Topology Seminar
Speaker: Aparajita Karmakar, IIT Bombay
Host: Rekha Santhanam
Title: Presheaves and Diagrams on the Burnside 2-categroy
Time, day and date: 11:00:00 AM – 12:00:00 PM, Friday, November 28
Venue: Room 113
Abstract: Mackey functors play a central role in equivariant algebraic topology: classically, they are additive (abelian-group-valued) functors on the Burnside category of a group. In this talk we introduce and explore higher analogues of Mackey functors. Concretely, we replace the Burnside category by the Burnside 2-category, and replace the target category of abelian groups by the 2-categorical world of strict symmetric monoidal (permutative) categories. Our objects of study are then enriched presheaves and diagrams on the Burnside 2-category.
Our principal goal is to establish an equivalence of homotopy theories between these enriched presheaf and diagram categories, paralleling the classical self-duality of the Burnside category. In the 1-categorical setting this self-duality yields both covariant and contravariant structure for Mackey functors. However, in the 2-categorical context the duality fails to hold strictly. To overcome this obstacle, we employ a strictification technique on coloured-operad pseudo-algebras restoring the duality up to equivalence. This approach builds on recent advances in the homotopy theory of enriched Mackey functors.
Thus, the talk will outline the construction of higher Mackey functors, explain why 2-categorical self-duality fails, and present the strictification via operadic methods that recovers an equivalence of homotopy theories — shedding light on how classical Mackey‐theoretic phenomena generalize in a higher-categorical setting.