Homotopy types and their linearization

I’ll introduce some foundational concepts in preparation for Benjamin Moldstad’s PhD defense on 15 April (the following Math Seminar slot).

Specifically, I’ll introduce the notion of a homotopy type, the spirit of which is a set whose elements are redundantly identified. This will be motivated via redundant equations in linear algebra, prompting consideration of chain-complexes. I’ll explain how a topological space determines a homotopy-type.

Then I’ll introduce some instances of linearizations of homotopy-types, matching with that motivation. I’ll state a theorem: in the zoo of linearizations of homotopy-types, there’s an initial one: spectra. I’ll indicate what a spectrum is and give some examples of such. I’ll supply a heuristic for why one might expect this theorem. The talk will conclude with some computations that reveal the rich structure of spectra — structure that’s therefore present in any linearization of homotopy-types.