topos theory

# Contents

## Definition

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

A morphism $(f^{-1}, f^\sharp):(S_X, O_X) \to (S_Y, O_Y)$ of ringed sites is a pair $(f^{-1},f^\sharp)$ where $f^{-1}:S_Y\to S_X$ is a functor representing a morphism $f:S_X\to S_Y$ of sites and $f^\sharp:O_Y\to f_* O_X$ 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, $S_X = Op(X)$ the category of open subsets of $X$ with its standard Grothendieck topology and $O_X := C(-,\mathbb{R})$ the sheaf of continuous functions with values in (say) the real numbers.

• A supermanifold is a ringed site where $X$ is the underlying manifold, $S_X = Op(X)$ the category of open subsets $U$ of $X$ such that for each contractible $U$ $O_X(U) \simeq C^\infty(U) \otimes_{\mathbb{R}} \Lambda^\bullet V$ for $V$ a fixed finite dimensional vector space, $\Lambda^\bullet V$ 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 (∞,1)-topos.