On the fractions of semi-Mackey and Tambara functors

TL;DR

This paper introduces the concept of fractions of semi-Mackey and Tambara functors, extending algebraic fraction constructions to these $G$-bivariant functors for finite groups.

Contribution

It generalizes the notion of fractions of rings to the setting of semi-Mackey and Tambara functors, providing a new algebraic framework for $G$-bivariant functors.

Findings

Defined the fraction of a Tambara functor by a multiplicative semi-Mackey subfunctor.

Extended algebraic fraction concepts to $G$-bivariant functors.

Provided foundational results for algebraic manipulations of these functors.

Abstract

For a finite group $G$, a semi-Mackey (resp. Tambara) functor is regarded as a $G$-bivariant analog of a commutative monoid (resp. ring). As such, some naive algebraic constructions are generalized to this $G$-bivariant setting. In this article, as a $G$-bivariant analog of the fraction of a ring, we consider {\it fraction} of a Tambara (and a semi-Mackey) functor, by a multiplicative semi-Mackey subfunctor.