Fibred Categories and the Six Functors Formalism

Abstract

In Section 1, we introduce the basic language used in this book, the so-called premotivic categories and their functoriality. This is an extension of the classical notion of fibered categories. They appear with different categorical structures. In Section2, the language of premotivic categories is specialized to that of triangulated categories and to algebraic geometry. We introduce several axioms of such categories which ultimately will lead to the full six functors formalism. An emphasis is given on the study of the main axioms, with a special care about the so-called localization axiom. Then in Section 3, the general theory of descent is formulated in the language of premotivic model categories. We study several particular case of descent, with an emphasis on Voevodsky’s h-topologies (cdh, qfh and h). In the last section, we introduce convenient finiteness conditions, generically called constructibility, on premotivic triangulated categories with the aim of giving sufficient conditions to obtain duality theorems. A general theorem on stability of constructibility under the six operations is shown. A general formalism of compatibility of premotivic categories with projective limits is introduced under the name of continuity.