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ringed topos

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Topos Theory

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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,π’ͺ X)(X,\mathcal{O}_{X}) is a topos XX equipped with a choice of ring object π’ͺ\mathcal{O}. If XX is a sheaf topos over a site CC then π’ͺ 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,π’ͺ 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) is

  • a topos 𝒳\mathcal{X}

  • equipped with a distinguished unital ring object π’ͺ π’³βˆˆπ’³\mathcal{O}_{\mathcal{X}} \in \mathcal{X}: a ring internal to the topos.

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

A morphism of ringed toposes (f,Ξ·):(𝒳,π’ͺ 𝒳)β†’(𝒴,π’ͺ 𝒴)(f, \eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) is

  • a geometric morphism

    (f *⊣f *):𝒳→𝒴 (f^* \dashv f_*) : \mathcal{X} \to \mathcal{Y}
  • and a morphism of ring objects in 𝒳\mathcal{X}

    Ξ·:f *π’ͺ 𝒴→π’ͺ 𝒳 \eta : 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˜:π’ͺ 𝒴→f *π’ͺ 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,π’ͺ 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) is a geometric morphism

    π’ͺ 𝒳:𝒳→PSh((CRing fg) op), \mathcal{O}_{\mathcal{X}} : \mathcal{X} \to PSh((CRing^{fg})^{op}) \,,
  • a morphism (f,Ξ·):(𝒳,π’ͺ 𝒳)β†’(𝒴,π’ͺ 𝒴)(f,\eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) is a diagram

    𝒳 β†’f *←f * 𝒴 β†˜β†– π’ͺ 𝒳 ⇙ Ξ· ↙↗ π’ͺ 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 π’ͺ X:Tβ†’X\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 J→RingedToposJ \to RingedTopos be a diagram of ringed toposes. Its limit exists and is given by

  • the limiting topos

    lim ← j(𝒳 j,π’ͺ 𝒳 j)β†’p j(𝒳 j,π’ͺ 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 J→RingedTopos→J \to RingedTopos \stackrel{}{\to} Topos;

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

    lim β†’ jp j *π’ͺ 𝒳 j∈lim ← j𝒳 j. {\lim_\to}_j p_j^* \mathcal{O}_{\mathcal{X}_j} \in {\lim_\leftarrow}_j \mathcal{X}_j \,.

In more detail: let

(𝒴,π’ͺ 𝒴) f i↙ ≃⇙ ρ β†˜ f j (𝒳 i,π’ͺ 𝒳 i) β†’h ij (𝒳 j,π’ͺ 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 *π’ͺ 𝒳 i ←f i *(h ij *π’ͺ 𝒳 jβ†’π’ͺ 𝒳 i) f j *h ij *π’ͺ 𝒳 j ←≃ρ π’ͺ 𝒳 j f j *π’ͺ 𝒳 j β†˜ ↙ π’ͺ 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,π’ͺ 𝒴)β†’f i(𝒳 i,π’ͺ 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:𝒴→lim ← j(𝒳 j,π’ͺ 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 β†’ jp j *π’ͺ 𝒳 j≃lim β†’ jh *p j *π’ͺ 𝒳 jβ†’π’ͺ 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 *π’ͺ 𝒳 jβ†’π’ͺ 𝒴 f_j^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{Y}}

    form a cocone under the diagram of ring objects f j *π’ͺ 𝒳 jβˆˆπ’΄f_j^* \mathcal{O}_{\mathcal{X}_j} \in \mathcal{Y}.

References

An original reference is

A systematic development of geometry internal to a ringed topos is discussed in

  • Monique Hakim, Topos annelΓ©s et schΓ©mas 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

See also

Revised on November 5, 2013 01:58:43 by Anonymous Coward (92.140.60.181)