The basic specific example a fractal infinity category of formal moduli problems for commutative rings of characteristic zero. We’d fundamentally like to understand extensions of this phenomenon to positive characteristic.
In one direction, staying in the setting of characteristic-zero geometry, we have relative versions of this by the work of Gaitsgory-Rozenblyum, asserting that
essentially asserting that a formal deformation of the stack can be recovered from the automorphism stack . This is the situation explored in constr - unpointed toposic Koszul duality.
We’re interested in applying these examples to formal stacks like As a group stack. This, for example, encode the Adams-Novikov spectral sequence. (see also Thm - Amitsur Cochain as E_1 cotangent complex). Our intent is to show that this provides a geometric setting for -based synthetic spectra, which then gives a geometric interpretation of the even/motivic filtration e.g. as in Notes from Raksit talk.
**What could the framework of fractal -categories do for us in understanding -based synthetic spectra?
In our hypothetical fractal setting over the sphere, the formal deformation viewed as a formal moduli problem under , is captured by the -group stack At the level of sheaves, this gives a comparison
We recognize this as the Amitsur descent along MU that gives rise to the Adams-Novikov spectral sequence.
Now we apply the fractal property to produce an -group stack that again fully captures the data of the deformation: this time as an (2-categorical) symmetry Unwinding definitions we see the RHS is the affine scheme . This formalizes a certain additional structure on
Question: How is the group stack structure related to the cyclotomic structure? Can the latter be recovered from the former?
An group likes to have 2-categorical representations
Question: How does all this interact with Cartier-Transforms?