Covanishing topos and generalizations

This chapter focuses on the Faltings topos, the general framework for Faltings' approach in p-adic Hodge theory. It presents a new approach to the Faltings topos based on a generalization of Deligne's covanishing topos. Along the way, it corrects the original definition of Faltings. After introducing the relevant notation and conventions, the chapter describes the oriented products of topos, covanishing topos and generalized covanishing topos, morphisms with values in a generalized covanishing topos, ringed total topos and ringed covanishing topos, and finite étale site and topos of a scheme. It concludes with a discussion of Faltings site and topos, along with the inverse limit of Faltings topos.