Stable Cohomotopy as $K \mathbb{F}_1$

Question

Various classical results suggest that stable cohomotopy may usefully be regarded as being the algebraic K-theory over the "field with one element" $\mathbb{F}_1$:

$$ K \mathbb{F}_1 \;\simeq\; \mathbb{S} $$

A summary table with pointers is on the nLab here.

Guillot has made this observation the starting point in his study of "Adams operations in cohomotopy" (arXiv:math/0612327), on which Jack Morava then based his note "Power operations and Absolute Geometry" (2012, pdf). Both of these seem to remain unpublished, though.

Has anyone picked this up? Are there any further developments along these lines?

Answer 1

Every symmetric monoidal category has a $K$-theory spectrum, and the $K$-theory spectrum of the category of finite sets and bijections is just the sphere spectrum. I believe that everything that you have mentioned can be done in a well-defined way form this point of view, without needing to refer to $\mathbb{F}_1$. I think that Stefan Schwede has written about $\beta$-rings in these terms, although I do not remember where. You pointed to a page on the nLab involving the Atiyah-Segal completion theorem and related things. As these can be formulated uniformly for a large class of groups $G$, it is best to discuss them in the context of global stable homotopy theory, which is another body of work of Schwede. I am not sure whether he has tied these things together.