Higher category theory

Working group on $\mathrm{\infty }\backslash infty$-categories that I organized with Johannes Huisman and Pierre Martinez at the LMBA in Brest. The sessions were held in Brest and broadcast via Zoom for interested participants in Vannes. Most of them were recorded: the videos are linked in the table below.

Aim

The aim was to explore the main models of $(\mathrm{\infty },1)(\backslash infty,\; 1)$-categories, with emphasis on the Boardman–Vogt–Joyal–Lurie model (quasi-categories), and then to move towards higher algebra [6]. We followed, more or less faithfully, Julia Bergner's book [2] from Chapter 4 on.

Prerequisites

Basic category theory, model categories and simplicial homotopy theory (see the references below). The introductory session motivated the study of $\mathrm{\infty }\backslash infty$-categories by going back to the modern conception of homotopy theory (“à la Quillen”), with reminders during the sessions when necessary. For those not yet familiar with these ideas, we recommended the first three chapters of [2], the beginning of [5] (up to page 23), or the survey [3].

Programme

1. Introduction and motivation; the language of $\mathrm{\infty }\backslash infty$-categories; simplicial categories and enrichment in Kan complexes.
2. Segal spaces.
3. Rezk spaces (complete Segal spaces).
4. The comparison theorems of Bergner, Joyal–Tierney, Lurie and Dugger–Spivak (and Barwick–Kan).
5. Quasi-categories: Joyal's theorem, Dwyer–Kan localization, Bousfield localization in the $\mathrm{\infty }\backslash infty$-categorical setting, Quillen's Theorem A, adjunctions and the Yoneda lemma for $\mathrm{\infty }\backslash infty$-categories, etc. (as in Bergner [2], Land [4] or Lurie [5]).

Lectures and videos

See also the working group on $\mathrm{\infty }\backslash infty$-categories at Paris 13, whose website is very complete.

References

1. J. C. Baez and M. Shulman, Lectures on n-categories and cohomology.
2. J. Bergner, The homotopy theory of $(\mathrm{\infty },1)(\backslash infty,\; 1)$-categories.
3. J. Bergner, A survey of $(\mathrm{\infty },1)(\backslash infty,\; 1)$-categories.
4. M. Land, Introduction to $\mathrm{\infty }\backslash infty$-categories.
5. J. Lurie, Higher topos theory.
6. J. Lurie, Higher algebra.

Other references:

7. D.-C. Cisinski, Higher categories and homotopical algebra.
8. P. Goerss and R. Jardine, Simplicial homotopy theory.
9. M. Hovey, Model categories.
10. A. Joyal, Notes on quasi-categories.
11. E. Riehl, Category theory in context.