I’ve been reading lately about the Sullivan conjecture and its proof (which is the subject of a course that Kirsten Wickelgren is teaching next semester). The resolution of this conjecture and work related to it led to a great deal of interesting algebra in the 1980s and 1990s, which I’ve been trying to understand a little about. Some useful references here are Haynes Miller’s 1984 paper, Lionel Schwartz’s book, and Jacob Lurie’s course notes.

**1. Motivation**

Let be a variety over the complex numbers. The complex points are a topological space that has a homotopy type, which is often of interest. Étale homotopy theory (a refinement of étale cohomology) allows one to give a purely algebraic description of the profinite completion of the homotopy type of . If is defined over the real numbers, though, then one can also study the topological space of real points of ; one has

for the conjugation action on . (more…)