**Research**

## I do algebraic topology. Specially, stable homotopy theory and triangulated categories.

**PhD**

## I finished my PhD in March 2011 in the University of Barcelona under the advisory of Carles Casacuberta and Fernando Muro. It can be downloaded in this link: Adams Representability in Triangulated Categories.

**Articles**

__Abstract__: The hammock localization provides a model for a homotopy function complex in any Quillen model category. We prove that a homotopy between a pair of morphisms induces a homotopy between the maps induced by taking the hammock localization. We describe applications of this fact to the study of homotopy algebras over monads and homotopy idempotent functors. Among other things, we prove that, under Vopnkaěs principle, every homotopy idempotent functor in a cofibrantly generated model category is determined by simplicial orthogonality with respect to a set of morphisms. We also give a new proof of the fact that left Bousfield localizations with respect to a class of morphisms always exist in any left proper combinatorial model category under Vopěnka's principle.

__Abstract__: In a well generated triangulated category T, given a regular cardinal a, we consider the following problems: given a functor from the category of a-compact objects to abelian groups that preserves products of <a objects and takes exact triangles to exact sequences, is it the restriction of a representable functor in T? Is every natural transformation between two such restricted representable functors induced by a map between the representatives? If the answer to both questions is positive we say that T satisfies a-Adams representability. A classical result going back to Brown and Adams shows that the stable homotopy category satisfies Adams representability for the first infinite cardinal. For that cardinal, Adams representability is well understood thanks to the work of Christensen, Keller and Neeman. In this paper, we develop an obstruction theory to decide when T satisfies a-Adams representability. We derive necessary and sufficient conditions of homological nature, and we compute several examples. In particular, we show that there are rings satisfying a-Adams representability for all non-countable cardinals a and rings which do not satisfy a-Adams representability for any infinite cardinal a. Moreover, we exhibit rings for which the answer to both questions is no for infinite many cardinals. As a side result, we give an example of an infinite phantom map.

with C. Casacuberta and A. Tonks

__Abstract__: Bousfield proved that localizations preserve loop spaces and infinite loop spaces. We study, to what extend, these results follow from the existence of an adjunction (in Bousfield cases (suspension,loops) and (infinite suspension, infinite loops)) and the fact that the image of the right adjoint are algebras of some sort (in Bousfield cases A_infinity spaces and E_infinity spaces). We give conditions on adjoint pairs so that we can compare localizations on both sides. We study the strict and the homotopic cases and give applications to modules and module spectra.

**Homotopical localizations**

with C. Casacuberta and J. J. Gutiérrez

__Abstract__: This work in progress clarifies the different interpretations of localizing in homotopy theory. The naïve approach defining a homotopy localization as a functor inducing a strict localization after inverting homotopy turns out to be difficult to use in practice. More concise approaches include left Bousfield localizations, Barwick's enriched localizations and Lurie's (infinity,1)-localizations. We are also interested in existence results.

**Miscellaneous Articles **

with C. Casacuberta

__Abstract__: We prove that the category of spectra localized with respect to a

Morava K-theory is not Quillen equivalent to the stabilization of the category of spaces localized with respect to the same homology theory. This shows that, in general, stabilization does not commute with homology localizations.