Morphism of locally ringed spaces

Definition

Let $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ be locally ringed spaces. A morphism of locally ringed spaces $f:X\to Y$ consists of

1. a continuous map $f:X\to Y$, and
2. a morphism of sheaves of rings into the direct image sheaf

such that for every $x\in X$, the induced map on stalks

is a local homomorphism of local rings. This means that the inverse image of the maximal ideal of $\mathcal O_{X,x}$ is the maximal ideal of $\mathcal O_{Y,f(x)}$.

The local condition ensures that functions vanishing at $f(x)$ pull back to functions vanishing at $x$. A morphism of schemes is precisely a morphism of locally ringed spaces between schemes.