Unstable synthetic deformations

This talk is on joint work with Piotr Pstragowski which develops synthetic deformations of unstable homotopy theories. In the motivating example, we construct a category of E-synthetic spaces, for a sufficiently nice homology theory E, which categorifies the unstable E-Adams spectral sequence and has as its special fiber a suitable derived category of unstable E${}_*$E-coalgebras. Other examples include categories of "normed E-synthetic ring spectra", deforming categories of structured ring spectra. I will sketch the construction of these deformations and describe some things that can be done with them.