nLab
ringed site

Definition

A ringed site is a site $S_X$ equipped with a sheaf $O_X$ of rings.

A morphism $(f^{-1},f^{♯}):(S_X,O_X)\to (S_Y,O_Y)$ of ringed sites is a pair $(f^{-1},f^{♯})$ where $f^{-1}:S_Y\to S_X$ is a functor representing a morphism $f:S_X\to S_Y$ of sites and $f^{♯}:O_Y\to f_{*}{O}_X$ is a morphism of sheaves of rings over $Y$ (also called a $f$-comorphism).

Examples

• The archetypical and motivating class of examples is: $X$ a topological space, $S_X=\mathrm{Op}(X)$ the category of open subsets of $X$ with its standard Grothendieck topology and $O_X:=C(-,ℝ)$ the sheaf of continuous functions with values in (say) the real numbers.

• A supermanifold is a ringed site where $X$ is the underlying manifold, $S_X=\mathrm{Op}(X)$ the category of open subsets $U$ of $X$ such that for each contractible $U$ $O_X(U)\simeq C^{\mathrm{\infty }}(U){\otimes }_{ℝ}{\Lambda }^{•}V$ for $V$ a fixed finite dimensional vector space, ${\Lambda }^{•}V$ its exterior algebra and the isomorphism being one of ${\mathbb{Z}}_2$-grading rings.

• The full generalization of the notion of a ringed site is that of a structured generalized space.