The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors

TL;DR

This paper investigates the right adjoint to a forgetful functor between incomplete Tambara functors associated with N_infinity operads, providing explicit computation for cyclic groups of prime order.

Contribution

It explicitly computes the right adjoint to the equivariant operadic forgetful functor for certain finite groups, clarifying its structure in specific cases.

Findings

Explicit computation of the right adjoint for cyclic groups of prime order.

Identification of the right adjoint as a functor related to norms in incomplete Tambara functors.

Enhanced understanding of adjoint functors in the context of equivariant operads.

Abstract

For $N_\infty$ operads $\mathcal O$ and $\mathcal O'$ such that there is an inclusion of the associated indexing systems, there is a forgetful functor from incomplete Tambara functors over $\mathcal O'$ to incomplete Tambara functors over $\mathcal O$. Roughly speaking, this functor forgets the norms in $\mathcal O'$ that are not present in $\mathcal O$. The forgetful functor has both a left and a right adjoint; the left adjoint is an operadic tensor product, but the right adjoint is more mysterious. We explicitly compute the right adjoint for finite cyclic groups of prime order.