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From Higher Invariants

HIOB 19/20: Étale Homotopy Type

The topic of this semester's HIOB (Higher Invariants Oberseminar) is the étale homotopy type of schemes.

The first few talks will be about the classical étale fundamental group of schemes its pro- étale cousin as developed by Bhatt and Scholze. These invariants can be seen as part of the homotopy type and already contain a lot (and sometimes even all) of the information. The second half is devoted to introducing the étale homotopy type of a scheme and finally prove the generalized Riemann existence theorem. For this we develop a very general form of Galois-Theory due to Lurie and Hoyois which might be of interest on its own.

This seminar is designed as a learning seminar for anyone interested. Prerequisites for the first half will only be very basic algebraic geometry. For the second half we chose a very modern approach which requires some fluency with infinity categories. Yet we hope that anyone who is willing to accept some blackboxes can follow the whole seminar. As we will see throughout the seminar, infinity-topoi are simply the most natural setting for defining and studying the étale homotopy type of schemes.

Dates and location

Mondays, 12-14, SFB Seminar Room.


The detailed program can be found here(updated 13.10.).


If you are interested in giving a talk in this seminar, please contact Benedikt Preis.

Talk   Date Title Speaker
1 14. October Introduction and overview Benedikt Preis
2 21. October Étale Fundamental Group I: Abstract Galois Theory Marco Volpe
3 28. October Étale Fundamental Group II: Definition Yassin Mousa
4 4. November Étale Fundamental Group III: Some key properties Johannes Sprang
5 11. November The Pro-Étale Topology and infinite Galois Categories Marta Barigozzi
6 18. November The Pro-Étale fundamental group Han-Ung Kufner
7 25. November Pro-∞-categories and Shape Benedikt Preis
8 2. December Shape and Torsors Charanya Ravi
9 9. December no seminar due to windberg no speaker
10 16. December Finite Galois Theory Maria Yakerson
11 13. January Cohomology in ∞-topoi Denis Nardin
12 20. January Infinite Galois Theory and the shape of nice topological spaces Marco Volpe
13 27. January Discussion : Next semester's topic Starts at 13:00
14 3. February Talk postponed:The Riemann Existence Theorem Denis-Charles Cisinski