A generalization of Ohkawa's theorem

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.Comment: 13 pages; consequences in motivic homotopy theory have been adde

Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This result follows from the fact that cohomology equivalences are \Sigma_2. In contrast with this fact, homology equivalences are \Sigma_1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies have been correcte

El proper Congrés Internacional dels Matemàtics, ICM 2006, Madrid

El proper Congrés Internacional dels Matemàtics, ICM 2006, MadridFactoria FM

On finite groups acting on acyclic complexes of dimension two

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

Comparing localizations across adjunctions

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 $T f$-localization on spaces underlying homotopy $T$-algebras, and similarly for cellularizations

Localization of algebras over coloured operads

We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.Comment: 34 page

Localizations of abelian Eilenberg-Mac Lane spaces of finite type

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