Burnside and Mackey functors revisited

Tuesday 28th September, hosted on Zoom by Lille. Local organiser: Ivo Dell'Ambrogio

Videos and slides from this meeting are now available below.

Timetable (Paris time)

• 14h15-14h55: Nicolas Lemoine
• 15h-15h40: Marco Praderio
• 15h45-16h25: Zac Hall
• 16h25-17h: social
• 17-17h40: Ilaria Castellano
• 1745-18h25: Jun Maillard

Titles & abstracts

Nicolas Lemoine, Generalized biset functors from a category of fusion systems (Lemoine slides)

The biset category is constructed by taking finite groups as objects and Burnside modules $B(G,H)$ as sets of homomorphisms between them. From two fusion systems $\mathcal{F}_1$ and $\mathcal{F}_2$ on respective $p$-groups $S_1$ and $S_2$, we can associate the submodule $B(\mathcal{F}_1,\mathcal{F}_2)$ of $B(S_1,S_2)$ consisting of elements that are stable through fusion. This leads to a new category whose objects are fusion systems on $p$-groups. As for biset functors, we can look at linear functors from this category to the category of modules, and especially simple functors among them.

Marco Praderio, Mackey functors on saturated fusion systems (Praderio slides)

Using results from Puig as well as from Diaz and Libman I propose a candidate definition of sets over a saturated fusion system (analogous to the one of G-sets) and use that in order to define Mackey functors on a saturated fusion system. I go on to prove that, analogously to the finite group case, this definition comes associated with a classification of the simple Mackey functors as well as a Mackey algebra which is semi-simple given some condition on the characteristic of the base field.

Zac Hall, On the Burnside ring of a pro-fusion system (Hall slides)

Using the definition for the Burnside ring of a finite group, we consider extensions to the theory in terms of the Burnside ring of a profinite group (Dress and Siebeneicher) and the Burnside ring of a fusion system (Reeh). In addition we combine these notions to establish the Burnside of a pro-fusion system and compare results found in the finite case with the profinite setting.

Ilaria Castellano, On Mackey functors for TDLC-groups (Castellano slides)

The study of totally disconnected locally compact (= TDLC) groups is essential in the process of understanding the structure of general locally compact groups. Many ideas and techniques from diverse areas - such as profinite group theory, geometric group theory and dynamics - has been transferred to TDLC-groups producing significant advances with their structure theory. In this talk, we will discuss the possibility of transferring Mackey functors from (pro)finite groups to TDLC-groups. This is a joint work in progress with Nadia Mazza and Brita Nucinkis.

Jun Maillard, A stable elements formula for cohomological Mackey 2-functors (Maillard slides)

Mackey 2-functors are a categorification of Mackey functors, used to study categories associated to finite groups, such as the categories of linear representations or their derived categories. They exhibit several properties analogous to those of classical Mackey functors. I will present a stable elements formula for cohomological Mackey 2-functors, which can be seen as a categorification of the Cartan-Eilenberg stable elements formula for cohomological Mackey functors.