Ringed topos

One of the key motivating examples of a ringed topos comes from topology. Consider the site ${\displaystyle {\text{Open}}(X)}$ of a topological space ${\displaystyle X}$, and the sheaf of continuous functions

${\displaystyle C_{X}^{0}:{\text{Open}}(X)^{op}\to {\text{CRing}}}$

sending an object ${\displaystyle U\in {\text{Open}}(X)}$, an open subset of ${\displaystyle X}$, to the ring of continuous functions ${\displaystyle C_{X}^{0}(U)}$ on ${\displaystyle U}$. Then, the pair ${\displaystyle ({\text{Sh}}({\text{Open}}(X)),C_{X}^{0})}$ forms a ringed topos. Note this can be generalized to any ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ where

${\displaystyle {\mathcal {O}}_{X}:{\text{Open}}(X)^{op}\to {\text{Rings}}}$

so the pair ${\displaystyle ({\text{Sh}}({\text{Open}}(X)),{\mathcal {O}}_{X})}$ is a ringed topos.

Another key example is the ringed topos associated to a scheme ${\displaystyle (X,{\mathcal {O}}_{X})}$, which is again the ringed topos associated to the underlying locally ringed space.

Recall that the functor of points view of scheme theory defines a scheme ${\displaystyle X}$ as a functor ${\displaystyle X:{\text{CAlg}}\to {\text{Sets}}}$ which satisfies a sheaf condition and gluing condition.[3] That is, for any open cover ${\displaystyle {\text{Spec}}(R_{f_{i}})\to {\text{Spec}}(R)}$ of affine schemes, there is the following exact sequence

${\displaystyle X(R)\to \prod X(R_{f_{i}})\rightrightarrows \prod X(R_{f_{i}f_{j}})}$

Also, there must exist open affine subfunctors

${\displaystyle U_{i}={\text{Spec}}(A_{i})={\text{Hom}}_{\text{CAlg}}(A_{i},-)}$

covering ${\displaystyle X}$, meaning for any ${\displaystyle \xi \in X(R)}$, there is a ${\displaystyle \xi |_{U_{i}}\in U_{i}(R)}$. Then, there is a topos associated to ${\displaystyle X}$ whose underlying site is the site of open subfunctors. This site is isomorphic to the site associated to the underlying topological space of the ringed space corresponding to the scheme. Then, topos theory gives a way to construct scheme theory without having to use locally ringed spaces using the associated locally ringed topos.

The category of sets is equivalent to the category of sheaves on the category with one object and only the identity morphism, so ${\displaystyle {\text{Sh}}(*)\cong {\text{Sets}}}$. Then, given any ring ${\displaystyle A}$, there is an associated sheaf ${\displaystyle {\text{Hom}}_{Sets}(-,A):{\text{Sets}}^{op}\to {\text{Rings}}}$. This can be used to find toy examples of morphisms of ringed topoi.