Ringed topos of a topological space

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.

Ringed topos of a scheme

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.

Ringed topos of sets

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.