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.

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