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1. Bi-incomplete Tambara functors

   Blumberg, Andrew J. ; Hill, Michael A. ( January 2021 , Equivariant Topology and Derived Algebra)

   Balchin, S. ; Barnes, D. ; Kędziorek, M. ; Szymik, M. (Ed.)

   For an equivariant commutative ring spectrum R, \pi_0 R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N_\infty ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and \pi_0 R has the structure of an incomplete Tambara functor. However, if R is an N_\infty ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures. 

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

   https://doi.org/10.1090/conm/729/14691  

   Blumberg, Andrew J. ; Hill, Michael A. ( January 2019 , Contemporary mathematics)

   For N∞ operads O and O' such that there is an inclusion of the associated indexing systems, there is a forgetful functor from incomplete Tambara functors over O' to incomplete Tambara functors over O. Roughly speaking, this functor forgets the norms in O' that are not present in 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. 

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3. Bi-incomplete Tambara functors as𝒪-commutative monoids

   https://doi.org/10.2140/tunis.2024.6.1  

   Chan, David ( January 2024 , Tunisian Journal of Mathematics)

   Tambara functors are an equivariant generalization of rings that appear as the homotopy groups of genuine equivariant commutative ring spectra. In recent work, Blumberg and Hill have studied the corresponding algebraic structures, called bi-incomplete Tambara functors, that arise from ring spectra indexed on incomplete G-universes. We answer a conjecture of Blumberg and Hill by proving a generalization of the Hoyer–Mazur theorem in the bi-incomplete setting. Bi-incomplete Tambara functors are characterized by indexing categories which parametrize incomplete systems of norms and transfers. In the course of our work, we develop several new tools for studying these indexing categories. In particular, we provide an easily checked, combinatorial characterization of when two indexing categories are compatible in the sense of Blumberg and Hill. 

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4. Examples of étale extensions of Green functors

   https://doi.org/10.1090/bproc/189  

   Lindenstrauss, Ayelet ; Richter, Birgit ; Zou, Foling ( July 2024 , Proceedings of the American Mathematical Society, Series B)

   We provide new examples of étale extensions of Green functors by transferring classical examples of étale extensions to the equivariant setting. Our examples are Tambara functors, and we prove Green étaleness for them, which implies Tambara étaleness. We show that every$C_2$-Galois extensions of fields gives rise to an étale extension of$C_2$-Green functors. Here we associate the constant Tambara functor to the base field and the fix-Tambara functor to the extension. We also prove that all$C_n$-Kummer extensions give rise to étale extensions for arbitrary finite$n$. Étale extensions of fields induce étale extension of$G$-Green functors for any finite group$G$by passing to the corresponding constant$G$-Tambara functors.

    

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5. COHOMOLOGICAL MACKEY 2-FUNCTORS

   https://doi.org/10.1017/S1474748022000408  

   Balmer, Paul ; Dell’Ambrogio, Ivo ( January 2024 , Journal of the Institute of Mathematics of Jussieu)

   We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

    

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