2 Separating subcategories and Birkhoff’s theorem

To avoid any confusion let us recall the well-known notion of separation.

For instance, the Yoneda functor, embedding a small category into the corresponding presheaf topos, is separating.

In what follows we consider a presheaf topos over a small category $\mathcal {D}$ , and we distinguish here classes and sets in the usual way. It is a well-known fact that a family of morphisms $\{ f_i:P_i\to P\}_{i\in I}$ , $I$ a set, is jointly epic if for every $x\in PD$ , $D\in \mathcal {D}$ , there exists an $i\in I$ such that $x$ is in the image of $(\varphi _i)_D:P_iD\to PD$ . This follows basically from the fact that the condition on the family is equivalent to the morphism $\sum _iP_i\to P$ , induced by the $f_i$ ’s, being epi, and epis in a presheaf category are pointwise onto. The same result for classes is surely folklore, but since we were unable to locate an appropriate reference in the literature, we provide a proof.

Proof. One direction is clear. For the other, assume that $F=\{f_i:P_i\to P\}_{i\in I}$ is jointly epic. Because we are in a presheaf topos there is, for each $i\in I$ , a jointly epic set $E_i=\{g:R_g\to P_i\}$ of maps with representable domain. The class

$$\begin{align*}H:=\{f_ig:R_g\to P|i\in I, g\in E_i\} \end{align*}$$

consists of maps with domain in the Yoneda embedding $\mathcal {D}\to \widehat {\mathcal {D}}$ , so the family $H$ is actually a set because $\mathcal {D}$ is small and $\widehat {\mathcal {D}}$ is locally small. Since jointly epic families compose, $H$ is set and it is jointly epic. The result then follows by the remarks previous to the statement of the lemma.

Proof. One direction follows from the well-known fact that representables separate. For the converse, let $\mathcal {D}$ be a small category and let ${\mathcal {G} \rightarrow \widehat {\mathcal {D}}}$ be a full and separating subcategory. Let $D$ be an object in $\mathcal {D}$ . Since ${\mathcal {G} \rightarrow \widehat {\mathcal {D}}}$ is separating, the class

$$\begin{align*}F = \{ f : G_f \rightarrow \mathcal{D}(- , D) \mid G_f \in \mathcal{G} \} \end{align*}$$

of maps to ${\mathcal {D}(- , D)}$ , with domain in the subcategory ${\mathcal {G} \rightarrow \widehat {\mathcal {D}}}$ , is jointly epic. By Lemma 2.2, it follows that there exists an $f:G_f\to \mathcal {D}(-,D)$ in $F$ such that id $_D$ is in the image of $f_D:G_fD\to \mathcal {D}(D,D)$ . Thus $f$ is epi, and since $\mathcal {D}(-,D)$ is projective, it follows that $\mathcal {D}(-,D)$ is a splitting of an idempotent on $G_f$ .

Coseparating subcategories will also play a role in the paper.

Fix a category $\mathcal {A}$ and let $\mathrm {B}$ be a class of maps in $\mathcal {A}$ .

To recall Birkhoff’s theorem it is convenient to introduce some terminology.

If $\mathcal {A}$ has a terminal object then it is not sdi because the empty family of quotients is a subdirect representation of $1$ .

We recall in passing that sdi algebras are directly indecomposable [Reference Burris and Sankappanavar3, Theorem 8.5].

3 Determination

Birkhoff’s theorem directs our attention to the class of regular epimorphisms with sdi codomain but, for several reasons, a different choice of maps could be interesting. Between the class of maps used in that result, and an arbitrary class of maps, there are classes that we could call ‘principal’. Indeed, any full subcategory ${\mathrm {P} \rightarrow \mathcal {A}}$ determines the ‘principal’ class $\mathrm {B}_{\mathrm {P}}$ of maps in $\mathcal {A}$ with codomain in $\mathrm {P}$ .

In other words, in the categories mentioned above, the initial object is a coseparator.

Let $\mathbf {Ring}$ be the coextensive category of (commutative and unitary) rings.

Birkhoff warns that the fact that sdi rings without nilpotents are fields [Reference Birkhoff2, lemma 2], does not imply that a ring without nilpotents is a subdirect product of fields. “It is not necessarily true that a homomorphic image of rings without nilpotent elements is itself without nilpotent elements.”

For the next example recall that the category of $C^\infty $ -rings is the category of models of the Lawvere theory of finite powers of $\mathbb {R}$ and smooth maps between them. Since polynomials with coefficients in $\mathbb {R}$ determine smooth functions, there is an obvious forgetful functor from the category of $C^\infty $ -rings to that of $\mathbb {R}$ -algebras. See [Reference Kock6, Reference Moerdijk and Reyes19] for detailed information.

Relying on [Reference Moerdijk and Reyes19, Proposition I.3.27], we may say that a $C^\infty $ -ring $A$ is Weil if, and only if, there is a unique map ${A \rightarrow \mathbb {R}}$ and $A$ is finite dimensional as a real vector space.Footnote ¹

W-determined $C^\infty $ -rings are used in [Reference Kock6, III.9] to construct models of SDG embedding manifolds with boundary. See also the related [Reference Menni18], where variants of these models are constructed in order to exhibit a bidirectional ring of Euler reals.

Two other full subcategories $\mathrm {P}$ of $C^\infty $ -rings, and their associated $\mathrm {P}$ -determined objects, are considered in [Reference Moerdijk and Reyes19, Definition I.4.1].

Let us consider now the category $\mathbb {C}/\mathbf {Ring}$ of $\mathbb {C}$ -algebras. Relying on [Reference Marmolejo and Menni13, lemma 6.2], we may say that a $\mathbb {C}$ -algebra is Weil if it is finitely generated and local (i.e., it is the domain of a unique map with $\mathbb {C}$ as codomain). We then have the associated notion of W-determined $\mathbb {C}$ -algebra.

The next corollary of Birkhoff’s theorem might be folk but we do not know a reference.

It should be remarked that, in contrast with the case of $k$ -algebras for a field, a finitely generated $C^\infty $ -ring need not be finitely presentable. Similarly, a finitely generated local $C^\infty $ -ring need not be Weil, and germ-determined $C^\infty $ -rings play an important role in the theory. There are indications, that similar distinctions seem to arise in the category of integral rigs [Reference Spada and St. John20].

Boolean algebras, distributive lattices, rings, $\mathbb {C}$ -algebras, $C^\infty $ -rings, and so on are all coextensive categories. So, as expected, some of the above concepts have a direct geometrical expression in extensive categories.

Let us consider an extensive category $\mathcal {E}$ with terminal object $1$ . As usual a point is a map with terminal domain. We may say that ${1 \rightarrow X}$ is a point of X.

If we let $\mathcal {E}$ be the opposite of the category of Boolean algebras, or that of distributive lattices, then Example 3.2 implies that 1 is a separator in $\mathcal {E}$ . Similarly, if $\mathcal {E}$ is the opposite of the category of $C^\infty $ -rings, Example 3.4 shows that manifolds determine objects in $\mathcal {E}$ whose points are jointly epic. For future reference we emphasize the following.

4 Orthogonality against a natural monomorphism

Here we introduce a slight generalization of Lawvere’s notion of Birkhoff object, in order to prove two basic results about them that do not need the background on pre-cohesive toposes. Recall that a geometric morphism is hyperconnected if its inverse image functor is fully faithful, and its counit is monic [Reference Johnstone5, A4.6.6].

Let ${t : \mathcal {E} \rightarrow \mathcal {F}}$ be a hyperconnected geometric morphism between toposes and denote its (monic) counit by ${\xi }$ . For brevity, let ${T = t^* t_* : \mathcal {E} \rightarrow \mathcal {E}}$ .

Definition 4.1. An object $R$ in $\mathcal {E}$ is orthogonal (to $\xi $ ) if for every ${X}$ in $\mathcal {E}$ and every map ${f : T X \rightarrow R}$ there exists a unique ${f' : X \rightarrow R}$ such that the following diagram

commutes. Similarly, we say that $R$ is weakly orthogonal (to $\xi $ ) if for every $X$ and $f$ as above there is at most one $f'$ making the triangle commute.

The definition is very similar to that of separated objects for a Lawvere–Tierney topology. The lemma below shows that weakly orthogonal objects share with separated objects many useful closure properties.

Proof. We prove only the exponential ideal property. Let $R$ be weakly orthogonal and let $Y$ be any object in $\mathcal {E}$ . Let ${g, h:X \rightarrow R^Y}$ be such that

commutes. Then

also commutes. Since $R$ is weakly orthogonal, ${ev (g\times id) = ev (h\times id)}$ , so ${g = h}$ .

But not everything can be weakly orthogonal. The following is an adaptation of the argument in the penultimate paragraph of [Reference Lawvere10].

5 The Leibniz core

Let ${p : \mathcal {E} \rightarrow \mathcal {S}}$ be a local and hyperconnected geometric morphism (recall that a geometric morphism is local if its direct image part has a fully faithful right adjoint [Reference Johnstone5, C3.6.1]). Denote the (monic) counit of ${p^* \dashv p_*}$ by $\beta $ . As observed in [Reference Lawvere8], there is a canonical monic natural transformation ${\phi : p^* \rightarrow p^!}$ (see also [Reference Lawvere and Menni11, lemma 2.1]). We picture $\mathcal {E}$ as a topos ‘of spaces’, ${p^* : \mathcal {S} \rightarrow \mathcal {E}}$ as the full subcategory of discrete spaces, its right adjoint $p_*$ as sending a space to the corresponding discrete space of points, and the counit ${\beta _X : p^* (p_* X) \rightarrow X}$ as the discrete subspace of points of $X$ . The (necessarily fully faithful) right adjoint ${p^! : \mathcal {S} \rightarrow \mathcal {E}}$ to $p_*$ is thought of as the subcategory of codiscrete spaces.

We say that $p$ is a quality type if the canonical ${\phi : p^* \rightarrow p^!}$ is an isomorphism. Also, an object $X$ in $\mathcal {E}$ will be called Leibniz if ${\beta _X : p^* (p_* X) \rightarrow X}$ has a retraction [Reference Lawvere10, Remark 4.2].

Proposition 5.1. The full subcategory ${s^* : \mathcal {L} \rightarrow \mathcal {E}}$ of Leibniz objects is the inverse image functor of a hyperconnected geometric morphism ${s : \mathcal {E} \rightarrow \mathcal {L}}$ . Also, the composite ${p_* s^* : \mathcal {L} \rightarrow \mathcal {S}}$ is the direct image functor of a quality type ${q : \mathcal {L} \rightarrow \mathcal {S}}$ and the following diagram

commutes.

Proof. This is essentially a strengthened version of [Reference Lawvere8, Theorem 2]. The details may be found in [Reference Marmolejo and Menni14]. For the purposes of the present paper it is enough to borrow from there that the comonad ${(L, \lambda )}$ corresponding to ${s^* \dashv s_*}$ is defined by declaring that the square below

is a pullback, where $\eta $ is the unit of ${p_* \dashv p^!}$ . Since $\phi $ is monic, so is $\lambda $ .

We refer to ${\lambda _X : L X \rightarrow X}$ as the Leibniz core of $X$ . The intuition is better described by a quotation from [Reference Lawvere10]: “The Leibniz Core of a space $X$ is the union ${L(X)}$ of all its generalized points; this is obtained as the right adjoint of the inclusion functor from the subcategory of those spaces that ‘look like clouds of Leibnizian monads’. The more general figures that substantiate cohesion between points are omitted in the reduction from $X$ to ${L(X)}$ , but each point may have self-cohesion (which is retained in ${L(X)}$ ).” This intuition also suggests that $X$ and ${L X}$ have the same points. More precisely we have the following lemma.

6 Birkhoff objects

Let ${p : \mathcal {E} \rightarrow \mathcal {S}}$ be a local and hyperconnected geometric morphism and let ${s : \mathcal {E} \rightarrow \mathcal {L}}$ be the canonical hyperconnected geometric morphism built in Proposition 5.1. As in the proof of that result, denote ${s^* s_*}$ by $L$ and the (monic) counit of ${s^* \dashv s_*}$ by $\lambda $ , the Leibniz core.

Definition 6.1. An object $R$ in $\mathcal {E}$ is called Birkhoff if it is weakly orthogonal to $\lambda $ (in the sense of Definition 4.1). More explicitly, $R$ is Birkhoff if, and only if, for every pair of maps ${f, g:X \rightarrow R}$ , commutativity of the diagram

implies that ${f = g}$ .

Birkhoff objects were introduced in [Reference Lawvere10] as those $R$ such that “for any $X$ , any ‘infinitesimal’ map ${L(X) \rightarrow R}$ can be integrated in at most one way to a global function ${X \rightarrow R}$ .” As an example, Proposition 5.1 loc. cit. considers the pre-cohesive topos ${p: \widehat {\Delta _1} \rightarrow \mathbf {Set}}$ of reflexive graphs (over $\mathbf {Set}$ ), recalls that “The Leibniz graphs are those consisting entirely of loops, so that the inclusion ${L(X) \rightarrow X}$ omits those arrows from $X$ whose source and target are distinct,” and states that Birkhoff objects coincide with the reflexive binary relations. (To see this notice that the Leibniz core of the reflexive graph $\rightarrow $ is the discrete graph with two nodes.) In other words, in the case of ${p: \widehat {\Delta _1} \rightarrow \mathbf {Set}}$ , Birkhoff objects coincide with the separated objects for the subtopos ${p_* \dashv p^!}$ . We will see that this is not true in general, but one implication lifts to a more general result.

Proof. Since Birkhoff objects are closed under subobjects by Lemma 4.2, it is enough to show that codiscrete objects are Birkhoff. So let ${g, h:X \rightarrow p^! S}$ be such that

commutes. Then

commutes, where $\varepsilon $ is the counit ${p_* \dashv p^!}$ . Since ${p_* \lambda }$ is an iso by Lemma 5.2, ${\varepsilon (p_* g) = \varepsilon (p_* h)}$ and so ${g = h}$ , showing that ${p^! S}$ is Birkhoff.

Alternatively, using the explicit construction of $\lambda $ recalled in Proposition 5.1, it is easily shown that ${\beta _X : p^* (p_* X) \rightarrow X}$ factors through $\lambda _X$ . Since ${\beta _X}$ is the least dense (w.r.t. the subtopos ${p_* \dashv p^!}$ ) subobject of $X$ by [Reference Menni17, lemma 3.2], the result follows.

On the other hand, in a comment after [Reference Lawvere10, proposition 5.1], Lawvere states that the subobject classifier cannot have the Birkhoff property. Indeed, the next result implies that if $p$ is sufficiently cohesive then the subobject classifier is not Birkhoff.

7 Birkhoff objects in pre-cohesive presheaf toposes

In this section we consider a small category $\mathcal {C}$ with a terminal object and such that every object has a point, so that the canonical geometric morphism ${p : \widehat {\mathcal {C}} \rightarrow \mathbf {Set}}$ is pre-cohesive, and we characterize the Birkhoff objects therein.

Lemma 7.1. An object $R$ in $\widehat {\mathcal {C}}$ is Birkhoff if, and only if, for every $C$ in $\mathcal {C}$ , and every pair ${s, t : \mathbf {y} C \rightarrow R}$ , commutativity of the following diagram

implies that ${s = t}$ .

Proof. One direction is immediate. For the converse assume that condition in the statement holds, and consider a pair of maps ${f, g:X \rightarrow R}$ , such that the following diagram

commutes. We need to check that ${f = g}$ . By Yoneda, it is enough to test that ${f x = g x}$ for every ${x : \mathbf {y} C \rightarrow X}$ and every object $C$ in $\mathcal {C}$ . For any such $x$ we have that the following diagram commutes:

so, by hypothesis, ${f x = g x}$ , as we needed to show.

Roughly speaking, by the Yoneda lemma, to test that an object in ${\widehat {\mathcal {C}}}$ is Birkhoff it is enough to test against Leibniz cores of representables.

In other words, the pseudo-constants are those morphisms that are constant on points. We think of a pseudo-constant as a morphism that factors through an object that has exactly one point. Notice that if $D$ has exactly one point then every map ${D \rightarrow C}$ is a pseudo-constant.

Lemma 7.3. For any $C$ in $\mathcal {C}$ , if we let ${\mathbf {y} C = \mathcal {C}(-, C)}$ and ${\lambda _{\mathbf {y} C} : L(\mathbf {y} C) \rightarrow \mathbf {y} C }$ be its Leibniz core, then for each ${D}$ in ${\mathcal {C}}$ , the inclusion

$$ \begin{align*}{\lambda_{\mathbf{y} C, D} : (L(\mathbf{y} C)) D \rightarrow (\mathbf{y} C) D = \mathcal{C}(D, C)}\end{align*} $$

may be identified with the subset of pseudo-constants from $D$ to $C$ .

Proof. As observed in [Reference Marmolejo and Menni14], it follows from the explicit construction of the Leibniz core ${\lambda : L X \rightarrow X}$ that

$$ \begin{align*}{ (L X)D = \{ x \in X D \mid \text{ for all } a, b: 1\to D, x \cdot a = x\cdot b \} \rightarrow X D}\end{align*} $$

for every $X$ in $\widehat {\mathcal {C}}$ and $D$ in $\mathcal {C}$ . Taking ${X = \mathbf {y} C = \mathcal {C}(-, C)}$ completes the proof.

In order to discuss a sufficient condition for Birkhoff objects to form a separating family we first need to characterize them more explicitly in the present context.

Proof. By Lemma 7.1 we have that an object $R$ in ${\widehat {\mathcal {C}}}$ is Birkhoff if, and only if, for every $C$ in $\mathcal {C}$ , and every ${s, t : \mathcal {C}(-, C) \rightarrow R}$ , commutativity of the following diagram

implies that ${s = t}$ . Now the fork above commutes if, and only if, for every $D$ in $\mathcal {C}$ , the following diagram

commutes. By Lemma 7.3, ${\lambda : (L(\mathbf {y} C))D \rightarrow \mathcal {C}(D, C)}$ is the subset of pseudo-constants from $D$ to $C$ , so the result follows.

Compare Proposition 7.4 with the analogous characterization of $\neg \neg $ -separated objects obtained by replacing ‘pseudo-constant’ by ‘point’. (We will recall the details in the proof of Theorem 7.9). Sometimes, separated objects for the $\neg \neg $ -topology are said to ‘live on their points’. Similarly, we may say that Birkhoff objects are those that ‘live on their Leibnizian monads’.

The following corollary is more directly applicable to the examples at hand.

Let $\mathrm {Aff}_{\mathbb {C}}$ be the extensive category of affine schemes (over $\mathbb {C}$ ). That is, the opposite of the category of finitely presentable $\mathbb {C}$ -algebras. (Objects in $\mathrm {Aff}_{\mathbb {C}}$ are sometimes called (complex-)affine schemes of finite type.) It is well-known that the topos of presheaves on $\mathrm {Aff}_{\mathbb {C}}$ classifies $\mathbb {C}$ -algebras.

Let $\mathrm {cAff}_{\mathbb {C}} \rightarrow \mathrm {Aff}_{\mathbb {C}}$ be the full subcategory of connected affine schemes. It is also well-known that the topos $\mathcal {G}_{\mathbb {C}}$ of presheaves on $\mathrm {cAff}_{\mathbb {C}}$ is pre-cohesive (over $\mathbf {Set}$ ), that it classifies $\mathbb {C}$ -algebras without idempotents, and that the restricted Yoneda embedding ${\mathrm {Aff}_{\mathbb {C}} \rightarrow \widehat {\mathrm {cAff}_{\mathbb {C}}} = \mathcal {G}_{\mathbb {C}}}$ preserves, not only finite limits, but also finite coproducts.

Notice that, for a $\mathcal {C}$ as in the statement of Theorem 7.5, we may extract the full subcategory ${\mathcal {C}'\rightarrow \mathcal {C}}$ consisting of the objects whose pseudo-constants are jointly epic. The category $\mathcal {C}'$ contains the terminal object, every object in $\mathcal {C}'$ has a point, and Theorem 7.5 implies that Birkhoff objects in the pre-cohesive ${\widehat {\mathcal {C}'} \rightarrow \mathbf {Set}}$ separate. Notice that this process is a dual analogue of extracting a class of $\mathrm {B}$ -determined algebras for some $\mathrm {B}$ . In fact, such a process is used in [Reference Menni18, sec. 5], in order to use the properties of W-determined $C^\infty $ -rigs to show that the ring of Euler reals in the smaller topos has a ‘good order’.

It is natural to wonder if there is an elementary version of the process sketched above. Given a pre-cohesive geometric morphism ${\mathcal {E} \rightarrow \mathcal {S}}$ , can we extract a subtopos ${\mathcal {E}^{\sharp } \rightarrow \mathcal {E}}$ such that the composite ${\mathcal {E}^{\sharp } \rightarrow \mathcal {S}}$ is pre-cohesive and $\mathcal {E}^{\sharp }$ has ‘enough’ Birkhoff objects?

Theorem 7.9 provides a general explanation of why $\neg \neg $ -separated and Birkhoff objects coincide in certain pre-cohesive toposes such as that of reflexive graphs considered in [Reference Lawvere10], and many other pre-cohesive toposes of combinatorial spaces: the classifier of distributive lattices without complemented elements, its subtopos of simplicial sets, the classifier of non-trivial Boolean algebras, the classifier of strictly bipointed objects and the topos of ball complexes. See [Reference Menni16, Examples 4.2 and 4.3] and references therein. (Notice that in order to prove that the initial object coseparates in the category of finitely presentable distributive lattices we are tacitly invoking Birkhoff.)

It seems worth mentioning that some of the examples, such as reflexive graphs, or simplicial sets, are weakly generated [Reference Lawvere8] by Birkhoff objects in the sense that the smallest subtopos containing the Birkhoff objects is the whole topos.

Finally, many of the examples satisfy that Birkhoff objects coincide with the separated objects for the ‘infinitesimal dimension’ level $\epsilon $ [Reference Marmolejo and Menni13]; but we will prove this elsewhere.