nLab locally algebra-ed topos

Idea

The notion of a topos $X$ that is equipped with a local algebra-object $\mathcal{O}_X$ is a generalization of the notion of a locally ringed topos. The algebra object $\mathcal{O}_X$ is then also called the structure sheaf.

For that reason in (Lurie) such pairs $(X, \mathcal{O}_X)$ are called structured toposes. But since the notion of locally ringed topos is a special case, maybe a more systematic and descriptive term is locally algebra-ed topos. Elsewhere this is called a locally $T$-modelled topos, where $T$ the given algebraic theory.

Definition

Let $\mathcal{C}_{\mathbb{T}}$ be the syntactic category of an essentially algebraic theory $\mathbb{T}$, hence any category with finite limits. Let $J$ be a subcanonical coverage on $\mathcal{C}_{\mathbb{T}}$. Notice that this makes $(\mathcal{C}_{\mathbb{T}}, J)$ be a standard site and every standard site will do.

Then the sheaf topos $Sh(\mathcal{C}_{\mathbb{T}}, J)$ is the classifying topos for the geometric theory of $\mathbb{T}$-local algebras.

For $\mathcal{E}$ any topos, a local $\mathbb{T}$-algebra object in $\mathcal{E}$ is a geometric morphism

$$(\mathcal{O}_X \dashv A_*) : \mathcal{E} \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{A_*}{\to}} Sh(\mathcal{C}_{\mathbb{T}}, J) \,.$$

By the discussion at classifying topos this is equivalently a functor

$$\mathcal{O}_X : \mathcal{C}_{\mathbb{T}} \to \mathcal{E}$$

such that

1. it preserves finite limits (and hence produces a $\mathbb{T}$-algebra in $\mathcal{E}$);

2. it sends $J$-coverings to epimorphisms; which makes it a local $\mathbb{T}$-algebra.

The pair $(\mathcal{E}, \mathcal{O}_X)$ is called a locally $\mathbb{T}$-algebra-ed topos.

Examples

All of the following notions are special cases of locally algebra-ed toposes:

• ringed space, locally ringed space;

• ringed site, locally ringed site

• ringed topos, locally ringed topos.

The (∞,1)-category theory-version is that of

• locally algebra-ed (∞,1)-topos.

Related concepts

• Cole's theory of spectrum

References

• Julian Cole, The Bicategory of Topoi, and Spectra , ms. (pdf)

• Jacob Lurie, Structured Spaces