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Model Structures on Commutative Monoids in General Model Categories. ArXiv eprints
, 2014
"... Abstract. We provide conditions on a monoidal model category M so that the category of commutative monoids in M inherits a model structure from M in which a map is a weak equivalence or fibration if and only if it is so in M. We then investigate properties of cofibrations of commutative monoids, re ..."
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Abstract. We provide conditions on a monoidal model category M so that the category of commutative monoids in M inherits a model structure from M in which a map is a weak equivalence or fibration if and only if it is so in M. We then investigate properties of cofibrations of commutative monoids, rectification between E ∞ algebras and commutative monoids, the relationship between commutative monoids and monoidal Bousfield localization functors, when the category of commutative monoids can be made left proper, and functoriality of the passage from a commutative monoid R to the category of commutative Ralgebras. In the final section we provide numerous examples of model categories satisfying our hypotheses.
Central extensions and generalized plusconstructions
 Trans. Amer. Math. Soc
, 2001
"... Abstract We describe the effect of homological plusconstructions on the homotopy groups of EilenbergMacLane spaces in terms of universal central extensions. 1 Introduction Higher algebraic Ktheory was introduced by Quillen [27] by means of the plusconstruction (a precursor of which goes back to ..."
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Abstract We describe the effect of homological plusconstructions on the homotopy groups of EilenbergMacLane spaces in terms of universal central extensions. 1 Introduction Higher algebraic Ktheory was introduced by Quillen [27] by means of the plusconstruction (a precursor of which goes back to Varadarajan [32, p. 368]). When applied to a space X, it yields a map X − → X+ which quotients out the maximal perfect subgroup of pi1X without changing the homology of X. In the case where X = BGL(R) is the classifying space of the general linear group of a ring R, Kn(R): = pin(K0(R)×X+).
Monoidal bousfield localizations and algebras over operads
 Ph.D.)–Wesleyan University. Department ofMathematics, Wesleyan University
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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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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].
HOMOLOGICAL LOCALIZATIONS OF EILENBERGMAC LANE SPECTRA
, 2005
"... We discuss the Bousfield localization LEX for any spectrum E and any HRmodule X, where R is a ring with unit. Due to the splitting property of HRmodules, it is enough to study the localization of Eilenberg–MacLane spectra. Using general results about stable flocalizations, we give a method to c ..."
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We discuss the Bousfield localization LEX for any spectrum E and any HRmodule X, where R is a ring with unit. Due to the splitting property of HRmodules, it is enough to study the localization of Eilenberg–MacLane spectra. Using general results about stable flocalizations, we give a method to compute the localization of an Eilenberg–MacLane spectrum LEHG for any spectrum E and any abelian group G. We describe LEHG explicitly when G is one of the following: finitely generated abelian groups, padic integers, Prüfer groups, and subrings of the rationals. The results depend basically on the Eacyclicity patterns of the spectrum HQ and the spectrum HZ/p for each prime p.
www.elsevier.com/locate/topol On cellularization for simplicial presheaves and motivic homotopy
"... We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky. ..."
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We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky.
ON LOCALIZATIONS OF TORSION ABELIAN GROUPS
"... Abstract. As it is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by T ℵ0 whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and ..."
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Abstract. As it is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by T ℵ0 whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, the relationship between localizations of abelian pgroups and their basic subgroups is completely characterized. 1.
ON LOCALIZATIONS OF TORSION ABELIAN GROUPS
"... As it is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by T ℵ0 whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investiga ..."
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As it is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by T ℵ0 whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, the relationship between localizations of abelian pgroups and their basic subgroups is completely characterized.