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We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if $T$ is the reduced monad associated with a simplicial operad and $f$ is any map of pointed simplicial sets, then $f$-localization coincides with $Tf$-localization on spaces underlying homotopy $T$-algebras, and similarly for cellularizations., 39 pages

A model category is called combinatorial if it is cofibrantly generated and its underlying category is locally presentable. As shown in recent years, homotopy categories of combinatorial model categories share useful properties, such as being well generated and satisfying a very general form of Ohkawa’s theorem.

This is an expository article about Ohkawa’s theorem stating that acyclic classes of representable homology theories form a set. We provide background in stable homotopy theory and an overview of subsequent advances in the study of Bousfield lattices. As a new result, we prove that there is a proper class of acyclic classes of nonrepresentable homology theories.

We prove that every homotopical localization of the circle $S^{1}$ is an aspherical space whose fundamental group $A$ is abelian and admits a ring structure with unit such that the evaluation map End $(A) \rightarrow A$ at the unit is an isomorphism of rings. Since it is known that there is a proper class of nonisomorphic rings with this property, and we show that all occur in this way, it follows that there is a proper class of distinct homotopical localizations of spaces (in spite of the fact that homological localizations form a set). This answers a question asked by Farjoun in the nineties. More generally, we study localizations $L_{f} K(G, n)$ of Eilenberg-Mac Lane spaces with respect to any map $f$, where $n \geq 1$ and $G$ is any abelian group, and we show that many properties of $G$ are transferred to the homotopy groups of $L_{f} K(G, n)$. Among other results, we show that, if $X$ is a product of abelian Eilenberg-Mac Lane spaces and $f$ is any map, then the homotopy groups $\pi_{m}\left(L_{f} X\right)$ are modules over the ring $\pi_{1}\left(L_{f} S^{1}\right)$ in a canonical way. This explains and generalizes earlier observations made by other authors in the case of homological localizations.

We prove that, under suitable assumptions on a category C, the existence of supercompact cardinals implies that every absolute epireflective class of objects of C is a small-orthogonality class. More precisely, if L is a localization functor on an accessible category C such that the unit morphism X \to LX is an extremal epimorphism for all X, and the class of L-local objects is defined by an absolute formula with parameters, then the existence of a supercompact cardinal above the cardinalities of the parameters implies that L is a localization with respect to some set of morphisms., Comment: 15 pages

We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S of T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets., 5 pages

The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an f-localization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable large-cardinal axiom from set theory (Vopěnka's principle). We also show that it is impossible to prove that all homotopy idempotent functors are f-localizations using the ordinary ZFC axioms of set theory (Zermelo–Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.

We extend arbitrary group completions to the category of pairs ( G , N ) where G is a group and N is a normal subgroup of G. Relative localizations are special cases. Our construction is a group-theoretical analogue of fibrewise completion and fibrewise localization in homotopy theory, and generalizes earlier work of Hilton and others on relative localization at primes. We use our approach to find conditions under which factoring out group radicals preserves exactness. This has implications in the study of the effect of plus-constructions on homotopy fibre sequences.

We prove that stable f f -localizations (where f f is any map of spectra) preserve ring spectrum structures and module spectrum structures, under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg–Mac Lane spectrum H Z H{\mathbb {Z}} . As a consequence of this study, we infer that localizations of stable GEMs are stable GEMs, and it also follows that there is a proper class of nonequivalent stable localizations.

We obtain a simplicial group model for localization of (not necessarily nilpotent) spaces at sets of primes by applying a suitable functor dimensionwise, as in earlier work of Quillen and Bousfield-Kan. For a set of primesP and any groupG, letG→L PG be a universal homomorphism fromG into a group which is uniquely divisible by primes not inP, and denote also byL P the prolongation of this functor to simplicial groups. We prove that, ifX is any connected simplicial set andJ is any free simplicial group which is a model for the loop space ΩX, then the classifying space $$\overline W LpJ$$ is homotopy equivalent to the localization ofX atP. Thus, there is a map $$X \to \overline W LpJ$$ which is universal among maps fromX into spacesY for which the semidirect products πκ(Y)⋊ π1(Y) are uniquely divisible by primes not inP. This approach also yields a neat construction of fibrewise localization.

Given a monad (also called a triple) T on an arbitrary category, an idempotent approximation to T is dened as an idempotent monad ^ T rendering invertible precisely the same class of morphisms which are rendered invertible by T. One basic example is homological localization with coecients in a ring R; which is an idempotent approximation to R-completion in the homotopy category of CW-complexes. We give general properties of idempotent approximations to monads using the machinery of orthogonal pairs, aiming to a better understanding of the relationship between localizations and completions. c 1999 Elsevier Science B.V. All rights reserved.

Given an integern>1 and any setP of positive integers, one can assign to each topological spaceX a homotopy universal mapX(P,n)→X whereX(P,n) is an (n−1)-connected CW-complex whose homotopy groups areP-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces.

A theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories., 13 pages; consequences in motivic homotopy theory have been added

Our object of study is the natural tower which, for any given map f : A → B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E∗, yielding e.g. an analogue of Quillen’s plus-construction for each E∗. We discuss in detail the case of ordinary homology with coefficients in Z/p or Z[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f : A → pt), which allows us, among other things, to generalize Neisendorfer’s observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(Z[1/p], 1).

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M p , N p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopenka's principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories., 29 pages; a few changes made in Section 2

We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected CW {\text {CW}} -complexes, some of which extend P P -localization of nilpotent spaces, at a set of primes P P . We focus our attention on one such functor, whose local objects are CW {\text {CW}} -complexes X X for which the p p th power map on the loop space Ω X \Omega X is a self-homotopy equivalence if p ∉ P p \notin P . We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield’s homology localization, Bousfield-Kan completion, and Quillen’s plus-construction.

In a series of papers [13, 14, 15], Hilton introduced the terminology relative group to denote a group epimorphism ∈:G↠Q, and relative space to denote a map ƒ:E→B between connected spaces inducing an epimorphism of fundamental groups. He pointed out the desirability of relativizing the theory of P-localization of nilpotent groups and spaces developed in [17], and carried out the algebraic part of this project in [14, 16]. The homotopy-theoretic part was settled by Llerena in [18, 19].

Data from a predominantly urban sample of 116 men and 130 women aged 25-64 years and collected in 1984/85 as a part of the Swiss WHO MONICA project, were analysed cross-sectionally to study the interrelationship between relative weight, subcutaneous fat and fat distribution, as well as the dependence of these anthropometric characteristics on behavioral and sociodemographic factors. Skinfold thicknesses were found to increase with age almost linearly in women, while in men they increased only before age 40 to 45. Subcutaneous fat was, but fat distribution was not, highly correlated with relative weight in both sexes. Alcohol consumption, healthy dietary habits (inversely), and exercise (inversely) were all significantly related to subcutaneous fat in men, while the relatively strongest predictors of female skinfold thicknesses were smoking (inversely), coffee consumption, and education (inversely). In multivariate analysis, environmental factors explained up to 10% of skinfold variance in male subjects and between 10 and 15% in females. Fat distribution was more influenced by environmental factors in men (about 8% of explained variance) than in women (about 4%). In men, truncal fat depended more on lifestyle that did upper arm fat, with smoking (directly) and exercise (inversely) being relatively most predictive of abdominal fat. We conclude that, although relative weight, subcutaneous fat, and fat distribution correlate intra-individually, they are not equivalent and interchangeable anthropometric characteristics. This is reflected by the varying associations of the three fatness indicators with age and environmental factors such as smoking, diet, exercise, and education. Gender seems to be an important modifying factor of environment-body fat-associations.(ABSTRACT TRUNCATED AT 250 WORDS)

In order to study functorial changes caused by homotopy localizations on the fundamental group of unbased simplicial sets, it is convenient to use groupoids instead of groups, and therefore localizations of groupoids become useful. In this article we develop homotopy localization techniques in the model category of groupoids, with emphasis on the relationship with homotopy localizations of simplicial sets and also with discrete localizations of groups.

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 Vopenka's principle.In this article we extend the validity of this result to any left proper, combinatorial, simplicial model category $\cat M$ and show that, under additional assumptions on $\cat M$, 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., Comment: 9 pages

X with f ∈ M, there is a unique morphism h : B → X with hf = g. The orthogonal subcategory problem [13] asks whether D is reflective in C, i.e., under which conditions the inclusion functor D → C admits a left adjoint E : C → D; see [17]. Many authors have given conditions on the category C and the class of morphismsM ensuring the reflectivity of D, sometimes even providing an explicit construction of the left adjoint E : C → D; see for example Adams[1], Bousfield [3],[4], Deleanu-Frei-Hilton [9][10], Heller [15], Yosimura [22], Dror-Farjoun [11], Kelly [12]. The functor E is often referred to as a localisation functor of C at the subcategory D. Most of the known existence results of left adjoints work well when the category C is cocomplete [12] or complete [19]. Unfortunately, these methods cannot be directly applied to the homotopy category of CW-complexes, as it

A further advance—though it was not presented as such—was contained in Theorem 1.6 of [1], which effectively gave an example showing that the answer to our question is not always affirmative. The main purpose of this paper is to generalize both Theorem 0.1 and the counterexample contained in Theorem 1.6 of [1]. Thus we prove (cf. Theorems 2.1 and 2.3, which reproduce part (b) and part (a) of Theorem 0.2 respectively)

We conjecture that every finite group C acting on a contractible CW-complex X of dimension 2 has at least one fixed point. We prove this in the case where G is solvable, and under this additional hypothesis, the result holds for X acyclic.