Bi-Incomplete Tambara Functors as Coherent Monoids

Abstract

By considering arbitrary finite G-sets for a finite group G rather than just the orbits G/H for subgroups H we realise the norm functor of Hoyer and the box product of Tambara functors as instances of the same construction. Using similar constructions and a new bicategorical framework we give characterisations of incomplete Mackey functors and bi-incomplete Tambara functors as coherent monoids. A recent conjecture of Blumberg and Hill is a corollary of our characterisation of bi-incomplete Tambara functors. In particular, our characterisation of (O'_a, O'_m)-Tambara functors as coherent (O'_a, O'_m)-monoids in (O_a,O_m)-Tambara functors allows both additive and multiplicative structure to be added simultaneously. We also provide a notion of module category over a tight symmetric bimonoidal category and use this to relate our characterisation to work of Hill and Hopkins on equivariant symmetric monoidal structures. Show more