Email this article On 2-categories of extensions
- Authors: Kaledin D.B.1,2
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
- National Research University Higher School of Economics, Moscow, Russia
- Issue: Vol 216, No 12 (2025)
- Pages: 57-78
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/358684
- DOI: https://doi.org/10.4213/sm10354
- ID: 358684
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Abstract
The paper is essentially an illustration to the general technique of homotopical enhancements developed recently in [6]. Taking a derived category of an abelian category we consider its full subcategory generated by complexes of length 2. It has a natural refinement to a 2-category, which we call the ‘2-category of extensions’. However, it is not possible to construct this refinement by only using a triangulated structure. In this short note, first we construct a 2-category of enhancements by hand, using the techniques of abelian categories, and then we show how it can quite easily and naturally be recovered in the framework of the enhanced formalism of [6].
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About the authors
Dmitry Borisovich Kaledin
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; National Research University Higher School of Economics, Moscow, Russia
Email: kaledin@mi-ras.ru
Scopus Author ID: 12790495100
ResearcherId: T-5886-2017
Doctor of physico-mathematical sciences, no status
References
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- A. Grothendieck, “Categories fibrees et descente”, Revêtements etales et groupe fondamental, Seminaire de geometrie algebrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Math., 224, Springer-Verlag, Berlin–New York, 1971, Exp. VI, 145–194
- D. Kaledin, “How to glue derived categories”, Bull. Math. Sci., 8:3 (2018), 477–602
- D. Kaledin, Trace theories, Bökstedt periodicity and Bott periodicity
- D. Kaledin, Enhancement for categories and homotopical algebra
- D. B. Kaledin, “How to enhance categories, and why?”, УМН, 80:2(482) (2025), 51–122
- J.-L. Verdier, Des categories derivees des categories abeliennes, Asterisque, 239, Soc. Math. France, Paris, 1996, xii+253 pp.
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