The theory and practice of $\infty$-categories
Abstract
Subscription
Subscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-carmona-s25-join@mis.mpg.de
Next lectures
23.04.2025, 10:30 (Live Stream)
30.04.2025, 10:30 (Live Stream)
07.05.2025, 10:30 (G3 10 (Lecture hall))
14.05.2025, 10:30 (Live Stream)
21.05.2025, 10:30 (Live Stream)
28.05.2025, 10:30 (G3 10 (Lecture hall))
04.06.2025, 10:30 (G3 10 (Lecture hall))
11.06.2025, 10:30 (G3 10 (Lecture hall))
18.06.2025, 10:30 (G3 10 (Lecture hall))
25.06.2025, 10:30 (G3 10 (Lecture hall))
02.07.2025, 10:30 (Live Stream)
09.07.2025, 10:30 (Live Stream)
16.07.2025, 10:30 (G3 10 (Lecture hall))
The aim of this course is to introduce participants to the language of higher categories.
The theory of ordinary categories, as a unifying and organizational language, has proven to be incredibly useful. For instance, how could we understand Grothendieck's revolution in algebraic geometry without relying on categories?
Nevertheless, even when the theory of categories was in its infancy, mathematicians like Adams, Boardman, Eilenberg, Kan, and even Grothendieck himself were already advocating for the development of a richer theory-one that would be perfectly suited for homotopy and homological algebra: the theory of higher categories.
Although the formal development of higher category theory has been latent almost since the early days of ordinary category theory, as mentioned above, it has experienced a surge since the early years of this century. Numerous current developments in various areas of mathematics now rely on this new language. A couple of remarkable examples are the (sketch of a) proof of Cobordism Hypothesis by Lurie and the recent proof of the Geometric Langlands Conjecture by Gaitsgory and collaborators.
The plan for these lectures is as follows: (1) revisit the minimum requirements of homotopical algebra and simplicial sets, (2) introduce the homotopy theory of quasicategories as a model for higher categories, (3) understand the Higher Yoneda Lemma through fibrations, and (4) interpret ordinary categorical concepts, such as (co)limits, within the framework of higher categories.
Keywords
higher categories, homotopical algebra, simplicial sets.
Prerequisites
Background in Algebraic Topology and Category Theory might be helpful.
