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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

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 prove that, under suitable restrictions, an idempotent monad t defined on a full subcategory A of a category C can be extended to an idempotent monad T on C in a universal (terminal) way. Our result applies in particular to the case when t is P-localization of nilpotent groups (where P denotes a set of primes) and C is the category of all groups. The corresponding monad T on C is, in a certain precise sense, the best idempotent approximation to the usual Zp-completion of groups; it turns out to be a strict epimorphic image of Bousfield's HZp-localization.

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.