Clarification and Coinduction of Tambara Functors

TL;DR

This paper introduces clarified Tambara functors, providing a new decomposition framework that enhances understanding of their structure, Morita invariance, and K-theory in equivariant algebraic contexts.

Contribution

It defines clarified Tambara functors and demonstrates their role in decomposing general Tambara functors, establishing a new perspective on their structure and invariance properties.

Findings

Every Tambara functor admits a decomposition into coinductions of clarified Tambara functors.

Projection onto the non-coinduced part defines a reflective localization called clarification.

Studied Morita invariance, K-theory, and properties of field-like and Nullstellensatzian 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 Morita invariance and $K$-theory of Tambara functors, field-like Tambara functors, and Nullstellensatzian clarified Tambara functors.