Hello @jens-hemelaer,

I study the axiom $2=0$. So we have two different topos that classify $\mathbb{F}_2$-algebra. The classical is $\left[ \mathbb{F}_2-\text{f.p.Alg}, \text{Set} \right]$ and the second is $\text{Sh}(\text{f.p.Ring}^\text{op},J)$ where $J$ is the topology generated by $\mathbb Z \to \mathbb{F}_2$.

I study a little $\text{Sh}(\text{f.p.Ring}^\text{op},J)$ in particular, unless i'm mistaken, the sheafification functor is given by $X \mapsto \left[ R \mapsto X(R/2R) \right]$. The key lemma is just to see $X$ is a $J$-sheaf if and only if for all f.p.ring $R$ the canonical map $\pi : R \to R/2R$ induce a bijection $X(\pi)  : X(R) \to X(R/2R)$.  (It's nice that we can give a concret description )

and we get an equivalence between the two topos (i do some proof in a pdf if you want  (but in french 😀 ).  (i will try to extract a more general context it's very baby example )

For the axiom $2 =0$ or  $2$ is invertible. For the moment a have no idea but i take this axiom because in commutative algebra (in some sense) $2$ is invertible is the " negation " of  $2 = 0$. So perhaps it's funny to look in details ! 

Hi @orlando,

I agree that the two toposes agree. An alternative proof would be to use the Comparison Lemma, as follows. The morphism $R \to R/2R$ generates a $J$-covering sieve, and $R/2R$ is an $\mathbb{F}_2$-algebra. So $\mathbf{Sh}(\mathrm{f.p.Ring}^\mathrm{op},J) \simeq \mathbf{Sh}(\mathbb{F}_2\mathrm{-f.p.Alg}^\mathrm{op},J')$, where $J'$ is the restriction of $J$ to the full subcategory $\mathbb{F}_2\mathrm{-f.p.Alg}^\mathrm{op} \subset \mathrm{f.p.Ring}^\mathrm{op}$. Then it remains to show that $J'$ is the trivial topology (with only maximal sieves as covering sieves). This last part is probably the most complicated.

Your proof works just as well, but maybe the Comparison Lemma is useful for other examples, such as when you add the axiom that $2=0$ or $2$ is invertible. With this axiom, $R \to R[1/2]$ and $R \to R/2R$ together generate a covering sieve, so using the Comparison Lemma, you can restrict to the full subcategory $\mathcal{C} \subset \mathrm{f.p.Ring}^\mathrm{op}$ that consists of both the $\mathbf{F}_2$-algebras and the $\mathbb{Z}[1/2]$-algebras. I think that the restriction of $J$ to $\mathcal{C}$ gives again the trivial topology, but I'm not sure yet.

hi @jens-hemelaer,

Thank for the comparaison theorem, i will read the Sga 4 version today. 

For the $\mathbb{F}_2$-story. I extract a more  general context.

Let $\mathcal{C}$ a category with finite limite, in particular a terminal objet $\star$. Let $U$ an objet of $\mathcal{C}$ and suppose $\iota : U \to \star$ is a monomorphism.  Let $J(\iota)$ the topology on $\mathcal{C}$ generated by $\iota$. Denote by $T := \left[ \mathcal{C}^\text{op}, \text{Set} \right]$. We have an equivalence of two toposes : $T / U \simeq \text{Sh}(\mathcal{C},J)$.

You can describe $J$-sheaf. A functor $X : \mathcal{C}^\text{op} \to \text{Set}$ is a $J$-sheave iff pour all $C \in \mathcal{C}$, the projection $\pi : U \times C \to C$ induce a bijection $X(\pi) : X(C) \to X(C\times U)$. The sheafification functor is given by $X \mapsto \left[ C \to X(C \times U) \right]$.

I check some details and a think it's ok. As example : $\mathcal{C} = \text{f.p.ring}^\text{op}$ and $\mathbb{Z} \to \mathbb{F}_2$ (in ring it's an epi)  or $\mathbb{Z} \to \mathbb{Z}[ 1/2]$ (localisation is epi).

But with the axiom $2=0$  or $2$ is invertible it's harder to describe ! So perhaps comparaison lemma is very useful here ! 

Another thing : Let $J$a topology on $\text{f.p.ring}^\text{op}$ suppose that $\mathbb{A}^1$ (so $\text{hom}_{\text{Ring}}( \mathbb{Z[X]}, \bullet)$ is a $J$-sheaf. Then for all ring $R$, we have $\text{Hom}(R,\bullet)$ is a $J$-sheaf.

Do you agree with this lemma ? Do you have a proof (more general not specific to ring) ? 

The more general context is interesting, in topos theory terminology you showed above that the classifying topos of $\mathbb{F}_2$-algebras is an open subtopos of the classifying topos of all rings.

More generally, a subtopos $\mathcal{T}' \hookrightarrow \mathcal{T}$ is open if we can write $\mathcal{T}'$ as $\mathcal{T}/U$, for some monomorphism $U \hookrightarrow 1$ (this time $U$ is a sheaf, not necessarily representable by an object in $\mathcal{C}$).

If you translate everything, then a monomorphism $U \hookrightarrow 1$ as above is the same thing as a sieve on the terminal object $\ast$. So maybe this can help for the axiom that $2=0$ or $2$ is invertible. Then you take the sieve generated by $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z} \to \mathbb{Z}[1/2]$.

I agree with your lemma at the end. We can write each ring $R$ as a colimit of rings that are isomorphic to $\mathbb{Z}[x]$. I will just write this as $R = \mathrm{colim}_i\; \mathbb{Z}[x]$. Then we find that:
$\mathrm{Hom}(R,-) = \mathrm{Hom}(\mathrm{colim}_i\; \mathbb{Z}[x],-) = \mathrm{lim}_i \; \mathrm{Hom}(\mathbb{Z}[x],-)$.

For both presheaves and sheaves we have $(\mathrm{lim}_i\; \mathcal{F}_i)(C) = \mathrm{lim}_i \; (\mathcal{F}_i(C))$. So it does not matter whether we are talking about limits in the category of sheaves or in the category of presheaves, and we do not have to worry about ambiguous notation. We have written $\mathrm{Hom}(R,-)$ as a limit of sheaves of the form $\mathbb{A}^1 = \mathrm{Hom}(\mathbb{Z}[x],-)$, so if $\mathbb{A}^1$ is a sheaf, then each $\mathrm{Hom}(R,-)$ is a sheaf as well.

More generally, suppose you have a category $\mathcal{C}$ and a Grothendieck topology $J$ on $\mathcal{C}$. Further, suppose there is a full subcategory $\mathcal{D} \subseteq \mathcal{C}$ such that every object in $\mathcal{C}$ is a colimit of objects in $\mathcal{D}$. Then if $\mathbf{y}D$ is a $J$-sheaf for all $D$ in $\mathcal{D}$, then we also have that $\mathbf{y}C$ is a $J$-sheaf for all $C$ in $\mathcal{C}$. I think the same proof will work here.

Thx !!! It's a very good situation ! Just the kind of example to study the definition !

For me with to deal with $X \mapsto \underline{Hom}(U,X)$  :  my candidate to sheafification functor ! (but for the moment, it's a little deep for me) ! I take some days to clearify my self !

hi @jens-hemelaer, some news !

For the moment, i'm not sure at all, but :

Let $\mathcal{C}$ a category (with a terminal objet $\star$), let $T = \left[ \mathcal{C}^\text{op}, \text{Set} \right]$, for an objet $p$ of  $\mathcal{C}$, i denote $\widehat{p}$ for the yoneda of $p$.

let $1$ the terminal objet of $T$ i.e $1 = \widehat{\star}$.  Let $U$ a subobjet of $1$ i.e the terminal morphism $U\to 1$ a mono, this subobjet is a sieve of $\star$. Let $J$ the topology on $\mathcal{C}$ generated by $U$.

For $X$ and $Y$ in $T$, denote by $\underline{\text{Hom}}(X,Y)$ the functeur $R \mapsto \text{Hom}(X \times \widehat{R}, Y)$. This construction is bifunctorial, contravariant in a the first argument.

key Lemma : an presheaf $X$ is a sheaf if and only if the canonical map $i^\star : \underline{\text{Hom}}(1,X) \to \underline{\text{Hom}}(U,X)$ is an isomorphism.

Here it's complicated to write realy a proof but  for any objet $p$ of $\mathcal{C}$, there is a cover $\widehat{p} \times U \to \widehat{p}$ (obtain by pulling back $U \to 1$ along $p \to \star$).The condition $i^star$ is an isomorphism say that $\text{hom}(1 \times \widehat{p}, X) \simeq \text{Hom}(U \times \widehat{p},X)$ is a bijection.

1. Suppose $X$ is a $J$-sheaf. Let $U \times \widehat{p} \to X$ a morphism, with this bijection  you can complete the morphism to a morphism $\widehat{p} \to X$. 

2. The converse is difficult, because we have to see that there is no other cover than $ \widehat{p} \times U \to \widehat{p}$, i can't really write a proof.

Now, if this lemma is ok.

lemma : Let $X$ a presheaf, then $\underline{\text{Hom}}(U,X)$ is a $J$-sheaf.

Because $\underline{\text{Hom}}(U,\underline{\text{Hom}}(U,X)) \simeq \underline{\text{Hom}}(U \times U,X)$ and $U \times U \simeq U$ (cause subobjet of $1$).

We have a map $X\to \underline{\text{Hom}}(U,X)$ it's just $i^\star$ ! In fact, $\underline{\text{Hom}}(1,X) \simeq X$.

Now $X$ a presheaf.

if i take, a morphism $f : X \to Y$ with $Y$ a $J$-sheaf. I have a morphism $\underline{\text{Hom}}(U,X) \to \underline{\text{Hom}}(U,Y)$ (functoriality of $\underline{\text{Hom}}(U, \bullet)$). And $Y$ is a sheaf so $\underline{\text{Hom}}(U,Y) \simeq Y$ and we get a map $\underline{\text{Hom}}(U,X)  \to  Y$, the commutativity of the triangle is ok !).

So $ X \mapsto \underline{\text{Hom}}(U,X)$ is the sheafification functor (if key lemma is ok). I think for example the sheafification of $\mathbf{A}^1$ (in the $2 =0$ or $2$ invertible) is $R \mapsto R/2R \times R[1/2]$ (not sure).

For the moment it's a little hard to finish proof !

Hi @orlando,

Great! Your calculations made me think that this Grothendieck topology (the smallest Grothendieck topology such that $U \to 1$ is a covering sieve) is precisely the Grothendieck topology corresponding to the open subtopos $T/U$ of $T$. The Grothendieck topology associated to an open subtopos is described in "Theories, Sites, Toposes", last paragraph of page 127. It is exactly the smallest Grothendieck topology such that $U \to 1$ is a covering sieve!

In what follows on page 128, the relation to logic is described. You get this open subtopos by adding a single axiom $\top \vdash \phi$, where $\phi$ is the geometric formula without free variables corresponding to the sieve $U \to 1$ (in this case, the formula $(2=0)\vee (\exists y,~ 2y=1)$).

Now you can compare your sheafification functor with the sheafification functor for open subtoposes in the literature. Here it says that the sheafification (they call it reflector) for a presheaf $X$ is exactly $X^U$. By definition, $X^U(R) = \mathrm{Hom}(U \times \hat{R}, X) = \underline{\mathrm{Hom}}(U,X)$ in your notation (so your computation of the sheafification functor is correct). Then the map from a presheaf to its sheafification is $p : X = \underline{\mathrm{Hom}}(1,X) \to \underline{\mathrm{Hom}}(U,X)$ from your key lemma, so this is indeed an isomorphism if and only if $X$ is a sheaf.

I think your direct proof (without looking at the literature) is also possible. The reason it is so hard is because you really need an explicit description of the Grothendieck topology (not just "it is the smallest one such that $U \to 1$ is a covering sieve"). This explicit description of the Grothendieck topology is given in "Theories, Sites, Toposes" at the end of page 127. In your situation, a sieve $S$ on an object $Y$ is a covering sieve if and only if it contains the map $Y \times U \to Y$.

This topic is very interesting to me... let me know if I push the conversation too much in a certain direction, or if you want to look at things from your own point of view first.

hi @jens-hemelaer,

Thx for the references, i take time to study a little. 

In algebraic geometry, a topological space is called a "scheme" if it can be covered by open sets that are isomorphic to affine schemes, which are the spectrum of a commutative ring.

If we consider a connected commutative ring R, we can classify the corresponding schemes as follows:

1. If R is a field, then the scheme Spec(R) is a point.

2. If R is a local ring, then the scheme Spec(R) is a point with a nilpotent element.

3. If R is a Dedekind domain, then the scheme Spec(R) is a disjoint union of a finite number of integral curves.

4. If R is a regular local ring of dimension n, then the scheme Spec(R) is a smooth n-dimensional variety.

5. If R is a finitely generated algebra over a field, then the scheme Spec(R) is an affine variety.

These are just a few examples of the classification of schemes for a connected ring. The classification becomes more complex for rings with more complicated properties.