What is $\mathrm{Sh}(\mathrm{CRing}^{op})$ a classifying topos of?

Question

$\mathrm{Sh}(\mathrm{CRing}_{fp}^{op})$(here $\mathrm{CRing}_{fp}^{op}$ is the opposite category of the category of finitely presented rings) will be the classifying topos of the theory of local rings. My question is: What is $\mathrm{Sh}(\mathrm{CRing}^{op})$ a classifying topos of? Thanks in advance.

Answer 1

I find that talking about universes (plural!) confuses people who don't already know how to handle them, so I won't use that concept here. Instead I will talk about small categories of rings in general.

Let $\mathcal{A}$ be any small full subcategory of $\textbf{CRing}^\textrm{op}$ with the following properties:

• $\mathbb{Z}$ is in in $\mathcal{A}$.

• Given ring homomorphisms $A \to B$ and $A \to C$, if $A$, $B$, and $C$ are in $\mathcal{A}$, then the pushout $B \otimes_A C$ is also in $\mathcal{A}$.

• If $A$ is an object in $\mathcal{A}$, then the polynomial ring $A [x]$ is also in $\mathcal{A}$.

Example. The skeleton of the opposite of the category of finitely presented rings satisfies the above hypotheses.

Proposition. Let $X : \mathcal{A} \to \mathcal{E}$ be a functor that preserves finite limits. For each object $A$ in $\mathcal{A}$, we have an $A$-algebra $O (A)$ in $\mathcal{E}_{/ X (A)}$, where:

• The underlying object of $O (A)$ is given by the morphism $X (A [x]) \to X (A)$ in $\mathcal{E}$ induced by the canonical $A$-algebra homomorphism $A \to A [x]$.

• For each $a \in A$, the constant $a$ of $O (A)$ is given by the morphism $X (A) \to X (A [x])$ in $\mathcal{E}$ induced by the $A$-algebra homomorphism $A [x] \to A$ sending $x$ to $a$.

• The addition (resp. multiplication) of $O (A)$ is given by the morphism $X (A [x_0, x_1]) \to X (A [x])$ in $\mathcal{E}$ induced by the $A$-algebra homomorphism $A [x] \to A [x_0, x_1]$ sending $x$ to $x_0 + x_1$ (resp. $x_0 x_1$).

Furthermore, for each ring homomorphism $A \to B$ where $A$ and $B$ are in $\mathcal{A}$, we get an $A$-algebra homomorphism $O (B) \to X (B) \times_{X (A)} O (A)$ in $\mathcal{E}_{/ X (B)}$, where the $A$-algebra structure on $O (B)$ is the one induced by the given ring homomorphism $A \to B$. This action is functorial in the appropriate sense.

The proposition justifies the following:

Definition. An $\mathcal{A}$-algebra in a Grothendieck topos $\mathcal{E}$ is a functor $\mathcal{A} \to \mathcal{E}$ that preserves finite limits.

Remark. Note that $X (\mathbb{Z}) \cong 1$, so $\mathcal{E}_{/ X (\mathbb{Z})}$ is equivalent to $\mathcal{E}$ itself and $O (\mathbb{Z})$ is just a ring in $\mathcal{E}$.

Example. The Yoneda embedding $Y : \mathcal{A} \to [\mathcal{A}^\textrm{op}, \textbf{Set}]$ preserves finite limits, so it is an $\mathcal{A}$-algebra in the presheaf topos $[\mathcal{A}^\textrm{op}, \textbf{Set}]$.

In fact:

Proposition. The Yoneda embedding $Y : \mathcal{A} \to [\mathcal{A}^\textrm{op}, \textbf{Set}]$, considered as an $\mathcal{A}$-algebra, is universal in the sense that for every $\mathcal{A}$-algebra $X : \mathcal{A} \to \mathcal{E}$, there is a geometric morphism $f : \mathcal{E} \to [\mathcal{A}^\textrm{op}, \textbf{Set}]$ such that $f^* Y \cong X$, and these are unique up to unique isomorphism in the appropriate sense. That is, the presheaf topos $[\mathcal{A}^\textrm{op}, \textbf{Set}]$ classifies $\mathcal{A}$-algebras.

Definition. A local $\mathcal{A}$-algebra in a Grothendieck topos $\mathcal{E}$ is a functor $X : \mathcal{A} \to \mathcal{E}$ with the following properties:

• $X$ preserves finite limits. (So $X$ is an $\mathcal{A}$-algebra, as defined previously.)

• For every object $A$ in $\mathcal{A}$ and every finite list of elements $a_0, \ldots, a_{n-1}$ in $A$ such that $a_0 + \cdots + a_{n-1} = 1$, the induced morphisms $X (A [a_0{}^{-1}]) \to X (A), \ldots, X (A [a_{n-1}{}^{-1}]) \to X (A)$ are jointly epimorphic in $\mathcal{E}$.

Proposition. Let $X : \mathcal{A} \to \mathcal{E}$ be an $\mathcal{A}$-algebra and let $O (A)$ be $X (A [x])$ considered as an $A$-algebra in $\mathcal{E}_{/ X (A)}$. The following are equivalent:

• $X : \mathcal{A} \to \mathcal{E}$ is a local $\mathcal{A}$-algebra.

• $O (\mathbb{Z})$ is a local ring in $\mathcal{E}$, i.e. the morphisms $X (\mathbb{Z} [x, x^{-1}]) \to X (\mathbb{Z} [x])$ and $X (\mathbb{Z} [x, (1 - x)^{-1}]) \to X (\mathbb{Z} [x])$ are jointly epimorphic in $\mathcal{E}$.

• For every object $A$ in $\mathcal{A}$, $O (A)$ is a local ring in $\mathcal{E}_{/ X (A)}$.

• The geometric morphism $\mathcal{E} \to [\mathcal{A}^\textrm{op}, \textbf{Set}]$ classifying $X$ factors through the subtopos of Zariski sheaves on $\mathcal{A}$.

Thus:

Theorem. The topos of Zariski sheaves on $\mathcal{A}$ classifies local $\mathcal{A}$-algebras.