Bi-directional models of
‘Radically Synthetic’ Differential Geometry

The origin of Synthetic Differential Geometry (SDG) may be traced back to certain 1967 lectures by Lawvere, later summarized in [Law79]. That summary includes, between brackets, some remarks based on developments that occurred since the original lectures. It postulates a locally catesian closed category ${\cal X}$, a ring $R$ therein, and an isomorphism ${R^{D}\cong R\times R}$, where ${D\rightarrow R}$ denotes the subobject of elements of square $0$. Between brackets, it is observed that such isomorphism could be interpreted to mean that the canonical ${R\times R\rightarrow R^{D}}$ is invertible, as had been done in several papers by Kock. See [Koc77] where this interpretation is introduced and where rings satisfying the resulting ‘Kock-Lawvere (KL)’ axiom are called of line type.

The summary also sketches the construction of models, including the role of the algebraic theories of real-analytic and of ${C^{\infty}}$ functions. Between brackets, it is mentioned that, in 1978, Dubuc succeeded in constructing a model of the axioms containing the category of real ${C^{\infty}}$-manifolds. See [Dub79] where ${C^{\infty}}$-rings play the prominent role.

The first book on SDG appeared in 1981 and it was reprinted as a second edition in [Koc06]. Several other books have appeared since 1990 giving respective accounts of the development of the subject [MR91, Lav96, Bel98, Koc10] and [BGSLF18].

Much can be done with the KL-axiom alone but, for some purposes, (e.g. integration), a pre-order on a ring of line type $R$, compatible with the ring structure, is also postulated. The typical models are ‘gros’ toposes such as those arising in classical Algebraic Geometry [Koc06] and those intentionally built to produce ‘well-adapted’ models embedding the category of manifolds [Dub79, Koc06, MR91]. See also [Men21a] for more recent examples, similar to those in Algebraic Geometry over $\mathbb{C}$, but over the simple rig with idempotent addition.

The ‘gros’ vs ‘petit’ distinction among toposes appears already in [AGV72] but the idea to axiomatize toposes ‘of spaces’ is from the early ‘80s [Law05]. See also [Law07] for a more recent formulation and [Men14] or [Men21b] for the definition of pre-cohesive geometric morphism. For instance, the ‘gros’ Zariski topos $\mathcal{E}$ determined by the field $\mathbb{C}$ of complex numbers is a well-known model of SDG and the canonical geometric morphism ${\mathcal{E}\rightarrow\mathbf{Set}}$ is pre-cohesive. The same holds for some of the models in [Men21a]. On the other hand, the well-adapted models of SDG are intuitively toposes of spaces but, as far as I know, very little work has been done to relate them with Axiomatic Cohesion. Perhaps the only exception are the results in [MR91] noting that the well-adapted models discussed there are local, which is one of the requirements in [Law05] (and also in the definition of pre-cohesive map). Another requirement is that the inverse image of a pre-cohesive geometric morphism ${\mathcal{E}\rightarrow\mathcal{S}}$ must have a finite-product preserving left adjoint ${\mathcal{E}\rightarrow\mathcal{S}}$ which, intuitively, sends a space $X$ to the associated set ${\pi_{0}X}$ of connected components of $X$. This cannot be true for the canonical geometric morphism from the ‘smooth Zariski topos’ [MR91, VI] to the topos of sets because, in constrast with the classical cases over a field, where every affine scheme is a finite coproduct of connected ones, there are affine ${C^{\infty}}$-schemes with infinite coproduct decompositions.

The paragraph above does not imply that ‘well-adaptation’ is incompatible with Cohesion. We will show in Sections 2 and 4 how to modify the techniques to construct well-adapted models of SDG so that they produce very simple (presheaf) pre-cohesive toposes (over $\mathbf{Set}$) with rings of line type. Moreover, these pre-cohesive toposes also model something more radical.

The radically synthetic foundation for smooth geometry formulated in [Law11] (and briefly recalled in Section 3) postulates a space $T$ with the property that it has a unique point and, out of the monoid $T^{T}$ of endomorphisms of $T$, it extracts a submonoid $R$ which, in many cases, is the (commutative) multiplication of a rig structure. The rig $R$ is said to be bi-directional if its subgroup $U$ of invertible elements is such that ${\pi_{0}U=\mathbb{Z}_{2}}$ the multiplicative group with two elements. In this case, $R$ may be equipped with a pre-order compatible with the rig structure. As explained loc.cit., this is ‘radically synthetic’ in the sense that all algebraic structure is derived from constructions on the geometric spaces rather than assumed. (See also [Law] which is unpublished but freely available from Lawvere’s webpage.)

We remark that no explicit models of bi-directional Radical SDG are considered in [Law11]. It is clear that the objects $D$ arising in SDG can play the role of $T$ and that they will give the expected result, but the issue of bi-directionality is more subtle. On the one hand, $R$ is not bidirectional in the models coming from Algebraic Geometry; on the other hand, we don’t know which of the known well-adapted models of SDG have a ‘$\pi_{0}$’ functor. Nevertheless, many well-adapted models at least satisfy that the subobject of invertibles in $R$ is not connected; in fact, in these cases, $U$ is representable by the ${C^{\infty}}$-ring of smooth $\mathbb{R}$-valued functions on the non-connected manifold ${(-\infty,0)+(0,\infty)}$ so, at least, they are ‘bi-directional’ in this sense. Essentially the same phenomenon implies that, in our pre-cohesive models, bi-directionality holds in the sense of the previous paragraph.

Finally we show in Section 5 that, in one of the pre-cohesive models, the pre-order on $R$, derived radically synthetically from bi-directionality, coincides with the pre-order that is defined in the analogous model of SDG.

Let $\mathbf{Ring}$ be the coextensive category of rings so that the slice ${\mathbb{R}/\mathbf{Ring}}$ is the category of $\mathbb{R}$-algebras. We next recall the algebraic theory of ${C^{\infty}}$-rings [Koc06, III.5]. For any finite set $n$, the $n$-ary operations are the smooth functions ${\mathbb{R}^{n}\rightarrow\mathbb{R}}$. The associated algebraic category of ${C^{\infty}}$-rings will be denoted by ${C^{\infty}}\textnormal{-}\mathbf{Ring}$. There is an evident algebraic functor ${{C^{\infty}}\textnormal{-}\mathbf{Ring}\rightarrow\mathbb{R}/\mathbf{Ring}}$ that has the following not quite so evident property.

As an application we prove the following (folk?) basic fact.

Let ${({C^{\infty}}\textnormal{-}\mathbf{Ring})_{fg}\rightarrow{C^{\infty}}\textnormal{-}\mathbf{Ring}}$ be the full subcategory of finitely generated ${C^{\infty}}$-rings.

In contrast with Lemma 2.1 the forgetful ${{C^{\infty}}\textnormal{-}\mathbf{Ring}\rightarrow\mathbb{R}/\mathbf{Ring}}$ does not create the universal solution to inverting an element. This will be of key importance throughout the paper.

Also, in contrast with the case of $k$-algebras for a field $k$: it is not the case that every finitely generated ${C^{\infty}}$-ring is finitely presentable [Koc06, Example III.5.5], and it is not the case that every finitely generated ${C^{\infty}}$-ring is a finite direct product of directly indecomposable ones. Moreover, by the Nullstellensatz, every non-final finitely generated $\mathbb{C}$-algebra has a (co)point, but see [Koc06, p. 165] for an example of a non-final finitely generated ${C^{\infty}}$-ring without points.

Let $\mathrm{Aff}_{{C^{\infty}}}$ be the (extensive) opposite of the category of finitely generated ${C^{\infty}}$-rings. Its objects might be called affine ${C^{\infty}}$-schemes.

Euler’s observation that real numbers are ratios of infinitesimals is the topic of [Law11]. To make that observation rigorous, Lawvere suggests some basic properties of the underlying category of spaces and postulates the existence of an object ‘of infinitesimals’ with the sole property that it has a unique point. Then he shows how some simple axioms of geometric nature allow to construct a pre-ordered ring ‘of Euler reals’. In this section, we briefly recall some of these ideas and relate them to SDG.

Let $\mathcal{E}$ be an extensive category with finite limits and let ${0:\mathbf{1}\rightarrow T}$ be a pointed object in $\mathcal{E}$ such that $T$ is exponentiable. For any object $X$ in $\mathcal{E}$, ${X^{T}}$ is called the tangent bundle of the space $X$, with evaluation at $0$ inducing the bundle map ${ev_{0}:X^{T}\rightarrow X}$.

The exponential $T^{T}$ carries a canonical monoid structure determined by composition in $\mathcal{E}$. This monoid structure restricts to $R$ and is called multiplication. The transposition ${\mathbf{1}\rightarrow T^{T}}$ of the evident composite ${T\rightarrow\mathbf{1}\rightarrow T}$ factors as a map ${0:\mathbf{1}\rightarrow R}$ followed by the inclusion ${R\rightarrow T^{T}}$. In this way, $R$ has the intrinsic structure of a monoid with $0$.

We are mainly interested in categories of spaces that are toposes, but some of the ideas may be directly illustrated at the level of sites.

The preservation properties of the Yoneda functor imply that, in the topos of presheaves on $\mathrm{Aff}_{k}$, the monoid of Euler reals determined by the presheaf representable by $T$ is representable by $R$. On the other hand, ${\mathrm{Aff}_{k}\rightarrow\widehat{\mathrm{Aff}_{k}}}$ does not preserve finite coproducts.

The similarity between the intuitions about $T$ and about the object $D$ in SDG is no accident. To give a rigorous comparison we prove the following result which has surely been known since the mid 60’s although I don’t think it appears explicitly in [Law79] or in [Koc06]. An ‘external’ form of the result is mentioned in the paragraph before Section 4 in [Ros84]. The present form is suggested in the last paragraph of p. 250 of [Law11] and it is stated explicitly just before Corollary 5.5 in [CC14].

In any case, regardless of the preferred style of proof, Proposition 3.5 shows, roughly speaking, that the ‘radical’ and the ‘conservative’ versions of SDG are compatible.

Recall that, in an extensive category, an object is said to be connected if it has exactly two complemented subobjects. For instance, the rings of line type in Examples 3.2 and 3.8 are connected. On the other hand:

(A different approach to connectedness is that in [MR91, Section III.3] using internal topological spaces in well-adapted models of SDG, but we will not deal with that here.)

Even assuming that $R$ is connected, the subobject ${U\rightarrow R}$ of invertibles may or may not be connected.

In contrast, consider the following.

In order to continue our discussion it is convenient to recall the following restricted version of [Yet87, Definition 0.2].

To capture the idea of a ‘discrete space of connected components’, the suggestion in [Law11] is to use a full reflective subcategory of $\mathcal{E}$ of $T$-discrete objects and such that the left adjoint preserves finite products. We will not make emphasis on $T$-discrete objects. We will simply assume that we have a reflective subcategory ${\mathcal{S}\rightarrow\mathcal{E}}$ whose left adjoint ${\pi_{0}:\mathcal{E}\rightarrow\mathcal{S}}$ preserves finite products. Given such a reflective subcategory we may say that an object $X$ in $\mathcal{E}$ is connected if ${\pi_{0}X=\mathbf{1}}$.

Example 3.13 may be lifted to toposes as in [Men14]. Again, see [Men21a] for analogous examples in the context of algebras with idempotent addition.

To recapitulate, let $\mathcal{E}$ be an extensive category with finite limits and let ${\mathcal{S}\rightarrow\mathcal{E}}$ be a full subcategory with finite-product preserving left adjoint $\pi_{0}$. Let ${0:\mathbf{1}\rightarrow T}$ be a pointed object in $\mathcal{E}$ with exponentiable $T$ and let ${R}$ be the associated monoid of Euler reals.

In other words, under mild assumptions, $R$ is connected if and only if, for every space $X$, the tangent bundle of $X$ has the same connected components as $X$. (Notice that if $R$ is a ring of line type then the subobject ${R\rightarrow D^{D}}$ appearing in the proof of Proposition 3.5 has a retraction needed to define derivatives.)

Rings of line type have always been commutative. Concerning monoids of Euler reals [Law11] says that “To justify that commutativity seems difficult, though intuitively it is related to the tinyness of $T$, in the sense that even for slightly larger infinitesimal spaces, the (pointed) endomorphism monoid is non-commutative”.

Also, [Law11] briefly discusses two ways to insure that monoids of Euler reals have a unique addition. One via Integration, the other via trivial Lie algebras. The first one is to consider the subobject ${\Phi(X)=\mathrm{Hom}_{R}(R^{X},R)\rightarrow R^{(R^{X})}}$ of the functionals ${\phi}$ such that ${\phi(\lambda f)=\lambda(\phi f)}$ for every ${\lambda\in R}$, and then require that ${\Phi(\mathbf{1}+\mathbf{1})=(\Phi\mathbf{1})^{2}}$, so that addition “emerges as the unique homogeneous map ${R\times R\rightarrow R}$ which becomes the identity when restricted to both $0$-induced axes ${R\rightarrow R\times R}$.” The second one is to consider the kernel ${\mathrm{Lie}(R)\rightarrow R^{T}}$ of ${ev_{0}:R^{T}\rightarrow R}$ which has a binary operation that may be called addition; “the space of endomorphisms of ${\mathrm{Lie}(R)}$ for that operation is a rig that contains (the right action of) $R$ as a multiplicative sub-monoid, so that if we postulate that $R$ exhausts the whole endomorphism space, then $R$ inherits a canonical addition”.

Assuming that the monoid $R$ underlies a ring structure, the simple result below is applied to derive, from a subgroup of $R$, a pre-order on $R$, meaning a subrig $M$ of ‘non-negative quantities’.

For example, if ${P\subseteq\mathbb{C}}$ is the subobject of invertible elements then ${A=\{0\}}$ and ${M=\mathbb{C}}$. On the other hand, if ${P=(0,\infty)\subseteq\mathbb{R}}$ then ${M=A=[0,\infty)\subseteq\mathbb{R}}$.

Let ${U\rightarrow R}$ be the subobject of invertible elements and let the following square

be a pullback, where ${1:\mathbf{1}\rightarrow U}$ is the unit of $R$ as a subobject of $U$. Since $U$ is a subgroup of $R$, and $\pi_{0}$ preserves products, ${\pi_{0}U}$ is also a group and ${U_{+}\rightarrow U}$ is the kernel of the group morphism ${U\rightarrow\pi_{0}U}$.

The map ${\pi_{0}1:\mathbf{1}\rightarrow\pi_{0}U}$ may be an isomorphism and, in this case, ${U_{+}\rightarrow U}$ is also an isomorphism, of course. We are mainly interested in contexts where this is not the case.

So, assuming that the monoid $R$ of Euler reals underlies a ring, we may apply Proposition 3.16 to the subgroup ${U_{+}\subseteq R}$ in order to obtain a subrig ${M\subseteq R}$. The ‘real’ intuition suggests that it is not unnatural to require or expect that ${1+U_{+}\subseteq U_{+}}$; in other words, ${1\in A\subseteq R}$. Then the first item of Lemma 3.17 implies that ${-1\not\in M}$, unless ${0=1}$ in $R$.

Readers are invited to compare the above with the efforts in [Koc06, MR91] to prove that the pre-orders defined on certain rings of line type there are compatible with the ring operations. We should also remark that some of those efforts will play a role below when the time comes to give a simple description of the subrigs ${M\rightarrow R}$ derived radically synthetically.

Let ${p:\mathcal{E}\rightarrow\mathcal{S}}$ be a geometric morphism. Recall that $p$ is hyperconnected if ${p^{*}:\mathcal{S}\rightarrow\mathcal{E}}$ is fully faithful and the counit $\beta$ of ${p^{*}\dashv p_{*}}$ is monic. Intuitively, $\mathcal{E}$ is a category of spaces, ${p^{*}}$ is the full subcategory of discrete spaces, and the right adjoint ${p_{*}:\mathcal{E}\rightarrow\mathcal{S}}$ sends a space to the associated discrete space of points. For a space $X$, the monic ${\beta_{X}:p^{*}(p_{*}X)\rightarrow X}$ may be thought of as the discrete subspace of points.

In perspective, we may say that $p$ is pre-cohesive if it is hyperconnected, $p^{*}$ is cartesian closed and ${p_{*}}$ preserves coequalizers. It follows [Men21b] that ${p_{*}}$ has a (necessarily fully faithful) right adjoint $p^{!}$ and that ${p^{*}}$ has a left adjoint ${p_{!}:\mathcal{E}\rightarrow\mathcal{S}}$ that preserves finite products. Hence, $p$ is pre-cohesive if and only if ${p^{*}\dashv p_{*}}$ extends to a string of adjoints ${p_{!}\dashv p^{*}\dashv p_{*}\dashv p^{!}}$ such that ${p^{*}}$ is fully faithful, the counit of ${p^{*}\dashv p_{*}}$ is monic and the leftmost adjoint $p_{!}$ preserves finite products. It follows that the reflector ${\sigma_{X}:X\rightarrow p^{*}(p_{!}X)=\pi_{0}X}$ is epic. Sometimes we say that $\mathcal{E}$ is pre-cohesive over $\mathcal{S}$.

For any pre-cohesive $p$, the fact that the canonical ${p_{*}\sigma:p_{*}\rightarrow p_{*}p^{*}p_{!}\cong p_{!}}$ is an epimorphism is consistent with the intuition that every connected component has a point. In particular, if ${p_{*}X=\mathbf{1}}$ then ${p_{!}X=\mathbf{1}}$. In other words, if the space $X$ has a unique point then it is connected.

The purpose of this section is to build a pre-cohesive topos ${p:\mathcal{E}\rightarrow\mathbf{Set}}$ and a (necessarily connected) object $T$ in $\mathcal{E}$ such that ${p_{*}T=\mathbf{1}}$, and such that the resulting monoid $R$ of Euler reals is bi-directional. Moreover, it will be evident from the construction that $R$ underlies a ring of line-type.

Pre-cohesion will follow from the next result borrowed from [Joh11]. (See also [Men14, Proposition 2.10] for a statement consistent with our terminology.)

Now recall Remark 3.4 and let ${\mathcal{C}\rightarrow\mathrm{Aff}_{{C^{\infty}}}}$ be the full subcategory of connected objects that have a point. The category $\mathcal{C}$ is essentially small but contains many objects of interest. It certainly has a terminal object. Also, germ-determined ${C^{\infty}}$-rings have a copoint almost by definition ([Koc06, Exercise III.7.1]), so every non-trivial germ-determined finitely-generated ${C^{\infty}}$-ring with exactly two idempotents determines an object in $\mathcal{C}$.

In particular, for any manifold $M$, ${{C^{\infty}}(M)}$ is finitely-presentable and germ-determined (in fact, point-determined) by [Koc06, Theorem III.6.6 and Corollary III.5.10]. Moreover, if $M$ is connected then ${{C^{\infty}}(M)}$ has exactly two idempotents by the Intermediate Value Theorem. Altogether, for any connected manifold $M$, ${{C^{\infty}}(M)}$ determines an object of $\mathcal{C}$.

Also, every Weil algebra (over $\mathbb{R}$) is a finitely presentable ${C^{\infty}}$-ring by [Koc06, Proposition III.5.11] and hence determines an object in $\mathcal{C}$.

Let $T$ be the object in $\mathcal{C}$ determined by the Weil algebra $\mathbb{R}[\epsilon]$. Of course, it has a unique point, and we denote it by ${0:\mathbf{1}\rightarrow T}$.

By the remarks above, the object ${R=\operatorname{Spec}({C^{\infty}}(\mathbb{R}))}$ is in the subcategory ${\mathcal{C}\rightarrow\mathrm{Aff}_{{C^{\infty}}}}$. The product

is also in $\mathcal{C}$, so $R$ is a ring object in $\mathcal{C}$.

It is clear from the proof of Proposition 4.4 that ${U_{+}=\operatorname{Spec}({C^{\infty}}(0,\infty))}$ in $\widehat{\mathcal{C}}$. Then ${1+U_{+}\subseteq U_{+}}$ and so, by the remarks following Lemma 3.17, we obtain a subrig ${M\subseteq R}$ inside the complement of ${-1:\mathbf{1}\rightarrow R}$. I have not found an illuminating expression of this subrig though. In the next section we consider a smaller topos, also with a bi-directional $R$, but where ${M\subseteq R}$ has a simple description.

In this section we construct another bi-directional ring of line type but in a topos where the induced pre-order is easier to describe explicitly in terms of the site.

A ${C^{\infty}}$-ring $A$ is W-determined if the family of maps ${A\rightarrow W}$ in ${C^{\infty}}\textnormal{-}\mathbf{Ring}$ with Weil codomain is jointly monic.

W-determined ${C^{\infty}}$-rings are called near-point determined in [MR91].

Let ${\mathcal{C}_{W}\rightarrow\mathcal{C}}$ be the full subcategory induced by the objects in $\mathcal{C}$ that are W-determined as ${C^{\infty}}$-rings. It follows from the discussion following [Koc06, Theorem III.9.4] that every connected manifold with boundary determines an object in $\mathcal{C}_{W}$. Moreover, this assignment is functorial.

Let ${{C^{\infty}}{[0,\infty)}={C^{\infty}}(\mathbb{R})/I}$ where ${I\subseteq{C^{\infty}}(\mathbb{R})}$ is the ideal of functions that vanish on ${[0,\infty)\subseteq\mathbb{R}}$.

The ${C^{\infty}}$-ring ${{C^{\infty}}{[0,\infty)}}$ is W-determined by the remarks in [Koc06, p.185]. The corresponding object in $\mathcal{C}_{W}$ will be denoted by $H$. So the quotient ${{C^{\infty}}(\mathbb{R})\rightarrow{C^{\infty}}[0,\infty)}$ induces a monomorphism ${H\rightarrow R}$ in $\mathcal{C}_{W}$.

Let ${\Gamma=\mathcal{C}_{W}(\mathbf{1},-):\mathcal{C}_{W}\rightarrow\mathbf{Set}}$ be the usual ‘points’ functor.

For a connected manifold $M$ we let ${\overline{M}=\operatorname{Spec}({C^{\infty}}(M))}$. To complete our calculation we need another standard result that we reformulate as follows.

Recall that ${U\rightarrow R}$ is the subobject of invertibles, that ${U_{+}\rightarrow U}$ is the kernel of ${U\rightarrow\pi_{0}U}$ and that ${M\subseteq R}$ is the subrig determined by bi-directionality of $R$.