Modern algebraic geometry in the tradition of Grothendieck and his school is typically formulated in terms of commutative rings, their modules, and things defined in terms of them, such as locally-ringed spaces, i.e. schemes. To a working mathematician for whom "ring" and "algebra" are roughly synonymous, this is satisfying. But for someone like me with a background in computer science and category theory, when I hear "algebra" I don't think "ring" — I think "monad."
For someone (such as myself) for whom "algebra" conjures images not of matrices and vector spaces, but of *universal algebra* in the tradition of Lawvere, a monad $M$ encodes an algebraic theory: a specification of some abstract operations (or *effects*) and the equations they satisfy. So, from this perspective, the question arises: how much of classical algebraic geometry can be formulated more generally in terms of (some subclass of) monads, their algebras, etc.?
This question is not merely of interest for philosophical or aesthetic reasons. Generalizing algebraic geometry to a more general class of algebraic objects is of critical interest for the programme of finding the so-called "*field with one element*" $\mathbb{F}_1$,[^1] which may offer a promising path to age-old questions such as the *Riemann Hypothesis*. The idea is this: Weil proved analogues of the Riemann Hypothesis for algebraic curves over finite fields, and Grothendieck and collaborators later expanded these results to arbitrary varieties over finite fields via cohomological machinery. If the Riemann zeta function could be realized as the *Hasse-Weil zeta function* of some algebraic curve/variety "over $\mathbb{F}_1$," one might be able to import that machinery and prove the Riemann Hypothesis itself. The search for $\mathbb{F}_1$ is in some sense a search for the right generalization of algebraic geometry beyond commutative rings that allows for such cohomological tools to be brought over from the traditional realm of commutative rings and schemes.
[^1]: Note that this name is decidedly metaphorical/evocative rather than literal — $\mathbb{F}_1$ is not expected to be a field in any traditional sense, since, by definition, all fields have at least two elements (0 and 1). Rather, $\mathbb{F}_1$ is expected to be some kind of "field-like" object that captures the combinatorial essence of algebraic geometry without the full structure of a field.
This post sketches what I've found in the course of trying to understand all this. The punchline — which I'll build up to over the course of the post — is that the *maybe monad* $\mathsf{mb}$, whose algebras are just pointed sets, emerges naturally as a candidate for $\mathbb{F}_1$ (though it's worth noting that, by now, most active research on the field with one element has moved beyond or generalized this idea; however, it remains foundational to these subsequent developments), and the Riemann zeta function indeed falls out as a Hasse-Weil zeta function in this setting.
---
# 1. Some Universal Algebra
## Finitary Monads and Lawvere Theories
I'll assume familiarity with monads and their Eilenberg-Moore algebras. Recall that a monad $M = (M, \eta, \mu)$ on $\mathbf{Set}$ is *finitary* if it preserves filtered colimits — equivalently, if it is determined by its restriction to finite sets. Finitary monads on $\mathbf{Set}$ correspond exactly to *Lawvere theories*: single-sorted algebraic theories presented by operations of finite arity and equational axioms. The Eilenberg-Moore algebras of $M$ are then the models of the corresponding theory, i.e. sets equipped with operations satisfying the given equations. E.g. there is a Lawvere theory of Abelian groups, which is (unsurprisingly) represented by the free abelian group monad $\mathsf{Mod}_\mathbb{Z}$ on $\mathbf{Set}$.
Monads in computer science are typically thought of as encoding *effects*. A *finitary* monad is, intuitively (and also precisely, via the correspondence with Lawvere theories), one that can be specified by some collection of operations or effects of finite arity, together with equations they must satisfy. Algebras of a given monad $M$ correspond to different ways of *running* or *interpreting* its effects.
Some running examples of finitary monads and their algebras:
| Monad $M$ | $M$-algebras ($=$ $M$-modules) |
|-----------|-------------------------------|
| Free $R$-module monad $\mathsf{Mod}_R$ | $R$-modules |
| Maybe monad $\mathsf{mb}$ | Pointed sets |
| Distribution monad $\mathcal{D}$ | Convex sets (sets with weighted averages) |
The *free $R$-module monad* $\mathsf{Mod}_R$ for a given ring $R$ maps a set $X$ to the free $R$-module generated by $X$. Its algebras are precisely $R$-modules, i.e. sets equipped with an $R$-action satisfying the usual module axioms. For $R$ a field, $R$-modules are vector spaces, while for $R = \mathbb{Z}$, $R$-modules are abelian groups, and hence $\mathsf{Mod}_\mathbb{Z}$ is the free abelian group monad/the monad corresponding to the Lawvere theory of abelian groups.
The *maybe monad* $\mathsf{mb}$ maps a set $X$ to $\mathsf{mb}(X) := X + 1$. Its algebras are precisely pointed sets, i.e. sets with a distinguished element. Note that $\mathsf{mb}$ is the only commutative exception monad — for cases where there is more than one exception value, the order of effects may cause a different exception to be thrown in different cases.
The *distribution monad* $\mathcal{D}$ maps a set $X$ to the set of finitely-supported probability distributions on $X$. Its algebras are *convex sets*: sets equipped with a way of taking formal weighted averages of their elements. It is a basic example of a *probability monad*.
## Commutative Monads
Intuitively, a commutative monad is one where the *order* in which effects apply doesn't matter. More precisely: for a monad $M$ on a monoidal category $(\mathcal{C}, \otimes, I)$, a *tensorial strength* is a natural transformation
$t_{A,B} : A \otimes MB \longrightarrow M(A \otimes B)$
satisfying coherence conditions. Intuitively, $t_{A,B}$ allows us to "pull" a pure value $A$ inside an $M$-computation over $B$. In $\mathbf{Set}$, every monad has a canonical strength.
Given a strength $t$ on a monad $M$ in a symmetric monoidal category, we can define the *costrength* $t'_{A,B} : MA \otimes B \to M(A \otimes B)$ by conjugating with the symmetry isomorphism. $M$ is *commutative* if the two composite maps
$MA \otimes MB \;\rightrightarrows\; M(A \otimes B)$
— one obtained by first applying the strength, the other by first applying the costrength — are equal. Spelling these out:
$MA \otimes MB \xrightarrow{t'_{MA,B}} M(MA \otimes B) \xrightarrow{M t_{A,B}} M^2(A \otimes B) \xrightarrow{\mu} M(A \otimes B)$
$MA \otimes MB \xrightarrow{t_{A,MB}} M(A \otimes MB) \xrightarrow{M t'_{A,B}} M^2(A \otimes B) \xrightarrow{\mu} M(A \otimes B)$
In other words: *the order in which independent effects are run does not matter*.
We typically think of commutative monads as encoding various notions of *nondeterminism*: the maybe monad encodes "might fail" nondeterminism, the distribution monad encodes probabilistic nondeterminism, and the free $R$-module monad encodes a kind of "weighted" nondeterminism where outcomes carry coefficients in $R$. In each case, commutativity reflects the fact that independent nondeterministic choices don't interfere with each other.
All of the monads in our running examples are commutative:
| Monad $M$ | Commutative? | Intuition |
|-----------|-------------|-----------|
| $\mathsf{mb}$ | **Yes** | Only one possible exception value; order irrelevant |
| Free $R$-module (for commutative $R$) | **Yes** | Scalar multiplication commutes |
| $\mathcal{D}$ (distribution) | **Yes** | Independent probabilistic choices commute |
### The Symmetric Monoidal Structure on M-algebras
Commutativity has a fundamental structural consequence for the theory of $M$-algebras. If $M$ is a commutative finitary monad on $\mathbf{Set}$, then its Eilenberg-Moore category $\mathbf{EM}(M)$ carries a closed symmetric monoidal structure given by the *tensor product*.
The tensor product $A \otimes_M B$ of $M$-algebras $A,B$ is characterized by the universal property that *$M$-bilinear maps* $A \times B \to C$, i.e. maps that are $M$-algebra homomorphisms in each argument separately, correspond naturally to homomorphisms $A \otimes_M B \to C$. The monoidal unit is $M(1)$, i.e. the free $M$-algebra on one generator.
In our running examples:
- **$M = \mathsf{Mod}_\mathbb{Z}$:** $\otimes_M$ is the usual tensor product of abelian groups.
- **$M = \mathsf{Mod}_R$:** $\otimes_M$ is the tensor product of $R$-modules.
- **$M = \mathsf{mb}$:** $A \otimes_{\mathsf{mb}} B$ is the *smash product* of pointed sets, which "deletes" (i.e., identifies with the basepoint) the points where either $A$ or $B$ is at the basepoint (the "zero" element), and otherwise pairs elements of $A$ and $B$.
## $M$-Monoids
With a symmetric monoidal structure on $\mathbf{EM}(M)$ in hand, we can ask for a *commutative monoid object* in $\mathbf{EM}(M)$ with respect to the tensor product, i.e. an $M$-algebra $N$ equipped with $M$-algebra maps
$m : N \otimes_M N \to N \qquad \text{and} \qquad e : M(1) \to N$
satisfying commutativity, associativity, and unit laws. I'll call such an object an *$M$-monoid* for short. The category of $M$-monoids, which I'll write $M\text{-}\mathbf{Monoid}$, is itself algebraic — it is presentable by a Lawvere theory extending that of $M$ with an additional binary operation satisfying the usual associativity, unitality, and commutativity axioms that additionally distributes over all of the operations of $M$. Informally, $M$-monoids are the "commutative ring-like objects" native to the algebraic theory encoded by $M$.
Here is how this plays out in our examples:
| Monad $M$ | $M$-monoids |
|-----------|-------------|
| $\mathsf{mb}$ | Commutative monoids with an absorbing zero element |
| $\mathsf{Mod}_\mathbb{Z}$ | Commutative rings |
| $\mathsf{Mod}_R$ | Commutative $R$-algebras |
| $\mathcal{D}$ | Commutative convex monoids (e.g. the unit interval $[0,1]$) |
Note that for $M = \mathsf{Mod}_\mathbb{Z}$, we recover exactly commutative rings, i.e. the objects of classical algebraic geometry. So the machinery of algebraic geometry over commutative rings is, in this language, the special case $M = \mathsf{Mod}_\mathbb{Z}$.
For $M = \mathsf{mb}$: an $\mathsf{mb}$-monoid is a commutative monoid $N$ with an element $0$ such that $0 \cdot n = 0$ for all $n$ — i.e., $0$ is *absorbing* for multiplication. These are sometimes called *multiplicative monoids* or *monoids with zero*, and include the multiplicative monoids underlying all (commutative) rings and fields as special cases.
### Finitely Presentable Algebras as Specifications
By general abstract nonsense, for any finitary monad $M$, the category of $M$-algebras is *locally finitely presentable,* and so in particular has a well-defined subcategory of *finitely-presentable $M$-algebras,* which contains the free $M$-algebras on finitely many generators and is closed under finite colimits (i.e. under "adding equations"). Intuitively, the finitely presentable $M$-algebras are those that can be described by a finite system of equations about the operations of $M$ on finitely many "generating" or "base" elements, whence the name "finitely presentable." A presentation $\mathfrak{F}[X_1, \ldots, X_n] / f_i = g_i$
in this sense, is given by its list of generators $X_1, \dots, X_n$, and the equations $f_i = g_i$ (where $f_i$ and $g_i$ are terms in the free $M$-algebra on $X_1, \dots, X_n$) they must satisfy.
From a computer science perspective, a finitely presentable $M$-monoid $A$ can be understood as a **specification** of program behavior, i.e. a finite system of equations about when combinations of (independent, hence commutative) effects produce equal results. An $M$-monoid homomorphism $A \to N$ is then a *model* or *implementation* of this specification in $N$, i.e. an assignment of concrete values in $N$ to the generators such that all the equations hold.
However, finitely presentable $M$-monoids are inherently *syntactic* objects: they are freely generated from the constructors of $M$ and then quotiented by equations. They tell us what equations hold, but not *where to look for solutions*. In the above, we could let $N$ be an arbitrary $M$-monoid, rather than a finitely presentable one, which broadens the scope of possible interpretations to more "semantic" objects. But this still assumes that the codomain $N$ is closed under the operations of $M$, which may not always be the case. There may only be some subset of the operations of $M$ that are well-defined in $N$—yet we still want to talk about when the equations of $A$ hold whenever their corresponding operations are well-defined.
A classical example from algebraic geometry is *projective space*. An $n$-dimensional projective space $\mathbb{P}^n$ over a field $k$ is defined by taking the free $k$-algebra on generators $X_0, \ldots, X_n$ and then quotienting by the equivalence relation that identifies *collinear* points, i.e. those that differ by a nonzero scalar multiple. The resulting object is not a $k$-algebra in the usual sense, since arbitrary polynomials may not be well-defined on it (e.g. the polynomial $X^2 + X + 1$ is not well-defined on $\mathbb{P}^n$ since it does not map collinear points to collinear results). However, *homogeneous* polynomials $p(X_0, \dots, X_n)$, i.e. those such that every summand has the same total degree $d$, are well-defined on $\mathbb{P}^n$ since for any nonzero scalar $\lambda$, we have $p(\lambda X_0, \dots, \lambda X_n) = \lambda^d p(X_0, \dots, X_n)$, so rescaling the inputs just rescales the output by a nonzero factor, which doesn't change the point in projective space. Hence it is perfectly sensible to talk about solutions to homogeneous polynomial equations in projective space, even though projective space is not a $k$-algebra in the usual sense.
To adapt this idea to the general setting of $M$-monoids, we need a more general notion of "space" in which to evaluate our specifications. This is where sheaves on the Zariski topology enter the picture.
---
# 2. The Zariski Topology on M-Monoids
## The Classical Zariski Topology
Classically, given a commutative ring $R$, the *spectrum* $\text{Spec}(R)$ is a topological space on the set of prime ideals of $R$, endowed with the [Zariski topology](https://en.wikipedia.org/wiki/Zariski_topology).
It turns out that this construction is enough to view the objects of $\mathbf{CRing}^{op}$ (and by the same token, $\mathbf{CRing}^{op}_{fp}$) as "spaces" in their own right, without any direct reference to prime ideals or Zariski open sets, etc. For commutative rings $A,B$, the homomorphisms $A \to B$ are in natural bijection with continuous maps $\text{Spec}(B) \to \text{Spec}(A)$. Moreover, since finitely presentable commutative rings are closed under finite colimits, it follows that the category $\mathbf{CRing}^{op}_{fp}$ of "ringed spaces" (aka "affine schemes") is closed under finite limits, allowing us to construct equalizers and pullbacks representing solution sets of systems of equations in these "spaces," which, as we have already seen, correspond to adding equations to the specifications of the underlying fp rings.
In order to be able to form more complex/"non-syntactic" spaces, we can then follow the usual Grothendieckian recipe and close $\mathbf{CRing}^{op}_{fp}$ up under arbitrary colimits, allowing us to "glue together" our basic "affine" spaces into more general ones. Doing this näively would give us the category of *presheaves* on $\mathbf{CRing}^{op}_{fp} = [\mathbf{CRing}_{fp}, \mathbf{Set}]$, but this fails to account for the geometry encoded by the Zariski topology and the "local" structure of affine schemes.
To fix this, we equip $\mathbf{CRing}^{op}_{fp}$ with the *Zariski coverage*, which is generated by certain finite families of localization maps $\{R \to R[f_i^{-1}]\}_{i \in I}$, where $R[f_i^{-1}]$ (the *localization* of $R$ at $f_i$) is the result of freely adjoining a multiplicative inverse to $f_i$ in $R$. Such a family $\{D(f_i)\}_{i \in I}$ is a Zariski cover of $\text{Spec}(R)$ if and only if the $f_i$ generate the unit ideal of $R$, i.e. there exist $\lambda_i \in R$ such that
$\sum_{i \in I} \lambda_i f_i = 1.$
This is the *Zariski coverage condition*.
Sheaves on the site given by $\mathbf{CRing}^{op}_{fp}$ equipped with the Zariski coverage are then the "spaces" of algebraic geometry, and, essentially by construction, the topos of all such sheaves has the universal property that it classifies the geometric theory *local rings*, i.e. rings in which any finite family of elements that sum to 1 must contain at least one invertible element.
## Localizations of M-Monoids
How much of the above can be generalized to an arbitrary (finitary) commutative monad $M$? The answer, as it turns out, is "most of it," as long as some care is taken in setting out the definitions in the correct form for generalization (which we have by-and-large already done). For an $M$-monoid $N$ and an element $n \in N$, the *localization* $N[n^{-1}]$ is the $M$-monoid obtained by freely adjoining a multiplicative inverse to $n$, i.e. the following pushout in $M\text{-}\mathbf{Monoid}$:
$\begin{array}{ccc}
\mathfrak{F}[X] & \xrightarrow{X \mapsto n} & N\\
_{X \mapsto X}\downarrow & & \downarrow\\
\mathfrak{F}[X, X^{-1}] & \xrightarrow{} & N[n^{-1}]
\end{array}$
where $\mathfrak{F}[X]$ denotes the free $M$-monoid on a single generator $X$, and $\mathfrak{F}[X, X^{-1}]$ is the free $M$-monoid on a generator $X$ with a chosen inverse. The right-hand vertical map in the above pushout gives the canonical localization map $\iota_n : N \to N[n^{-1}]$. It follows that if $N$ is finitely presentable, then so is $N[n^{-1}]$, since the category of finitely presentable $M$-monoids is closed under finite colimits and freely generated $M$-monoids on finitely many generators are finitely presentable. In terms of effects, we can think of localization as *adding the ability to undo a computation* $n$. For $M = \mathsf{Mod}_\mathbb{Z}$, this recovers the classical ring-theoretic localization $R \to R[f^{-1}]$.
## The Coverage Condition
To define covers, we need a generalization of the Zariski coverage condition $\sum_i \lambda_i f_i = 1$. For this we need an appropriate notion of an $M$-linear combination.
For an fp $M$-monoid $N$ and a finite set $I$, the *free $N$-module on generators $\{X_i\}_{i \in I}$* is the coproduct $\sum_{I} N$ in the category $\mathbf{EM}(M)$ with $N$-action $N \otimes \sum_I N \to \sum_I N$ given by distributing $N \otimes -$ over $\sum_I N$ and applying the monoid structure on $N$ to the resulting $N \otimes N$ terms as follows: $
N \otimes \sum_I N \cong \sum_I (N \otimes N) \xrightarrow{} \sum_I N
$ Given a tuple $\{k_i : M(1) \to N\}_{i \in I}$ of elements of $N$ (recalling that $M(1)$ is the monoidal unit for the tensor product in $\mathbf{EM}(M)$), we obtain an *evaluation map* $\sum_I N \to N$ as follows: $
\sum_I N \cong \sum_I N \otimes M(1) \xrightarrow{\sum_I N \otimes k_i} \sum_I N \otimes N \xrightarrow{[m]_{i \in I}} N
$ In other words, taking elements of $\sum_I N$ as "formal $N$-linear combinations" of the generators $X_i$, we can evaluate such a formal combination at the tuple $(k_i)_{i \in I}$ to get an actual element of $N$ by substituting $k_i$ for $X_i$ and applying the monoid structure to combine the resulting terms.
Applying this definition to some of our examples:
- **$M = \mathsf{Mod}_R$:** $\sum_I N = \bigoplus_{i \in I} N = N^I$. A formal combination is $\sum_i \lambda_i X_i$ with $\lambda_i \in N$, and it evaluates to $\sum_i \lambda_i k_i \in N$.
- **$M = \mathsf{mb}$:** $\sum_I N = \{0\} \sqcup (N \times I)$, i.e., either the zero element, or a single pair $(\lambda, i)$ representing $\lambda \cdot X_i$. Evaluating $(\lambda, i)$ at $(k_j)_{j \in J}$ gives $\lambda \cdot k_i$, while evaluating $0$ gives $0$.
With this in hand, we can say that a finite family of localization maps $\{N \to N[k_i^{-1}]\}_{i \in I}$ forms a *Zariski cover* of $N$ if there exists an element of the free $N$-module $\sum_I N$ that evaluates to $1 \in N$ when applied to the tuple $(k_i)_{i \in I}$.
Intuitively, the coverage condition says that the effects $k_i$ *jointly account for all possibilities*, in that every effectful computation in $N$ can be reconstructed as some "$N$-linear combination" of the $k_i$.
In the above cases, this translates as follows:
- **$M = \mathsf{Mod}_\mathbb{Z}$:** a cover requires $\sum_i \lambda_i k_i = 1$ for some $\lambda_i \in N$, which is just the unit ideal condition from above.
- **$M = \mathsf{mb}$:** a cover requires $\lambda \cdot k_i = 1$ for some single index $i$ and coefficient $\lambda \in N$, i.e., *at least one $k_i$ must already be invertible*.
## Local M-Monoids
An $M$-monoid $N$ is *local* if, whenever a finite family $(k_i)_{i \in I}$ satisfies the Zariski covering condition for $N$, at least one $k_i$ is invertible in $N$.
For $M = \mathsf{Mod}_\mathbb{Z}$, this is a *local ring* in the sense given above. For $M = \mathsf{mb}$, since a Zariski cover already requires some $k_i$ to be invertible, *every* $\mathsf{mb}$-monoid is automatically local, a fact that simplifies the "algebraic geometry" of $\mathsf{mb}$ considerably.
The topos of sheaves on the Zariski site of finitely presentable $M$-monoids (with the Zariski coverage) is then the classifying topos of local $M$-monoids. This is the direct generalization of the classical result for rings.
Working through the $M = \mathsf{mb}$ case explicitly as a sanity check, we have the following:
Since every $\mathsf{mb}$-monoid is local, and the Zariski coverage requires an invertible element to already be present, the Zariski topology on $\mathsf{mb}$-$\mathbf{Monoid}^{op}$ is *trivial*—every generating cover must already contain the identity morphism. Consequently:
- The sheaf topos on the Zariski site is just the *presheaf topos* $\mathbf{PSh}(\mathsf{mb}\text{-}\mathbf{Monoid}^{op}_{fp})$.
- The classifying topos classifies all $\mathsf{mb}$-monoids (not just "local" ones, since they all are).
This makes $M = \mathsf{mb}$ an exceptionally simple case to work with. Moreover, this triviality of the Zariski topology has a geometric interpretation consistent with the $\mathbb{F}_1$ intuition: over a field with one element, there is no interesting local-global distinction — everything is already global.
---
# 3. Pointed Monads and Field-Like M-Monoids
## Pointed Monads
A commutative monad $M$ is *pointed* if it comes equipped with a monad morphism $\mathsf{mb} \to M$, i.e., a natural transformation $\mathsf{mb}(X) \to M(X)$ compatible with the unit and multiplication of both monads. Such a morphism says that $M$ *supports a notion of failure*: every $M$-computation can "go to zero."
At the level of Lawvere theories, this is an extension: the theory of $M$ contains the theory of pointed sets as a sub-theory. Concretely:
- Every $M$-algebra $A$ acquires a distinguished element $0_A$ (the image of the unique non-basepoint of $\mathsf{mb}(1)$ under the algebra structure).
- Every $M$-algebra map preserves $0$.
For any pointed commutative monad $M$, in every $M$-monoid $N$, the element $0$ is absorbing for multiplication: $0 \cdot n = 0$ for all $n \in N$. The reason is that the multiplication map $m: N \otimes_M N \to N$ is an $M$-algebra map, hence in particular an $\mathsf{mb}$-algebra map (via the monad morphism), hence a pointed map. Put another way, the element $0 \otimes n$ corresponds to the basepoint in $N \otimes_{\mathsf{mb}} N$ (the smash product), which maps to $0 \in N$.
So for any pointed commutative monad, $M$-monoids automatically have an additive-flavored zero that annihilates multiplication.
## Field-Like M-Monoids
For a pointed commutative monad $M$, an $M$-monoid $N$ is *field-like* if every nonzero element of $N$ is invertible. In effects terms: every non-failing computation is reversible.
This generalizes fields to the setting of arbitrary pointed commutative monads:
| Monad $M$ | Field-like $M$-monoids |
|-----------|----------------------|
| $\mathsf{mb}$ | Sets of the form $G^+ = G \sqcup \{0\}$, for $G$ an abelian group ($\simeq \mathbf{Ab}$) |
| $\mathsf{Mod}_\mathbb{Z}$ | Fields (in the usual sense) |
| $\mathsf{Mod}_F$ (for a field $F$) | Field extensions of $F$ |
In particular:
- Every finite field $\mathbb{F}_{p^n}$ is a field-like $\mathsf{Mod}_\mathbb{Z}$-monoid.
- The finite fields of characteristic $p$, namely $\mathbb{F}_{p^n}$ for $n \geq 1$, are all field-like $\mathsf{Mod}_{\mathbb{F}_p}$-monoids.
- The underlying multiplicative monoid $(\mathbb{F}_{p^n}^\times \cup \{0\},\, \cdot,\, 1,\, 0)$ of any finite field is a field-like $\mathsf{mb}$-monoid. Concretely, it can be presented as the free $\mathsf{mb}$-monoid on a generator $X$ subject to the equation $X^{p^n - 1} = 1$, reflecting the fact that $\mathbb{F}_{p^n}^\times$ is cyclic of order $p^n - 1$.
Note in particular that the category of field-like $\mathsf{mb}$-monoids is equivalent to $\mathbf{Ab}$, the category of abelian groups (via $G \mapsto G^+$). Continuing the analogy between $\mathsf{mb}$ and the "field with one element," this implies that "fields over $\mathbb{F}_1
quot; are "just" abelian groups with freely added zero elements, in this sense.
---
# 4. Synthetic Algebraic Geometry Over M
## The Universal Local M-Monoid
The classifying topos $\mathcal{T}_M$ of local $M$-monoids contains, by definition of a classifying topos, a *universal local $M$-monoid* $\mathfrak{M}$: an object of $\mathcal{T}_M$ carrying the structure of a local $M$-monoid, with the universal property that geometric morphisms $\mathcal{E} \to \mathcal{T}_M$ from any Grothendieck topos $\mathcal{E}$ correspond naturally to local $M$-monoid objects of $\mathcal{E}$.
The internal language of $\mathcal{T}_M$ is thus a form of dependent type theory in which $\mathfrak{M}$ plays the role of a "generic" local $M$-monoid.
In the case of commutative rings/local rings, this is the starting point for *synthetic algebraic geometry*, itself a species of [[Synthetic Mathematics]]. Rather than building geometric spaces externally (by specifying local patches and gluing), we work *internally* in the Zariski topos and formulate geometric properties as statements about the universal local ring $\mathfrak{R}$.
This approach has been developed for the classical case ($M = \mathsf{Mod}_\mathbb{Z}$, $\mathfrak{M}$ the generic local ring) by Ingo Blechschmidt, and subsequently in a homotopy-type-theoretic framework by Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn, Hugo Moeneclaey, and others. The approach sketched here generalizes this framework to an arbitrary commutative finitary monad $M$.
## M-Algebras and Spectra
Working internally in $\mathcal{T}_M$, an *$\mathfrak{M}$-algebra* is an $M$-monoid $A$ together with an $M$-monoid homomorphism $\alpha : \mathfrak{M} \to A$ — equivalently, an object of the under-category $\mathfrak{M} / M\text{-}\mathbf{Monoid}$. The $M$-monoid $\mathfrak{M}$ itself can then be viewed as an $\mathfrak{M}$-algebra via the identity on $\mathfrak{M}$. An $\mathfrak{M}$-algebra homomorphism is an $M$-monoid map $A \to B$ commuting with the maps from $\mathfrak{M}$; write $\mathfrak{M}\text{-}\mathbf{Alg}$ for the resulting category.
The *spectrum* of a finitely presentable $\mathfrak{M}$-algebra $A$ is the type
$\operatorname{Spec} A \;:=\; \operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}(A,\, \mathfrak{M}).$
A point of $\operatorname{Spec} A$ is a "way of evaluating the algebra $A$ in the universal local $M$-monoid $\mathfrak{M}quot;, or put another way, an $\mathfrak{M}$-valued point of the variety presented by $A$.
### Synthetic Quasi-Coherence
The key axiom of synthetic algebraic geometry is:
**Axiom (Synthetic Quasi-Coherence, SQC).** *For any finitely presentable $\mathfrak{M}$-algebra $A$, the type of functions $\operatorname{Spec} A \to \mathfrak{M}$ is equivalent to (the underlying type of) $A$ itself:*
$\mathfrak{M}^{\operatorname{Spec} A} \;\simeq\; A$
SQC is the precise statement that **syntax and semantics are dual to one another** for finitely presentable $\mathfrak{M}$-algebras, with $\mathfrak{M}$ serving as the dualizing object. The specification $A$ is equivalent to the $M$-monoid of $\mathfrak{M}$-valued functions on its solution space $\operatorname{Spec} A$.
In CS terms: SQC says that every *observable behavior* of solutions to $A$ is already captured by $A$ itself. There are no "hidden" properties of the solution space that escape the specification. This is why the synthetic approach is so powerful: algebraic manipulations on $A$ faithfully represent geometric operations on $\operatorname{Spec} A$, and SQC guarantees this.
An immediate consequence: for finitely presentable $\mathfrak{M}$-algebras $A$ and $B$,
$\operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}(A,\, B) \;\simeq\; (\operatorname{Spec} B \to \operatorname{Spec} A).$
This follows from the universal property of $A$ and SQC:
$\operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}(A, B) \simeq \operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}\!\bigl(A,\, \mathfrak{M}^{\operatorname{Spec} B}\bigr) \simeq \bigl(\operatorname{Spec} B \to \operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}(A, \mathfrak{M})\bigr) = \bigl(\operatorname{Spec} B \to \operatorname{Spec} A\bigr).$
In particular, for the $\mathfrak{M}$-algebra $\mathfrak{M}[X_1, \dots, X_n]$ freely generated by $n$ elements $X_1, \dots, X_n$, we have $
\operatorname{Spec} \mathfrak{M}[X_1, \dots, X_n] \simeq \mathfrak{M}^n
$ which in particular implies that the function space $
\mathfrak{M}^n \to \mathfrak{M}
$ is equivalent to $\mathfrak{M}[X_1, \dots, X_n]$, or in other words: the only $\mathfrak{M}$-valued functions on $\mathfrak{M}^n$ are the "generalized polynomials" in the free $\mathfrak{M}$-algebra on $n$ generators. For the classical case of rings, this implies that the only functions $\mathfrak{R}^n \to \mathfrak{R}$ in the internal language of the Zariski topos are the polynomial functions.
---
# 5. Varieties
Classically, an algebraic variety is a subspace of some ambient space (typically affine or projective) cut out by a system of polynomial equations (subject to additional conditions concerning irreducibility, etc., which we will ignore here). In the above, we motivated *spaces* as the semantic counterpart to *specifications*. SQC tells us that specifications (fp $\mathfrak{M}$-algebras) and their solution spaces ($\operatorname{Spec}$) are two dual views of the same object. What remains is to explain how solution sets embed into ambient spaces, i.e. how a specification $A$ carves out a subspace of (e.g.) affine or projective space in our abstract framework.
## Affine Varieties
For any finitely presentable $\mathfrak{M}$-algebra $A$ with presentation $\mathfrak{M}[X_1, \ldots, X_n] \twoheadrightarrow A$ (i.e. a surjective map quotienting the free algebra on some finite set of generators by finitely many relations), the quotient map induces (contravariantly by precomposition) an embedding
$\operatorname{Spec} A \hookrightarrow \operatorname{Spec} \mathfrak{M}[X_1, \ldots, X_n] \simeq \mathfrak{M}^n.$
In other words, the solution space of $A$ is a subspace of $\mathfrak{M}^n$ cut out by the equations that define $A$ as a quotient of the free algebra on $n$ generators. Hence $\operatorname{Spec} A$, viewed as a subspace of $\mathfrak{M}^n$, is the *affine variety defined by $A$*.
To generalize this, we should understand the abstract properties of such subspaces/varieties, in order to translate them from affine space $\mathfrak{M}^n$ to more general spaces. The key property for this purpose turns out to be the following:
* A subtype embedding $X \hookrightarrow Y$, viewed as a family of propositions $U : Y \to \mathbf{Prop}$, is *closed* if for all $y : Y$ there (merely) exists a finite family of elements $x_i, y_i : \mathfrak{M}$ such that $U(y)$ is equivalent to $\bigwedge_i x_i = y_i$In other words, the property of being in $X$ can, for each $y : Y$, be expressed as a finite system of equations in $\mathfrak{M}$.
Hence we will say that a "generalized variety" in a type $Y$ (viewed as a "space") is a subtype $X \hookrightarrow Y$ that is closed in the above sense.
## Projective Varieties
Classical projective space $\mathbb{P}^n$ is the space of lines through the origin in $(n+1)$-dimensional affine space. In our setting, the *projective $n$-space* $\mathbb{P}^n$ is the quotient
$\mathbb{P}^n \;:=\; \bigl(\mathfrak{M}^{n+1} \setminus \{0\}\bigr) / {\sim},$
where $(x_0, \ldots, x_n) \sim (y_0, \ldots, y_n)$ iff there exists an invertible $\lambda \in \mathfrak{M}$ such that $x_i = \lambda y_i$ for all $i$. In other words, two nonzero tuples are identified if one is an invertible scalar multiple of the other.
Per our above definition, the "projective varieties" of our framework are the closed subspaces of $\mathbb{P}^n$.
For instance, given a homogeneous element $f : \mathfrak{M}[X_0, \dots, X_n]$, i.e. such that $f(\lambda X_0, \dots, \lambda X_n) = \lambda^d f(X_0, \dots, X_n)$ for some $d$ and all $\lambda : \mathfrak{M}$, we can define the *zero-locus* of $f$ in $\mathbb{P}^n$ as the subspace $V(f) := \{[x_0, \dots, x_n] \in \mathbb{P}^n : f(x_0, \dots, x_n) = 0\}.$ Note that:
1. this is well-defined since $f(\lambda x_0, \dots, \lambda x_n) = \lambda^d f(x_0, \dots, x_n)$ and $\lambda^d$ is invertible whenever $\lambda$ is, so the property of being a zero of $f$ is invariant under rescaling by $\lambda$ (0 absorbs multiplication), and
2. it is a closed subspace since the property of being a zero of $f$ can be expressed as the single equation $f(x_0, \dots, x_n) = 0$ in $\mathfrak{M}$.
By the same token, given a finite family of homogeneous elements $f_i$, we can define the zero-locus of the family as the intersection $\bigcap_i V(f_i)$, which is again a closed subspace. In the case $M = \mathsf{Mod}_\mathbb{Z}$, these are precisely the classical projective varieties (modulo the usual technicalities about irreducibility, etc. that we are ignoring here).
---
# 6. The Riemann Zeta Function as a Hasse-Weil Zeta Function
## Hasse-Weil Zeta Functions
Classically, for a variety $V$ defined over a finite field $\mathbb{F}_p$, the *Hasse-Weil zeta function* packages the point counts $|V(\mathbb{F}_{p^n})|$ into a generating function:
$Z(V/\mathbb{F}_p,\, s) \;=\; \exp\!\left(\sum_{n=1}^{\infty} \frac{|V(\mathbb{F}_{p^n})|}{n}\, p^{-sn}\right).$
The global zeta function is the Euler product of these generating functions over all primes $p$:
$\zeta(V, s) \;=\; \prod_{p\,\text{prime}} Z(V/\mathbb{F}_p,\, s).$
The Weil conjectures, proved by Grothendieck and Deligne, give deep structural information about $\zeta(V, s)$, including an analogue of the Riemann Hypothesis for its zeros.
### Finite Fields as Field-Like mb-Monoids
For the remainder of this section, take $M = \mathsf{mb}$. As established earlier, for each prime power $p^n$, the pointed multiplicative monoid of $\mathbb{F}_{p^n}$ is a finitely presentable field-like $\mathsf{mb}$-monoid, which I'll denote $B_{p^n}$:
$B_{p^n} \;=\; \mathsf{mb}\text{-}\mathbf{Monoid}[X] / (X^{p^n - 1} = 1).$
These are the "$\mathbb{F}_{p^n}$-points" available to us as test objects in the $\mathsf{mb}$-setting.
## The Terminal Variety and the Riemann Zeta Function
Consider the simplest possible specification: no generators or equations at all. This corresponds to taking $A = \mathfrak{M}$ itself, with its structure map being the identity $\mathfrak{M} \xrightarrow{\text{id}} \mathfrak{M}$. This is the *initial* $\mathfrak{M}$-algebra: every $\mathfrak{M}$-algebra $B$ receives a unique $\mathfrak{M}$-algebra map $\mathfrak{M} \to B$ (the algebra structure map $\alpha : \mathfrak{M} \to B$). By duality, $\operatorname{Spec} \mathfrak{M}$ is the *terminal (affine) variety*.
The $B_{p^n}$-points of $\operatorname{Spec}\mathfrak{M}$ are:
$\operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}(\mathfrak{M},\, B_{p^n}).$
Since $\mathfrak{M}$ is the initial $\mathfrak{M}$-algebra, any $\mathfrak{M}$-algebra $B$ has *exactly one* $\mathfrak{M}$-algebra map from $\mathfrak{M}$:
$\operatorname{Hom}_{\mathfrak{M}\text{-}\mathbf{Alg}}(\mathfrak{M},\, B_{p^n}) \;\simeq\; 1.$
Thus $N_{p^n} := |\operatorname{Spec}\mathfrak{M}(B_{p^n})| = 1$ for every prime $p$ and every $n \geq 1$.
Substituting into the Hasse-Weil formula:
$Z(\operatorname{Spec}\mathfrak{M}/B_p,\, s) \;=\; \exp\!\left(\sum_{n=1}^{\infty} \frac{1}{n}\, p^{-sn}\right).$
Recognizing the Taylor series $\log\frac{1}{1-x} = \sum_{n=1}^\infty \frac{x^n}{n}$ with $x = p^{-s}$:
$Z(\operatorname{Spec}\mathfrak{M}/B_p,\, s) \;=\; \exp\!\left(-\log(1 - p^{-s})\right) \;=\; \frac{1}{1 - p^{-s}}.$
Taking the Euler product over all primes:
$\zeta(\operatorname{Spec}\mathfrak{M},\, s) \;=\; \prod_{p\,\text{prime}} \frac{1}{1 - p^{-s}} \;=\; \zeta(s),$
which is precisely the Euler product formula for the Riemann zeta function.
---
# 7. Conclusion
Although one might hope that the above calculation could straightforwardly be used to adapt Grothendieck and Deligne's proofs of the Weil conjectures to the $\mathsf{mb}$-setting and thereby deduce the Riemann Hypothesis, this is sadly not the case. The above is more-or-less a reconstruction of Deitmar's theory of "monoid schemes" (called "the very core of $\mathbb{F}_1$ geometry" by Lorscheid) in terms more familiar to category theorists, type theorists, and computer scientists. Despite its elegance, Deitmar's theory on its own does not appear to have the machinery needed to prove a Riemann Hypothesis analogue, nor to support all of the expected properties of a "geometry over $\mathbb{F}_1$." As such, although the above provides a clean and conceptually satisfying framework for understanding the field with one element and its associated geometry, it is only a first step in a much larger programme, which has been the subject of much speculation and many different approaches over the past few decades. See the references below for some of the most influential work in this area.
In particular, beyond Deitmar's theory of monoid schemes, the framework described above is perhaps closest in spirit to Durov's approach to Arakelov geometry via "generalized rings" (which are essentially finitary commutative monads on sets); Durov, however, defines an analogue of the Zariski topology directly on the category of such generalized rings itself, rather than on the category of monoids over the modules of such rings/monads. Other recent proposals in this same vein include those of Connes and Consani, and Borger, all of which are related to each other in various ways but differ in their technical details and emphases.
However, I hope that the above can serve as an illuminating introduction to some of the general ideas and concepts of algebraic geometry for those (such as myself, prior to writing this) who may not be familiar with the subject, but for whom algebra and its various geometric incarnations are of interest. In particular, the generalization to arbitrary commutative monads $M$ is perhaps less directly relevant to the field with one element, but it serves to clarify the essential features of the theory and to situate the $\mathsf{mb}$-case within a broader landscape of "synthetic algebraic geometry" that may have applications beyond $\mathbb{F}_1$, and perhaps even in other areas of mathematics and theoretical computer science where the notion of a "geometry of computations" may be fruitful.
---
# References
**Synthetic algebraic geometry:**
- I. Blechschmidt, *Using the internal language of toposes in algebraic geometry*, PhD thesis, University of Augsburg (2017). arXiv: [2111.03685](https://arxiv.org/abs/2111.03685)
- F. Cherubini, T. Coquand, M. Hutzler, *A foundation for synthetic algebraic geometry*, arXiv: [2307.00073](https://arxiv.org/abs/2307.00073) (2024)
- F. Cherubini, T. Coquand, M. Hutzler, D. Wärn, *Projective space in synthetic algebraic geometry*, arXiv: [2405.13916](https://arxiv.org/abs/2405.13916) (2024)
- T. Coquand, H. Moeneclaey, _Chatelet’s Theorem in Synthetic Algebraic Geometry_ arXiv: [2504.08326](https://arxiv.org/abs/2504.08326) (2025)
**The field with one element:**
* A. Deitmar, *Schemes over $\mathbb{F}_1$*, in van der Geer, G.; Moonen, B.; Schoof, R. (eds.), _Number Fields and Function Fields: Two Parallel Worlds_, Progress in Mathematics, vol. 239
* N. Durov, *New approach to Arakelov geometry*, arXiv: [0704.2030](https://arxiv.org/abs/0704.2030) (2008)
- A. Connes and C. Consani, *Schemes over $\mathbb{F}_1$ and zeta functions*, arXiv: [0903.2024](https://arxiv.org/abs/0903.2024) (2010)
- J. Borger, *Lambda rings and the field with one element*, arXiv: [0906.3146](https://arxiv.org/abs/0906.3146) (2009)