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Higher structures in homotopy theory

2nd July 2018 to 6th July 2018
Stefan Schwede Rheinische Friedrich-Wilhelms-Universität Bonn
Clark Barwick Massachusetts Institute of Technology
Julie Bergner University of Virginia
Ieke Moerdijk Radboud Universiteit Nijmegen

Workshop Theme

Homotopy theory has covered a long distance since its origins, the classification of spaces up to homotopy equivalence. Over the years, various kinds of mathematical structures have been investigated from a homotopical perspective, such as equivariant spaces, rings, C^*-algebras, or varieties. Many different approaches of how to formalize what a “homotopy theory” is were proposed, the most prominent ones being the notions of model category and ∞-caterroy.

The relationship between the different ways to formalize a homotopy theory is now well understood; indeed, for comparing different concepts of homotopy theories, one often wants to consider all of them together as another homotopy theory, i.e., a ‘homotopy theory of homotopy theories’. Somewhat surprisingly, most of the concepts organize themselves into a Quillen model category, and the various approaches are Quillen equivalent. After these individual comparison results, Töen was even able to axiomatically characterize a homotopy theory of homotopy theories.

The homotopy theory of homotopy theories is only the first step in a hierarchy of interesting structures, namely the homotopy theoretic approach to higher categories. From this broader perspective, homotopy theories are just (∞, 1)-categories, where the ∞ indatices a structure with higher morphisms of all levels, and the 1 refers to the fact that all 1-morphisms and higher morphisms are weakly invertible. There are now ways to give rigorous meaning to the notion of (∞, n)-categories i.e., where only higher morphisms in level n and above are invertible. Having a rigorous model category of (∞,n)-categories is a cornerstone for the modern approach to topological field theory, thereby unifying categorical considerations with those of homotopy and manifold theory.

This workshop consists of lecture series as well as individual research talks. The introductory series will explain some of the key methods relevant to many parts of the overarching program; they are intended to invite graduate students and postdocs into the field, as well as to strengthen the common ground of the program participants. The individual talks will inform us about recent developments about higher structures in homotopy theory.

Deadline for applications: 02 March 2018

Apply now

Please note members of Cambridge University are welcome to turn up and sign in as a non-registered attendee on the day(s) during the workshop and attend the lecture(s). Please note that we cannot provide you with any support including name badge, meals or accommodation.

In addition to visiting the INI, there are multiple ways in which you can participate remotely.



Registration Only    
  • Registration Package: £227
  • Student Registration Package: £177

The Registration Package includes admission to all seminars, lunches and refreshments on the days that lectures take place (Monday - Friday), wine reception and formal dinner, but does not include other meals or accommodation.

Registration and Accommodation
  • Accommodation Package: £581

The Accommodation Package includes a registration fee, bed and breakfast accommodation at TBC from the evening of Sunday 1st July 2018 to breakfast on Saturday 7th July 2018, together with lunches and refreshments during the days that lectures take place (Monday - Friday). The formal dinner is also included, but no other evening meals.

Formal Dinner Only
  • Formal Dinner: £50

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event. For all remaining participants who would like to attend, such as programme participants, the above charge will apply.


Accommodation in single study bedrooms with shared facilities, breakfast and lunches are provided at TBC,



Lunch will be served at Wolfson Court in the Cafeteria from 12:30 to 13:30 on days that lectures take place.

  • Accommodation Package and Registration Package participants should present their badge as payment for their meal
  • Those issued with a blue Institute door entrance card can add money onto the card via the Porters' Lodge at Wolfson Court
  • Other participants must purchase their meal using their dining card via the Porters' Lodge (forms can be found on the registration desk or at the Porters' Lodge)

Evening Meal

Participants are free to make their own arrangements for dinner.

The Formal Dinner date, time and venue will be confirmed shortly.  Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event.  All remaining participants are encouraged to attend at a charge (price £50).

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons