Could not include topos theory - contents
A ringed space is a pair $(X,O_X)$ where $X$ is a topological space and $O_X$ is a sheaf of unital rings. The sheaf $O_X$ is called the structure sheaf of the ringed space $(X,O_X)$.
If all stalks of the structure sheaf are local rings, it is called a locally ringed space.
A morphism of ringed spaces $(f,f^\sharp):(X,O_X)\to (Y,O_Y)$ is a pair where $f:X\to Y$ is a continuous map and the comorphism $f^\sharp : O_Y\to f_* O_X$ is a morphism of sheaves of rings over $Y$. Here $f_*$ denotes the direct image functor for sheaves. Any sheaf of abelian modules $\mathcal{M}$ equipped with actions $O_X(U)\times\mathcal{M}(U)\to\mathcal{M}(U)$ making $\mathcal{M}(U)$ left $O_X$-modules, and such that the actions strictly commute with the restrictions, is called a sheaf of left $O_X$-modules.
Every ringed space induces a ringed site: To a ringed space $(X,O_X)$ assign the ringed site $(Op_X,O_X)$ where $Op_X$ is the category of open sets and inclusions equipped with the pretopology of open covers and $O_X$ is just viewed as a sheaf of rings on $Op_X$.
In toric geometry and sometimes in relation to the “absolute” algebraic geometry over $F_1$, one talks about monoided or monoidal space (Kato; Deitmar); which is a topological space together with a sheaf of monoids. N. Durov on the other hand develops a generalized algebraic geometry based on a notion of generalized ringed space, which is a space equipped with a sheaf of (commutative) generalized rings, which are finitary (= algebraic) monads in $\mathrm{Set}$ with a commutativity condition (which are related to higher analogues of Eckmann-Hilton argument).
ringed space, locally ringed space,
Section 6.25 of