Abstract: Pro objects, such as the absolute Galois group of a field, are pervasive in algebra. They formed the basis for the original applications of homotopy theory in geometry and number theory, via étale homotopy theory. For some time, local homotopy theory and étale homotopy theory were almost orthogonal as theories, but the relationship between the two is much better understood now, and there is a theory which engulfs both.

Perhaps there is an even more general theory that is not based on pro objects, with potential geometric applications which are not bound to the étale topology. I will describe a candidate in this talk, after some teaching moments.