The seminar has already finished.
The seminar is scheduled on Tuesdays, from 2pm, in the seminar room of the Dept. of Algebra and currently runing in a hybrid mode (contact me if you do not have Zoom coordinates).
Here, we will focus on selected topics (see the list of goals below) rather then on a general introduction to the theory. The latter was the main theme of a former seminar ∞-categories, running in 2018-2020, which unfortunately became a victim of the Covid pandemic.
Here we a list goals of the seminar, as it evolves, and possibly indicate when the goals were ticked as achieved:
- Undestand relations between ∞-categories and others (sometimes more restrictive) homotopical/homological settings such as simplicial categories, dg categories or A∞-categories.
- Understand the relation of the coherent nerve construction to Koszul duality between dg categories and pointed curved conilpotent dg-coalgebras in the sense of [HL21].
A brief overview of what has been told in the seminar will be updated below.
- November 2, 2021
- A reminder of the classical Koszul duality, a bird's-eye view of the picture obtained in [HL21], where the Koszul duality is viewed as a linear version of the coherent nerve adjunction.
- November 9, 2021
- A crash course on dg categories, dg modules, derived categories of dg modules and pretriangulated dg categories.
- November 16, 2021
- Simplicial sets as models for homotopy types of topological spaces, the nerve of an ordinary category, simplicial categories and the homotopy coherent nerve functor (see [Ker, §2.4] for the latter).
- November 23, 2021
- The Koszul duality for augmented dg algebras and coaugmented dg coalgebras over a field (following [P11]). The bar and cobar constructions for (co)algebras and (co)modules.
- November 30, 2021
- Curved dg (co)algebras and (co)modules over them. Derived categories of the second kind (in the sense of [P11]). Koszul duality between dg algebras and coaugmented curved dg coalgebras.
- December 7, 2021
- Koszul duality for small dg categories and pointed curved dg coalgebras (and encoding small dg categories into semialgebras, following [HL21]).
- December 15, 2021
- A crash course on ∞-categories.
- January 4, 2022
- The dg nerve functor - motivation from [HL21] and a construction from [Ker, §2.5.3].
- January 26, 2022
- The left adjoint of the dg nerve functor (mostly about how to extract its description from [Ker, §2.5.3], but also about relations to [F17,COS19, RZ18] and to [HL21]).
- March 3, 2022
- Contramodules over (curved dg) coalgebras, Koszul triality for dg algebras and comodules and contramodules over curved dg coalgebras (following [P11]).
- March 10, 2022
- Introduction to the ∞-categorical version of simplicial presheaves on Cartesian spaces and glimpses of higher differential geometry.
A lot of sources can be found on the home page of the old seminar from 2018-2020. From the introductory sources, we just mention here
|[Ker]||Kerodon, an online resource for homotopy-coherent mathematics. [link]|
As a rough guide to the topic, one can also use
|[C13]||O. A. Camarena, A whirlwind tour of the world of (∞,1)-categories, preprint, 2013. [arXiv]|
There are various sources comparing ∞-categories, simplicial categories, dg categories and A∞-categories. Some guidance on how to use the sources can be found in the following post on StackExchance, while the comparison between ∞- and dg categories can be found in [Ker, §2.5]. In [H15], we may be interested in Example 5.11. In [RZ18], §6 is useful.
|[B18]||J. Bergner, The homotopy theory of (∞,1)-categories, LMS Student Texts 90, Cambridge University Press, Cambridge, 2018.|
|[COS19]||A. Canonaco, M. Ornaghi, P. Stellari, Localizations of the category of A∞ categories and internal Homs, Doc. Math. 24 (2019), 2463-2492. [link] [arXiv]|
|[F17]||G. Faonte, Simplicial nerve of an A∞-category, Theory Appl. Categ. 32 (2017), Paper No. 2, 31-52. [link] [arXiv]|
|[H15]||R. Haugseng, Rectification of enriched ∞-categories, Algebr. Geom. Topol. 15 (2015), no. 4, 1931-1982. [link] [arXiv]|
|[LM20]||W. Lowen, A. Mertens, Linear quasi-categories as templicial modules, preprint, 2020. [arXiv]|
|[O18]||M. Ornaghi, Comparison results about dg-categories, A∞-categories, stable ∞-categories and noncommutative motives, Ph.D. thesis, Università degli Studi di Milano, 2018. [link]|
|[RZ18]||M. Rivera, M. Zeinalian, Cubical rigidification, the cobar construction and the based loop space, Algebr. Geom. Topol. 18 (2018), no. 7, 3789-3820. [link] [arXiv]|
The relation between the coherent nerve functor and Koszul duality is explained in
|[HL21]||J. Holstein, A. Lazarev, Categorical Koszul duality, preprint, 2021. [arXiv]|
Other sources for the Koszul duality on which [HL21] builds include
|[K06]||B. Keller, Koszul duality and coderived categories (after K. Lefèvre), a short note, 2006. [link]|
|[LH03]||K. Lefèvre-Hasegawa, Sur les A∞-catégories, Ph.D. thesis, Universite Paris 7 - Denis Diderot, 2003. [link]|
|[P11]||L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc. 212 (2011), no. 996. [link] [arXiv]|
The idea of modeling categories by a semialgebras and functors by semimodules comes from Aguiar's thesis. For a comprehensive source for semiinfinite homological algebra as such, we refer to the monograph [P10].
|[A97]||M. Aguiar, Internal categories and quantum groups, Ph.D. thesis, Cornell University, 1997. [link]|
|[P10]||L. Positselski, Homological Algebra of Semimodules and Semicontramodules, Semi-infinite Homological Algebra of Associative Algebraic Structures, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series), Birkhäuser/Springer Basel AG, Basel, 2010. [link] [arXiv]|