Could not include topos theory - contents
A morphism of ringed spaces is a pair where is a continuous map and the comorphism is a morphism of sheaves of rings over . Here denotes the direct image functor for sheaves. Any sheaf of abelian modules equipped with actions making left -modules, and such that the actions strictly commute with the restrictions, is called a sheaf of left -modules.
Every ringed space induces a ringed site: To a ringed space assign the ringed site where is the category of open sets and inclusions equipped with the pretopology of open covers and is just viewed as a sheaf of rings on .
In toric geometry and sometimes in relation to the “absolute” algebraic geometry over , 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 with a commutativity condition (which are related to higher analogues of Eckmann-Hilton argument).
ringed space, locally ringed space,
Section 6.25 of