The "traditional" approach to the so-called "field with one element" $\mathbb{F}_{1}$ is by using monoids, or, to put it in another way, by forgetting the additive structure of rings. In Deitmar's approach to the subject, $\mathbb{F}_{1}$ is declared to be the trivial monoid $\{1\}$. Furthermore, $\mathbb{F}_{1}$-modules (or vector spaces, since we are calling $\mathbb{F}_{1}$ a "field" after all, although it's not really a field) are pointed sets, and $\mathbb{F}_{1}$-algebras are commutative monoids. The motivations for these ideas may predate Deitmar's work, and these ideas may also be found in this unpublished work by Kapranov and Smirnov.
Deitmar's approach is kind of a "template" for later approaches to the field with one element, such as the one by Toen and Vaquie. However, one approach that is different is the one by Borger. In this approach, $\mathbb{F}_{1}$-algebras are lambda rings, which are rings together with Frobenius lifts. The idea is that the extra structure provided by the Frobenius lifts serves as descent data to $\mathbb{F}_{1}$, while the forgetful functor that forgets that extra structure is the base change to $\mathbb{Z}$, i.e. it is the functor $-\otimes_{\mathbb{F}_{1}}\mathbb{Z}$.
In other words, the $\mathbb{F}_{1}$-algebras have less structure than rings ($\mathbb{Z}$-algebras) in Deitmar's approach, while they have more structure in Borger's approach. But it is often said that the two are related; in fact the nLab article on the field with one element claims that Borger's approach subsumes many aspects of previous approaches. For instance, in Lieven Le Bruyn's paper, Absolute Geometry and the Habiro Topology, it is shown that in Borger's approach one can arrive at the idea of Soule (also apparently going back to Kapranov and Smirnov) that
$\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[\mu_{n}]$.
In what other ways is Borger's approach related to these other approaches? With ideas similar to lambda rings apparently finding their way into algebraic topology and homotopy theory (e.g. Adams Operations in Cohomotopy by Guillot and Power Operations and Absolute Geometry by Morava and Santhanam), do we have an analogue of Toen and Vaquie's "schemes over $\mathbb{S}$" and related homotopy-theoretic constructions? What about Connes and Consani's "arithmetic site?" Can we obtain analogous constructions but with lambda rings instead of commutative monoids playing the role of $\mathbb{F}_{1}$-algebras?
Best Answer
Given a monoid $M$, the ring $\mathbb{Z}.M$ of $\mathbb{Z}$-linear combinations of elements of $M$ has a natural $\lambda$-ring structure given by setting the elements $m \in M$ to have rank $1$ (i.e. $\lambda^k(m)=0$ for all $k>1$). We then have functors $$ \text{Monoids} \xrightarrow{\mathbb{Z}.-} \lambda\text{-rings} \xrightarrow{\text{forget}} \text{Rings} $$ (monoid and ring structures assumed commutative) with right adjoints $$ \text{Monoids} \xleftarrow{\text{Rank } 1 \text{ elements}} \lambda\text{-rings} \xleftarrow{W} \text{Rings}, $$ where $W$ denotes the big Witt vectors.
Descent data from rings $A$ to monoids take the form of ring homomorphisms $A \to \mathbb{Z}.A$ for which the two compositions $A \to \mathbb{Z}.A \to \mathbb{Z}.(\mathbb{Z}.A)$ agree. On the other hand, a $\lambda$-ring structure is a ring homomorphism $A \to WA$ for which the two compositions $A \to WA \to W(WA)$ agree. As for instance in Cuntz-Deninger arXiv:1410.5249, WA is often a natural completion of $\mathbb{Z}.A$.
If we think of $\mathbb{F}_1$-algebras as monoids, then this suggests we should think of $\lambda$-rings as rings with formal descent data to $\mathbb{F}_1$, or as affine schemes over an analogue $\mathrm{dR}(\mathbb{Z}/\mathbb{F}_1)$ of Carlos Simpson's de Rham stack.
$\lambda$-rings don't easily fit into the Toën-Vaquié setting, since they don't come from an obvious monoidal category. However, to build schemes (resp. DM stacks, resp. Artin stacks) from affine objects, you only really need a good notion of open immersions (resp. étale maps, resp. smooth maps). Things probably work satisfactorily if you just require the underlying morphism of affine schemes to be an open immersion. There are good notions of modules $M$ over $\lambda$-rings $A$ (just put operations $\lambda^k$ on $M$ such that $A \oplus M$ becomes a $\lambda$-ring, which amounts to saying something like $\lambda^k$ is $\Psi^k$-semilinear), giving you a theory of quasi-coherent sheaves as well.