9:00-9:40

**Jiří Rosický **
(Masaryk University, Brno)
*Combinatorial model categories *

The talk will survey some results and open problems concerning combinatorial model categories. Above all, the conjecture of J. H. Smith about model structures determined by a set of morphisms and my problem of generalized Brown representability will be discussed. In addition, the concept of a class-combinatorial model category (introduced in my joint work with B. Chorny) will be mentioned.

9:50-10:30

**Oriol Raventos **
(Universitat de Barcelona)
*Representability theorems for well generated triangulated categories *

Let *S* be a triangulated subcategory of a triangulated category *T*. A contravariant functor from *S* to the category of abelian groups is representable if it is the restriction to *S* of a functor of the form *Hom*(-,*X*), for *X* in *T*. We discuss some conditions on *T* and *S* that ensure that cohomological functors as above that send coproducts in *S* to products are representable.

11:00-11:40

**George Raptis **
(University of Osnabrück)
*Properness in the Bousfield Localisation of Model Categories *

The purpose of this talk will be to discuss the relevance of the property of (left) properness in the (left) Bousfield localisation of model categories. The Bousfield localisation of a model category at a set of morphisms is known to exist for the left proper cellular model categories and the left proper combinatorial model categories. The main result of the talk will say that the Bousfield localisation of a well-behaved cofibrantly generated model category exists iff the candidate class of trivial cofibrations is cofibrantly closed. This result will be presented as an application of the various known facts about combinatorial model categories, that will be briefly reviewed.

11:50-12:30

**Alexandru Stanculescu **
(Masaryk University, Brno)
*A homotopy theory for comodules and coalgebras *

We construct a Quillen model category structure on the categories of comodules and comonoids in certain monoidal model categories.

14:00-14:40

**Damir Franetic **
(University of Ljubljana)
*Decompositions of p-local spaces *

A ring is called semi-perfect if the identity 1 can be (uniquely) decomposed as a sum of orthogonal local idempotents. When *G* is a *p*-local H-space, there is a unit-reflecting homomorphism from [*G*,*G*] into some semi-perfect ring. Unique decomposition of 1 as a sum of local idempotents then implies unique decomposition of *G* as a product of indecomposable *p*-local H-spaces. Also, a straightforward dualization of this argument gives a decomposition for *p*-local coH-spaces. This proof is much shorter than the original proof given by C. Wilkerson.

14:50-15:30

**Javier J. Gutierrez **
(Centre de Recerca Matematica)
*Cellularization of structures in triangulated categories *

We describe the formal properties of cellularization and nullification functors in triangulated categories. We give sufficient conditions for cellularization functors to preserve ring structures and module structures in stable homotopy categories. As an example, we compute some cellularizations of Eilenberg-Mac Lane objects in the homotopy category of spectra.

16:00-16:40

**Carles Casacuberta **
(Universitat de Barcelona)
*Singly generated ideals and coideals in monoidal triangulated categories *

In joint work with Gutiérrez and Rosický, we show that Vopěnka's Principle implies that localizing ideals are coreflective and colocalizing coideals are reflective in homotopy categories of stable monoidal combinatorial model categories. We also prove that every localizing ideal is singly generated under Vopěnka's Principle. In this talk we will outline the arguments in our proofs and give some evidence that the statement that colocalizing coideals are singly generated might fail to be true in general.

16:50-17:30

**Lukáš Vokřínek **
(Masaryk University, Brno)
*Fibrations up to an equivalence and homotopy colimits *

Both quasifibrations and local homology fibrations behave nicely with respect to homotopy colimits. We give conditions on a class *H* of continuous maps (with examples being weak equivalences and homology equivalences as above) that determine a theory of fibrations up to an equivalence from *H* that has similar properties. Slight variations of these, universal *H*-fibrations, behave nicely with respect to pullbacks. We will also give a classification result for *H*-fibrations up to a natural notion of their equivalence.