How to glue derived categories
type: Item
instance of: scholarly article
title: How to glue derived categories
zbMATH Open document ID: 1435.18017
published in: Bulletin of Mathematical Sciences
publication date: 2018-12-10T00:00:00Z
review text: 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 gi…
Mathematics Subject Classification ID: 18G80
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 18N60
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 6991233
zbMATH Keywords: derived category
zbMATH Keywords: triangulated category
zbMATH Keywords: gluing
author: Dmitry Kaledin
reviewed by: Tanya Kaushal Srivastava
MaRDI profile type: Publication
full work available at URL: s13373-018-0119-z
OpenAlex ID: W2790965530
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DOI: 10.1007/S13373-018-0119-Z
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description: scientific article; zbMATH DE number 6991233