UCLA

We present a definition of Bi-Incomplete Generalized Mackey and Tambara Functors, which in special cases reduces to both the notion of (bi-incomplete) G-Mackey and G-Tambara functors and the notion of motivic Mackey and Tambara functors (as defined in [2]). We then prove a foundational theorem about these generalized objects, whose incarnation for G-Mackey and G-Tambara functors is due to Mazur [18], Hoyer [14], and Chan [10].

A G-Mackey functor is a product-preserving functor $\mathcal{A}_{G{-}\mathsf{set}} \to \mathsf{Set}$ satisfying a certain additivity condition (G-Tambara functors have a similar definition). Here $\mathcal{A}_{G{-}\mathsf{set}}$ is a certain category constructed from the category $G{-}\mathsf{set}$ of finite G-sets. The perspective we take is that the category G-set may be replaced here by another category C to obtain a generalized notion of Mackey/Tambara functor. We furthermore generalize the notion of bi-incompleteness introduced by Blumberg and Hill [4] to our setting. We spell out precisely the conditions needed on C to interpret the definitions of bi-incomplete Mackey and Tambara functors and prove the generalized Hoyer-Mazur theorem in question. Finally, we discuss applications of this generalized theorem to computations with motivic Tambara functors.