A morphism of ringed sites is a pair where is a functor representing a morphism of sites and is a morphism of sheaves of rings over (also called a -comorphism).
The archetypical and motivating class of examples is: a topological space, the category of open subsets of with its standard Grothendieck topology and the sheaf of continuous functions with values in (say) the real numbers.
A supermanifold is a ringed site where is the underlying manifold, the category of open subsets of such that for each contractible for a fixed finite dimensional vector space, its exterior algebra and the isomorphism being one of -grading rings.
The full generalization of the notion of a ringed site is that of a structured (∞,1)-topos.
ringed site, locally ringed site