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LeftDetermined Model Categories and Universal Homotopy Theories
"... We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense ..."
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We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural leftdetermined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. 1
The strength of Mac Lane set theory
 ANNALS OF PURE AND APPLIED LOGIC
, 2001
"... SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, ..."
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Cited by 18 (1 self)
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SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing,
The orthogonal subcategory problem in homotopy theory
 Proceedings of the Arolla conference on Algebraic Topology 2004
"... Abstract. It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a largecardinal axiom called Vopěnka’s principle. In this article we extend the validity of this result to any left proper, combinat ..."
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Abstract. It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a largecardinal axiom called Vopěnka’s principle. In this article we extend the validity of this result to any left proper, combinatorial, simplicial model category M and show that, under additional assumptions on M, every homotopy idempotent functor is in fact a localization with respect to some set of maps. These results are valid for the homotopy category of spectra, among other applications.
LOCALIZATIONS OF GROUPS
, 2000
"... A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings ..."
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Cited by 7 (2 self)
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A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introduction. For localizations η: A → H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and nonequalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (nonabelian) group. Libman showed that An → SOn−1(R) for a natural embedding of the alternating group An is a localization if n even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] that any nonabelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of A’s. The new techniques exploit abelian centralizers of free (nonabelian) subgroups of H which constitute a rigid system of cotorsionfree abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].
Localization theory for triangulated categories
 In Triangulated categories, volume 375 of London Math. Soc. Lecture Note Ser
, 2010
"... 2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 14 ..."
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2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 14
Localization for triangulated categories
"... 2. Categories of fractions and localization functors 4 3. Calculus of fractions 11 4. Localization for triangulated categories 16 ..."
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2. Categories of fractions and localization functors 4 3. Calculus of fractions 11 4. Localization for triangulated categories 16
Definable orthogonality classes in accessible categories are small
 Journal of the European Mathematical Society
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Localization with respect to a class of maps. II. Equivariant cellularization and its application
 Israel J. Math
"... Abstract. We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result. ..."
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Abstract. We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result.
CONSTRUCTING SIMPLE GROUPS FOR LOCALIZATIONS
, 2001
"... A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concept, obviously not depending on the notion of groups, extends classical localizations as known for rings a ..."
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Cited by 2 (0 self)
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A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concept, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introduction. For localizations η: A → H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and nonequalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (nonabelian) group. Libman 739 revision:20010220 modified:20010223 showed that An → SOn−1(R) for a natural embedding of the alternating group An is a localization if n is even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] that any nonabelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of A’s. The new techniques exploit abelian centralizers of free (nonabelian) subgroups of H which constitute a rigid system of cotorsionfree abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].