Workshop on Gray Products and Lax Constructions

Picture of Raitenhaslach

Time and Place

This workshop will take place February 24 - 27, 2025 at the TUM Akademiezentrum Raitenhaslach , a former Cistercian monastery.

Speakers and Format

We will have two experts who will be mentoring us during the workshop: Fernando Abellán and Hadrian Heine. Additionally, we will have David Gepner as a special guest for the last two days.

Mathematical Program

Participant List (click to expand)

Fernando Abellán

Victor Carmona

David Gepner

Johannes Glossner

Jonte Gödicke

Ben Haïoun

Hadrian Heine

Patrick Kinnear

Naruki Masuda

Angus Rush

Claudia Scheimbauer

Pelle Steffens

Walker Stern

Will Stewart

Anja Svraka

Jackson van Dyke

Tashi Walde

Yang Yang

Markus Zetto

Logistical Information

The conference location is a bit remote and not trivial to reach with public transport. For this reason we have booked busses from Garching (suburb of Munich, easily reachable by train and not far from the airport) to Raitenhaslach, and vice versa.

Departure on Monday (February, 24th) will be at 10:00. We arrive in Raitenhaslach for lunch and the mathematical program will start afterwards. On Thursday (February, 27th), we leave after lunch and the busses will reach Garching around 15:00. Unfortunately this means that if you are travelling from outside of Bavaria, you might need to find accommodation for one night (23-24) in Munich to take the bus in the morning. Hopefully the return trip is such that most of you will be able to return home on the same day.

Schedule

FEBRUARY 24
10:00-11:30 BUS: Garching → Monastery (DEPARTS 10:00)
12:30-13:30 Lunch
13:45-14:45 Speaker 1
Gray's original tensor product

The Gray tensor product was first introduced in the setting of strict 2- categories, where it is far more amenable to direct computation. This talk will survey the original definition and construction from [Gra74] and its first properties - in particular, the connection to lax transformations. Time permitting, the talk may also cover homotopical aspects of Gray's construction (e.g., the Quillen bifunctoriality of [Lac04]) or variants (such as the variant representing pseudonaturality of [GPS95]).

15:00-16:00 Ben Haïoun
The scaled simplicial set approach

This talk should briefly introduce scaled simplicial sets, and the construction of their Gray product, following section 2 of [GHL21]. It should then survey the relation to functor categories with lax (or oplax) transformations, and, time permitting, describe the universal property in terms of oplax normal functors.

16:00-16:30 Coffee
17:45-18:00 BUS: Monastery → Hotel
18:30-19:30 Dinner
FEBRUARY 25
9:10-9:30 BUS: Hotel → Monastery
9:45-10:45 Markus Zetto
The model-independent approach

The second talk of this series should survey the results of [CM23] providing a model-independent characterization of the Gray product of (∞, 2)-categories in terms of cubes. In particular, the talk should cover the universal properties of Corollary 3.5 in op. cit., which characterize the Gray product in terms of its operation on cubes.

11:15-12:15 Speaker 4
The Gray tensor product of (∞, n)-categories

This talk should introduce cubical sets and marked cubical sets, and discuss the way in which they are expected to model (∞, ∞)-categories. The speaker should then cover the construction of the Gray tensor products of marked cubical sets in section 2.2 of [CKM20]. Time permitting, the talk can also discuss some of the model-categorical aspects of the Gray product of marked cubical sets.

12:30-13:30 Lunch
13:45-14-45 Jonte Gödicke
The Gray tensor product of (Steiner) ω-categories I

This talk will survey the theory of Steiner ω-categories and their relation to augmented directed chain complexes, following the background in section 1 of [AGOR23], supplemented by the original reference [Ste04].

15:00-16:00 Naruki Masuda
The Gray tensor product of (Steiner) ω-categories II

Using the definitions and machinery established in the previous talk, this talk should introduce the tensor products of augmented directed complexes and strong Steiner ω-categories (section 7 of [Ste04]). It should discuss the monoidal closedness of this product, and, time permitting, its relation to the Gray product constructed in [AS93].

16:00-16:30 Coffee
17:45-18:00 BUS: Monastery → Hotel
18:30-19:30 Dinner
FEBRUARY 26
9:10-9:30 BUS: Hotel → Monastery
9:45-10:45 Hadrian Heine
Talk I

11:15-12:15 Hadrian Heine
Talk II

12:30-13:30 Lunch
13:45-14:45 Hadrian Heine
Talk III

15:00-16:00 Fernando Abellán
Talk I

16:00-16:30 Coffee
17:45-18:00 BUS: Monastery → Hotel
18:30-19:30 Dinner
FEBRUARY 27
9:10-9:30 BUS: Hotel → Monastery
9:45-10:45 Fernando Abellán
Talk II

11:00-12:00 Fernando Abellán
Talk III

12:00-13:00 Lunch
13:00-15:00 BUS: Monastery → Garching (ARRIVING AROUND 15:00)