Lecture Notes

Abstract

It is VERY hard to compute homotopy groups. We want to put as much algebraic structure as possible in order to make computation easier. You can’t add maps in HoTop but you can in Spectra (i.e. Spectra is an Ab-category). The motivation to go to S−Alg = HoS then becomes that you want an abelian category We want to study the ring-like objects that arise in this category. “Ring-like” means ring-object, i.e. using the lens of category theory. They have no points, so you can’t do traditional algebra. To measure complexity of these we’ll use dimension.