Algebra/Topology Seminar

Speaker: Sil Linskens

Title: Higher parametrized semiadditivity

Abstract: Given a semiadditive category, every object uniquely obtains the
structure of a commutative monoid. This simple fact turns out to be
crucial in the higher categorical context, where endowing an object with
the structure of a commutative monoid can be highly nontrivial endeavour.
More recently, the notion of higher semiadditivity has attracted
considerable attention, primarily because many categories appearing in
chromatic homotopy theory are higher semiadditive. Higher semiadditivity
again automatically endows objects with considerable algebraic structure
which has been crucial for many recent breakthroughs in chromatic homotopy
theory, for example the recent disproof of the telescope conjecture.
However to leverage this algebraic structure we require a
"generator-and-relations style" description of the operations it entails.
This is given by a theorem of Harpaz, which gives a formula for the
universal higher semiadditive category on a point. In this talk I will
discuss a parametrised analogue of higher semiadditivity, which
simultaneously subsumes higher semiadditivity and the Wirthmüller
isomorphisms in equivariant homotopy theory. I will then present the
analog of the theorem of Harpaz. This is joint work with Bastiaan Cnossen
and Tobias Lenz.