Maltsiniotis's first conjecture for K1

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona

Entrevista a Isabel Fernández, conferenciante invitada al ICM 2010

Isabel Fernández, de la Universidad de Sevilla, es conferenciante invitada junto con Pablo Mira (Universidad Politécnica de Cartagena) a la sección de Geometría del próximo ICM, que se celebrará en agosto en Hyderabad (India). Son los únicos matemáticos españoles invitados a este congreso. Isabel, que realizó su tesis en la Universidad de Granada, es, además, la primera española invitada a un ICM

Homotopy units in A-infinity algebras

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.Junta de AndalucíaMinisterio de Educación y CienciaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny

On the unit of a monoidal model category

In this paper we show how to modify cofibrations in a monoidal model category so that the tensor unit becomes cofibrant while keeping the same weak equivalences. We obtain aplications to enriched categories and coloured operads in stable homotopy theory.Junta de AndalucíaMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Cylinders for non-symmetric DG-operads via homological perturbation theory

We construct small cylinders for cellular non-symmetric DG-operads over an arbitrary commutative ring by using the basic perturbation lemma from homological algebra. We show that our construction, applied to the A-infinity operad, yields the operad parametrizing A-infinity maps whose linear part is the identity. We also compute some other examples with non-trivial operations in arities 1 and 0.Junta de AndalucíaMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Moduli spaces of algebras over non-symmetric operads

In this paper we study spaces of algebras over an operad (nonsymmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of To¨en and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel

Homotopy theory of bicomplexes

We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure’s weak equivalences are detected by the E2-term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes.Ministerio de Economía y Competitivida

K-theory of derivators revisited

We define a K-theory for pointed right derivators and show that it agrees with Waldhausen K-theory in the case where the derivator arises from a good Waldhausen category. This K-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator K-theory, as originally defined, is the best approximation to Waldhausen K-theory by a functor that is invariant under equivalences of derivators.Junta de Andalucía (Consejería de Innovación, Ciencia y Empresa)Ministerio de Educación y CienciaGeneralitat de Cataluny

Los cementerios en el contexto urbano : el cementerio de Tolosa

Los cementerios constituyen uno de los usos urbanos consumidores de grandes espacios. Su localización y consideración dentro del entramado urbano ha ido modificándose conforme la sociedad evoluciona. Los cementerios son un elemento bastante reciente. El artículo analiza la evolución histórica que sufre el cementerio de Tolosa, analizando su localización geográfica y su estructuración interio

Torsion homology and cellular approximation

We describe the role of the Schur multiplier in the structure of the p-torsion of discrete groups. More concretely, we show how the knowledge of H2G allows us to approximate many groups by colimits of copies of p-groups. Our examples include interesting families of noncommutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of Emmanuel Farjoun.Fondo Europeo de Desarrollo RegionalMinisterio de Ciencia e InnovaciónConsejería de Economía, Innovación y Ciencia (Junta de Andalucía