So after the sentence

The perspective that the K-theory $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be stable Cohomotopy has been highlighted in (Deitmar 06, p. 2, Guillot 06).

I have added this sentence:

Generalized to equivariant stable homotopy theory, the statement that equivariant K-theory $K_G \mathbb{F}_1$ over $\mathbb{F}_1$ should be equivariant stable Cohomotopy is discussed in Chu-Lorscheid-Santhanam 10, 5.3.

Am adding this now also to the respective points at *stable Cohomotopy* and *equivariant stable Cohomotopy*.

added pointer to

- Chenghao Chu, Oliver Lorscheid, Rekha Santhanam,
*Sheaves and K-theory for $\mathbb{F}_1$-schemes*(https://arxiv.org/abs/1010.2896)

Yes p.92.

]]>Thanks for the pointer to Berman (p. 92 I suppose?) I have added pointers here and here

]]>Using cached pages, I see even I knew it in 2005. In 2002, (TWF184) John was speaking of the projective spaces over $F_1$ as finite sets. But as you say in #26, it’s folklore.

On my travels, I see that John Berman in his thesis (Chap. 4) – Categorified algebra and equivariant homotopy theory – is after

a picture that unites noncommutative motivic homotopy theory (arising from the Eilenberg-Watts theorem) with global equivariant homotopy theory

which pairs ’Algebraic K-theory’ with ’Equivariant sphere spectrum’.

]]>We anticipate that this analogy could be made rigorous by an understanding of noncommutative motives over the field with one element.

added pointer also to

- Jack Morava, Rekha Santhanam,
*Power operations and absolute geometry*, 2012 (pdf)

which sort of builds on

- Pierre Guillot,
*Adams operations in cohomotopy*(arXiv:0612327)

and his interpretation of stable cohomotopy as algebraic K-theory over $\mathbb{F}_1$ (and which seems to be the only article citing it?!)

]]>Gee, that sounds bad. Hopefully they have a backup…

]]>According to John on Azimuth, yesterday:

You may have noticed that all the pictures in my lectures are gone. That’s because they’re held on the U.C. Riverside math department server math.ucr.edu, and this server is down. It’s been down for several days since a power outage on campus. I’m trying to get people to restart it, but so far with no luck!

This explains the TWF observation of David_R.

]]>TWF 187 perhaps but for some reason I see a blank at John’s page.

]]>Thanks. I suppose the statement is folklore, but it would be nice to find some “canonical” reference that could be cited in publications.

]]>John Baez told me about modules over $F_1$ being pointed sets at least as far back as 2006. I’ll see if I can trace back.

]]>Hi David,

so Priddy in his article, when he says “K-theory”, means *K-theory of a permutative category*, and the line you quote then refers to this result by Segal: $K(FinSet) \simeq \mathbb{S}$.

What I am trying to write out is how this is the first step in the “proof” that $\mathbb{S} \simeq K \mathbb{F}_1 \coloneqq K(\mathbb{F}_1 Mod)$, where on the right I mean the K-theory of the permutative category of $\mathbb{F}_1$-modules, in accord with this example.

Therefore my question for a good source of the claim or definiton that $\mathbb{F}_1 Mod = FinSet$ (or maybe $= FinSet^{\ast/}$, but that doesn’t make a difference for the K-theory of a permutative category, since it only sees the isomorphisms).

Now looking more closely at Noah Snyder’s old post (here) I get the impression that his argument is meant to be a review of Durov 07.

Now in Durov 07 I see mentioning of this statement as a parenthetical inside remark 2.5.6. Does Durov’s article make this more explicit in the sections before?

]]>The latter from 1973 starts out with

]]>K-theory of the category of finite sets and permutations together with the composition law of disjoint union corresponds to stable cohomotopy theory.

David, our messages overlapped! Thanks for the pointer to Deitmar’s article!

]]>am starting a section “Algebra over $\mathbb{F}_1$” (here), to record some basics.

What’s a canonical source to cite for the idea or fact that modules over $\mathbb{F}_1$ are pointed finite sets (to go here)?

Sometimes this is apparently attributed to

- Christophe Soulé,
*Les varietes sur le corps a un element*Mosc. Math. J., 4(1):217-244, 312, 2004 (pdf)

but I didn’t spot that exact statement in there. Wikipedia credits Noad Snyder’s secret seminar here (at the bottom). (?)

]]>You saw this MO question? Peter Arndt’s answer mentions Deitmar’s role.

Riepe suggests Priddy “Transfer, symmetric groups, and stable homotopy theory” for an early appearance.

]]>Which authors admit that algebraic K-theory over the field with one element is… stable cohomotopy?

I see Guillot 06 “Adams operations in cohomotopy” says this

Apparently also Deitmar 06, but now I don’t find this.

]]>Fixed a link

]]>I added a reference

- Oliver Lorscheid,
*$\mathbb{F}_1$ for everyone*, 2018 (arXiv:1801.05337)

I have added pointer to Durov’s text and in particular pointer to equation (0.4.24.2) there, where the definition of modules over $\mathbb{F}_1$ being modules over the maybe monad finally appears. (Also added the analogous pointer to *maybe monad*.)

Does what Connes writes on p. 3 of this help us a little with the conundrum of #14?

$|\mathbb{P}^d(\mathbb{F}_1)| = \frac{3^{d-1} - 1}{2} = |\mathbb{P}^d(\mathbb{F}_3)|.$Some restriction of points to degree zero gets us to the $d + 1$ expected.

]]>Hm, actually Michael Rosen’s book on the analogy “Number theory in function fields” starts with saying on its pages 1 and 2 that $\mathbb{Z}$ is like $F_q[x]$ for q=3. Er? Maybe it’s time for me to get another coffee.

]]>The function field analogy starts with saying that $Z$ is like $F_q[x]$ for $q\to 1$.

]]>It would be very nice if the affine line over $\mathbb{F}_1$ “is” $\operatorname{Spec} \mathbb{Z}$… but that seems unlikely to me, because $\operatorname{Spec} \mathbb{Z}$ is somehow not very “homogeneous”. I also would not say that the function field analogy starts with $\mathbb{F}_1$; if anything, it is $\mathbb{F}_1$ that starts with the function field analogy!

]]>