The Functor Construction of F Applications Summary An “almost” full embedding of the category of graphs into the category of groups Adam J. Przeździecki Warsaw University of Life Sciences CAT, 2009 Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary The Functor Choice of categories F : Graphs → Groups I full and faithful: Hom(X , Y ) ∼ = Hom(FX , FY ) I “almost” full: ∼ = HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY ) where Rep(FX , FY ) = Hom(FX , FY )/FY Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary The Functor Choice of categories F : Graphs → Groups I full and faithful: ( (( hh hhh( (( ∼ h Hom(X , Y()h Hom(FX ( =( hhh , FY ) ((( h h I “almost” full: ∼ = HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY ) where Rep(FX , FY ) = Hom(FX , FY )/FY Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary The Functor Choice of categories F : Graphs → Groups Choice of categories: target I Groups - is interesting in itself I Groups −→ Ho (unpointed homotopy category) yields, up to constant maps, a full embedding B BF : Graphs −→ Ho Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary The Functor Choice of categories Choice of categories: source Graphs is very comprehensive and well researched. Many “non-homotopy” categories are contained in Graphs as full subcategories: I category of groups I category of fields I category of R-modules I category of Hilbert spaces I category of partially ordered sets I category of simplicial sets I category of metrizable spaces and continuous maps I category of CW-complexes and continuous maps I category of models of some first order theory I many more Tool: Adámek, Rosický Locally presentable and accessible categories, Theorem 2.65. Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees Bass-Serre theory on groups acting on trees c QQ Q Q Qc M c I I I c G = M ∗N P N c Q P Q Q QQ c If A ⊆ G is finite then it stabilizes a vertex of the tree hence is conjugated to a subgroup of M or P. Take A = M finite, s.t. Hom(M, P) = ∗ and M → M is either trivial or an inner automorphism. Let NM (N) = N, N ⊆ M does not extend to P → M, N ⊆ P extends uniquely to P → P then M ⊆ G uniquely extends to G → G. Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees Bass-Serre theory on groups acting on trees c QQ Q Q Qc M c I I I c G = M ∗N P N c Q P Q Q QQ c If A ⊆ G is finite then it stabilizes a vertex of the tree hence is conjugated to a subgroup of M or P. Take A = M finite, s.t. Hom(M, P) = ∗ and M → M is either trivial or an inner automorphism. Let NM (N) = N, N ⊆ M does not extend to P → M, N ⊆ P extends uniquely to P → P then M ⊆ G uniquely extends to G → G. Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees Bass-Serre theory on groups acting on trees c QQ Q Q Qc M c I I I c G = M ∗N P N c Q P Q Q QQ c If A ⊆ G is finite then it stabilizes a vertex of the tree hence is conjugated to a subgroup of M or P. Take A = M finite, s.t. Hom(M, P) = ∗ and M → M is either trivial or an inner automorphism. Let NM (N) = N, N ⊆ M does not extend to P → M, N ⊆ P extends uniquely to P → P then M ⊆ G uniquely extends to G → G. Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees G = (M ∗N P) ∗N P c QQ Q Q c M Q c !! ! ! c! P a aa N aa c Q c c !! Q ! N QQ !! cP aa aa ac Q Rep(G, G) = {∗} ∪ Hom(2, 2) Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees Start with a graph Γ '$ V ∅ q ? e &% Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees '$ V c M q ? e &% Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees '$ P0 c ? e N c M &% Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees Obtain graph of groups GΓ, define F Γ = colim GΓ P4 N4 chh N3 hhh hhhh P3 hc P0 c c Z Z Z Z N Z N0 Z Z M N2 c (((( ( ( ( P2 Z c((( N 1 P1 Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Bass-Serre theory Construction of the functor F Reduction to trees Example A11 ⊕ A10 chh hh hhh A10 N ⊕ A(S3 ⊕ S7 ) A11 hhh c N ⊕ A11 c Z11 o Z5 Z N c M23 Z Z Z Z N ⊕ A(S4 ⊕ S6 ) Z Z A10 (((( c (( ( Z c((( A11 A11 ⊕ A10 Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Reduction to trees Bass-Serre theory Construction of the functor F Reduction to trees P4 N4 chh N3 hhh P hhh hh c 3 P0 c N N2 Z Z Z Z Z N0 Z c c ((( Z ((( P2 Z c(((( N 1 M P1 c F1 Γ c F0 Γ F2 Γ F Γ = F 1 Γ ∗ F0 Γ F 2 Γ Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem Two definitions of a localization L : C → C 1. L is a left adjoint of an inclusion D ⊆ C of some subcategory. 2. L is a functor with coaugmentation η : Id → L such that ηLX = LηX : LX → LLX is an isomorphism Localizations may be viewed as projections onto the class of local objects D along the class of L-equivalences E = {f | Lf is an equivalence} For every f : A → B in E, an L-equivalence and Z in D, an L-local object ∼ we have: = Hom(B, Z ) −→ Hom(A, Z ) ' map(B, Z ) −→ map(A, Z ) Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem Orthogonality classes If for f : A → B and Z we have ∼ = Hom(B, Z ) −→ Hom(A, Z ) ' map(B, Z ) −→ map(A, Z ) then we say that f is orthogonal to Z and write f ⊥ Z . A pair (E, D) is orthogonal if E = D⊥ and D = E ⊥ . A localization always yields an orthogonal pair. Whether every orthogonal pair yields a localization depends on set theory in Graphs: NO is consistent with ZFC weak Vopěnka’s principle is equivalent to YES Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem Orthogonality classes If for f : A → B and Z we have ∼ = Hom(B, Z ) −→ Hom(A, Z ) ' map(B, Z ) −→ map(A, Z ) then we say that f is orthogonal to Z and write f ⊥ Z . A pair (E, D) is orthogonal if E = D⊥ and D = E ⊥ . A localization always yields an orthogonal pair. Whether every orthogonal pair yields a localization depends on set theory in Graphs: NO is consistent with ZFC weak Vopěnka’s principle is equivalent to YES Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem More properties of F : Graphs → Groups ∼ = 1. HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY ) 2. f ⊥ Z if and only if Ff ⊥ FZ 3. F preserves directed colimits 4. F preserves intersections and countably co-directed limits 5. ∆ ⊆ Γ implies F ∆ ⊆ F Γ 6. for every g ∈ F Γ there exists a finite subgraph ∆ ⊆ Γ s.t. g ∈ F∆ 7. F does not preserve products. Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem Large localizations of finite groups Theorem There exist localizations L : Groups → Groups whose values LM on a finite group M have arbitrarily large cardinalities. Proof. Vopěnka (1965): there exist arbitrarily large graphs Γ s.t. Hom(Γ, Γ) = {id}. The inclusion i : ∅ ⊆ Γ is orthogonal to Γ. Fi ⊥ F Γ If L = LFi then LM = F Γ. Proved by Shelah and Göbel (2002) on 28 pages. Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem Theorem The following are equivalent: 1. Every orthogonal pair (E, D) in Groups is associated with a localization. 2. Every orthogonal pair (E, D) in Graphs is associated with a localization (weak Vopěnka’s principle). 2 =⇒ 1 was proved by Adámek and Rosický (1994) 1 =⇒ 2 follows from properties of F Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Localizations More properties of F Large localizations of finite groups Orthogonal subcategory problem Theorem The following are equivalent: 1. For every orthogonal pair (E, D) in Groups there exists a homomorphism f such that D = {f }⊥ . 2. For every orthogonal pair (E, D) in Graphs there exists a map f such that D = {f }⊥ (Vopěnka’s principle). 3. For every orthogonal pair (E, D) in Ho there exists a map f such that D = {f }⊥ . 2 =⇒ 3 was proved by Casacuberta, Scevenels, Smith (2005) 1 =⇒ 2 and 3 =⇒ 1 follow from properties of F Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Summary An almost full embedding F : Graphs → Groups ∼ = HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY ) is a “black box” tool translating some categorical constructions from many point-set categories to the category of groups (or to the homotopy category). I Question Is there an embedding F : Graphs → Ab − Groups such that f ⊥ Z if and only if Ff ⊥ FZ . Adam J. Przeździecki An “almost” full embedding ... The Functor Construction of F Applications Summary Summary thank you An almost full embedding F : Graphs → Groups ∼ = HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY ) is a “black box” tool translating some categorical constructions from many point-set categories to the category of groups (or to the homotopy category). I Question Is there an embedding F : Graphs → Ab − Groups such that f ⊥ Z if and only if Ff ⊥ FZ . Adam J. Przeździecki An “almost” full embedding ...