How to glue derived categories
The author writes an exceptional expository article on gluing of derived categories (triangulated categories) using the formulation of stable model pairs, which is more general than the enhancement by stable model categories, thereby increasing the applicability of the techniques to more than just algebraic setting. The article begins with an historical overview of derived/triangulated categories and their enhancements and compares the approach to be taken via stable model categories with other known approaches (via DG categories) or upcoming approaches (via $\infty$-categories), and puts the current developments in perspectives along with mentioning the major question which needs more investigation. The main aim of the article is to glue the triangulated categories of a Grothendieck precofibration $\mathcal{C} \rightarrow I$, over a small category $I$, which comes from a stable model pair. The gluing category is constructed as a certain decorated derived category of cosection of the precofibration, in Section 9. The article also deals with another example of a gluing situation arising from a (stable) comonad $\Phi$ over a category $\mathcal{C}$. The gluing in this situation is given by the category of derived coalgebras, in Section 10. The proofs and constructions are developed and written in a very pedagogical manner starting from the examples of elementary gluing and increasing the complexity of the situation step by step. All the main technical tools of stable model categories and pairs, families of categories (Grothendieck (co)-fibrations), simplicial replacements, comonads and coalgebras, and Reedy categories are well explained and carefully constructed. Moreover, the author devotes a full section to treat the case of algebraic triangulated categories and provides ample examples. The author ends the article by giving a construction analogous to Verdier localization for the stable model pairs using inductive completion of a right proper model category and he gives an explanation for his choice of definitions and compares them with the one currently used. On the way to it, he also proves a new result that the inductive completion of a right proper model category has a natural model structure.
scientific article; zbMATH DE number 6991233
scientific article