topos theory

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 $(X,\mathcal{O}_{X})$ 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

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

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

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

References

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