Abstract. Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by

These notes explore equivariant homotopy theory from the perspective of model categories in the case of a discrete group G. Section 2 reviews the situation for topological spaces, largely following [May]. In section 3, we discuss two approaches to equivariant homotopy theory in more general model categories. Section 4 discusses

C to S with a model structure, defining weak equivalences and fibrations objectwise but only on D. Our first concern is the effect of moving C, D and S. The main notion introduced here is the “D-codescent ” property for objects in S C. Our long-term program aims at reformulating as codescent statements the Conjectures of Baum-Connes and Farrell-Jones, and at tackling them with new methods. Here, we set the grounds of a systematic theory of codescent, including pull-backs, push-forwards and various invariance properties. 1.

Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of S-valued C-indexed diagrams with D-weak equivalences and D-fibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category and D ⊂ C are small categories. An application to the notion of homotopy colimit is presented. 1.

We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.

Let I be a small indexing category, G: I op → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define

We use the language of homotopy topoi, as developed by Lurie [17], Rezk [21], Simpson [23], and Töen-Vezossi [24], in order to provide a common foundation for equivariant homotopy theory and derived algebraic geometry. In particular, we obtain the categories of G-spaces, for a topological group G, and E-schemes, for an E∞-ring spectrum E, as full topological subcategories of the homotopy topoi associated to sheaves of spaces on certain small topological sites. This allows for a particularly elegant construction of the equivariant elliptic cohomology associated to an oriented elliptic curve A and a compact abelian Lie group G as an essential geometric morphism of homotopy topoi. It follows that our definition satisfies a conceptually simpler homotopy-theoretic analogue of the Ginzburg-Kapranov-Vasserot axioms [8], which allows us to calculate the cohomology of the equivariant G-spectra S V associated to representations V of G. iii To my parents. iv Acknowledgments I would like to thank the many people who have been both directly and indirectly involved with this project.

We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.

Abstract. We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U ∗ → X is a weak equivalence. This fact is used to construct topological realization functors for the A1-homotopy theory of schemes over real and complex fields. 1.

We show that hermitian K-theory and Witt groups are representable both in the unstable and in the stable A¹-homotopy category of Morel and Voevodsky. In particular, Balmer Witt groups can be nicely expressed as homotopy groups of a topological space. The proof and its consequences include other new results related to Nisnevish-Mayer-Vietoris squares, Laurent polynomials, the projective line, blow ups and homotopy purity. The results will hopefully become part of a proof of Morel’s conjecture on certain A¹-homotopy groups of spheres.