Definition:Morphism of Ringed Spaces

Definition
Let $(X, \mathcal O_X)$ and $(Y, \mathcal O_Y)$ be ringed spaces.

Definition 1
A morphism of ringed spaces from $(X, \mathcal O_X)$ to $(Y, \mathcal O_Y)$ is a pair $(f, f^\sharp)$ where:
 * $f : X \to Y$ is continuous
 * $f^\sharp : \mathcal O_Y \to f_* \mathcal O_X$ is a morphism of sheaves to the direct image sheaf of $\mathcal O_X$ via $f$

Definition 2
A morphism of ringed spaces from $(X, \mathcal O_X)$ to $(Y, \mathcal O_Y)$ is a pair $(f, f^\sharp)$ where:
 * $f : X \to Y$ is continuous
 * $f^\sharp : f^{-1}\mathcal O_Y \to \mathcal O_X$ is a morphism of sheaves from the inverse image sheaf of $\mathcal O_Y$ via $f$

Also see

 * Definition:Category of Ringed Spaces
 * Definition:Composition of Morphisms of Ringed Spaces
 * Definition:Morphism of Locally Ringed Spaces
 * Definition:Spectrum of Ring Functor