Shape Theory in Homotopy Theory and Algebraic Geometry

This work defines the étale homotopy type in the context of non-archimedean geometry, in both Berkovich’s and Huber’s formalisms. To do this we take the shape of a site’s associated hypercomplete 1-topos. This naturally leads to discussing localizations of the category of pro-spaces. For a prime number p, we introduce a new localization intermediate between profinite spaces and {p}`-profinite spaces. This new category is well suited for comparison theorems when working over a discrete valuation ring of mixed characteristic. We prove a new comparison theorem on the level of topoi for the formalisms of Berkovich and Huber, and prove an analog of smooth-proper base change for nonarchimedean analytic spaces. This provides a necessary result for the non-archimedean analog of Friedlander’s homotopy fiber theorem, which we prove. For a variety over a non-archimedean field, we prove a comparison theorem between the classical étale homotopy type and our étale homotopy type of the variety’s analytification. Finally, we examine certain log formal schemes over the formal spectrum of a complete discrete valuation ring, and compare their Kummer étale homotopy type with the étale homotopy type of the associated non-archimedean analytic space.