Clarification and Coinduction of Tambara Functors

Abstract: Tambara functors are equivariant analogues of rings arising in representation theory and equivariant homotopy theory. We introduce the notion of a clarified Tambara functor and show that under mild conditions every Tambara functor admits a decomposition as a product of coinductions of clarified Tambara functors; projection onto the non-coinduced part defines a reflective localization we call clarification. Through this perspective we study Nakaoka spectra, Morita invariance and $K$-theory of Tambara functors, field-like and domain-like Tambara functors, $G$-$\mathbb{E}_\infty$-rings, Nullstellensatzian clarified Tambara functors, and transversal Tambara functors.