Equivariant Topology and Derived Algebra

This volume contains eight research papers inspired by the 2019 'Equivariant Topology and Derived Algebra' conference, held at the Norwegian University of Science and Technology, Trondheim in honour of Professor J. P. C. Greenlees' 60th birthday. These papers, written by experts in the field, are intended to introduce complex topics from equivariant topology and derived algebra while also presenting novel research. As such this book is suitable for new researchers in the area and provides an excellent reference for established researchers. The inter-connected topics of the volume include: algebraic models for rational equivariant spectra; dualities and fracture theorems in chromatic homotopy theory; duality and stratification in tensor triangulated geometry; Mackey functors, Tambara functors and connections to axiomatic representation theory; homotopy limits and monoidal Bousfield localization of model categories.

Model Structures for the 𝔸1-homotopy Category

This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM eff nis ≤0 of DM eff nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.

Morita homotopy theory for (∞,1)-categories and ∞-operads

We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of {(\infty,1)} -categories and {\infty} -operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.