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How comprehensive is the category of abelian groups? Adam J. Przez´dziecki WarsawUniversityofLifeSciences-SGGW Large-Cardinal Methods in Homotopy, 2011 PreprintavailableonarXiv. Introduction ConstructionofthefunctorG Alg-universalcategories Applications TheEmbedding Questions When a category is comprehensive? When the category of graphs fully embeds into it. Why graphs? Every accessible category admits a full embedding into graphs (Ada´mek-Rosicky´, 1994) Assuming that the measurable cardinals are bounded above every concrete category fully embeds into graphs (Hedrl´ın-Kucˇera, 1969 unpublished) 2/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Introduction ConstructionofthefunctorG Alg-universalcategories Applications TheEmbedding Questions Some categories admit a full embedding of Graphs: (cid:73) Category of semigroups (Hedrl´ın-Lambek, 1969) (cid:73) Category of integral domains (Fried-Sichler, 1977) Some more categories admit an “almost” full embedding of Graphs: (cid:73) Category of metric spaces (uptoconstantmaps – Trnkova´, 1972) up to constant maps (cid:73) Category of paracompact spaces ( ) –Koubek,1974 (cid:73) Category of groups ( up to trivial homomorphisms ) andconjugationinthetargets (cid:73) The unpointed homotopy category (uptonull-homotopicmaps) (cid:73) Category of abelian groups (upto???) 3/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Introduction ConstructionofthefunctorG Alg-universalcategories Applications TheEmbedding Questions Some categories admit a full embedding of Graphs: (cid:73) Category of semigroups (Hedrl´ın-Lambek, 1969) (cid:73) Category of integral domains (Fried-Sichler, 1977) Some more categories admit an “almost” full embedding of Graphs: (cid:73) Category of metric spaces (uptoconstantmaps – Trnkova´, 1972) up to constant maps (cid:73) Category of paracompact spaces ( ) –Koubek,1974 (cid:73) Category of groups ( up to trivial homomorphisms ) andconjugationinthetargets (cid:73) The unpointed homotopy category (uptonull-homotopicmaps) (cid:73) Category of abelian groups (upto???) 3/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Introduction ConstructionofthefunctorG Alg-universalcategories Applications TheEmbedding Questions G : Graphs → Ab (cid:73) full embedding: ∼= Hom(X,Y) −→ Hom(GX,GY) (cid:73) “almost” full for abelian groups: Z[Hom (X,Y)] −∼=→ Hom (GX,GY) Graphs Ab where Z[S] is the free group with basis S. 4/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Introduction ConstructionofthefunctorG Alg-universalcategories Applications TheEmbedding Questions G : Graphs → Ab (cid:73) full embedding: Ho(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)m((cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)X(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104),Y(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104))(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)−∼=→(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)H(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)om(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(G(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)X(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40)(cid:104)(cid:40),GY) (cid:73) “almost” full for abelian groups: Z[Hom (X,Y)] −∼=→ Hom (GX,GY) Graphs Ab where Z[S] is the free group with basis S. 4/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Natural completion of abelian groups η : A → A(cid:98). A (cid:73) A(cid:98) ∼= limn∈NA/nA (cid:73) A(cid:98) ∼= (cid:81) A∧ p p (cid:73) kerη is the divisible subgroup of A. A The completion is an idempotent functor (localization, reflector), that is: (cid:73) η : A(cid:98) −∼=→ A(cid:98)(cid:98) (cid:98)A (cid:73) For all groups A, B the map η Hom(A(cid:98),B(cid:98)) −→A Hom(A,B(cid:98)) is an isomorphism. If A ⊆ C ⊆ A(cid:98) and C is pure in A(cid:98) then C(cid:98) = A(cid:98). In particular the inclusion A ⊆ C induces an isomorphism ∼ Hom(A,A(cid:98)) = Hom(C,A(cid:98)) Introduction NaturalcompletionandtheCornermethod ConstructionofthefunctorG ConstructionofthefunctorG Applications Proofof Z[Hom(X,Y)]∼=Hom(GX,GY) Questions Finiteapproximation Theorem (Corner, 1963) Let A be a ring of cardinality at most continuum, whose additive group is free. Then there exists a group A such that (a) A ⊆ A ⊆ A(cid:98) as left A-modules. (b) A ∼= Hom(A,A) (c) |A| = |A| The construction: Choose elements αa, βa (a ∈ A) of Z(cid:98) that are algebraically independent over Z. Define elements ea (a ∈ A) of A(cid:98) as e = α ·1+β ·a a a a and take A to be the pure subgroup of A(cid:98) generated by A and Ae (a ∈ A). a 6/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Introduction NaturalcompletionandtheCornermethod ConstructionofthefunctorG ConstructionofthefunctorG Applications Proofof Z[Hom(X,Y)]∼=Hom(GX,GY) Questions Finiteapproximation Theorem (Corner, 1963) Let A be a ring of cardinality at most continuum, whose additive group is free. Then there exists a group A such that (a) A ⊆ A ⊆ A(cid:98) as left A-modules. (b) A ∼= Hom(A,A) (c) |A| = |A| The construction: Choose elements αa, βa (a ∈ A) of Z(cid:98) that are algebraically independent over Z. Define elements ea (a ∈ A) of A(cid:98) as e = α ·1+β ·a a a a and take A to be the pure subgroup of A(cid:98) generated by A and Ae (a ∈ A). a 6/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups? Introduction NaturalcompletionandtheCornermethod ConstructionofthefunctorG ConstructionofthefunctorG Applications Proofof Z[Hom(X,Y)]∼=Hom(GX,GY) Questions Finiteapproximation Construction of the functor G Let Γ be a full subcategory of Graphs whose objects are representatives of countable graphs. Let A = Z[Γ] be the ring whose additive group is free with the basis consisting of the identity 1 and the maps ϕ : X → Y in Γ. If ψ : M → N is another map in Γ then (cid:73) the product ϕψ in A is the composition ϕψ if N = X (cid:73) ϕψ = 0 if N (cid:54)= X 7/31 AdamJ.Przez´dziecki Howcomprehensiveisthecategoryofabeliangroups?

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How comprehensive is the category of abelian groups? Adam J. Przezdziecki´ Warsaw University of Life Sciences - SGGW Large-Cardinal Methods in Homotopy, 2011
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