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
a topos
equipped with a distinguished unital ring object : a ring internal to the topos.
If all stalks of are local rings, is a called a locally ringed topos.
A morphism of ringed toposes is
and a morphism (“comorphism”) of ring objects in
which is equivalently, by the -adjunction, a morphism of ring objects
The usual variants apply: we can speak of toposes equipped with, specifically, commutative ring objects, unital/nonunital ring objects, ring objects under other ring objects, hence associative algebra objects.
Let be the classifying topos for the Lawvere theory of rings. Then
a ringed topos is a geometric morphism
a morphism is a diagram
in the 2-category Topos.
So the 2-category of ringed toposes is the lax slice 2-category .
More generally:
For a Lawvere theory, a -ringed topos is a topos together with a product-preserving functor .
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”.
Let be a diagram of ringed toposes. Its limit exists and is given by
the limiting topos
of the underlying diagram Topos;
equipped with the colimiting ring object of all the inverse image rings
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 .
ringed topos, locally ringed topos,
Original references:
Alexander Grothendieck, Jean-Louis Verdier, Topos annelés, localisation dans les topos annelés, Section 11 in Exposé iv in: M. Artin et al., Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics 269, Springer (1972) [doi:10.1007/BFb0081551, pdf]
Monique Hakim, Chapitre III in: Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete 64, Springer (1972) [doi:10.1007/978-3-662-59155-0]
See also:
The generalization to structured (infinity,1)-toposes is due to
See also references at ringed space, such as
Last revised on April 16, 2023 at 09:04:35. See the history of this page for a list of all contributions to it.