Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

A naive approach to crystalline cohomology is to lift a smooth ring to characteristic zero and then to take de Rham cohomology. However, due to the non-canonical local nature of this process, it isn't easy to glue in order to define crystalline cohomology globally by this method. We consider a variant of this approach by taking higher homotopies into account, making the lift canonical and gluing easy. (Joint work with G. Tamme.)