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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.

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.

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 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.

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

Let S+n denote the n-sphere with a disjoint basepoint. We give conditions ensuring that a map h: X → Y that induces bijections of homotopy classes of maps [S+n, X] ≅ [S+n, Y] for all n ⩾ 0 is a weak homotopy equivalence. For this to hold, it is sufficient that the fundamental groups of all path-connected components of X and Y be inverse limits of nilpotent groups. This condition is fulfilled by any map between based mapping spaces h: map∗(B, W) → map∗(A, V) if A and B are connected CW-complexes. The assumption that A and B be connected can be dropped if W = V and the map h is induced by a map A → B. From the latter fact we infer that, for each map ƒ, the class of ƒ-local spaces is precisely the class of spaces orthogonal to ƒ and ƒ λ S + n for n ⩾ 1 i homotopy category. This has useful implications in the theory of homotopical localization.

We give an example showing that, for a nilpotent group G G and a set of primes P P , the P P -localization homomorphism l : G → G P l:G \to {G_P} need not induce an isomorphism in cohomology with arbitrary (twisted) Z P {{\mathbf {Z}}_P} -module coefficients. From this fact we infer that, in the pointed homotopy category of connected CW-complexes, the inclusion of the subcategory of spaces whose higher homotopy groups are Z P {{\mathbf {Z}}_P} -modules and whose fundamental group is uniquely P ′ {P’} -radicable does not admit a left adjoint.