Cohomology and homotopy
In higher category theory
Since a topos is a cartesian monoidal category, the notion of a unital ring and commutative unital ring can be defined internal to it.
A ringed topos is a topos equipped with a choice of ring object . If is a sheaf topos over a site then is a sheaf of rings on : a structure sheaf.
The notion of ringed topos makes sense for the theory of rings replaced by any Lawvere theory. Moreover, it makes sense for higher toposes such as (∞,1)-toposes. This is described at structured (∞,1)-topos.
A ringed topos is
If all stalks of are local rings, is a called a locally ringed topos.
A morphism of ringed toposes is
a geometric morphism
and a morphism of ring objects in
which is equivalently, by the -adjunction, a morphism of ring objects
See locally algebra-ed topos for more on this.
In order to say what locally -ringed means, one needs the extra structure of a geometry on . See there for details.
A ringed space induces the ringed localic topos of sheaves on the category of open subsets of the topological space .
Similarly but more generally a ringed site induces the ringed Grothendieck topos .
In some applications the ringed topos is refined to a lined topos when instead of a ring object an algebra-object is required.
For any (possibly non-commutative) algebra, let be its poset of commutative subalgebras. The presheaf topos naturally carries the commutative ring object . This example appears in the description of states in quantum mechanics after “Bohrification”.
Limits and colimits
Let be a diagram of ringed toposes. Its limit exists and is given by
In more detail: let
be a cone in , then this induces the cocone of ring objects in
whose commutativity may be understood as being the 2-commutativity of the prism in Topos over the classifying topos with rear side faces and , with front face (corresponding to ) and top face .
We check the universal property of the limit:
for any cone over the given diagram, we have by the definition of morphisms of ringed toposes:
an essentially unique geometric morphism
a unique morphism of ring objects
induced from the fact that the inverse image preserves colimits and that the morphisms
form a cocone under the diagram of ring objects .
An original reference is
A systematic development of geometry internal to a ringed topos is discussed in
- Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).
A textbook source is section 16.7 of
The generalization to structured (infinity,1)-toposes is due to