# Infinity categories (2021)

We would first like to explore the main models of $(\infty, 1)$-categories with emphasis on the Boardman-Vogt-Joyal-Lurie model (the quasi-categories) and then go in the direction of higher algebra [6]. We will follow a priori more or less faithfully Julia Bergner’s book [2] from chapter 4. The courses will be held in Brest but it is planned that they will be broadcasted via Zoom for the people of Vannes who would be interested.

As far as the prerequisites are concerned, it would be good to know the basic notions of category theory, model categories and simplicial homotopy (references for these can be found below). However there will be an introductory session in which we will motivate the study of $\infty$-categories by going back to the modern conception of homotopy theory (“à la Quillen”). Moreover, some reminders will be made during the sessions when necessary and questions, even those that may seem naive, are obviously encouraged. Finally, for those who wish to participate but are not really familiar with these ideas, we can only advise them to read the first three chapters of [2], the beginning of [5] (up to page 23) or even the article [3].

The program of the working group (which may change along the semester) :

1. Introduction, motivations, language of infinite-categories, simplicial / complex enrichment of Kan.

2. Segal spaces.

3. Rezk spaces.

4. Bergner-Joyal-Tierney-Lurie-Dugger-Spivak(-Barwick-Kan) comparison theorems.

5. Quasi-categories: Joyal’s theorem, Dwyer-Kan’s localization, Bousfield’s in the infinite-category frame, Quillen’s A theorem, Yoneda’s adjunction theorem and lemma in the infinite-category version, etc. (what we find in Bergner, Land [3] or Lurie [4] for example).

6. Stable quasi-categories and higher algebra (essentially the first chapter of Higher Algebra).

Notes for the lectures will be added below.

There’s also a website of a working group organized in Paris 13 on the same subject, which by the way is very complete and contains a lot of stuff (if someone wants to talk about one of the topics treated at the time, he is strongly invited to do so). You can have access here.

References:

1. Lectures on n-categories and cohomology”, John C. Baez, Michael Shulman

2. “The homotopy theory of $(\infty, 1)$-categories”, Julia Bergner

3. A survey of (\infty, 1)-categories”, Julia Bergner

4. “Introduction to $\infty$-categories”, Markus Land

5. Higher topos theory”, Jacob Lurie

6. Higher algebra”, Jacob Lurie

Other references:

1. “Higher categories and homotopical algebra”, Denis-Charles Cisinski.

2. “Simplicial homotopy theory”, Paul Goerss, Rick Jardine

3. Model categories”, Mark Hovey

4. Notes on quasi-categories”, André Joyal

5. “Category theory in context”, Emily Riehl

We will provide lecture notes here soon.

LecturesSpeakerVideos of the lectures
Introduction, motivation (English)Santiago ToroMotivation
Simplical categories I (French)Pierre MartinezSimplicialCatsI
Interlude: Simplicial objects and Hodge theory (French)Johannes HuismanInterlude
Simplicial categories II (English)Santiago ToroSimplicialCatsII
Simplicial categories III (French)Pierre MartinezSimplicialCatsIII
Segal spaces I (English)Santiago ToroSegalSpcI
Segal spaces II (French)Pierre MartinezSegalSpcII
Interlude: Cohomological descent (French)Johannes Huisman-
Segal spaces III (English)Santiago ToroSegalSpcIII
Interlude: Deformation theory (French)Grégoire MarcDeformations
Complete Segal spaces I (French)Pierre MartinezCompleteSegalSpc
Interlude: Good complexifications (French)Johannes Huisman-
Quasicategories I (English)Santiago Toro-
Quasicategories II (French)Pierre MartinezQuasicatsII
Interlude: Rigidification of quasicategories (French)Johannes Huisman-
Quasicategories III (English)Santiago ToroQuasicatsIII