On Colimits and Elementary Embeddings

Bagaria, Joan; Brooke-Taylor, Andrew D.

doi:10.2178/jsl.7802120

Cited by 10 publications

(44 citation statements)

References 13 publications

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“…Vopěnka's principle implies that every full embedding between accessible categories is accessible. The same conclusion can be inferred from the existence of sufficiently large C(n)-extendible cardinals [8].…”

Section: Accessible Categoriessupporting

confidence: 62%

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: Accessible Categoriessupporting

confidence: 62%

Definable orthogonality classes in accessible categories are small

Bagaria¹,

Casacuberta²,

Mathias³

et al. 2015

J. Eur. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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“…For the purpose of this discussion, say a cardinal κ is λ-L µ,ω -compact if any κ-complete filter generated by at most λ many sets extends to a µ-complete ultrafilter. In recent work [3,5,6], L µ,ω -compactness has been referred to as "µstrong compactness", mostly with µ = ℵ 1 . This fits with a tradition in which λ-L κ,ω -compactness of κ was called λ-compactness (indeed, this is the terminology of the standard text [13]), with "strongly" potentially thought of as indicating "for all λ".…”

Section: Preliminariesmentioning

confidence: 99%

“…Let α = |I|, and let j : V → M be an elementary embedding as in Theorem 2.5 for our κ, α and µ = µ L . Note that because M is closed under < µ L -tuples, M correctly computes whether an object is in Red Σ 1 ,Σ (T ): being a model for the theory T is ∆ 1 -definable from the "set of all < µ-tuples" function P µ , and hence is absolute between models of set theory that agree on P µ (see [3,Proposition 16], [4,Proposition 3.3]). Consider the diagram j(D) in M. It is a j(κ)-directed diagram with index category j(I) of cardinality j(α).…”

Section: Powerful Imagesmentioning

confidence: 99%

“…We present two different ways of employing our large cardinal assumption to obtain our result. One follows [15] directly and the other uses elementary embeddings of models of set theory (see [3]). The first step in each case is to reduce to the case of a suitable reduct functor Red Σ 1 ,Σ (T ) → Str(Σ) where T is a theory in infinitary logic with signature Σ 1 ⊇ Σ.…”

Section: Introductionmentioning

confidence: 99%

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Accessible images revisited

Brooke-Taylor

Rosický

2016

Proc. Amer. Math. Soc.

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Abstract. We extend and improve the result of Makkai and Paré [15] that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of L µ,ω -compact cardinals for sufficiently large µ, and also show that under this assumption the λ-pure powerful image of F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result -one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the settheoretic universe.

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“…Also, several other implications of the principle were proved long ago, especially in category theory (see [1]). Recently there has been a revived interest in VP through new set-theoretic proofs of category-theoretic results (see Bagaria and Brooke-Taylor [2]). Furthermore, Brooke-Taylor [5] showed the relative consistency of VP with almost all usual ZFC-independent statements, like GCH and diamond principles (see the Introduction of [5]).…”

mentioning

confidence: 99%

Consequences of Vopěnka’s Principle over weak set theories

Tzouvaras

2016

Fund. Math.

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It is shown that Vopěnka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, ∆0-Separation and Induction along ω, then EST + VP proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous ones (2014), as well as by H. Friedman's (2015), where a distinction is made among various forms of VP. As a corollary, EST + Foundation + VP = ZF + VP and EST + Foundation + AC + VP = ZFC + VP. Also, it is shown that the Foundation axiom is independent of ZF − {Foundation} + VP. It is open whether the Axiom of Choice is independent of ZF + VP. A very weak form of choice follows from VP, and some other similar forms of choice are introduced.

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On Colimits and Elementary Embeddings

Cited by 10 publications

References 13 publications

Definable orthogonality classes in accessible categories are small

Definable orthogonality classes in accessible categories are small

Accessible images revisited

Consequences of Vopěnka’s Principle over weak set theories

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