Preliminaries on Filtered Derived Categories and Topoi

In this chapter, we prove some preliminary results on filtered derived categories and topoi which we need in later chapters. In the first four sections, we recall several notions and basic properties concerning filtered derived categories: filtered complex, filtered quasi-isomorphism, filtered derived category, filtered injective resolution, filtered flat resolution, filtered derived functor and filtered adjunction formula. It is important for us because our weight-filtered complexes will be defined as objects in certain filtered derived categories. In the last two sections, we recall the notion of topoi associated to diagrams of topoi and prove several basic properties of the topoi associated to diagrams of (restricted) log crystalline topoi.