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,

We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined 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

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 large-cardinal 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.

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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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 non-equalities. 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 (non-abelian) group. Libman 739 revision:2001-02-20 modified:2001-02-23 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 non-abelian 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 (non-abelian) subgroups of H which constitute a rigid system of cotorsion-free 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].

The aim of this paper is to describe the concept of localization, as it usually comes up in topology, and give some examples of it. Many of the results we will describe are due to Bousfield.

Abstract. We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S ⊂ T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither

Abstract. Given a simplicial idempotent augmented endofunctor F on a simplicial combinatorial model category M, under the assumption of Vopěnka’s principle, we exhibit a set A of cofibrant objects in Msuch that F is equivalent to CWA, the cellularization with respect to A.

Homotopical localizations at a space ⋆