nLab ringed site



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




A ringed site is a site S XS_X equipped with a sheaf O XO_X of rings.

A morphism (f 1,f ):(S X,O X)(S Y,O Y)(f^{-1}, f^\sharp):(S_X, O_X) \to (S_Y, O_Y) of ringed sites is a pair (f 1,f )(f^{-1},f^\sharp) where f 1:S YS Xf^{-1}:S_Y\to S_X is a functor representing a morphism f:S XS Yf:S_X\to S_Y of sites and f :O Yf *O Xf^\sharp:O_Y\to f_* O_X is a morphism of sheaves of rings over YY (also called a ff-comorphism).


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

  • A supermanifold is a ringed site where XX is the underlying manifold, S X=Op(X)S_X = Op(X) the category of open subsets UU of XX such that for each contractible UU O X(U)C (U) Λ VO_X(U) \simeq C^\infty(U) \otimes_{\mathbb{R}} \Lambda^\bullet V for VV a fixed finite dimensional vector space, Λ V\Lambda^\bullet V its exterior algebra and the isomorphism being one of 2\mathbb{Z}_2-grading rings.

  • The full generalization of the notion of a ringed site is that of a structured (∞,1)-topos.


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