nLab
ringed topos

Definition

A topos can be considered a symmetric monoidal category with respect to the cartesian product. Thus the notion of a unital ring, and commutative unital ring can be defined in that monoidal category.

A ringed topos is a topos $X$ with a distinguished unital ring $𝒪$ in $X$, usually, but not necessarily commutative.

The standard references are SGA IV and

• M. Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

J. Lurie is also making a modern exposition of this notion along with $(\mathrm{\infty }\mathrm{,1})$-version; see also HAG and DAG by Toen et al.

Examples

• A ringed space $(X,𝒪)$ induces the ringed localic topos $(\mathrm{Sh}(X),𝒪)$ of sheaves on the category of open subsets of that space. Similarly but more generally a ringed site $(S,𝒪)$ induces the ringed Grothendieck topos $(\mathrm{Sh}(S),𝒪)$.

• In some appllcations the ringed topos is refined to a lined topos when instead of a ring object an algebra-object is required.