For an equivariant commutative ring spectrum R, ?0R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N? ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and ?0R has the structure of an incomplete Tambara functor. However, if R is an N? ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.

We survey some methods developed in a series of papers, for classifying localising subcategories of tensor triangulated categories. We illustrate these methods by proving a new theorem, providing such a classification in the case of the stable module category of a unipotent finite supergroup scheme.

In their work on the period map and the dualising sheaf for Lubin–Tate space, Gross and the second author wrote down an equivalence between the Spanier–Whitehead and Brown–Comenetz duals of certain type n-complexes in the K(n)-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin–Tate space, two important invertible sheaves become isomorphic modulo p.

We give conditions on a monoidal model category M and on a set of maps C so that the Bousfield localisation of M with respect to C preserves the structure of algebras over various operads. This problem was motivated by an example that demonstrates that, for the model category of equivariant spectra, preservation does not come for free. We discuss this example in detail and provide a general theorem regarding when localisation preserves P-algebra structure for an arbitrary operad P. We characterise the localisations that respect monoidal structure and prove that all such localisations preserve algebras over cofibrant operads. As a special case we recover numerous classical theorems about preservation of algebraic structure under localisation, in the context of spaces, spectra, chain complexes, and equivariant spectra. To demonstrate our preservation result for non-cofibrant operads, we work out when localisation preserves commutative monoids and the commutative monoid axiom. Finally, we provide conditions so that localisation preserves the monoid axiom.

In this note, we construct a higher-dimensional version of the chromatic fracture square. We then categorify the resulting chromatic fracture cubes obtaining a decomposition of the category of E(n)-local spectra into monochromatic pieces.

We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making systematic use of finite groupoids. This provides a ‘road map’ for the various approaches to the axiomatic representation theory of finite groups, as well as some details that are hard to find in the literature.

The project of Greenlees et al. on understanding rational G-spectra in terms of algebraic categories has had many successes, classifying rational G-spectra for finite groups, SO(2), O(2), SO(3), free and cofree G-spectra as well as rational toral G-spectra for arbitrary compact Lie groups. This chapter provides an introduction to the subject in two parts. The first discusses rational G-Mackey functors, the action of the Burnside ring and change of group functors. It gives a complete proof of the well-known classification of rational Mackey functors for finite G. The second part discusses the methods and tools from equivariant stable homotopy theory needed to obtain algebraic models for rational G-spectra. It gives a summary of the key steps in the classification of rational G-spectra in terms of a symmetric monoidal algebraic category. Having these two parts in the same place allows one to see clearly the analogy between the algebraic and topological classifications.

The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this chapter we review its definition and summarise some of these applications. We conclude with a discussion of why we could work with right Quillen functors instead, but cannot work with a combination of the two.