nLab ringed topos

Redirected from "ringed toposes".
Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

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

  • a topos 𝒳\mathcal{X}

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

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

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

  • a geometric morphism

    (f *f *):𝒳𝒴 (f^* \dashv f_*) : \mathcal{X} \to \mathcal{Y}
  • and a morphism (“comorphism”) of ring objects in 𝒳\mathcal{X}

    η:f *𝒪 𝒴𝒪 𝒳 \eta \colon f^* \mathcal{O}_{\mathcal{Y}} \to \mathcal{O}_{\mathcal{X}}

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

    η˜:𝒪 𝒴f *𝒪 𝒳. \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)PSh((CRing^{fg})^{op}) be the classifying topos for the Lawvere theory of rings. Then

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

    𝒪 𝒳:𝒳PSh((CRing fg) op), \mathcal{O}_{\mathcal{X}} : \mathcal{X} \to PSh((CRing^{fg})^{op}) \,,
  • a morphism (f,η):(𝒳,𝒪 𝒳)(𝒴,𝒪 𝒴)(f,\eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) is a diagram

    𝒳 f *f * 𝒴 𝒪 𝒳 η 𝒪 𝒴 PSh((CRing fg) op) \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)Topos/PSh((CRing^{fp})^{op}).

More generally:

Definition

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

See locally algebra-ed topos for more on this.

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

Examples

Properties

Limits and colimits

Proposition

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

  • the limiting topos

    lim j(𝒳 j,𝒪 𝒳 j)p j(𝒳 j,𝒪 𝒳 j) {\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 JRingedToposJ \to RingedTopos \stackrel{}{\to} Topos;

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

    lim jp j *𝒪 𝒳 jlim j𝒳 j. {\lim_\to}_j p_j^* \mathcal{O}_{\mathcal{X}_j} \in {\lim_\leftarrow}_j \mathcal{X}_j \,.

In more detail: let

(𝒴,𝒪 𝒴) f i ρ f j (𝒳 i,𝒪 𝒳 i) h ij (𝒳 j,𝒪 𝒳 j) \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 RingedToposRingedTopos, then this induces the cocone of ring objects in 𝒴\mathcal{Y}

f i *𝒪 𝒳 i f i *(h ij *𝒪 𝒳 j𝒪 𝒳 i) f j *h ij *𝒪 𝒳 j ρ 𝒪 𝒳 j f j *𝒪 𝒳 j 𝒪 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)PSh(CRing_{fg}^{op}) with rear side faces η i\eta_i and η j\eta_j, with front face η ij\eta_{i j} (corresponding to h ijh_{i j}) and top face ρ\rho.

Proof

We check the universal property of the limit:

for (𝒴,𝒪 𝒴)f i(𝒳 i,𝒪 𝒳 i)(\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:𝒴lim j(𝒳 j,𝒪 𝒳 j); h : \mathcal{Y} \to {\lim_\leftarrow}_j (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j});
  2. a unique morphism of ring objects

    h *lim jp j *𝒪 𝒳 jlim jh *p j *𝒪 𝒳 j𝒪 𝒴 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 *h^* preserves colimits and that the morphisms

    f j *𝒪 𝒳 j𝒪 𝒴 f_j^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{Y}}

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

References

Original references:

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.