Parametrized homotopy theory

University of Copenhagen
4-8 September 2023

The goal of this workshop is to introduce and share new techniques in parametrized homotopy theory, developed in recent years.

This theory has been made more user-friendly thanks, among other things, to Lurie's groundbreaking foundational works in higher category theory. This has allowed it to be used in fields such as equivariant homotopy theory and motivic homotopy theory, and to unify aspects of these fields. The time seems right to bring these foundational methods to a broader audience.

The workshop will consist in three lecture series, as well as discussion sessions among participants. It will feature a foundational lecture series by Louis Martini and Sebastian Wolf, and lecture series about applications to motivic homotopy theory and equivariant homotopy theory by, respectively, Tom Bachmann and Emanuele Dotto.



    • Louis Martini and Sebastian Wolf: Internal higher category theory
      Abstract: Internal higher category theory is the study of categories internal to an ∞-topos B, or equivalently of sheaves of ∞-categories on B. Such structures often arise in geometry, for example in the form of categorified invariants or in situations where one wishes to keep track of additional (topological or algebraic) data, such as in condensed mathematics. Studying these objects through the lens of internal higher category theory essentially allows one to treat them in the same fashion as ordinary ∞-categories. In our talks we give an overview of the theory of internal higher categories, discussing both the core features such as internal adjunction, (co)limits and Yoneda’s lemma as well as more advanced topics such as internal presentability and the theory of internal topoi.

    • Tom Bachmann: Motivic colimits and free normed spectra

      Abstract: Parameterized colimits in motivic homotopy theory are sometimes called motivic colimits. They were arguably first discovered in the guise of (generalized) Thom spectra, though they can also be viewed as extending other previous constructions. One of their early successes was to describe free normed spectra in a way that looks very similar to classical free E-infinity ring spectra. This allows us to construct power operations on normed spectra that are closely related to Voevodsky's steenrod operations.

      I will give an introduction to these topics.

    • Emanuele Dotto: Calculus in equivariant homotopy theory

      Abstract: The aim of the lectures is to present equivariant homotopy theory as an example of a parametrized homotopy theory. We will start with a gentle introduction to equivariant homotopy theory and equivariant homology theories, and explain which issues can be approached naturally in the parametrized setting, in particular by employing parametrized limits and colimits. We will then use this theory to set up a version of functor calculus in equivariant homotopy theory









Monday Tuesday Wednesday Thursday Friday
Lecture hall: Aud 2
Lecture hall: Aud 8 Lecture hall: Aud 5 Lecture hall: Aud 2 Lecture hall: Aud 2
11:00-12:00 Registration 10:00-12:00 Discussion session in rooms 04.04.01 and 04.04.20 10:00-12:00 Discussion session in rooms 04.04.01 and 04.04.20 10:00-12:00 Discussion session in rooms 04.04.01 and 04.04.20 10:30-11:30 Bachmann 3
12:00 Lunch 12:00 Lunch 12:00 Lunch 12:00 Lunch 12:00 Lunch
13:30 Coffee 13:30-14:30 Bachmann 1 13:30-14:30 Martini/Wolf 3 14:00-15:00 Martini/Wolf 4 13:30-14:30 Dotto 4
15:00-16:00 Martini/Wolf 1 15:00-16:00 Martini/Wolf 2 15:00-16:00 Bachmann 2 15:30-16:30 Dotto 3
16:30-17:30 Dotto 1 16:30-17:30 Dotto 2
18:00 Pizza reception
Probably outside the HCØ building where the conference is hosted.

18:15 Walk from the Department to ...

19:00 Dinner
Food Club Nørrebro. Sortedam Dossering 7C
2200 København N











The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.







We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.













Registration has ended.