only morphim in hom(a p, Y ). In fac, due o he added edge 1 0 a morphim f : A p Y necearily ake {0, 1} A p ono {0, 1} A q for ome q. Since p and q are prime and f reric o a mapping of a p-cycle ino a q-cycle, i i obviou ha p = q. And i i alo obviou ha A p ha no endomorphim mapping {0, 1} ino ielf excep he ideniy conequenly, f = i p. 3 General Orhogonaliy Logic 3.1. Remark (i) Recall our anding aumpion ha A i a locally preenable caegory. We will now preen a (non-finiary) logic for orhogonaliy and prove ha i i alway ound, and ha for e of morphim i i alo complee. We will acually prove he compleene no only for e, bu alo for clae Σ of morphim which are preenable, i.e., for which here exi a cardinal λ uch ha every member : A B of Σ i a λ-preenable objec of he lice caegory A A. The compleene of our logic for all clae Σ of morphim i he opic of he nex ecion. (ii) We recall he concep of a ranfinie compoiion of morphim a ued in homoopy heory. Given a ordinal α (conidered, a uual, a he chain of all maller ordinal), an α-chain in A i imply a funcor C from α o A. I i called mooh provided ha C preerve direced colimi, i.e., if i < α i a (ii) limi ordinal hen C i = colim j<i C j Definiion Le α be an ordinal. A morphim h i called an α-compoie of morphim h i (i < α), where α i an ordinal, provided ha here exi a mooh α-chain A i (i α) uch ha h i he connecing morphim A 0 A α and each h i i he connecing morphim A i A i+1 (i < α) Example (1) An ω-compoie of a chain A p A 0 h 0 A 1 h 1 A 2 h 2... i, for any colimi cocone c i : A i C (i < ω) of he chain, he morphim c 0 : A 0 C. (2) A 2-compoie i he uual concep of a compoie of wo morphim. (3) Any ideniy morphim i he 0-compoie he 0-chain Definiion The Orhogonaliy Deducion Syem coni of he following deducion rule. 10

11 ranfinie compoiion puhou coequalizer weak cancellaion i (i < α) u v u if i an α-compoie of he i if if f g i a puhou i a coequalizer and f = g We ay ha a morphim can be proved from a cla Σ of morphim in he Orhogonaliy Logic, in ymbol Σ provided ha here exi a formal proof of from Σ uing he above deducion rule Remark (1) The deducion rule ranfinie compoiion i, in fac, a cheme of deducion rule: one for every ordinal α. (2) A proof of from Σ i a collecion of morphim i (i α) for ome ordinal α uch ha = α and for every i α eiher i Σ, or i i he concluion of one of he deducion rule above whoe aumpion lie in he e { j } j<i. (3) The λ-ary Orhogonaliy Deducion Syem i he deducion yem obained from 3.4 by rericing ranfinie compoiion o all ordinal α < λ. We obain he λ-ary Orhogonaliy Logic by applying hi deducion yem o λ-ary morphim, ee 2.1. In he λ-ary Orhogonaliy Logic he proof are alo rericed o hoe of lengh α < λ. Example: if λ = ω we ge preciely he Finiary Orhogonaliy Logic of Secion Example Oher ueful ound rule for orhogonaliy conequence can be derived from he above deducion yem. Here are ome example: (i) The 2-ou-of-3 rule: in a commuaive riangle B A u any morphim can be derived from he remaining wo. In fac and o prove {, u} {u, } {, } u C by compoiion, by weak cancellaion (pu v = id), by weak cancellaion (pu v = id) 11

12 form a puhou of and : A B u C D r id C C We obain a unique morphim r a indicaed. Oberve ha due o r = id he diagram D r id D r C i a coequalizer wih he parallel pair equalized by. Thu we have r u = r puhou coequalizer compoiion (ii) A coproduc + : A + B A + B can be derived from and. Thi follow from he puhou along coproduc injecion (denoed by ): A A B B A + B +id B A + B A + B ida + A + B Thu we have puhou + id B id A + compoiion + = (id A + ) (+id B ) (iii) More generally: i I i can be derived from { i } i I. Thi follow eaily from (ii) and ranfinie compoiion. 12

13 (iv) In a commuaive diagram A 1 1 f g A 1 p q 2 A 2 A 2 A 3 3 A 3 q p ḡ P f P where he ouer and inner quare are puhou, he morphim (a colimi of he naural ranformaion wih componen 1, 2, 3 ) can be derived from { 1, 2, 3 }. Thi follow from (ii) and he following puhou A 1 + A 2 + A A 1 + A 2 + A 3 [ḡf,ḡ, f] P P [ qp, q, p] (v) More generally: For any mall caegory D, given diagram D 1, D 2 : D A and given a naural ranformaion beween hem X : D 1 X D 2 X for X objd hen i colimi : colim D 1 colim D 2 can be derived: { X } X objd. Thi i analogou o (iv) above: expre colim D i a a coequalizer of a coproduc in he andard way (ee [11]), hen i a puhou of a coproduc of componen of he given naural ranformaion. (vi) The following (rong) cancellaion propery u hold for all epimorphim. In fac, he quare u u 13 id

14 i a puhou, hu, from u we derive u via puhou, and hen we ue (i). (vii) A wide puhou = i i of morphim i (i I) i A i A i B can be derived from hoe morphim : { i } i I If I i finie, hi follow eaily from puhou, ideniy and compoiion. For I infinie ue ranfinie compoiion. (viii) coequalizer ha he following generalizaion: given parallel morphim g j : A B (j J) uch ha a morphim : A A equalize he whole collecion, hen he join coequalizer : B B of he collecion fulfil. In fac, for every (j, j ) J J a coequalizer jj of g j and g j fulfil jj. By (vii), we have ince i a wide puhou of all jj Obervaion In every cocomplee (no necearily locally preenable) caegory he Orhogonaliy Logic i ound: for every cla Σ of morphim a morphim which ha a proof from Σ i an orhogonaliy conequence of Σ: Σ implie Σ = The verificaion ha ranfinie compoiion i ound i rivial: given a mooh chain C : α A and an objec X orhogonal o h i : C i C i+1 for every i < α, hen X i orhogonal o h : C 0 C α. In fac, for every morphim u : C 0 X here exi a unique cocone u i : C i X of he chain C wih u 0 = u: he iolaed ep are deermined by X h i and he limi ep follow from he moohne of C. I i eay o ee ha u α : C α X i he unique morphim wih u = u α h Definiion (ee [9]) A morphim : A B of A i called λ-preenable if, a an objec of he lice caegory A A, i i λ-preenable Remark (i) Thi i cloely relaed o a λ-ary morphim: i λ-ary (i.e., A and B are λ-preenable objec of A) iff i a λ-preenable objec of he arrow caegory A, ee [3]. (ii) Unlike he λ-ary morphim (which are he morphim of he mall caegory A λ ) he λ-preenable morphim form a proper cla: for example all ideniy morphim are λ-preenable. (iii) A imple characerizaion of λ-preenable morphim wa proved in [9]: 14

15 f i λ-preenable f i a puhou of a λ-ary morphim (along an arbirary morphim). (iv) The λ-ary morphim are preciely he λ-preenable one wih λ-preenable domain (ee [9]). Tha i, given f : A B λ-preenable, hen A λ-preenable B λ-preenable. (v) For every objec A he cone of all λ-preenable morphim wih domain A i eenially mall. Thi follow from (iii), or direcly: ince A A i a locally preenable caegory, i ha up o iomorphim only a e of λ-preenable objec Example A regular epimorphim which i he coequalizer of a pair of morphim wih λ-preenable domain i λ-preenable. Tha i, given a coequalizer diagram K f g A B hen K i λ-preenable i λ-preenable. In fac, given a λ-filered diagram in A A wih objec d i : A D i and wih a colimi cocone c i : (d i, D i ) (d, D) = colim i I (d i, D i ), hen for every morphim h : (, B) (d, D) of A A we find an eenially unique facorizaion hrough he cocone a follow: K f g A B d h d i D c i D i The morphim d = h merge f and g. Oberve ha c i merge d i f and d i g for any i I. Since K i λ-preenable and D = colim D i i a λ-filered colimi in A, i follow ha ome connecing map d ij : (d i, D i ) (d j, D j ) of our diagram merge d i f and d i g. Thi implie d j f = d j g, hence, d j facor hrough : d j = k for ome k : B D j. Then k : (, B) (d j, D j ) i he deired facorizaion. epimorphim. I i unique becaue i an Definiion A cla Σ of morphim i called preenable provided ha here exi a cardinal λ uch ha every member of Σ i a λ-preenable morphim Example Every mall cla i preenable. In hi cae here even exi λ uch ha all member are λ-ary morphim. Thi follow from he fac ha every objec of a locally preenable caegory i λ-preenable for ome λ, ee [3]. 15

16 3.13. Remark We will prove ha he Orhogonaliy Logic i complee for preenable clae of morphim. Thi harply conra wih he following: if A i a locally finiely preenable caegory and Σ i a cla of finiely preenable morphim, he Finiary Orhogonaliy Logic need no be complee: Example (ee [4]) Le A be he caegory of algebra on counably many nullary operaion (conan) a 0, a 1, a 2,... Denoe by I = {a n } n N an iniial algebra, by 1 a erminal algebra, and by k he congruence on I merging ju a k and a k+1. The correponding quoien morphim e k : I I/ k i clearly finiely preenable, and o i he quoien morphim f : C 1 where C = {0, 1} i he algebra wih a 0 = 0 and a i = 1 for all i 1. I i obviou ha {e 1, e 2, e 3,... } {f} = e 0. Neverhele, a verified in [4], e 0 canno be proved from {e 1, e 2, e 3,... } {f} in he Finiary Orhogonaliy Logic. Oberve ha hi doe no conradic Theorem 2.16: he morphim f above i no finiary Conrucion of a Reflecion Le Σ be a cla of λ-preenable morphim in a locally λ-preenable caegory A. For every objec A of A a reflecion r A : A Ā of A in he orhogonal ubcaegory Σ i conruced a follow: We form he diagram D A : D A A of all λ-preenable morphim : A A provable from Σ wih domain A. Le Ā be a colimi of D A wih he colimi cocone : A Ā. We how ha he morphim r A = : A Ā (independen of ) i he deired reflecion. The precie definiion of D A i a follow: we denoe by Σ λ he cla of all λ- preenable morphim wih Σ. Le D A be he full ubcaegory of he lice caegory A A on all objec lying in Σ λ. By 3.9 (v) he diagram i eenially mall. D A : D A A, D A ( A A ) = A Propoiion For every objec A he diagram D A i λ-filered and r A : A Ā i a reflecion of A in Σ ; moreover, Σ r A. 16

17 Proof (1) The diagram D A i λ-filered: From coequalizer and 3.6(vii), Σλ i cloed under coequalizer of Σ λ -equalized λ-mall e of morphim and under λ-wide puhou. Thi aure ha A Σ λ i cloed under λ-mall colimi in A A, hu he caegory D A i λ-filered. (2) We prove Σ r A and Σ for all in D A. Thi follow from 3.6(v) applied o he naural ranformaion from he conan diagram of value A o D A wih componen : A A : I colimi i r A. Hence, by 3.6(v), we have Σ r A. Now oberve ha he rule 2-ou-of-3, 3.6(i), alo yield ha Σ for all in D A. (3) Given a morphim : R Q in Σ we prove ha every morphim f : R Ā ha a facorizaion hrough. R Q u v g R Q f f g A Ā ˆ ˆP q P By 3.9(iii) here exi a λ-ary morphim : R Q uch ha i a puhou of (along a morphim u). Due o (1) and ince R i a λ-preenable objec, he morphim f u : R Ā = colim A facor hrough one of he colimi morphim: f u = g for ome : A A in D A and ome g : R A. We denoe by ˆ a puhou of along f u. We furher denoe by Thi lead o he unique morphim a puhou of along g. q : P ˆP wih q = ˆ and q g = f v. By (2) we know ha Σ. Conequenly, compoiion yield Σ q 17

18 ince q = ˆ, and Σ ˆ by puhou. Nex, we oberve ha Σ q by 3.6(iv): apply i o he puhou P and ˆP and he naural ranformaion wih componen id R, and id Q. Now he 2-ou-of-3 rule yield Σ. Moreover, i λ-preenable ince i λ-ary, ee 3.9(iii). Therefore, he morphim p = : A P i alo λ-preenable, and Σ p by compoiion. Thu, p : A P i an objec of D A. The correponding colimi morphim p : P Ā fulfil r A = p p. Furher, ince i a connecing morphim of he diagram D A from o p, i follow ha Conequenly, = p. ( p g) = p g = g = f u and he univeral propery of he puhou Q of and u yield a unique h : Q Ā wih f = h and p g = h v. Thi i he deired facorizaion of f hrough. (4) Ā lie in Σ : We prove he uniquene of he facorizaion. Given h, k : Q Ā equalized by, we prove h = k. R Q h u v k R Q c A C r h k c Ā C Since Q i λ-preenable, he morphim h v, k v : Q Ā boh facor hrough ome of he colimi morphim of he λ-filered colimi Ā = colim D A: h v = h and k v = k for ome h, k : Q A. Form coequalizer c = coeq(h, k) and c = coeq(h, k ) 18

19 From h = k coequalizer yield Σ c and hen (2) above and compoiion yield Σ c. From he equaliy (c ) h = (c ) k we conclude ha c facor hrough c. Since c i an epimorphim, 3.6(vi) yield Σ c. Moreover, c i a λ-preenable morphim ince c = coeq(h, k ) and Q i λ-preenable, ee Example The morphim r = c : A C i hu alo a λ-preenable morphim wih Σ r, in oher word (r, C ) i an objec of D A, and c : (, A S ) (r, C ) i a morphim of D A. Thi implie ha he colimi map fulfil = r c. We are ready o prove h = k: by he univeral propery of he puhou Q we only need howing h v = k v: h v = h = r c h and analogouly k v = r c k, hu c h = c k finihe he proof. (5) The univeral propery of r A : Le f : A B be a morphim wih B orhogonal o Σ. Thu B i orhogonal o all morphim wih Σ, ee 3.7. r A A f B f A g Ā For every objec : A A of D A le f : A B be he unique facorizaion hrough. Thee morphim clearly form a compaible cocone of D A, and he unique facorizaion g : Ā B fulfil, for any objec of D A, f = f = g = g r A. Converely, uppoe g r A = f, hen g = g becaue for every objec of D A we have g = f = g ; hi follow from B due o (g ) = f = f. 19

20 3.17. Theorem The Orhogonaliy Logic i complee for all preenable clae Σ of morphim: every orhogonaliy conequence of Σ ha a proof from Σ in he Orhogonaliy Deducion Syem. Shorly, Σ = implie Σ. Proof Given an orhogonaliy conequence : A B of Σ, form a reflecion r A : A Ā of A in Σ a in Then Σ = implie ha Ā i orhogonal o, hu we have u : B Ā wih r A = u. Since Σ r A, by 3.15(2), we conclude Σ u. Now we have ha Σ = u (= r A ) and Σ =, and hi rivially implie ha Σ = u. Thu by he ame argumen wih replaced by u here exi a morphim v uch ha The la ep i weak cancellaion : Σ v u. u v u Corollary The Orhogonaliy Logic i complee for clae Σ of morphim of he form Σ = Σ 0 Σ 1, Σ 0 mall and Σ 1 RegEpi. Proof Le λ be a regular cardinal uch ha A i locally λ-preenable, and all morphim of Σ 0 are λ-preenable. We will ubiue Σ 1 wih a cla Σ 1 of λ-preenable morphim a follow: for every member : A B of Σ 1 chooe a pair f, g : A A wih = coeq(f, g). Expre A a a λ-filered colimi of λ-preenable objec A i wih a colimi cocone a i : A i A (i I ). Form a coequalizer i : A B i of f a i, g a i : A i B for every i I. Then we obain a filered diagram wih he objec B i (i I ) and he obviou connecing morphim. The unique b i : B i B wih = b i i form a colimi of ha diagram. Moreover, an objec X i orhogonal o iff i i orhogonal o i for every i I : f A i a i A g u A B b i i B i v v i X Le Σ 1 be he cla of all morphim i for all Σ 1 and i I. Then he cla Σ = Σ 0 Σ 1 20

21 coni of λ-preenable morphim, ee Example 3.10, and Σ = Σ. Given an orhogonaliy conequence of Σ, we hu have a proof of from Σ, ee Theorem I remain o prove i for every Σ and i I ; hen Σ implie Σ. In fac, ince i i an epimorphim, apply 3.6(vi) o = b i i Remark Since all λ-ary morphim form eenially a e (ince A λ i mall), he λ-ary Orhogonaliy Logic (ee 3.5) i complee for clae of λ-ary morphim he proof i analogou o ha of Theorem Vopěnka Principle 4.1. Remark The aim of he preen ecion i o prove ha he Orhogonaliy Logic i complee (for all clae of morphim) in all locally preenable caegorie iff he following large-cardinal Vopěnka principle hold. Throughou hi ecion we aume ha he e heory we work wih aifie he Axiom of Choice for clae Definiion Vopěnka Principle ae ha he caegory Rel(2) of graph (or binary relaional rucure) doe no have a large dicree full ubcaegory Remark (1) The following fac can be found in [3]: (i) Vopěnka Principle i a large-cardinal principle: i implie he exience of meaurable cardinal. Converely, he exience of huge cardinal implie ha Vopěnka Principle i conien. (ii) An equivalen formulaion of Vopěnka Principle i: he caegory Ord of ordinal canno be fully embedded ino any locally preenable caegory. (2) The following proof i analogou o he proof of Theorem 6.22 in [3] Theorem Auming Vopěnka Principle, he Orhogonaliy Logic i complee for all clae of morphim (of a locally preenable caegory). Proof (1) Every cla Σ can be expreed a he union of a chain Σ = Σ i (Σ i Σ j if i j) i Ord of mall ubclae hi follow from he Axiom of Choice. We prove ha every objec A ha a reflecion in Σ by forming reflecion r i (A) : A A i in Σ i for every i Ord, ee 2.2. Thee reflecion form a ranfinie chain in he lice caegory A A: for i j he fac ha Σ i Σ j implie he exience of a unique a ij : A i A j forming a commuaive riangle A r i (A) rj(a) A i a ij A j 21

22 We prove ha hi chain i aionary, i.e., here exi an ordinal i 0 uch ha a i0 j i an iomorphim for all j i 0 i will follow immediaely ha r i0 (A) i a reflecion of A in Σ. (2) Auming he conrary, we have an objec A and ordinal i(k) for k Ord wih i(k) < i(l) for k < l uch ha none of he morphim a i(k),i(l) wih k < l i an iomorphim. We derive a conradicion o Vopěnka Principle: he lice caegory A A i locally preenable, and we prove ha he funcor E : Ord A A, k r i(k) (A) i a full embedding. In fac, for every morphim u uch ha he diagram r i(k) (A) A A i(k) u r i(l) (A) A i(l) commue, we have k l and u = a k,l. The laer follow from he univeral propery of r i(k) (A). Thu, i i ufficien o prove he former: auming k l we how k = l. In fac, he morphim u i invere o a i(l),i(k) becaue (u a i(l),i(k) ) r i(l) (A) = r i(l) (A) implie u a i(l),i(k) = id and analogouly for he oher compoie. Our choice of he ordinal i(k) i uch ha whenever a i(l),i(k) i an iomorphim, hen k = l. (3) Every orhogonaliy conequence : A B of Σ ha a proof from Σ. The argumen i now preciely a in Theorem 3.17: we ue he above reflecion r A and he fac ha Σ r A (ee Propoiion 3.16 and he above fac ha r A = r i0 (A) for ome i 0 ) Example (under he aumpion of he negaion of Vopěnka Principle). In he caegory Rel(2, 2) of relaional rucure on wo binary relaion α, β we preen a cla Σ of morphim ogeher wih an orhogonaliy conequence which canno be proved from Σ: Σ = bu Σ. We ue he noaion of Example The negaion of he Vopěnka Principle yield graph (X i, R i ) in Rel(2) for i Ord, forming a dicree caegory. For every i le A i be he objec of Rel(2, 2) on X i whoe relaion α i R i and β i a clique (ee 2.16). Our cla Σ coni of he morphim, u of 2.18 and A i for all i Ord. 22

23 We claim ha he morphim : 1 i an orhogonaliy conequence of Σ. In fac, le B be an objec orhogonal o Σ and le i be an ordinal uch ha A i ha cardinaliy larger han B. We have a (unique) morphim h : A i B, and ince h canno be monic, he relaion β of B conain a loop (recall ha β i a clique in A i ). Thi implie ha B ha a unique join loop of α and β, herefore, B. To prove Σ i i ufficien o find a caegory A in which (i) Rel(2, 2) i a full ubcaegory cloed under colimi and (ii) ome objec K of A i orhogonal o Σ bu no o. From (ii) we deduce ha canno be proved from Σ in he caegory A, ee Obervaion 3.6. However, (i) implie ha every formal proof uing he Orhogonaliy Deducion Syem 3.4 in he caegory Rel(2, 2) i alo a valid proof in A. Togeher, hi implie Σ in Rel(2, 2). The imple approach i o chooe A = REL(2, 2), he caegory of all poibly large relaional yem on wo binary relaion, i.e., riple (X, α, β) where X i a cla and α, β are ubclae of X X. Morphim are cla funcion preerving he binary relaion in he expeced ene. Thi caegory conain Rel(2, 2) a a full ubcaegory cloed under mall colimi, and he objec K = i Ord i no orhogonal o : 1 ince none of A i conain a join loop of α and β. However, i i eay o verify ha K i orhogonal o Σ. A more economical approach i o ue a A ju he caegory Rel(2, 2) wih he unique objec K added o i, i.e., he full ubcaegory of REL(2, 2) on {K} Rel(2, 2) Corollary Vopěnka Principle i equivalen o he aemen ha he Orhogonaliy Logic i complee for clae of morphim of locally preenable caegorie. A i 5 A counerexample The Orhogonaliy Logic can be formulaed in every cocomplee caegory, and we know ha i i alway ound, ee 3.6. Bu ouide of he realm of locally preenable caegorie he compleene can fail (even for finie e Σ): 5.1. Example We ar wih he caegory CPO of ric CP O : objec are poe wih a lea elemen and wih direced join, morphim are ric coninuou funcion (preerving and direced join). Thi caegory i well-known o be cocomplee. We form he caegory CPO (1) 23

24 of all unary algebra on ric CP O : objec are riple (X,, α), where (X, ) i a ric CP O and α : X X i an endofuncion of X, morphim are he ric coninuou algebra homomorphim. I i eay o verify ha he forgeful funcor CPO (1) CPO i monoopological, hu, by and in [2] he caegory CPO (1) i cocomplee. We preen morphim 1, 2 and of CPO (1) uch ha an algebra A i orhogonal o (a) 1 iff i operaion α ha a mo one fixed poin (b) 2 iff i operaion α fulfil x αx for all x and (c) iff α ha preciely one fixed poin. We hen have { 1, 2 } = In fac, if an algebra A fulfil (b), we can define a ranfinie chain a i (i Ord) of i elemen by a o = a i+1 = αa i, and a j = i<j a i for all limi ordinal j. Thi chain canno be 1 1, hu, here exi i < j wih a i = a j and we conclude ha a i i a fixed poin of α. The fixed poin i unique due o (a), hu, A i orhogonal o. On he oher hand { 1, 2 } The argumen i analogou o ha in Example 4.6: The caegory A of poibly large CP O wih a unary operaion conain CPO (1) a a full ubcaegory cloed under mall colimi. And he following objec K i orhogonal o 1 and 2 bu no o : K = (Ord,, ucc) where i he uual ordering of he cla of all ordinald, and ucc i = i + 1 for all ordinal i. Thu, i remain o produce he deired morphim 1, 2 and. The morphim 1 i he following quoien x α y α α α... 1 x=y α α α... where boh he domain and codomain are fla CP O (all elemen excep are pairwie incomparable). The morphim 2 i carried by he ideniy homomorphim x α α... α α... id x α α... α α... 24

25 where he domain i fla and he codomain i fla excep for he unique comparable pair no involving being x < αx. Finally, i he embedding α α... α α α... wih boh he domain and he codomain fla. 6 Injeciviy Logic A menioned in he Inroducion, for he injeciviy logic he following Injeciviy Deducion Syem wa formulaed in [1]: ranfinie compoiion (ee 3.4) puhou (ee 3.4) u cancellaion We proved here ha i i ound and complee for e Σ of morphim. In conra o Theorem 4.4 hi deducion yem fail o be complee for clae of morphim in general, independenly of e heory: 6.1. Example Le Rel(2) be he caegory of graph. For every cardinal n le C n denoe a clique (2.16) on n node. Then he morphim i an injeciviy conequence of he cla : 1 Σ = { C n ; n Card}. In fac, given a graph X injecive w.r.. Σ, chooe a cardinal n > cardx. We have a morphim f : C n X which canno be monomorphic. Conequenly, X ha a loop. Thi prove ha X i injecive w.r... The argumen o how ha canno be proved from Σ i compleely analogou o 5.1: he caegory REL(2) of poenially large graph conain Rel(2) a a full ubcaegory cloed under mall colimi. The objec K = C n i injecive w.r.. Σ bu no n Card injecive w.r... Therefore, doe no have a formal proof from Σ in he Injeciviy Deducion Syem above applied in REL(2). Conequenly, no uch formal proof exi in Rel(2). Inead of REL(2) we can, again, ue he full ubcaegory on Rel(2) {K} for our argumen. 25

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