2. Rigs and the KL Axiom(s)

Recall that a rig is a set $A$ equipped with two commutative monoid structures $(A, \cdot , 1)$ and $(A, +, 0)$ such that $x \cdot ( y + z) = x\cdot y + x \cdot z$ and $0\cdot x = 0$ , for every $x, y, z \in A$ . The coextensive category of rigsFootnote ¹ and homomorphisms between them will be denoted by $\mathbf{Rig}$ . We sometimes omit the dot and simply write $ab$ instead of $a \cdot b$ .

For any rig $K$ , the objects in the coslice $K/\mathbf{Rig}$ will be called $K$ -rigs as in (Lawvere Reference Lawvere2008). For instance, taking $K = {\mathbb{Z}}$ , the canonical functor ${\mathbb{Z}}/\mathbf{Rig} \rightarrow \mathbf{Rig}$ from the category of $\mathbb{Z}$ -rigs coincides with the full subcategory $\mathbf{Ring} \rightarrow \mathbf{Rig}$ of rings, that is, rigs whose additive monoid is a (necessarily abelian) group.

Almost disjoint from the subcategory of rings lies the full subcategory $\mathbf{dLat} \rightarrow \mathbf{Rig}$ of distributive lattices, considered as rigs with addition and multiplication given by joins and meets, respectively. If we let $\unicode{x1D7DA}$ be the initial distributive lattice then the canonical functor ${\unicode{x1D7DA}}/\mathbf{Rig} \rightarrow \mathbf{Rig}$ from the category of $\unicode{x1D7DA}$ -rigs may be identified with the full subcategory of rigs such that $1 + 1 = 1$ ; equivalently, rigs with idempotent addition. Notice that the embedding $\mathbf{dLat} \rightarrow \mathbf{Rig}$ factors through ${\unicode{x1D7DA}}/\mathbf{Rig} \rightarrow \mathbf{Rig}$ . Between distributive lattices and $\unicode{x1D7DA}$ -rigs there is the variety of those rigs such that $1 + x = 1$ , and that are called integral in (Castiglioni et al. Reference Castiglioni, Menni and Zuluaga Botero2016), following a custom in the community of residuated lattices. The Gaeta and Zariski toposes for integral rigs are proved to be pre-cohesive in (Menni Reference Menni2021).

Example 2.1. [The $K$ -rig of dual numbers.] For any rig $K$ , the additive monoid $K\times K$ may be equipped with a multiplication defined by

\begin{equation*} (a, b) \cdot (u, v) = (a u, a v + b u) \end{equation*}

and it is straightforward to check that distributivity holds so that $K\times K$ becomes a rig that we denote by $K[\varepsilon ]$ . If we let $\varepsilon = (0, 1) \in K[\varepsilon ]$ and, for every $a \in K$ , we let $a = (a, 0) \in K[\varepsilon ]$ then, it follows that $(a, b) = a + b\varepsilon \in K[\varepsilon ]$ for each $a, b\in K$ and that $\varepsilon ^2 = 0$ . Moreover, the function that sends $a \in K$ to $a \in K[\varepsilon ]$ is a rig morphism $K \rightarrow K[\varepsilon ]$ and it satisfies the universal property of $K\rightarrow K[x]/(x^2 = 0)$ in the category $\mathbf{Rig}$ .

For general reasons (well explained in (Kock Reference Kock2006, Part II), for example), we may consider rigs and constructions among them in any topos. For instance, for a rig $R$ in a topos, we can define the subobject

\begin{equation*} D = \{ x \in R \mid x^2 = 0 \}\rightarrow R \end{equation*}

and the map

which, in the internal language, sends $(a,b, d) \in R\times R \times D$ to $a + b \cdot d \in R$ . Since toposes are cartesian closed, we have the transposition $R \times R \rightarrow R^D$ of the map above which, in the internal language, is sometimes described as sending $(a, b) \in R\times R$ to $(\varepsilon \mapsto a + b\varepsilon ) \in R^D$ .

The condition defining the Kock-Lawere (KL) axiom is called “Axiom 1” in (Kock Reference Kock2006, p.3) where $R$ is assumed to be a ring. After formulating Axiom 1, Kock immediately observes that the abelian group $R \times R$ can be equipped with the multiplication of Example 2.1 so that it becomes a ring $R[\varepsilon ]$ and moreover:

For instance, it follows from (Kock Reference Kock2006, Theorem III.1.2) that the generic model in the classifier of $K$ -algebras satisfies the KL axiom for any ring $K$ in the base topos of sets and functions. It also follows that the KL axiom holds in many subtoposes containing the generic model.

The same proof shows that the KL axiom holds if we let $R$ be the generic model of the classifier of $K$ -rigs for any rig $K$ . For instance, the KL axiom holds if $R$ is the generic $\unicode{x1D7DA}$ -rig with exactly two Boolean elements (Menni Reference Menni2021, Proposition 7.3). The reason is that, as emphasized in Example 2.1, for any rig $K$ , the universal map $K \rightarrow K[\varepsilon ]$ may be constructed as among rings; but this is not the case for every variety of rigs, and that is why the KL axiom fails in the classifier of integral rigs and also in its Gaeta and Zariski subtoposes.

Failure of the KL axiom leads us to consider variants suggested by Corollary 2.3. We are not sure what is the right generality but, to be more precise, let $\mathfrak{Q}$ be a quotient algebraic theory of the theory of rigs. Let $R$ be a model of $\mathfrak{Q}$ in a topos $\mathcal{E}$ , and let $R \rightarrow R\{\varepsilon \}$ be the universal map $R \rightarrow R[x]/(x^2 = 0)$ in the category of $\mathfrak{Q}$ -models in $\mathcal{E}$ (where $ R[x]/(x^2 = 0)$ might not coincide with the $R[\varepsilon ]$ built as in Example 2.1). We could then say that $R$ satisfies that KL axiom relative to $\mathfrak{Q}$ if there is an isomorphism $R\{\varepsilon \} \rightarrow R^D$ of rigs such that the following diagram commutes

where the diagonal map is the transposition of the projection.

3. Adding an Element of square 0 to an Integral Rig

As we mentioned above, a rig is integral if it satisfies $1 + x = 1$ for every $x$ . It easily follows that integral rigs satisfy $1 + 1 = 1$ . We let $\mathbf{iRig} \rightarrow {\unicode{x1D7DA}}/\mathbf{Rig}$ be the variety of integral rigs. As far as we know, the explicit interest in the category $\mathbf{iRig}$ is very recent. It is proposed as interesting case study of a core variety in (Lawvere Reference Lawvere2008, Section 8). Categories for the associated “rig geometry” were set up in (Menni Reference Menni2021) where it is shown that the Gaeta and Zariski toposes determined by the theory of integral rigs are pre-cohesive. Further results, also motivated by geometric considerations, have recently appeared in (Spada and St. John Reference Spada and St. John2025), where several interesting examples are presented. Apart from specific examples, it is useful to bear in mind that the set of ideals of a ring, equipped with addition and multiplication of ideals, is an integral rig. Moreover, the assignment that sends a ring to its integral rig of ideals underlies a functor $\mathbf{Ring} \rightarrow \mathbf{iRig}$ thanks to the behavior of ideal extension (Atiyah and Macdonald Reference Atiyah and Macdonald1969, Exercise 1.18, column 1, lines 1 and 3). This functor also preserves finite products by coextensivity.

Examples aside, we develop, in this section, some basic commutative algebra that will be needed for the rest of the paper. As a byproduct of this development, it will become evident that the variety of integral rigs is not closed under the “dual numbers” construction of Example 2.1.

The additive structure of any rig $A$ determines a canonical pre-order on $A$ defined by

\begin{equation*} x\leq y \quad \Leftrightarrow \quad (\exists d \in A)( x + d = y ) \end{equation*}

for all $x, y \in A$ . If $A$ is a ring, then this pre-order is degenerate in the sense that $x \leq y$ for every $x , y \in A$ . If $A$ is a $\unicode{x1D7DA}$ -rig, then the canonical pre-order is a partial order.

If $A$ is an integral rig, then we let $A \rightarrow A\{x\}$ be the free integral rig on one generator $x$ , and we let $A \rightarrow A\{\varepsilon \}$ be the universal solution (in $\mathbf{iRig}$ ) to adjoining an element of square zero denoted by $\varepsilon$ . A concrete description of $A \rightarrow A\{\varepsilon \}$ may be obtained by reflecting the rig $A[\varepsilon ]$ of dual numbers into $\mathbf{iRig}$ . An explicit construction of this reflection may be found in (Castiglioni et al. Reference Castiglioni, Menni and Zuluaga Botero2016, Lemma 3.4). We give here an alternative direct construction of $A\{\varepsilon \}$ .

Let $A$ be a rig and let $A_{\varepsilon } = \{ a + a' \varepsilon \mid a \leq a' \} \subseteq A[\varepsilon ]$ . This subobject clearly contains $0 \in A[\varepsilon ]$ and is closed under binary addition. It is also easy to see that it is closed under binary multiplication, but it contains $1 \in A[\varepsilon ]$ if, and only if, $1 \leq 0 \in A$ , or equivalently, $A$ is a ring. In this case, the subobject $A_{\varepsilon } \subseteq A[\varepsilon ]$ is the whole of $A[\varepsilon ]$ .

Proof. If $1 + \varepsilon$ is a unit for the restricted multiplication, then $(1+\varepsilon )^2 = 1 + \varepsilon$ . That is, $1 + (1 + 1)\varepsilon = 1 + \varepsilon \in A_{\varepsilon }$ , so $1 + 1 = 1 \in A$ . Conversely, if $A$ has idempotent addition then, for $b + b'\varepsilon$ in $ A_{\varepsilon }$ ,

\begin{equation*} (1 + \varepsilon ) (b + b'\varepsilon ) = 1 b + (b + b')\varepsilon = b + b'\varepsilon \end{equation*}

using that, in a $\unicode{x1D7DA}$ -rig, $b \leq b'$ if and only if $b + b' = b'$ .

For the second part, just notice that $A_\varepsilon \subseteq A[\varepsilon ]$ preserves the multiplicative unit if, and only if, $1 + \varepsilon = 1 \in A_\varepsilon$ if, and only if, $1 = 0 \in A$ .

For the third part, assume first that $A$ is integral. Then

\begin{equation*} (1 + \varepsilon ) + (a + a'\varepsilon ) = (1 + a) + (1 + a') \varepsilon = 1 + \varepsilon \in A_{\varepsilon } \end{equation*}

so $A_\varepsilon$ is integral. On the other hand, if $A_\varepsilon$ is integral then, for any $a \in A$ ,

\begin{equation*} 1 + \varepsilon = (1 + \varepsilon ) + a \varepsilon = 1 + (1+a) \varepsilon \in A_\varepsilon \end{equation*}

so $1 = 1 + a \in A$ .

There is an obvious function $A \rightarrow A_\varepsilon$ that sends $a \in A$ to $a + a \varepsilon \in A_\varepsilon$ . This function clearly preserves addition.

Proof. It is easy to check that if $A$ is a $\unicode{x1D7DA}$ -rig then $A \rightarrow A_\varepsilon$ preserves the multiplicative structure. If $A$ is integral then so is $A_\varepsilon$ by Lemma 3.1. To prove the universal property let $B$ be an integral rig and let $\rho \in B$ be such that $\rho ^2 = 0$ . Assume now that there exists a map $\overline {f} : A_\varepsilon \rightarrow B$ in $\mathbf{iRig}$ such that the following diagram commutes

and $\overline {f} \varepsilon = \rho$ . Then, for every $a + a'\varepsilon \in A_\varepsilon$ ,

\begin{equation*} \overline {f} (a + a'\varepsilon ) = \overline {f} (a + (a + a')\varepsilon ) = \overline {f}(a + a\varepsilon ) + \overline {f}(a' \varepsilon ) = \end{equation*}

\begin{equation*} f a + \overline {f}((a' + a'\varepsilon )\varepsilon ) = f a + (\overline {f}(a' + a'\varepsilon ) )(\overline {f}\varepsilon ) = f a + (f a') \rho \end{equation*}

so there is at most one such $\overline {f}$ . To complete the proof it is enough to check that, so defined, the function $\overline {f}$ is a rig morphism. Preservation of the additive structure is easy, and so is preservation of binary multiplication. To prove that $\overline {f}$ preserves the multiplicative unit notice that

\begin{equation*} \overline {f} (1 + \varepsilon ) = 1 \in B \end{equation*}

by the triangle above.

Example 3.3. It is immediate that ${\unicode{x1D7DA}}_{\varepsilon } = \{ 0 , \varepsilon , 1 + \varepsilon \} \subseteq {\unicode{x1D7DA}}[\varepsilon ]$ , but it is convenient to identify

\begin{equation*} {\unicode{x1D7DA}}\{\varepsilon \} = \{ 0 \lt \varepsilon \lt 1 \} \end{equation*}

with multiplication determined by $\varepsilon ^2 = 0$ .

From now on, we will abandon the notation $A_\varepsilon$ and simply use $A\{\varepsilon \}$ , invoking the concrete description when necessary. For instance, let us consider maps ${\unicode{x1D7DA}}\{\varepsilon \} \rightarrow B\{\varepsilon \}$ for any object $B$ in $\mathbf{iRig}$ .

Recall that, for an arbitrary rig $A$ , an element $b \in A$ is called Boolean if there is a (necessarily unique) $b' \in A$ such that $b b' = 0$ and $b + b' = 1$ . Boolean elements of $A$ are in bijective correspondence with direct product decompositions of $A$ , and that is why the following result will play a role when we will briefly discuss subtoposes of the classifier of integral rigs.

Proof. Assume that $p = x + y\varepsilon$ has complement $q = u + v\varepsilon$ in $B\{\varepsilon \}$ . That is, $p + q = 1$ and $pq = 0$ . Then

\begin{equation*} x + u + (y + v) \varepsilon = 1 + \varepsilon \quad \text{ and } \quad x u + (xv + y u)\varepsilon = 0 \end{equation*}

so $x + u = 1 = y + v$ and $x u = 0 = x v + y u$ in $B$ . In particular, $u$ is the complement of $x$ so, by hypothesis, $x = 0$ or $x = 1$ .

If $x = 0$ then $u = 1$ so $0 = x v + y u = 0v + y 1 = y$ , and hence $p = 0 \in B\{\varepsilon \}$ .

If $x = 1$ then $u = 0$ so $0 = x v + y u = v$ , and then $1 = y + v = y$ . So $p = 1 \in B\{\varepsilon \}$ .

4. Derivations

We continue with the development of the basic commutative algebra of integral rigs, emphasizing the differences with the case of rings. Specifically, we study the classical notion of derivation. See, for example, (Kock Reference Kock2006, Section I.11) for a reference in the context of SDG. The classical definition among rings and algebras may be immediately lifted to rigs as follows.

Definition 4.1. For a map $f : A \rightarrow B$ in $\mathbf{Rig}$ , a derivation on the $A$ -rig $f$ is a morphism $\phi : A \rightarrow B$ of additive monoids such that

\begin{equation*}\phi (x y) = (f x)(\phi y) + (\phi x)(f y) \end{equation*}

for every $x, y \in A$ .

Between rings, the calculation

\begin{equation*} \phi 1 = \phi (1\cdot 1) = 1 (\phi 1) + (\phi 1) 1 = \phi 1 + \phi 1\end{equation*}

implies that $\phi 1 = 0$ ; but this is not the case in general rigs.

Proof. If $\phi 1 = 1$ then calculate:

\begin{equation*} \phi x = \phi (x 1) = (f x) (\phi 1) + (\phi x) (f 1) = f x + \phi x \end{equation*}

to prove the first part of the statement. Also, if $f x \leq \phi x$ , then $1 = f 1 \leq \phi 1$ . Hence, if $B$ is integral, $1 = \phi 1$ .

Example 4.4 (Derivations above an element of square 0). Let $B$ be an integral rig and let $f : {\unicode{x1D7DA}}\{\varepsilon \} \rightarrow B$ be a rig morphism. Using the notation in Example 3.3, a derivation $\phi : {\unicode{x1D7DA}}\{\varepsilon \} \rightarrow B$ above $f$ is uniquely determined by the element $\phi \varepsilon$ satisfying $(f\varepsilon ) (\phi \varepsilon ) = 0$ . To prove this, notice first that, for any $b \in B$ , there exists a unique map of additive monoids ${\unicode{x1D7DA}}\{\varepsilon \} \rightarrow B$ that preserves $1$ and sends $\varepsilon$ to $b$ . Also, a morphism $\phi : {\unicode{x1D7DA}}\{\varepsilon \} \rightarrow B$ of additive monoids is a derivation above $f$ if, and only if,

\begin{equation*} 0 = \phi 0 = \phi (\varepsilon ^2) = (f\varepsilon ) (\phi \varepsilon ) \end{equation*}

so the claim is proved.

For any integral rig $B$ , let $\pi _0 : B\{\varepsilon \} \rightarrow B$ be the unique morphism that sends $\varepsilon$ to $0$ . The function $\pi _1 : B\{\varepsilon \} \rightarrow B$ that sends $x + y\varepsilon$ to $y$ is a derivation above $\pi _0$ .

Proof. The definition of derivation above a map allows us to execute a variation of a standard argument, but in the context of integral rigs. Simply by composition, $f = \pi _0 g : A \rightarrow B$ is a map in $\mathbf{iRig}$ and $\phi = \pi _1 g$ is a morphism of additive monoids. The function $g$ may be described as $g x = (f x) + (\phi x) \varepsilon \in B\{\varepsilon \}$ so, certainly $f x \leq \phi x \in B$ by our explicit description of $B\{\varepsilon \}$ . Also,

\begin{equation*} f(x y) + (\phi (xy)) \varepsilon = g(xy) = (g x) (g y) \end{equation*}

\begin{equation*} = ( f x + (\phi x)\varepsilon ) ( f y + (\phi y)\varepsilon ) = (f x)(f y) + [(f x)(\phi y) + (\phi x)(f y)]\varepsilon \end{equation*}

so $\phi (x y) = (f x)(\phi y) + (\phi x)(f y) \in B$ as we needed to prove.

In the opposite direction, given an $f : A \rightarrow B$ in $\mathbf{iRig}$ and a derivation $\phi$ above $f$ , $g x = (f x) + (\phi x) \varepsilon$ is the unique map from $A$ to the underlying object of $B\{\varepsilon \}$ such that $f = \pi _0 g$ and $\phi = \pi _1 g$ . Moreover, the resulting map $g : A \rightarrow B\{\varepsilon \}$ clearly preserves the additive structure, and a simple reorganization of the calculation above shows that $g$ preserves multiplication.

It should be interesting to study the notion of neighbor (in the sense of (Kock Reference Kock2017)) in the context of $\mathbf{iRig}$ . Unfortunately, we need to postpone that.

5. The KL Axioms in the Classifier of Integral Rigs

In this section, we identify, in the classifying topos for the theory of integral rigs, an object $T$ with a unique point, we calculate the object of Euler reals determined by $T$ , showing that it coincides with the generic integral rig, and we show that it satisfies one form of the KL axiom but not the other. Many of the calculations are done in the extensive site for the classifying topos, and that is why we concentrate on finitely presentable integral rigs.

Let $\mathbf{iAff}$ be the extensive opposite of the category of finitely presentable integral rigs as in (Menni Reference Menni2021, p.459). For any finitely presentable integral rig $A$ , we let $\mathrm{Spec}\, A$ be the corresponding object in $\mathbf{iAff}$ . In particular, the terminal object $\mathbf{1}$ in $\mathbf{iAff}$ is $\mathrm{Spec}\, {\unicode{x1D7DA}}$ .

We define $T = \mathrm{Spec}\, ({\unicode{x1D7DA}}\{\varepsilon \})$ in $\mathbf{iAff}$ . The object $T$ has a unique point denoted by $0 : \mathbf{1} \rightarrow T$ . This is, of course, the point $\mathrm{Spec}\,{\unicode{x1D7DA}} \rightarrow \mathrm{Spec}\, ({\unicode{x1D7DA}}\{\varepsilon \})$ determined by the unique map $! : {\unicode{x1D7DA}}\{\varepsilon \} \rightarrow {\unicode{x1D7DA}}$ in $\mathbf{iRig}$ . To further illustrate the notation, we state a simple but relevant analog of a classical fact in the context of rings and $C^\infty$ -rings (Kock Reference Kock2006, Part III).

For general reasons (Johnstone Reference Johnstone2002, Corollary D3.1.2), the topos $\widehat {\mathbf{iAff}}$ of presheaves on $\mathbf{iAff}$ classifies integral rigs, and $\mathrm{Spec}\,({\unicode{x1D7DA}}\{x\})$ is the generic integral rig. As is customary, we confuse, in the notation, objects with the corresponding representables. So $\mathrm{Spec}\, A$ denotes both the object in $\mathbf{iAff}$ and the associated presheaf in $\widehat {\mathbf{iAff}}$ . For instance,

\begin{equation*} (\mathrm{Spec}\, A)(\mathrm{Spec}\, B) \cong \mathbf{iAff}(\mathrm{Spec}\, B, \mathrm{Spec}\, A) \cong \mathbf{iRig}(A, B) \end{equation*}

for any finitely presentable integral rig $B$ .

Following the intuition in SDG, the “evaluation at 0” map $ev_0 = T^0 : X^T \rightarrow X$ will be called the tangent bundle of $X$ .

Proof. Again a standard argument:

\begin{equation*} X^T (\mathrm{Spec}\, B) \cong \widehat {\mathbf{iAff}}(\mathrm{Spec}\, B,X^T) \cong \mathbf{iAff}( (\mathrm{Spec}\, B) \times T, X) \cong \end{equation*}

\begin{equation*} \cong \mathbf{iAff}(\mathrm{Spec}\,(B\{\varepsilon \}), X) = X (\mathrm{Spec}\, (B\{\varepsilon \}) ) \end{equation*}

using the Yoneda Lemma and Lemma 5.1.

Additional information may be extracted from tangent bundles of representables.

Lemma 5.3. For any pair $A$ , $B$ of finitely presentable integral rigs, the set $(\mathrm{Spec}\, A)^T (\mathrm{Spec}\, B)$ is canonically isomorphic to $\mathbf{iRig}(A, B\{\varepsilon \})$ and so, also to the set of pairs $(f, \phi )$ with $f :A \rightarrow B$ in $\mathbf{iRig}$ and $\phi$ a derivation above $f$ . The tangent bundle

sends the pair $(f, \phi )$ to $f \in \mathbf{iRig}(A, B)$ .

Proof. Calculate using Lemma 5.2:

\begin{equation*} (\mathrm{Spec}\, A)^T (\mathrm{Spec}\, B) \cong (\mathrm{Spec}\, A)(\mathrm{Spec}\, B\{\varepsilon \}) \cong \mathbf{iRig}(A, B\{\varepsilon \}) \end{equation*}

and then apply Lemma 4.5.

The following is the analog of (Menni Reference Menni2024, Example 3.2) describing the tangent bundle of $T$ in the classical context of affine schemes over a field.

Example 5.4 (The tangent bundle of $T$ ). For $A = {\unicode{x1D7DA}}\{\varepsilon \}$ , Lemma 5.3 implies that, for each $\mathrm{Spec}\, B$ in $\mathbf{iAff}$ ,

\begin{equation*} T^T (\mathrm{Spec}\, B) \cong \mathbf{iRig}({\unicode{x1D7DA}}\{\varepsilon \}, B\{\varepsilon \}) \end{equation*}

may be identified with the set of pairs $(f, \phi )$ with $f : {\unicode{x1D7DA}}\{\varepsilon \} \rightarrow B$ and $\phi$ a derivation above $f$ , and that the tangent bundle

\begin{equation*} T^T (\mathrm{Spec}\, B) \rightarrow T(\mathrm{Spec}\, B) \cong \mathbf{iRig}({\unicode{x1D7DA}}\{\varepsilon \}, B) \end{equation*}

sends $(f, \phi )$ to $f$ . Using Example 4.6 and Lemma 3.4, we can actually take $(T^T) (\mathrm{Spec}\, B)$ to be the set of “affine functions” $b + c\varepsilon$ with $b, c\in B$ such that $b \leq c$ and $bc = 0$ . The bundle sends such an “affine function” to its “evaluation at $\varepsilon = 0$ ,” that is, to $b \in B$ . In other words, this calculation shows that, as in the standard models of SDG, $T^T$ is representable; in the present case, by $\mathrm{Spec}\,({\unicode{x1D7DA}}\{b, c\}/(b + c = c , b c = 0))$ and the tangent bundle

\begin{equation*} T^T = \mathrm{Spec}\, ({\unicode{x1D7DA}}\{b, c\}/(b + c = c , b c = 0)) \rightarrow \mathrm{Spec}\,({\unicode{x1D7DA}}\{\varepsilon \}) =T \end{equation*}

corresponds to the unique ${\unicode{x1D7DA}}\{\varepsilon \} \rightarrow {\unicode{x1D7DA}}\{b, c\}/(b + c = c , b c = 0)$ in $\mathbf{iRig}$ that sends $\varepsilon$ to $b$ .

The idea to start with the object $T$ with its single point, and build more complex spaces using the operations available in the topos, is promoted in (Lawvere Reference Lawvere2011) where the following special (p. 251 op. cit.) case plays a central role.

Definition 5.5. The monoid of Euler reals is the object $R$ in the pullback

equipped with the structure inherited from the monoid $T^T$ of endomorphisms.

The following lemma shows that the phenomenon familiar from classical affine schemes or $C^\infty$ -schemes (Menni Reference Menni2024, Examples 3.2 and 3.7) persists in the classifier of integral rigs, even though, as we will see, the KL axiom fails in its standard form.

Proof. At stage $\mathrm{Spec}\, B$ in $\mathbf{iAff}$ , we have the pullback

so, by Example 5.4, we may calculate its vertex as the affine functions $c\varepsilon$ with $c \in B$ . We may conclude that $R (\mathrm{Spec}\, B) \cong B$ . In other words, $R \cong \mathrm{Spec}\,({\unicode{x1D7DA}}\{x\})$ .

Lemma 5.6 may be strengthened to a coincidence between the monoid of Euler reals and the multiplicative monoid of the generic integral rig. Leaving this issue aside and considering $R = \mathrm{Spec}\,({\unicode{x1D7DA}}\{x\})$ equipped with the latter multiplication, it follows that

\begin{equation*} T = D = \{ x \in R \mid x^2 = 0 \} \rightarrow R \end{equation*}

so the canonical $R^2 \times D \rightarrow R$ in $\mathbf{iAff}$ is determined by the unique map

\begin{equation*} {\unicode{x1D7DA}}\{x\} \rightarrow {\unicode{x1D7DA}}\{y, z\} + {\unicode{x1D7DA}}\{\varepsilon \} = {\unicode{x1D7DA}}\{y, z, \varepsilon \} \end{equation*}

in $\mathbf{iRig}$ that sends $x$ to $y + z\varepsilon$ . Regardless of any preferred concrete description of free integral rigs, we have the canonical transposition $R^2 \rightarrow R^D$ in $\widehat {\mathbf{iAff}}$ and we can ask if it is an isomorphism.

Example 5.7 (The tangent bundle of $R$ , and why the KL axiom fails). For $A = {\unicode{x1D7DA}}\{x\}$ , Lemma 5.3 implies that

\begin{equation*} R^T (\mathrm{Spec}\, B) \cong \mathbf{iRig}({\unicode{x1D7DA}}\{x\}, B\{\varepsilon \}) \cong B\{\varepsilon \} \end{equation*}

for each $\mathrm{Spec}\, B$ in $\mathbf{iAff}$ ; but $(R\times R)(\mathrm{Spec}\, B) \cong B\times B$ .

Example 5.7 also suggests the following which is, in a sense, the main result of the paper.

Proof. Our elementary description of $A\{\varepsilon \}$ in Section 3 lifts to the topos $\widehat {\mathbf{iAff}}$ providing an integral rig $R\{\varepsilon \}$ in that topos and such that

\begin{equation*} (R\{\varepsilon \}) (\mathrm{Spec}\, B) = B\{\varepsilon \} \end{equation*}

for each finitely presentable integral rig $B$ .

It should be possible to mimic (Menni Reference Menni2024, Proposition 3.5) in order to prove that the relative KL axiom for $R$ implies that $R$ is the monoid of Euler reals determined by $0 : \mathbf{1} \rightarrow D$ . For the moment, though, there are not many examples, so we will not attempt to prove it here.

6. Connected, Local and “W-determined” integral Rigs

It follows from the main results in (Menni Reference Menni2021) that the Gaeta and Zariski subtoposes of $\widehat {\mathbf{iAff}}$ are pre-cohesive over $\mathbf{Set}$ . The reason is that every object of $\mathbf{iAff}$ is a finite coproduct of connected objects, so the Gaeta topos of $\mathbf{iAff}$ coincides with the topos of presheaves on the full subcategory of connected objects in $\mathbf{iAff}$ . Hence, by Lemma 3.5, all relevant calculations in $\mathbf{iAff}$ restrict to the subcategory, so the phenomenon described in 5.7 and 5.8 also manifests in the pre-cohesive Gaeta and Zariski toposes determined by the theory of integral rigs.

We expect these toposes to be useful instruments to explore Axiomatic Cohesion in general, and variations of SDG in particular, and probably also tangent structures (Cockett and Cruttwell Reference Cockett and Cruttwell2014), because although integral rigs look very different from $C^\infty$ -rings, the category of integral rigs has important similarities with that of $C^\infty$ -rings. The two mentioned categories are coextensive (Lawvere Reference Lawvere2008; Menni Reference Menni2021, Reference Menni2024), and we have discussed in previous sections the issue of the validity of the KL axiom. In the rest of this final section, we briefly comment on the suggestive behavior of Weil objects, assuming familiarity with (Marmolejo and Menni Reference Marmolejo and Menni2019) for some of the statements.

Although we are mainly interested in the toposes, some of the key issues may be appreciated, as in previous sections, at the level of sites. So, we consider the objects in $\mathbf{iAff}$ that have a unique point. Dually, the finitely presentable integral rigs $A$ such that there is a unique map $A \rightarrow {\unicode{x1D7DA}}$ .

For a variety ${\mathcal{V}}$ of rigs, the objects in $\mathcal{V}$ for which there is a unique map with initial codomain are called Weil (in $\mathcal{V}$ ) in (Spada and St. John Reference Spada and St. John2025, Definition 3.1). Bear in mind that, for finitely generated $\mathbb{C}$ -algebras $A$ , there is a unique map $A \rightarrow \mathbb{C}$ if, and only, if $A$ is Weil in the classical sense. Moreover, in this case, $A$ is finite dimensional as $\mathbb{C}$ -vector-space (Marmolejo and Menni Reference Marmolejo and Menni2019, Lemma 6.2).

Another relevant property of the category of affine schemes over the field $\mathbb{C}$ is that every pseudo-constant factors through an object with a unique point. Recall that a pseudo-constant is a map $ f :X \rightarrow Y$ such that, for every $a, b : \mathbf{1} \rightarrow X$ , $f a = f b$ . That property allowed the authors of (Marmolejo and Menni Reference Marmolejo and Menni2019) to identify the “infinitesimal dimension” (level $\varepsilon$ ) of the classical Gaeta and Zariski toposes over $\mathbb{C}$ .

As a corollary of (Spada and St. John Reference Spada and St. John2025, Corollary 3.9) it follows that, in ${{\mathbf{iRig}}^{\mathrm{op}}}$ , every pseudo-constant factors through an object with a unique point. Unfortunately, if we start with a map in the subcategory $\mathbf{iAff} \rightarrow {{\mathbf{iRig}}^{\mathrm{op}}}$ , we do not know if the factorization above lies in $\mathbf{iAff}$ . So, we still do not have a nice description of level $\varepsilon$ in the Gaeta topos associated to the theory of integral rigs.

An important difference with the classical complex case is that finitely generated Weil integral rigs do not coincide with finitely generated subdirectly irreducible integral rigs. See (Spada and St. John Reference Spada and St. John2025, Remark 3.2) for deceptively simple examples. This suggests that, as in the case of $C^\infty$ -rings (Kock Reference Kock2006, III.9), it may be useful to consider “W-determined” integral rigs.