An “almost” full embedding of the category of graphs into the category of groups

Przeździecki, Adam J.

doi:10.1016/j.aim.2010.04.015

Cited by 6 publications

(9 citation statements)

References 32 publications

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“…The same line of argument provides an answer to Farjoun's question in [20] of whether all homotopy reflections are f -localizations for some map f . It was shown in [17] that the answer is affirmative under Vopěnka's principle, and Przeździecki proved in [42] that an affirmative answer is in fact equivalent to Vopěnka's principle. Here we prove an analogue of Theorem 8.4.…”

Section: Consequences In Homotopy Theorymentioning

confidence: 99%

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Definable orthogonality classes in accessible categories are small

Bagaria¹,

Casacuberta²,

Mathias³

et al. 2015

J. Eur. Math. Soc.

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects S ⊥ is a small-orthogonality class (hence reflective) can be proved in ZFC if S is Σ1, while it follows from the existence of a proper class of supercompact cardinals if S is Σ2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is Σn+2 for n ≥ 1. These cardinals form a new hierarchy, and we show that Vopěnka's principle is equivalent to the existence of C(n)-extendible cardinals for all n.As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This follows from the fact that the class of E * -equivalences is Σ2-definable, where E denotes a spectrum treated as a parameter. In contrast with this fact, the class of E * -equivalences is Σ1-definable, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.

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Section: Consequences In Homotopy Theorymentioning

confidence: 99%

“…However, the precise consistency strength of many implications of this axiom in category theory or homotopy theory is not known, and in some cases the question of whether such statements are provable in ZFC remains unanswered. A relevant step in this direction was made in [42].…”

Section: Introductionmentioning

confidence: 99%

Definable orthogonality classes in accessible categories are small

Bagaria¹,

Casacuberta²,

Mathias³

et al. 2015

J. Eur. Math. Soc.

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“…It is natural to ask if an isomorphism GX ∼ = GY implies X ∼ = Y . The affirmative answers to the analogous questions in the cases of full embeddings and almost full embeddings in the sense of [25] or [20] are clear. We don't know the answer for the functor G considered here.…”

Section: The Isomorphism Problemmentioning

confidence: 90%

“…Proof. Since the functor G preserves countably directed colimits the proof of [20,Proposition 8.8] applies.…”

Section: Orthogonal Subcategory Problem In the Category Of Abelian Grmentioning

confidence: 99%

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

Przeździecki

2014

Advances in Mathematics

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We construct an embedding G : Graphs → Ab of the category of graphs into the category of abelian groups such that for X and Y in Graphs we havethe free abelian group whose basis is the set Hom Graphs (X, Y ). The isomorphism is functorial in X and Y . The existence of such an embedding implies that, contrary to a common belief, the category of abelian groups is as complex and comprehensive as any other concrete category. We use this embedding to settle an old problem of Isbell whether every full subcategory of the category of abelian groups, which is closed under limits, is reflective. A positive answer turns out to be equivalent to weak Vopěnka's principle, a large cardinal axiom which is not provable but believed to be consistent with standard set theory. We obtain some consequences to the Hovey-Palmieri-Stricland problem about existence of arbitrary localizations in a stable homotopy category.Several known constructions in the category of abelian groups are obtained as quick applications of the embedding.MSC: 18B15 (20K20,20K30,18A40,03E55)

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“…For all commutative, cotorsion-free rings R we construct embeddings G of the category of graphs into categories of R-modules which are almost full in the sense that the maps induced by the functoriality of G R[Hom Graphs (X, Y )] −→ Hom R (GX, GY ) (1.1) some subcategory of Topological Spaces -it was clear that the "metadefinition" was: an embedding is almost full if it is full up to "obvious sacrifices" imposed by the target category. In this broader sense the term was used in [19] and [20]. In this paper the target category is R-Modules, possibly with some additional structure.…”

Section: Introductionmentioning

confidence: 99%

An axiomatic construction of an almost full embedding of the category of graphs into the category of -objects

Göbel

Przeździecki

2014

Journal of Pure and Applied Algebra

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We construct embeddings G of the category of graphs into categories of Rmodules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of Gare isomorphisms. The symbol R[S] above denotes the free R-module with the basis S. This implies that, for any cotorsion-free ring R, the categories of Rmodules are not less complicated than the category of graphs. A similar embedding of graphs into the category of vector spaces with four distinguished subspaces (over any field, e.g. F 2 = {0, 1}) is obtained.

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An “almost” full embedding of the category of graphs into the category of groups

Cited by 6 publications

References 32 publications

Definable orthogonality classes in accessible categories are small

Definable orthogonality classes in accessible categories are small

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

An axiomatic construction of an almost full embedding of the category of graphs into the category of -objects

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