This is an informal advanced seminar aimed at learning various aspects of ∞-categories and their applications (not to be confused with a previous seminar on ∞-categories, running in the years 2018-2020).

Basic information

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).

The main aim is to learn more about the phenomenon of ∞-categories (also very briefly introduced in the following article in the Notices of the AMS) and their applications.

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:

  1. Undestand relations between ∞-categories and others (sometimes more restrictive) homotopical/homological settings such as simplicial categories, dg categories or A-categories.
  2. 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]