An "almost" full embedding of the category of graphs into the category of groups

AN “ALMOST” F ULL EMBEDDING OF THE CA TEGOR Y OF GRAPHS INTO THE CA TEGOR Y OF GR OUPS AD AM J. PRZE ´ ZDZIECKI 1 No v em b er 2, 2018 Abstract. W e construct a functor F : G raphs → G roups whic h is faithful and “almost” full, in the sense that every nont rivial group homomor phism F X → F Y is a co mpos itio n of an inner automorphism of F Y and a homomo rphism of the form F f , for a unique ma p of graphs f : X → Y . When F is comp osed with the Eilenberg -Mac Lane space construction K ( F X , 1) we obtain an embedding of the category of graphs int o the unpointed ho motopy category which is full up to n ull-homoto pic maps. W e provide several applications o f this construction to lo caliza- tions (i.e. idempotent functors); we show that the questions: (1) Is every or thogonality clas s reflec tiv e? (2) Is every or thogonality clas s a small- orthogonality class ? hav e the same a nsw ers in the category of gro ups as in the cate- gory o f g raphs. In o ther w ords they dep end on set theory: (1) is equiv alent to weak V op ˇ enk a’s principle and (2) to V op ˇ enk a’s princi- ple. Additionally , the se c ond question, considered in the homotopy category , is a lso equiv ale nt to V opˇ enk a’s principle. MSC: 1 8A40; 20 J15; 55P6 0; 18A22 ; 03E5 5 Keywor ds: Categor y of gro ups ; Lo calization; Lar ge cardina ls 1. Introduction Matumoto [17 ] pro v ed that fo r an y graph Γ there exists a group G whose outer automorphism group is isomorphic to the gr o up of au- tomorphisms of Γ. His r esult r eceiv ed a considerable atten t ion since ev ery group can b e r ealized as the group of automorphisms of some graph. The main result of this article ma y b e view ed as a functorial ve rsion of t he ab ov e. W e construct a functor F from t he categor y o f graphs to the category of groups whic h is faithful and “almost” full, in the sense that the maps F X,Y : Hom G rap h s ( X , Y ) → Hom G rou ps ( F X , F Y ) 1 The author w as partially suppo rted b y g r ant N N20 1 38 7034 of the Polish Ministry of Science and Highe r Education. 1 induce bijections F X,Y : Hom G rap h s ( X , Y ) ∪ {∗} → Rep( F X , F Y ) . Here Rep( A, B ) = Hom G rou ps ( A, B ) /B where B acts on Hom G rou ps ( A, B ) b y conjugation and ∗ is an additional p oint which w e send to the trivial elemen t of Rep. A gr aph is a set with a binary relation. F ull and faithful functors are con v enient tools that allo w o ne to transfer constructions and prop erties b etw een categories. The cate- gory of gra phs is v ery comprehensiv e and w ell researc hed. Ad´ amek and Rosic k´ y prov ed in [1, Theorem 2.65] that every accessible categor y has a full embedding in to the category of graphs. Inste ad of quot- ing the complete definition of accessible categories let us men tio n that these contain, as full sub categories, “most” of the “non- homotop y” categories: the categories of groups, fields, R -mo dules, Hilb ert spaces, p osets (i.e. partially ordered sets), simplicial sets, metrizable spaces or CW-spaces a nd con tin uo us maps, the category of mo dels of some first- order theory , and man y more. In fact, under a large cardinal hy p othesis that the measurable cardinals are bo unded ab ov e, an y concretiz able category fully em b eds into the category of gra phs [19, Chapter I I I, Corollary 4.5]. In this article w e describe sev eral a pplications of the functor F , con- structed in Section 4; the c hoice of the applications is strongly affected b y the in terests of the autho r . A lo c alization ma y b e defined as a functor from a category C to itself that is a left adjo int to inclusion of a sub category D ⊆ C ; it is an idempo t ent functor whic h ma y b e view ed as a pro jection of C on to the sub category D . A more common definition of lo calization can b e found in Section 8. Libman [16] inspired a question o f whether the v alues of lo calization functors at finite groups can ha v e arbitrarily large cardinalities. F or all finite simple gr oups suc h lo calizations w ere constructed b y G¨ ob el, Ro dr ´ ıguez, Shelah in [10], [11], and for some suc h groups by t he aut ho r in [18]. In Section 10 w e see tha t the functor F immediately pro duces ye t another such construction. This article w as motiv ated by another application. Ad´ amek and Rosic k ´ y prov ed in [1, Chapter 6] that large cardinal axioms called V op ˇ enk a’s principle and w eak V op ˇ enk a’s principle (b oth form ulated in the category of g raphs) hav e many implications related to lo calizations and the structure of accessible categories. These axioms are b eliev ed to b e consisten t with the standard set theory ZF C while their nega- tions are kno wn to b e consisten t with ZFC. Casacub erta, Scev enels and Smith [5] extended some of these implicatio ns to the homotop y category . In Section 9 w e see that a functor whic h sends a g raph Γ to the Eilen b erg-Mac Lane space K ( F Γ , 1) is, up to null-homotopic 2 maps, a full embedding of the catego r y of gra phs in to the (unp oin ted) homotop y categor y . W e strengthen the results of [5 ] by show ing that V op ˇ enk a’s principle is actually equiv alen t to its fo rm ulation in the ho- motop y category: ev ery orthogonality class in the homotopy category is a small-orthogonality class in the homo t op y category ( i.e. it is asso- ciated with an f -lo calization of Bo usfield and Dror F arj oun [9]) if and only if this is the case in the category of gr aphs. On the other hand, it w a s hop ed that some consequences of V op ˇ enk a’s principles in the category of gro ups might b e prov a ble in Z F C. Casacu- b erta and Scev enels [4] hin t tha t this might b e the case for a “long standing op en question in categorical group theory” that asks if ev ery orthogonality class D , in the category of groups, is reflectiv e – that is, if the inclusion functor D → G r oups has a left adjo in t. In Section 8 w e find that this ques tion is actually equiv alen t to w eak V opˇ enk a’s principle. The w ork presen ted in this pap er has b egun during the a utho r’s visit to Cen tre de Recerca Mathem` atica, Bellaterra, at the inspiration of Carles Casacub erta. 2. Definitions A gr aph Γ is a set o f vertic es , vert Γ, together with a set of e d ges , whic h is a binary relation edge Γ ⊆ v ert Γ × vert Γ. A morphism Γ → ∆ b et wee n graphs is an edge preserving function vert Γ → v ert ∆. The category of graphs is denoted G r aphs . An m-gr aph (m for m ulti- edge) is a category Γ whose ob jects form a disjoin t unio n o f a set of v ertices, v ert Γ, and a set of edges, edge Γ. Eac h noniden tity morphism of an m-graph Γ ha s its source in edge Γ and its target in vert Γ. Each edge e ∈ edge Γ is a source of t wo noniden tit y morphisms: one lab elled ι e whose target is the initial vertex of e , and the ot her lab elled τ e whose t a rget is the termin al vertex of e . Morphisms b et wee n m-graphs are f unctors that preserv e the edges, the v ertices a nd the lab elling: f ( ι e ) = ι f ( e ) and f ( τ e ) = τ f ( e ) . The category of m-g raphs is denoted m - G r aphs . A u-gr aph (u for undirected-edge) is an m-graph without the lab elling of morphisms. The category of u-graphs is denoted u - G r aphs . A u- graph is usually visualized as in (4.1) where the nonidentit y morphisms a re represen t ed b y incidence b et w een edges (interv a ls) and v ertices (small circles). A graph or an m- graph is similarly visualized, with arrow s on its edges. W e hav e an ob vious full a nd faithf ul inclusion functor I : G r aphs → m - G r aphs whic h has a left adjoint (the edge colla psing functor J : 3 m - G r aphs → G r aphs ), that is, Hom G rap h s ( J Γ , ∆) ∼ = Hom m - G r aphs (Γ , I ∆) where Γ is in m - G r aphs and ∆ is in G r aphs . A gr aph of g r o ups is a functor G : Γ → G r oups where Γ is a u-graph and for each morphism i in Γ, G ( i ) is a monomorphism. Γ is called the underlying u-gra ph of G . Convention. If G : Γ → G r oups is a graph o f groups and a , b a re ob jects in Γ, w e consider the v alues o f G on a and b , that is, G a and G b , to b e differen t whenev er a and b a re differen t, and G tak es morphisms to inclusions. In short, w e t reat G as the image of an inclusion of Γ in to G r oups all of whose morphisms are inclusions. The ob jects o f G are called the e dge and the vertex gr oups . A tr e e (a tr e e of gr oups ) is a connected u- graph (gra ph of gr o ups) without circuits, that is, closed paths without back trac king. If G is a g r o up, g ∈ G and A ⊆ G then g A denotes g Ag − 1 . 3. Bass- Serre the or y In this se ction w e collect facts concerning groups acting on t rees, whic h will b e used later. The k ey reference is [20]. The sym b ol ∗ A G i denotes the amalgam of groups G i along the common subgroup A , and colim G denotes the c olim it of a gra ph of groups G . Lemma 3.1. L et H 1 ⊆ G 1 and H 2 ⊆ G 2 and A b e a c omm o n sub gr oup of G 1 and G 2 . If H 1 ∩ A = B = H 2 ∩ A then the homo morphism h : H 1 ∗ B H 2 → G 1 ∗ A G 2 induc e d by the inclusions is inje ctive. Pr o of. See [20, § 1.3, Prop osition 3].  As a consequence w e obta in Lemma 3.2. L et G b e a gr aph of gr oups c on sisting o f on e c entr al vertex gr oup C and vertex gr oups B i , i ∈ I , attache d to C alo n g e dge gr oups A i , i ∈ I : ❝ C · · · ✑ ✑ ✑ ✑ ✑ A i ❝ B i ◗ ◗ ◗ ◗ ◗ A j ❝ B j If H i ⊆ B i ar e s ub gr oups such that H i ∩ A i is trivial for i ∈ I then the homomorph ism h : ∗ i ∈ I H i → colim G ind uc e d by the inclusions is inje ctive and its image trivial ly interse cts C . 4 Pr o of. W e iden tify I with an ordinal and pro ceed b y induction. The case when I is a singleton is ob vious, as is the case when I is a limit ordinal and the result is established for all I 0 < I . Supp ose that I = I 0 ∪ { i 0 } and the result is established fo r I 0 . Let G 0 b e the g r a ph of groups obtained fr o m G by deleting B i 0 and A i 0 . W e ha ve colim G = B i 0 ∗ A i 0 colim G 0 . By the inductiv e assumption, h is injectiv e on ∗ i ∈ I 0 H i and h ( ∗ i ∈ I 0 H i ) ∩ C is trivial, and therefore Lemma 3.1 implies the result for I .  The most p o w erful elemen t o f the Bass-Serre theory is the f ollo wing. Theorem 3 .3 ([20, § 4.5, Theorem 9]) . L et G b e a tr e e of gr oups and T the underlying u-gr a ph. Ther e exists a u-gr aph X c o n taining T and an action of G T = colim G on X which is char acterize d (up to i somor- phism) by the f o l lowi n g pr op erties: (a) T is the fundamen tal domain for X mo d G T and (b) for any v in vert T (r esp. e in edge T ) the stabilizer of v (r esp. e ) in G T is G v (r esp. G e ). Mor e over, X is a tr e e. As a corollary of Theorem 3.3 w e immediately o bta in: R emark 3.4 . Let X and G b e as ab ov e. (a) Eac h v ertex group of G is a subgroup of colim G . (b) The stabilize rs of the v ertices and edges of X are resp ectiv ely the colim G conjuga t es of the v ertex and edge groups of G . (c) If a subgroup H of colim G stabilizes t wo v ertices v and w in X then it stabilizes the shortest pa t h from v to w and therefore H is con tained in all the v ertex and edge stabilizers of this path. (d) F or any edge ❝ v e ❝ w in G w e hav e G v ∩ G w = G e in colim G . L emma 3.5 . If G is a tr e e of gr oups and H ⊆ colim G is a finite sub- gr oup then H is c onjugate in colim G to a s ub gr oup of some vertex gr oup G v . 4. Construction of the functor F W e start with the follo wing graph of groups, where some edge to v ertex incidences are lab elled with c : 5 (4.1) ❝ M N ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❝ P 0 c N 0 ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ N 4 ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❝ P 1 c N 1 ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ❝ P 4 c N 3 ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❝ P 2 c ❝ P 3 c N 2 W e assume the following conditions: C1 M is finite, cen terless and an y homomorphism f : M → M is either trivial or an inner automorphism. C2 M admits no non t r ivial homomorphisms to P i for i = 0 , 1 , . . . , 4. C3 If an inclus ion A ⊆ B in (4.1) is lab elled c and f : B → B is a homomorphism whic h is the iden tit y on A then f is the iden tity . C4 If A 1 and A 2 are edge groups ( A 1 6 = N 2 ) adjacen t to the common v ertex gro up B then A 1 is not conjugate in B to a subgroup o f A 2 . If A 1 = A 2 w e r equire that N B ( A 1 ) = A 1 . C5 N 1 ∩ N 2 and N 2 ∩ N 3 are trivial. C6 N 1 ∩ N 0 and N 3 ∩ N 4 are trivial. C7 If A ⊇ C ⊆ B is an edge in (4.1 ) and C ⊆ B is lab elled c then no homomorphism f : B → A is the iden tit y o n C . C8 If an inclusion A ⊆ B in (4.1) is lab elled c and K ⊆ B is a normal subgroup whic h contains A then K = B . L emma 4.2 . Ther e exists a gr aph of gr oups (4.1) s a tisfying c onditions C1–C8. Pr o of. W e hav e: 6 ❝ M 23 N ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❝ A 12 ⊕ A 11 N ⊕ A ( S 3 ⊕ S 8 ) ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ N ⊕ A ( S 4 ⊕ S 7 ) ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❝ A 11 ⊕ A 12 A 11 ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ❝ A 11 ⊕ A 12 A 11 ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❝ A 12 ❝ A 12 Z 12 Here M 23 is the Mathieu simple group, N ∼ = Z 11 ⋊ Z 5 is the no r ma lizer of the Sylo w 11- subgroup in M 23 [12, pag e 265], S n and A n denote the n -th sy mmetric and the n -th alternating groups. A ( S p ⊕ S q ) is the in tersection of S p ⊕ S q and A 12 in S 12 . The inclusions are as fo llo ws: (1) N ⊆ A 12 ⊕ A 11 is determined by any inclusions N ⊆ A 12 and N ⊆ A 11 . (2) N ⊕ A ( S p ⊕ S q ) ⊆ A 12 ⊕ A 11 equals ( N ⊆ A 12 ) ⊕ (natural inclusion A ( S p ⊕ S q ) ⊆ A 11 ). (3) N ⊕ A ( S p ⊕ S q ) ⊆ A 11 ⊕ A 12 equals ( N ⊆ A 11 ) ⊕ ( A ( S p ⊕ S q ) ⊆ A 12 ). (4) A 11 ⊆ A 12 is the inclusion of a maximal subgroup. (5) A 11 ⊆ A 11 ⊕ A 12 is determined b y id A 11 and A 11 ⊆ A 12 . (6) Z 12 ⊆ A 12 is the inclusion of a tr a nsitiv e subgroup. W e know [12, page 26 5] that M 23 has no outer automorphisms a nd has an elemen t of order 23. The order of M 23 is not divisible b y 25. Also all the automorphisms of A 11 and A 12 come fro m S 11 and S 12 . This a nd w ell known prop erties of symmetric groups mak e it straightforw ard to v erify that all the conditions C1–C8 are satisfied.  The c onstruction of G Γ and F Γ . Let Γ b e an m- g raph. W e construct a u-graph A Γ as follows . Replace eac h v ertex v in Γ with a vertex P 0 ,v , add a new verte x M , connect M to ev ery P 0 ,v with an edge N v , and finally replace ev ery subgraph ❝ P 0 ,v ✲ e ❝ P 0 ,w where e ∈ Γ with a subgraph 7 (4.3) ❝ P 0 ,v N 0 ,e ❝ P 1 ,e N 1 ,e ❝ P 2 ,e N 2 ,e ❝ P 3 ,e N 3 ,e ❝ P 4 ,e N 4 ,e ❝ P 0 ,w W e sa y that M , N , N i , P i for i = 0 , 1 , . . . , 4 are typ es of ob jects M , N a , N i,a , P i,a for i = 0 , 1 , . . . , 4 and a in vert Γ o r edge Γ, resp ectiv ely . W e see that the resulting functor A preserv es colimits of connected diagrams. W e construct a graph of groups G Γ b y taking A Γ as the underlying u-graph and sending eac h ob ject P of A Γ to a group isomorphic t o the group in (4.1) lab elled with the type of P . W e send morphisms in A Γ to t he corresp onding inclusions in (4.1). W e lab el c those inclusions in G Γ whic h corresp ond to sim ilarly lab elled inclus ions in (4.1). The isomorphisms b etw een the groups in G Γ and the groups in (4.1), t heir in vers es and comp ositions are referred to as standar d isom orphisms . If f : Γ → Γ ′ is a morphism of m- graphs then w e define Gf : G Γ → G Γ ′ in the ob vious w a y using standard isomorphisms. W e see that the resulting functor G , from m-gra phs to gr a phs of gr o ups, preserv es colimits of connected diag rams. W e define F Γ = colim G Γ , in particular F ∅ = M . W e obtain F f : F Γ → F Γ ′ as the colimit homomorphism. R emark 4.4 . Since colimits comm ute w e see that F also preserv es col- imits of connected diagrams. 5. Proper ties of the functor F In order to a pply Bass-Serre theory we need to construct F Γ using colimits of trees o f groups rather tha n colimits of g eneral graphs of groups. Let G 1 Γ b e t he subgraph of groups of G Γ consisting of the v ertices of ty p es M , P 0 , P 1 , P 4 and the edges of t yp es N , N 0 , N 4 . Let G 2 Γ be the subgraph of G Γ consisting of the v ertices of types P 2 , P 3 and the edges o f ty p e N 2 . Without changing the colimit, we can make G 2 Γ a tree of groups b y adding a trivial v ertex group and connecting it to ev ery v ertex group of t yp e P 2 with a trivial edge group. Let G 0 Γ b e the su b diagram of G Γ consisting of the edges of t ype N 1 and N 3 . Then G Γ is the colimit, in the category of diagra ms, o f the follo wing: G 1 Γ ← G 0 Γ → G 2 Γ . Let F i Γ = colim G i Γ for i = 1 , 2 , 3 . Since colimits commute, w e see that F Γ is the colimit of F 1 Γ ← F 0 Γ → F 2 Γ . 8 It is clear that F 0 Γ = ∗ e ∈ edge Γ ( N 1 ,e ∗ N 3 ,e ) and F 2 Γ = ∗ e ∈ edge Γ ( P 2 ,e ∗ N 2 ,e P 3 ,e ) . L emma 5.1 . The homomorph i s m s F 0 Γ → F i Γ for i = 1 , 2 ar e inje ctive. Pr o of. This is a consequence of Conditions C6 and C5 and Lemma 3.2.  L emma 5.2 . The vertex gr oups of G Γ ma p inje ctive l y in to F Γ . Pr o of. This fo llo ws from Remark 3.4(a) and the construction of F Γ b y means of colimits of trees, including Lemma 5.1.  W e need an analo g ue of Theorem 3.3: L emma 5.3 . L et Γ b e an m-gr aph and A Γ b e the underlying u-gr aph of G Γ . Ther e exists a u-gr aph X and an action of F Γ on X which is char acterize d ( up to isomorph ism) by the fol lowing pr op erties: (a) A Γ is the fundamen tal domai n for X mo d F Γ and (b) for any v in v ert A Γ ( r e s p. e in edge A Γ ) the stabilizer of v (r esp. e ) in F Γ is G Γ v (r esp. G Γ e ). Pr o of. The pro of is similar to the pro of of [20, § 4.5, Theorem 9]: Since w e kno w f r o m Lemma 5.2 that the v ertex groups G Γ v em b ed into the colimit group F Γ, it is clear that vert X (resp. edge X ) is the disjoin t union of the F Γ · v ∼ = F Γ /G Γ v for v ∈ v ert A Γ (resp. the F Γ · e ∼ = F Γ /G Γ e for e ∈ edge A Γ). The noniden tit y morphisms a r e defined b y means of the inclusions G Γ e ⊆ G Γ target of ι e and G Γ e ⊆ G Γ target of τ e . This define s a graph on which the group F Γ acts (on the left) in the ob vious wa y , and all the assertions of the lemma are immediate.  R emark 5.4 . A subgroup of F Γ stabilize s a v ertex or an edge of X if and only if it is conjugate in F Γ to a subgroup of a v ertex group or an edge group of G Γ. L emma 5.5 . If H ⊆ F Γ is a fini te sub gr oup then it stabil i z es a vertex of X . Pr o of. A t the b eginning of this section w e ha ve presen ted F Γ as the colimit of the following tr ee of g roups: ❝ F 1 Γ F 0 Γ ❝ F 2 Γ 9 Lemma 3.5 implies that H is conjugat e in F Γ to a subgroup of F 1 or F 2 , whic h again ar e colimits of t rees of groups. Remark 5.4 completes the pro of.  L emma 5.6 . L et X b e the u-gr ap h as in L emm a 5.3. I f N is a sub gr oup of F Γ which stabil i z e s two vertic es P and Q i n X then N stabilizes some p ath c onne cting these vertic es. Pr o of. Let ˜ X b e the tree as in Theorem 3 .3 f or the graph of groups G b elo w: ❝ F 1 Γ F 0 Γ ❝ F 2 Γ Then (cf. pro of of Lemma 5.3) vert ˜ X is the disjoin t union of the F Γ · v ∼ = F Γ /F i Γ for i = 1 , 2, and edge ˜ X = F Γ · e ∼ = F Γ /F 0 Γ. W e ha v e an F Γ-equiv arian t “map” of u-graphs f : X → ˜ X induced by the inclusions G Γ v ⊆ F 1 Γ or G Γ v ⊆ F 2 Γ for v ∈ vert X and G Γ e ⊆ F 0 Γ for e in edge X and of ty p e N 1 or N 3 . W e write “map” in quotation marks since it takes edges of type other than N 1 or N 3 to v ertices – it is a map of diagrams but not of u-graphs. If e ∈ edge ˜ X then f − 1 ( e ) is a set of disjoint edges in X . If v ∈ vert ˜ X then f − 1 ( v ) is a tree isomorphic to the underlying tree o f either G 1 Γ or G 2 Γ. No w N stabilizes f ( P ) and f ( Q ), and since ˜ X is a tree, it stabilizes the shortest path L in ˜ X , connecting f ( P ) to f ( Q ). If e ∈ edge L t hen the stabilizer of e is g F 0 Γ for some g ∈ F Γ, hence N ⊆ g F 0 Γ = ∗ a ∈ edge Γ ( g N 1 ,a ∗ g N 3 ,a ). Since the v ertex groups of G Γ a re finite, Remark 5.4 implies that N is finite, henc e N ⊆ g N i,a for i = 1 or i = 3 and some a ∈ edge Γ. This means that N stabilizes some edge in f − 1 ( e ) ⊆ X . If v ∈ vert L then the stabilizer of v is g F 1 Γ or g F 2 Γ for some g ∈ F Γ, hence N ⊆ g F i Γ for i = 1 or i = 2 and N stabilizes the tree f − 1 ( v ) ⊆ X . W e kno w that N stabilizes t w o v ertices in f − 1 ( v ): if v is a n inner v ertex of L these are ends of the edges in X , mapp ed b y f to t he edges adjacen t to v in L , and stabilized b y N as seen ab ov e; if v = f ( P ) or v = f ( Q ) is a n end of L then o ne o r b oth of these t w o v ertices is P or Q resp ectiv ely . Since f − 1 ( v ) is a t r ee w e see that N stabilizes the shortest path connecting these t wo v ertices. By concatenating the paths and edges described a b ov e, w e obta in the required path that connects P and Q , and is stabilized b y N .  L emma 5 .7 . L et A ⊆ B b e an e dge-to-vertex inclusion lab el le d c in (4.1) . L et X b e the u-gr aph as in L emma 5.3 and 10 ❝ P c A ′ ❝ B ′ b e an e dge in G Γ ⊆ X wher e A ′ and B ′ ar e of typ e A and B r esp e ctivel y. The standar d isom orphism f : A → A ′ extends uniquely to f : B → F Γ , an d this extensi o n is the standar d iso morphism onto B ′ . Pr o of. Only the uniquenes s needs to b e prov ed. Lemma 5.5 implies that f ( B ) stabilizes a v ertex V of X . Condition C7 excludes the case V = P . Lemma 5.6 implies that A ′ stabilizes some path connecting V to P . If V 6 = B ′ then A ′ stabilizes t wo differen t edges adja cen t to P or to B ′ . This is excluded b y Condition C4 as the stabilizers of edges in X adjacen t to a v ertex W in G Γ are the W -conjug ates of edges in G Γ a djacen t to W . W e are left with V = B ′ , that is, f ( B ) ⊆ B ′ , and Condition C3 completes the pr o of.  L emma 5 .8 . L et Γ and ∆ b e m-gr aphs. If h : F Γ → F ∆ is a homo- morphism whi c h r estricts to the identity on M = F ∅ then ther e exists a uniq ue f : Γ → ∆ such that h = F f . Pr o of. Lemma 5.7, applied to N ⊆ P 0 in (4.1), implies that for any v ertex v in Γ there exists a v ertex w in ∆ suc h that h takes P 0 ,v in G Γ to P 0 ,w in G ∆ via a standa r d isomorphism. This allo ws us to define f ( v ) = w . Lemma 5.7, a pplied to the remaining inclusions, lab elled c in (4.1), implies that for any edge e = ( v 1 , v 2 ) in Γ there exist edges e ′ = ( f ( v 1 ) , w 2 ) and e ′′ = ( w 1 , f ( v 2 )) in ∆ suc h that h tak es, via standard isomorphisms, the “half edge subgraphs” of G Γ to the “half edge subgraphs” of G ∆ as indicated b elow: ❝ P 0 ,v 1 N 0 ,e ❝ P 1 ,e N 1 ,e ❝ P 2 ,e to ❝ P 0 ,f ( v 1 ) N 0 ,e ′ ❝ P 1 ,e ′ N 1 ,e ′ ❝ P 2 ,e ′ and ❝ P 3 ,e N 3 ,e ❝ P 4 ,e N 4 ,e ❝ P 0 ,v 2 to ❝ P 3 ,e ′′ N 3 ,e ′′ ❝ P 4 ,e ′′ N 4 ,e ′′ ❝ P 0 ,f ( v 2 ) If e ′ 6 = e ′′ then P 2 ,e ∩ P 3 ,e = N 2 ,e in G Γ go es to P 2 ,e ′ ∩ P 3 ,e ′′ whic h is trivial, and w e hav e a contradiction. Th us e ′ = e ′′ and f prese rv es the edges.  L emma 5.9 . If Γ 0 is a sub-m-gr a ph of Γ then F Γ 0 is a sub gr oup of F Γ . Pr o of. It is clear that F i Γ 0 is a f ree factor of F i Γ for i = 0 and i = 2. It is also clear that G 1 Γ 0 is a subtree of groups of G 1 Γ; hence, inductiv ely applying Lemma 3.1 w e see that F 1 Γ 0 is a subgroup of F 1 Γ. W e complete the pro of b y applying Lemma 3.1 to t he inclusions F i Γ 0 ⊆ F i Γ for i = 1 , 2.  11 L emma 5.10 . L et Γ b e an m -gr aph. F or any g ∈ F Γ ther e e x ists a finite sub gr aph Γ 0 ⊆ Γ such that g ∈ F Γ 0 . Pr o of. This is clear since F Γ is generated b y the v ertex groups of G Γ and eac h of those comes fro m a single v ertex or edge in Γ .  L emma 5.11 . L et Γ b e an m -gr aph. F or any nontrivial homomorphism f : M → F Γ ther e exists an inner automorphism c g of F Γ such that the c om p osition c g f is the identity on M . Pr o of. Lemma 5.5 a nd Remark 5.4 imply that f ( M ) is conjug a te in F Γ to a subgroup of a v ertex group V in G Γ. Condition C2 and the construction of G Γ imply that V = M , th us c g f ( M ) ⊆ M fo r some g in F Γ . Condition C1 completes the pro of.  L emma 5.12 . If Γ is an m-gr aph, A is a gr oup and f : F Γ → A is a homomorphism w hich is trivial on M then f is trivial. Pr o of. The result follo ws from Condition C8 since F Γ is generated b y the v ertex groups connected to M b y pa t hs whose edges are la b elled c as in (4.1).  If A and B are groups then we define Rep( A, B ) = Hom ( A, B ) /B , that is, w e iden tify t w o homomorphisms f , h : A → B if there exists an inner automorphism c g of B suc h that f = c g h . The set Rep ( A, B ) con ta ins a trivial elemen t corresp onding t o the trivial homomorphism. The or em 5.13 . F or a l l m-gr aphs Γ , ∆ the c omp os i tion Hom m - G r aphs (Γ , ∆) ∪ {∗} → Hom G rou ps ( F Γ , F ∆) → Rep( F Γ , F ∆) , wher e ∗ is sent to the trivial homomorphism, is bije ctive . The isomor- phism is functorial in Γ and ∆ . Pr o of. This is immediate from Lemmas 5.12, 5.11 and 5.8 .  Let Hom( A, B ) denote the set of non trivial homomor phisms from A to B . R emark 5.14 . Hom( F Γ , F ∆) is functorial in Γ and ∆ since Hom( F Γ , F ∆) is and Lemmas 5 .11 and 5.12 imply that if f : F Γ → F ∆ and h : F ∆ → F Φ are non t r ivial homomorphisms then hf is also non tr ivial. R emark 5.15 . No t e that Hom( ∅ , ∆) = Hom G rap h s ( ∅ , ∆) is a p oint. Lem- mas 5.11 and 5.8 imply that f o r ev ery f : Hom( ∅ , ∆) → Hom( F ∅ , F ∆) w e hav e a pullback diagram: Hom(Γ , ∆) / /   Hom( F Γ , F ∆)   Hom( ∅ , ∆) f / / Hom( F ∅ , F ∆) 12 That is, Hom( F Γ , F ∆) ∼ = Hom( F ∅ , F ∆) × Hom(Γ , ∆) . The follo wing theorem puts t o gether Remarks 5.14 and 5.15. The or em 5.16 . F or m -gr aphs Γ and ∆ we ha v e a bije ction Hom( F Γ , F ∆) ∼ = Hom( F ∅ , F ∆) × Hom(Γ , ∆) ∪ {∗} , which i s functorial in Γ and ∆ . The ∗ c o rr esp onds to the trivial homo- morphism. A nontrivial homom orphism h : F Γ → F ∆ c orr esp onds to a p air h | F ∅ and f : Γ → ∆ such that F f = h . 6. Colimits and limits In this section w e pro v e that the functor F preserv es directed colimits and coun tably co directed limits. W e sa y that a poset X is dir e cte d (r esp. c ountably dir e cte d) if an y finite subse t (resp. an y countable subset) of X has an upp er b ound in X . A p oset is view ed as a category where a ≤ b corresp onds to a morphism a → b . A diagram (i.e. functor) Γ : X → C and its colimit colim Γ are called dir e cte d if X is directed. A diagram Γ a nd its limit lim Γ are called c ountably c o dir e cte d if the opp osite category X op is coun t ably directed. The results of this section ar e stated and prov ed for (coun tably) directed diagrams, but [1, Theorem 1.5 ] and [1, R emark 1.21] yield immediate generalizations to the ( countably) filtered case. In this article w e use Remark 6.1 only; the remainder o f this section is pro vided for the sake of completeness. Colimits W e ha v e noticed in Remark 4 .4 that F : m - G raphs → G r oups pre- serv es colimits of connected diagrams. Since the inclusion functor I : G r aphs → m - G r aphs preserv es directed colimits w e obtain R emark 6.1 . The composition F I : G r aphs → G r oups preserv es di- rected colimits. Limits The inclusion functor I preserv es all limits. W e in v estigate preser- v ation of limits by F . L emma 6.2 . If Γ 1 and Γ 2 ar e sub gr ap h s of an m-gr aph Γ then F (Γ 1 ∩ Γ 2 ) = F Γ 1 ∩ F Γ 2 . Pr o of. Lemma 5.9 implies that the statement of t he lemma mak es sense. Since Γ 1 ∪ Γ 2 = colim(Γ 1 ⊇ Γ 1 ∩ Γ 2 ⊆ Γ 2 ) Remark 4.4 im- plies that F (Γ 1 ∪ Γ 2 ) = F Γ 1 ∗ F (Γ 1 ∩ Γ 2 ) F Γ 2 hence the result follow s from Remark 3.4(d).  13 L emma 6.3 . If { Γ α } α ∈ A is a c ountably c o dir e cte d diagr am of finite m- gr aphs then ther e exist α 0 and β in A such that (a) the pr oje ction p 0 : lim Γ α → Γ α 0 is inje ctive, (b) the images of p 0 and p β α 0 : Γ β → Γ α 0 c oincide. Pr o of. If S is a set of ob jects in Γ = lim Γ α then for any pair s 6 = t in S there exists α s,t in A suc h that the pro jection p s,t : Γ → Γ α s,t is injectiv e on { s, t } . If S is at most coun ta ble then there exists α 0 suc h that eac h p s,t factors through p 0 : Γ → Γ α 0 , hence p 0 is injectiv e on S . But Γ α 0 is finite, hence Γ is finite, and b y taking S to b e the set of ob jects o f Γ w e complete the pro of of (a). If B = { β ∈ A | β → α 0 } then lim α ∈ A Γ α → lim β ∈ B Γ β is an isomor- phism. Clearly im p 0 ⊆ im p β α 0 for β ∈ B . Let K β = ( p β α 0 ) − 1 (im p β α 0 \ im p 0 ) b e view ed as a set of ob jects. If eac h K β is nonempt y then, as a co directed limit of finite sets, lim K β is nonempt y , whic h is a contra- diction since lim K β ⊆ lim Γ β and p 0 (lim K β ) ∩ p 0 (lim Γ β ) = ∅ .  L emma 6.4 . I f { Γ α } α ∈ A is a c o untably c o dir e cte d diag r a m of m-gr aphs and ∆ α ⊆ Γ α ar e finite sub gr aphs such that for al l structur e map s p β α : Γ β → Γ α we have ∆ α ⊆ p β α (∆ β ) then ther e exist fi nite sub gr aphs ∆ α ⊆ Γ α such that ∆ α ⊆ ∆ α for al l α and { ∆ α } α ∈ A is a diagr am, that is, p β α ( ∆ β ) ⊆ ∆ α . Pr o of. Define ∆ α as the union of p β α (∆ β ) ov er all structure maps p β α whose targ et is Γ α . Only the finiteness of ∆ α needs pro of. Supp ose that S = { s 0 , s 1 , . . . } is an infinite subset o f ob j ects in ∆ α . Then there exist α 0 , α 1 , . . . suc h that s i ∈ p α i α (∆ α i ) fo r i ∈ N . Since { Γ α } α ∈ A is coun ta bly co directed there exists α ∗ in A suc h that Γ α ∗ maps to ev ery Γ α i for i ∈ N , hence ∆ α i ⊆ p α ∗ α i (∆ α ∗ ) implies p α i α (∆ α i ) ⊆ p α ∗ α (∆ α ∗ ) for i ∈ N , whic h is a contradiction since ∆ α ∗ is finite.  Pr op osition 6 .5 . The functor F c onstructe d in Se ction 4 pr eserves c ount- ably c o dir e cte d li m its. Pr o of. Let { Γ α } α ∈ A b e a countably co directed diagr am of m-gra phs. W e obtain an extended dia g ram (6.6) { F Γ α } α ∈ A F lim Γ α o o h x x p p p p p p p p p p p lim F Γ α O O where h comes from the univ ersal prop ert y of the limit. W e need to pro v e that h is a bijection. Inje ctivity of h . Let g b e a no nidentit y elemen t of F lim Γ α . Lemma 5.10 implies the existence of a finite subgraph Γ 0 ⊆ lim Γ α suc h that 14 g ∈ F Γ 0 . W e lo ok at the diagram formed b y the images of Γ 0 in Γ α for α ∈ A , and b y Lemma 6.3(a) w e obta in α 0 suc h that Γ 0 maps injectiv ely to Γ α 0 ; hence Lemma 5.9 implies that F Γ 0 → F Γ α 0 is one-to-one and therefore h ( g ) is non trivial, whic h pro v es the injectivit y of h . Surje ctivity of h . Let g ∈ lim F Γ α and let g α b e the image of g in F Γ α . Let Γ g α ⊆ Γ α b e a finite subgraph suc h that g α ∈ F Γ g α for α ∈ A . Lemma 6.2 implies that w e ma y require Γ g α to b e the smallest subgraph with g α ∈ F Γ g α . The minimality implies that Γ g α ⊆ p β α (Γ g β ) for all structure maps p β α , hence by Lemma 6.4 w e obtain a diagram { Γ g α } α ∈ A of finite subgraphs suc h tha t Γ g α ⊆ Γ g α ⊆ Γ α . Lemma 6.3(a) giv es us α 0 suc h that p 0 : lim Γ g α → Γ g α 0 ⊆ Γ α 0 is injectiv e. Let Γ 0 b e the image of p 0 . W e put the ab o ve into the follow ing diagram, whic h is a mo dification of (6.6). (6.7) g α 0 ∈ F Γ 0 ⊆ F lim Γ g α h 0 v v n n n n n n n n n n n n n n n n n n n n n n n n n n n F p 0 ∼ = o o F Γ α 0 F lim Γ α g _ O O ∈ lim F Γ g α q 0 O O lim F Γ α ⊆ ⊆ o o h u u l l l l l l l l l l l l l l l l O O One easily deduce s from Lemma 6.3(b) that the image of lim F Γ g α in F Γ α 0 is con tained in F Γ 0 , hence q 0 is w ell defined. F p 0 is an isomor- phism since p 0 is an isomorphism, and therefore q 0 is on to. T o complete the pro of it is enough to sho w that q 0 is one-to- one. Supp ose that k er q 0 con ta ins a no niden tity elemen t k . Then w e hav e a structure map Γ α 1 → Γ α 0 suc h that k is not in the k ernel of lim F Γ g α → F Γ α 1 . As a b o ve , p 1 : lim Γ g α → Γ g α 1 is injectiv e a nd if Γ 1 = im p 1 then the image of lim F Γ g α in F Γ α 1 is con tained in F Γ 1 . W e obtain a mo dification of ( 6 .7): (6.8) F Γ 0 ⊆ F lim Γ g α ∼ = F p 0 o o ∼ = F p 1 = q 1 h 0 t t j j j j j j j j j j j j j j j j j j j j F Γ 1 O O ⊆ F Γ α 0 F Γ α 1 O O lim F Γ g α q 1 O O q 0 E E 15 and k ∈ k er q 0 \ k er q 1 , whic h is a con tra diction, since p 1 : lim Γ g α → Γ 1 is an isomorphism.  R emark 6 .9 . The functor F do es not preserv e codirected limits: Let Γ n = N for p ositiv e integers n . F or n < m define p m n : Γ m → Γ n as p m n ( k ) = max { 0 , k − ( m − n ) } . Then it is easy to see that lim Γ n is coun ta ble while lim F Γ n is uncoun ta ble. 7. Ap pr o xima tions of gro ups by gr aphs Pr op osition 7.1 . L et G b e a gr oup and M = F ∅ b e as in Se ction 4. F or every in c lusion i : M → G ther e exists an m-gr aph C i and a diagr am F ∅ ⊆ / / F C i a   M i / / G such that for every m-gr a p h Γ and f as b el o w F ∅ ⊆ / / F C i a   M i / / G F ∅ ⊆ / / F Γ F f _ _     ) 2 < f O O ther e ex i sts a unique f : Γ → C i for wh ich the diagr am ab ov e c om- mutes. Pr o of. The construction of C i is taut o logical: Let N ⊆ P 0 b e the inclusion as in (4.1). The v ertices of C i are homomorphisms v : P 0 → G suc h that v | N = i | N . The edges v → w of C i are those maps, of the graph of gro ups pictured in (4.3 ) to G , whose restrictions to P 0 ,v and to P 0 ,w are v and w resp ectiv ely . The existence and uniqueness of f is immediate.  8. Or thogonal subca te gor y pr oblem in the ca tegor y of gro ups In this section w e apply Theorem 5.16 to pro v e (Prop osition 8.7) t ha t if there exists an ortho g onal pair in the category of graphs whic h is not asso ciated with a lo calization then there exists an orthogonal pair in the category of gro ups whic h is not a ssociated with a lo calization. The premise of the implication ab ov e is consisten t with the standa r d set theory ZFC , in fact it is equiv alen t to the negation of we ak V o p ˇ enk a’s 16 principle. W e conclude this section with Prop osition 8.8. The con- v erses of Prop ositions 8.7 and 8.8 follow from [1, Theorem 6 .2 2] and [1, Corollary 6.24(iii) ]. In order to make the pap er self-con t a ined w e b egin with a collection of definitions and preliminary facts, most of them extracted from [4]. Ortho gonal p a irs Let C b e a categor y (here G r oups or G r aphs ). A morphism f : A → B is ortho gonal to an ob ject C (w e write f ⊥ C ) if f induce s a bijection (8.1) Hom C ( B , C ) → Hom C ( A, C ) . If M is a class of morphisms and O is a class of ob j ects in C then M ⊥ = { C ∈ C | f ⊥ C for ev ery f ∈ M } a nd O ⊥ = { f : A → B | f ⊥ C for ev ery C ∈ O } . An ortho gon al p air ( S , D ) consists of a class S of morphisms a nd a class D of o b jects suc h that S ⊥ = D and D ⊥ = S . If ( S , D ) is an orthogonal pair then D is called an ortho gonality class , D is closed under limits and S is closed under colimits. If M is a class of morphisms and O is a class of ob jects then ( M ⊥⊥ , M ⊥ ) and ( O ⊥ , O ⊥⊥ ) are orthogonal pairs. L o c alizations A lo c alization is a functor L : C → C tog ether with a natural trans- formation η : I d → L suc h that η LX : LX → LLX is an isomorphis m for ev ery X a nd η LX = Lη X for all X . Ev ery lo calization f unctor L giv es rise to an orthogonal pa ir ( S , D ) where S is the class of morphisms f suc h that Lf is an isomorphism and D is the class o f ob jects isomorphic to LX for some X . A class D is called r efle ctive if it is part of an orthogonal pair ( S , D ) whic h is asso ciated with a lo calizatio n. R emark 8.2 . Let C be a category and ( S , D ) an orthogonal pair in C . If for each ob ject X in C there exists a morphism η X : X → LX in S with LX in D then the assignmen t X 7→ LX define s a lo calization functor asso ciated with ( S , D ); this was o bserv ed in [3, 1 .2]. We ak V opˇ enka’s Prin c iple W eak V op ˇ enk a’s principle is a large cardinal axiom equiv alen t to the follo wing stat ements : (WV1) Ev ery orthogonal pair in G raphs is asso ciated with a lo caliza- tion. (WV2) Ev ery orthogonal pa ir in a lo cally presen table category ( G r oups is suc h a category) is asso ciated with a lo calization. The equiv alence to (WV1) is prov ed in [1, Theorem 6.22] and [1, Ex- ample 6.2 3]. The equiv alence t o (WV2) is prov ed in [1, Example 6.2 5] 17 and stated in Remark that precede s it. W eak V o pˇ enk a’s principle is b eliev ed to b e consisten t with t he standard set theory (ZFC ), but it is not prov able in ZFC: the nega t io n of w eak V o pˇ enk a’s principle is con- sisten t with ZFC. Prop osition 8.7 and (WV2) imply a new equiv alen t form ulation of w eak V opˇ enk a’s pr inciple: (WV3) Ev ery orthog o nal pair in G r ou ps is asso ciated with a lo caliza- tion. More details and an in teresting historical essay on V op ˇ enk a’s princi- ple and its w eak v ersion can b e f ound in [1]. Ortho gonal sub c a te gory pr o blem in the c ate gory of gr oups L emma 8 .3 . L et f : Γ → Φ b e a morphi s m and ∆ b e an ob j e ct in m - G r aphs . Then f ⊥ ∆ if and only if F f ⊥ F ∆ . Pr o of. Theorem 5.16 yields Hom( F Φ , F ∆) ∼ =   Hom( F ∅ , F ∆) × Hom(Φ , ∆) ∪ {∗}   Hom( F Γ , F ∆) ∼ = Hom( F ∅ , F ∆) × Hom(Γ , ∆) ∪ {∗} whic h implies the claim (see (8.1) for definition of ort ho g onalit y).  R emark 8.4 . Throughout the remainder of this section, f o r a give n orthogonal pa ir ( S , D ) in m - G r aphs we fix an or thogonal pa ir ( S , D ) in G r oups suc h that F S ⊆ S and F D ⊆ D . Suc h a pair ( S , D ) exists since b y Lemma 8.3 w e may tak e S = F D ⊥ and D = S ⊥ . L emma 8 .5 . L et G b e a gr oup in D whic h admits an emb e dding i : F ∅ → G . If C i is the m-gr aph descri b e d in Pr op osition 7. 1 then C i is in D . Pr o of. Let f : Γ → Φ b e in S and h : Γ → C i b e any map in m - G r aphs . Then the comp osition F ∅ ⊆ F Γ → F C i a − → G equals i , and so w e obtain F Γ F f   F h / / F C i a   F Φ t / / _ _ _ _ F s ; ; x x x x G The unique homomorphism t exists since F f ⊥ G . The lift F s exists b y Prop osition 7.1. Then aF sF f = tF f = aF h and the uniqueness in Propo sition 7 .1 implies F sF f = F h , hence by The orem 5.16 w e ha v e sf = h . If s, s ′ : Φ → C i are t wo maps suc h that sf = h = s ′ f then aF sF f = aF s ′ F f ; hence, as F f ⊥ G , w e ha ve aF s = aF s ′ . Uniqueness in Prop osition 7.1 yields F s = F s ′ , and hence b y Theorem 18 5.16 w e obtain s = s ′ . Th us f ⊥ C i for an y f in S and therefore C i is in D .  L emma 8.6 . If the ortho gonal p air ( S , D ) is asso ciate d with a lo c aliza- tion L then the p air ( S , D ) is also asso c iate d with a lo c aliz a tion. Pr o of. Remark 8.2 implies that it is enough to find for ev ery m- g raph Γ a map η Γ : Γ → ∆ in S suc h that ∆ is in D . W e lo ok at the dia gram F C i a   F ∅ ⊆ F Γ F f 9 9 r r r r r r r r r r r η F Γ / / F h * * T T T T T T T T T T T T T T T T T T T T T LF Γ # # F F F F F F Φ F or ev ery map h : Γ → Φ with Φ in D the g roup F Φ is in D , hence w e hav e a fa cto r izat io n of F h through η F Γ and therefore a factoriza- tion of h throug h f : Γ → C i . How ev er, the uniqueness of the map C i → Φ under Γ is problematic. W e remedy this thr o ugh an inductiv e construction. Let ∆ 0 = C i . If w e can c ho ose Φ in D and tw o differen t maps g 1 , g 2 : ∆ 0 → Φ suc h tha t g 1 f = g 2 f then w e define ∆ 1 to b e the limit of the diagra m ∆ 0 g 1 / / g 2 / / Φ W e view ∆ 1 as a subgraph of ∆ 0 , and corresp o ndingly w e obtain f 1 : Γ → ∆ 1 . W e rep eat this construction a lo ng some ordinal λ whose cofinalit y exceeds the cardinality of ∆ 0 ; for limit ordinals γ < λ we define ∆ γ to b e the limit, that is, the in tersection, of { ∆ α } α<γ . Since { ∆ α } is a strictly decreasing se quence of subgraphs of ∆ 0 it has to stabilize at some ∆ β , whic h implies that ev ery map Γ → Φ with Φ in D factors uniquely thro ugh f β : Γ → ∆ β , hence f β is in S . Also ∆ β is in D since C i is in D (b y Lemma 8.5) a nd D is closed under limits. Therefore η Γ = f β is the map w e w ere lo oking for.  Pr op osition 8.7 . Assumi n g the ne gation of we ak V opˇ enka’s principle, ther e exi s ts an ortho gon al p air in the c ate gory of gr oups which is not asso ciate d w i th a ny lo c alization. Pr o of. The negatio n of (WV1) implies the existence of an orthogonal pair ( S 0 , D ) in G r aphs whic h is not asso ciated with any lo calization. W e view S 0 and D as classe s of morphisms and ob jects in m - G r aphs . Let S = D ⊥ ; since S 0 ⊆ S and D = S ⊥ w e see that the orthogonal pair ( S , D ) is not asso ciated with an y lo calization in m - G raphs . Lemma 8.6 19 implies that no pair ( S , D ) as describ ed in Remark 8.4 is asso ciated with a lo calization in G r oups .  V opˇ enk a’s pri n ciple an d the existenc e of ge ner ators W e sa y tha t an orthogona l pair ( S , D ) is gener ate d b y a set of mor- phisms S 0 if D = S ⊥ 0 . If such a set S 0 exists then w e sa y that D is a smal l-ortho gonal i ty class . A class of graphs is rigid if it admits no morphisms except the identit y morphisms (i.e. the corresponding full sub category is discrete). A class is lar ge if it ha s no cardinality ( i.e. it is bigger than any cardinal num b er). V op ˇ enk a’s principle is another large cardinal axiom whic h influences the theory of lo calizations. Among man y equiv alent form ulations of this principle w e ha v e the following ones: (V1) There exists no la rge rigid class of graphs. (V2) Ev ery orthogonality class o f graphs is a small-orthogo na lit y class. (V3) Ev ery ortho g onalit y class of ob jects in any lo cally presen table category (among those is G r oups ) is a small-or t ho gonalit y class. Equiv alence b etw een these statemen ts follows from [1, Corollary 6.24] and [1, Example 6.12 ]. The next pro p osition is a nonconstructiv e but stronger, in terms of the large cardinal hierarc h y [14, page 472], ve rsion of [5, Theorem 6.3]. T ogether with (V3) it yields another c haracterization o f V op ˇ enk a’s principle: (V4) Ev ery orthogonality class o f groups is a small-orthogo na lit y class. Pr op osition 8 .8 . Assumin g the ne gation of V op ˇ enka’s pri n ciple ther e exists an ortho gonal p air ( S , D ) in the c ate gory of gr oups such that D is n ot a smal l-ortho go n ality class. Pr o of. Negation of (V2) implies the existence of a n orthogonal pair ( S , D ) in G r aphs such that D is no t a small-o rthogonalit y class. As in R emark 8.4, w e ha v e an o rthogonal pair ( S , D ) in G r oups suc h that F S ⊆ S and F D ⊆ D . Supp ose that D is a small-orthogonality class, that is, there exists a set S 0 ⊆ S suc h that D = S ⊥ 0 . Then there exists an uncoun table cardinal λ suc h that D is closed under λ -directed colimits; it is enough that the cofinality of λ is gr eater than all the cardinalities of domains and targets o f maps in S 0 . Since D = F − 1 ( D ) Remark 6.1 implies that D is closed under λ - directed colimits. As the orthogo na lit y class D is closed under arbitrary limits, b y [13, Corollary] it is a λ - orthogonality class and th us a small-orthogonality class [1, 1.3 5 a nd the follow ing]; this contradiction completes the pro of.  20 9. Homotopy ca tegor y W e translate the results of the preceding section to the homotopy category H o and to the pointed homotop y category H o ∗ . In this sec- tion w e obtain an ortho g onalit y preserving em b edding of G r aphs into H o and a c har acterization of V opˇ enk a’s principle in terms of the ho- motop y theory . Results of [5] w ere close to such a c haracterization. In this section sp ac e means simplicial set; whenev er a space X is a right argumen t of a Hom or of a mapping space functor we a ssume that X is fibran t. The functor B : G r oups → H o ∗ whic h sends a group G to the Eilen b erg–Mac Lane space K ( G, 1) is full and faithful. Since Hom H o ( X , Y ) = Hom H o ∗ ( X , Y ) /π 1 ( Y ) Theorem 5.13 implies that t he comp osition B F follo w ed by the forgetful functor H o ∗ → H o induces the bijections (9.1) B F X,Y : Hom m - G r aphs ( X , Y ) ∪ {∗} → Hom H o ( B F X, B F Y ) where ∗ is sen t to the constant map. W e sa y that a morphism f : A → B is ortho go n al to an o b ject X in H o if it induces an equiv alence of the mapping spaces map( B , X ) → map ( A, X ) This notion of orthogo na lit y is used, as in Section 8, to define or- thogonal pairs ( S , D ) whose righ t members D a re called orthog onal- it y classes. Analogously we define orthog onalit y in H o ∗ b y means o f the p ointed mapping spaces map ∗ ( C , X ). The fibrat io n ma p ∗ ( C , X ) → map( C , X ) → X for an y C shows that for X connected w e ha v e f ⊥ X in H o if and only if f ⊥ X in H o ∗ for an y c hoice of base p oin ts [9 , Chapter 1, A.1]. If X is an Eilen b erg–Mac Lane space then map( A, X ) is homotop y equiv alen t to a discrete space whose underlying set is Hom H o ( A, X ). Th us (9.1) yields the following. L emma 9 .2 . L et f : Γ → Φ b e a morphi s m and ∆ b e an ob j e ct in m - G r aphs . Then f ⊥ ∆ if and only if B F f ⊥ B F ∆ . The following strengthens t he r esult o f [5]. The or em 9.3 . The fol lowing c ondition s ar e e quivalent: (V2) Every ortho gonality class of gr ap hs is a sm al l-o rtho gonality class. ( ho V) Every ortho gonality class in the hom otopy c ate gory is a smal l- ortho gonality class. Pr o of. The implication (V2 ) = ⇒ ( ho V) is [5, Theorem 5.3]. Assuming the negation of (V2), Prop o sition 8 .8 yields an orthogonal pair ( S , D ) in the catego ry of groups suc h that D is not of the fo rm S ⊥ 0 for any set of morphisms S 0 . Let f : S 2 → ∗ b e a map f r o m a 2 - sphere 21 to a p oint. It is clear that a space X is orthogo nal to f if and only if all the connected comp onen ts of X are Eilen b erg–Mac Lane spaces. Th us f ∈ B D ⊥ and B D ⊥⊥ is the class consisting of those spaces all of whose connected comp onen ts are homotop y equiv alen t to a mem b er of B D . The remainder of the pro o f is similar to the pro o f of Prop osition 8 .8. If B D ⊥⊥ is a small orthogo nalit y class t hen it is closed under λ -directed homotop y colimits, for some ordinal λ of sufficien t ly large cofinality . But then B D is closed under λ - directed homotop y colimits, hence D is closed under λ -directed colimits, hence D is a small o r t hogonalit y class, whic h is a contradiction.  10. Large localiza tions of finite g r oups In this section w e obtain a third construction o f a class of localiza- tions whic h send a finite simple gro up to groups of arbitrarily large cardinalities. Previous examples of suc h lo calizations are describ ed in [10], [11] and [18]. Let M b e a group that is part of a g r aph of groups satisfying con- ditions C1– C8 stated b efore Lemma 4.2; we may take M = M 23 , the Mathieu group. The or em 10.1 . F or a n y infinite c ar d i n al κ ther e exists a lo c alization L in the c ate gory of gr oups such that LM has c ar d i n ality κ . Pr o of. Let F b e the functor constructed in Section 4. W e ha v e M = F ∅ . W e know [22] that fo r ev ery infinite cardina l κ there exists a graph Γ of cardinalit y κ suc h that the iden tit y is the unique morphism Γ → Γ. Let i : ∅ → Γ b e t he inclusion of the empt y set. Clearly i is orthogona l to Γ. Let η = F i : F ∅ → F Γ. Lemma 8.3 implies that η ⊥ F Γ. By [2, Lemma 2.1] t here exis ts a lo calization L in the category of gr o ups suc h that LF ∅ = F Γ, whic h completes the pro of.  11. Closing remarks It is in triguing to ask the following. Question: Do es there exist a faithful functor F from the category of graphs t o the category of ab elian g roups suc h that f ⊥ Γ in the category of graphs if and only if F f ⊥ F Γ in the category of ab elian groups? 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