Kathryn Hess

Jérôme Scherer

Wolfgang Lück

Brenda Johnson

Christian Ausoni

This volume contains the proceedings of the Fourth Arolla Conference on Algebraic Topology, which took place in Arolla, Switzerland, from August 20-25, 2012. The papers in this volume cover topics such as category theory and homological algebra, functor homology, algebraic -theory, cobordism categories, group theory, generalized cohomology theories and multiplicative structures, the theory of iterated loop spaces, Smith-Toda complexes, and topological modular forms.

An Alpine Expedition through Algebraic Topology

This volume contains the proceedings of the Third Arolla Conference on Algebraic Topology, which took place in Arolla, Switzerland, on August 18-24, 2008. This volume includes research papers on stable homotopy theory, the theory of operads, localization and algebraic K-theory, as well as survey papers on the Witten genus, on localization techniques and on string topology - offering a broad perspective of modern algebraic topology.

Third Arolla Conference on Algebraic Topology, August 18-24, 2008, Arolla, Switzerland

Alpine Perspectives on Algebraic Topology

This volume contains the proceedings of the Alpine Algebraic and Applied Topology Conference, held from August 15–21, 2016, in Saas-Almagell, Switzerland. The papers cover a broad range of topics in modern algebraic topology, including the theory of highly structured ring spectra, infinity-categories and Segal spaces, equivariant homotopy theory, algebraic -theory and topological cyclic, periodic, or Hochschild homology, intersection cohomology, and symplectic topology.

An Alpine Bouquet of Algebraic Topology

Wojciech Chacholski

Wojciech Chachólski

In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. A model approximation of a category $\mathcal{C}$ with a given class of weak equivalences is a model category $\mathcal{M}$ together with a pair of adjoint functors $\mathcal{M} \rightleftarrows \mathcal{C}$ which satisfy certain properties. Our key result says that if $\mathcal{C}$ admits a model approximation then so does the functor category $Fun(I, \mathcal{C})$. From the homotopy theoretical point of view categories with model approximations have similar properties to those of model categories. They admit homotopy categories (localizations with respect to weak equivalences). They also can be used to construct derived functors by taking the analogs of fibrant and cofibrant replacements. A category with weak equivalences can have several useful model approximations. We take advantage of this possibility and in each situation choose one that suits our needs. In this way we prove all the fundamental properties of the homotopy colimit and limit: Fubini Theorem (the homotopy colimit -respectively limit- commutes with itself), Thomason's theorem about diagrams indexed by Grothendieck constructions, and cofinality statements. Since the model approximations we present here consist of certain functors ``indexed by spaces'', the key role in all our arguments is played by the geometric nature of the indexing categories.

Jérôme Scherer

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