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locally algebra-ed topos

Idea

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

For that reason in (Lurie) such pairs $(X,{𝒪}_X)$ 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.

Definition

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

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

For $ℰ$ any topos, a local $𝕋$-algebra object in $ℰ$ is a geometric morphism

$$({𝒪}_X⊣A_{*}):ℰ\stackrel{\stackrel{{𝒪}_X}{\leftarrow }}{\underset{A_{*}}{\to }}\mathrm{Sh}({𝒞}_{𝕋},J).$$(\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

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

such that

1. it preserves finite limits (and hence produces a $𝕋$-algebra in $ℰ$);

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

The pair $(ℰ,{𝒪}_X)$ is called a locally $𝕋$-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.

References

• Jacob Lurie, Structured Spaces