Green 2-functors

Abstract

We extend the theory of Mackey 2-functors provided by Balmer and Dell’Ambrogio [Mackey 2-functors and Mackey 2-motives, European Mathematical Society (EMS), Zürich, 2020] by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show how all classical Green functors familiar from representation theory and topology arise by decategorification, in various ways, of some Green 2-functor occurring in Nature.

Cohomological Mackey 2-Functors

“…Remark 5.9. Following up on the previous remark, suppose that M is a Green 2-functor in the sense of [7]. In particular, this means that the categories M(G) are monoidal, the restriction functors are strong monoidal and there is a projection formula i * (i * X ⊗ Y ) ∼ = X ⊗ i * Y for faithful i : H G. After applying K 0 , one gets an ordinary Mackey functor satisfying I G H R G H = [i * (1)] • id, where 1 ∈ M(H) denotes the tensor unit.…”

“…Example 3.9. Suppose that M is a Mackey 2-functor for (G; J) taking values in monoidal categories M(G) and strong monoidal functors u * (for instance, M could be a Green 2-functor in the sense of [7]). Then we may take X G = Y G := 1 to be the tensor unit of M(G) and λ u ,ρ u to be the coherent isomorphisms of u * .…”