Seminar: Parametrized semiadditivity

Date, time, location

The seminar takes place weekly on Tuesday from 16:15 - 17:45 in M101, starting on Tuesday October 15th.

The seminar will also be live-streamed via Zoom: you may find the link here . Alternatively, use the following:

Meeting ID: 612 3361 0250 

Passcode: 832594

The program for the last couple of weeks looks as follows:

• January 14th: No seminar

• January 21st: The universal bi-adjointable functor (Bastiaan Cnossen)

• January 28th: Higher semiadditivity for (infty,k)-categories (Tashi Walde)

• February 4th: Lax (semi)additivity (Johannes Gloßner)

Notes for some of the talks

Here are notes for some of the talks that the speakers shared with me:

• Here are some notes of a talk I once gave, which very roughly cover talks 1-3 of the seminar.

• Talk 5: Higher semiadditivity in chromatic homotopy theory (Sil Linskens) 

• Talk 6: Cardinalities in chromatic homotopy theory (Pier Federico Pacchiarotti) 

• Talk 7: Semiadditive height (Tashi Walde) 

• Talk 10: Ambidexterity and transchromatic homotopy theory (Sil Linskens) 

• Talk 11: Universal property of span 2-categories (Bastiaan Cnossen)

• Talk 12: m-semiadditive (oo,n)-categories (Tashi Walde) 

Content

We will give an introduction to the theory of parametrized semiadditivity, following the treatment of [CLL24]. We will further discuss various examples and applications of this theory. The program may be found here, and consists of:

• The definition of parametrized semiadditivity, characterizations and examples;

• The definition of the parametrized span category and a proof of its universal property;

• The definition of parametrized commutative monoids, their universal properties, and the relation with Mackey functors / sheaves with transfers;

• A discussion of equivariant semiadditivity;

• A discussion of higher semiadditivity in chromatic homotopy theory (see here for Sil's notes);

• The notions of cardinality and semiadditive height;

• The semiadditive Fourier transform;

• A discussion of parametrized semiadditivity in the Kasparov K-theory categories;

• Ambidexterity and transchromatic character theory;

• Higher semiadditivity for (infty,k)-categories.

Program

You may find the most recent version of the program here.