Definition 6.25.1. A ringed space is a pair $(X, \mathcal{O}_ X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_ X$ on $X$. A morphism of ringed spaces $(X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ is a pair consisting of a continuous map $f : X \to Y$ and an $f$-map of sheaves of rings $f^\sharp : \mathcal{O}_ Y \to \mathcal{O}_ X$.
6.25 Ringed spaces
Let $X$ be a topological space and let $\mathcal{O}_ X$ be a sheaf of rings on $X$. We are supposed to think of the sheaf of rings $\mathcal{O}_ X$ as a sheaf of functions on $X$. And if $f : X \to Y$ is a “suitable” map, then by composition a function on $Y$ turns into a function on $X$. Thus there should be a natural $f$-map from $\mathcal{O}_ Y$ to $\mathcal{O}_ X$, see Definition 6.21.7 and Lemma 6.21.8. For a precise example, see Example 6.25.2 below. Here is the relevant abstract definition.
Example 6.25.2. Let $f : X \to Y$ be a continuous map of topological spaces. Consider the sheaves of continuous real valued functions $\mathcal{C}^0_ X$ on $X$ and $\mathcal{C}^0_ Y$ on $Y$, see Example 6.9.3. We claim that there is a natural $f$-map $f^\sharp : \mathcal{C}^0_ Y \to \mathcal{C}^0_ X$ associated to $f$. Namely, we simply define it by the rule
\begin{eqnarray*} \mathcal{C}^0_ Y(V) & \longrightarrow & \mathcal{C}^0_ X(f^{-1}V) \\ h & \longmapsto & h \circ f \end{eqnarray*}
Strictly speaking we should write $f^\sharp (h) = h \circ f|_{f^{-1}(V)}$. It is clear that this is a family of maps as in Definition 6.21.7 and compatible with the $\mathbf{R}$-algebra structures. Hence it is an $f$-map of sheaves of $\mathbf{R}$-algebras, see Lemma 6.23.1.
Of course there are lots of other situations where there is a canonical morphism of ringed spaces associated to a geometrical type of morphism. For example, if $M$, $N$ are $\mathcal{C}^\infty $-manifolds and $f : M \to N$ is a infinitely differentiable map, then $f$ induces a canonical morphism of ringed spaces $(M, \mathcal{C}_ M^\infty ) \to (N, \mathcal{C}^\infty _ N)$. The construction (which is identical to the above) is left to the reader.
It may not be completely obvious how to compose morphisms of ringed spaces hence we spell it out here.
Definition 6.25.3. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ and $(g, g^\sharp ) : (Y, \mathcal{O}_ Y) \to (Z, \mathcal{O}_ Z)$ be morphisms of ringed spaces. Then we define the composition of morphisms of ringed spaces by the rule
\[ (g, g^\sharp ) \circ (f, f^\sharp ) = (g \circ f, f^\sharp \circ g^\sharp ). \]
Here we use composition of $f$-maps defined in Definition 6.21.9.
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