Online Talk

Homotopical structure can often be viewed as deforming algebraic structure. For example, the Postnikov tower of a connective ring spectrum R interpolates between the spectrum R and its 0th homotopy ring. Each map in this tower is a square-zero extension; this realizes R as a "nilpotent thickening" of π_0(R), and leads to a deformation theory for lifting algebraic things over π_0(R) to homotopical things over R. Similarly, the (décalage of the) Adams tower of a spectrum roughly interpolates between the spectrum and its homology; the associated deformation theory is the Adams spectral sequence, whose higher structure is the subject of synthetic homotopy theory.

I will talk about joint work with Piotr Pstrągowski that develops a theory of unstable deformations extending these stable examples. In the basic case, we develop a categorified deformation theory which replaces connective ring spectra with certain higher categorical algebraic theories, providing more insight into Blanc-Dywer-Goerss style decompositions of moduli spaces in homotopy theory. Time permitting, I will sketch how this allows us to define categories of synthetic spaces which encode the higher structure of the unstable Adams spectral sequence.