Idea
Let’s begin with a sketch/roadmap of ideas that will be made precise in this entry.
A spectrum determines a certain deformation of In the sense of deforming an object within a linear 2-category. This object denotes the map on 2-categorical structure sheaves induced by the map of stacks It turns out that this construction retains all information about spectra and their morphisms. More precisely, it determines a fully faithful embedding The image can be characterized as those deformations that are “periodic” in a certain sense. Hence, this fashions a description of the -category of spectra as periodic deformations of an object in an category.
In later entries we will use this picture to interpret even-periodic synthetic spectra and the even prismatic filtration. We will also spell out how this interacts with the Landweber correspondence.
Koszul dual description of 2-deformations of -reps
Endomorphisms of as a -linear category forms a monoidal category: Remark here we’ve made a specific choice to view as a -linear stable -category. We could have chosen e.g. to remember linearity over , or to work with a plain category. The reason for this setting will become clear later in the text.
Looping once more gives us the categorical center Again, our choice to remember linearity over graded spectra shows up here: we’re taking a -valued mapping spaces.
Theorem (Waldhaus.pdf, 3.4.8)
There’s an equivalence of -monoidal -linear stable -categories