nLab ringed topos

Idea

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 $(X,\mathcal{O}_{X})$ is a topos $X$ equipped with a choice of ring object $\mathcal{O}$. If $X$ is a sheaf topos over a site $C$ then $\mathcal{O}_X$ is a sheaf of rings on $C$: 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.

Definition

Definition

A ringed topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is

• a topos $\mathcal{X}$

• equipped with a distinguished unital ring object $\mathcal{O}_{\mathcal{X}} \in \mathcal{X}$: a ring internal to the topos.

If all stalks of $\mathcal{O}_{\mathcal{X}}$ are local rings, $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a called a locally ringed topos.

A morphism of ringed toposes $(f, \eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})$ is

• a geometric morphism

  $$(f^* \dashv f_*) : \mathcal{X} \to \mathcal{Y}$$

• and a morphism (“comorphism”) of ring objects in $\mathcal{X}$

  $$\eta \colon f^* \mathcal{O}_{\mathcal{Y}} \to \mathcal{O}_{\mathcal{X}}$$

  which is equivalently, by the $(f^* \dashv f_*)$-adjunction, a morphism of ring objects

  $$\tilde \eta : \mathcal{O}_{\mathcal{Y}} \to f_* \mathcal{O}_{\mathcal{X}} \,.$$

Remark

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.

Remark

Let $PSh((CRing^{fg})^{op})$ be the classifying topos for the Lawvere theory of rings. Then

• a ringed topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a geometric morphism

  $$\mathcal{O}_{\mathcal{X}} : \mathcal{X} \to PSh((CRing^{fg})^{op}) \,,$$

• a morphism $(f,\eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})$ is a diagram

  $$\array{ \mathcal{X} &&\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}&& \mathcal{Y} \\ & {}_{\mathllap{}}\searrow \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}} &\swArrow_{\eta}& \swarrow \nearrow_{\mathcal{O}_{\mathcal{Y}}} \\ && PSh((CRing^{fg})^{op}) }$$

  in the 2-category Topos.

So the 2-category of ringed toposes is the lax slice 2-category $Topos/PSh((CRing^{fp})^{op})$.

More generally:

Definition

For $T$ a Lawvere theory, a $T$-ringed topos is a topos $X$ together with a product-preserving functor $\mathcal{O}_X : T \to X$.

See locally algebra-ed topos for more on this.

In order to say what locally $T$-ringed means, one needs the extra structure of a geometry on $T$. See there for details.

Examples

• A ringed space $(X,\mathcal{O})$ induces the ringed localic topos $(Sh(X),\mathcal{O})$ of sheaves on the category of open subsets of the topological space $X$.

  Similarly but more generally a ringed site $(S, \mathcal{O})$ induces the ringed Grothendieck topos $(Sh(S), \mathcal{O})$.

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

• For $A \in Alg$ any (possibly non-commutative) algebra, let $\mathcal{C}(A)$ be its poset of commutative subalgebras. The presheaf topos $[\mathcal{C}(A), Set]$ naturally carries the commutative ring object $\underline A : (C \in \mathcal{C}(A)) \mapsto C$. This example appears in the description of states in quantum mechanics after “Bohrification”.

Properties

Limits and colimits

Proposition

Let $J \to RingedTopos$ be a diagram of ringed toposes. Its limit exists and is given by

• the limiting topos

  $${\lim_\leftarrow}_j (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j}) \stackrel{p_j}{\to} (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j})$$

  of the underlying diagram $J \to RingedTopos \stackrel{}{\to}$ Topos;

• equipped with the colimiting ring object of all the inverse image rings

  $${\lim_\to}_j p_j^* \mathcal{O}_{\mathcal{X}_j} \in {\lim_\leftarrow}_j \mathcal{X}_j \,.$$

In more detail: let

$$\array{ && (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) \\ & {}^{\mathllap{f_i}}\swarrow &{}^{\mathllap{\simeq}}\swArrow_{\rho}& \searrow^{\mathrlap{f^j}} \\ (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i}) &&\underset{h_{i j}}{\to}&& (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j}) }$$

be a cone in $RingedTopos$, then this induces the cocone of ring objects in $\mathcal{Y}$

$$\array{ f_i^* \mathcal{O}_{\mathcal{X}_i} & \stackrel{f_i^*(h_{i j}^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{X}_i} )}{\leftarrow} & f_j^* h_{i j}^* \mathcal{O}_{\mathcal{X}_j} &\underoverset{\simeq}{\rho_{\mathcal{O}_{\mathcal{X}_j}}}{\leftarrow}& f_j^* \mathcal{O}_{\mathcal{X}_j} \\ & \searrow && \swarrow \\ && \mathcal{O}_{\mathcal{Y}} }$$

whose commutativity may be understood as being the 2-commutativity of the prism in Topos over the classifying topos $PSh(CRing_{fg}^{op})$ with rear side faces $\eta_i$ and $\eta_j$, with front face $\eta_{i j}$ (corresponding to $h_{i j}$) and top face $\rho$.

Proof

We check the universal property of the limit:

for $(\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) \stackrel{f_i}{\to} (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$ any cone over the given diagram, we have by the definition of morphisms of ringed toposes:

1. an essentially unique geometric morphism

   $$h : \mathcal{Y} \to {\lim_\leftarrow}_j (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j});$$

2. a unique morphism of ring objects

   $$h^* {\lim_\to}_j p_j^* \mathcal{O}_{\mathcal{X}_j} \simeq {\lim_\to}_j h^* p_j^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{Y}}$$

   induced from the fact that the inverse image $h^*$ preserves colimits and that the morphisms

   $$f_j^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{Y}}$$

   form a cocone under the diagram of ring objects $f_j^* \mathcal{O}_{\mathcal{X}_j} \in \mathcal{Y}$.

Related concepts

• ringed space, locally ringed space

• ringed site, locally ringed site

• ringed topos, locally ringed topos,

• locally algebra-ed topos

• structured (∞,1)-topos

References

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:

• Aise Johan de Jong, section 18.7 of: The Stacks Project (pdf) (project website)

The generalization to structured (infinity,1)-toposes is due to

• Jacob Lurie, Structured Spaces

See also references at ringed space, such as

• eom, Ringed spaces