DubTop Seminar: Deformations of stable infinity categories

To a stable infinity category C equipped with a well-behaved homology functor to an abelian category A, Patchkoria--Pstrągowski associate a derived category which, in a precise sense, interpolates between objects in C and Adams spectral sequences in A abutting to them. In this talk, we will begin by showing how this derived category arises naturally when trying to categorify Adams resolutions of spectra and then take a close look at its construction which axiomatizes the deformation-theoretic ideas of synthetic spectra. As applications, we will catch a glimpse of a few very strong algebraicity results--that is, results identifying stable infinity categories of interest with the derived categories of certain abelian categories.