nLab
locally algebra-ed topos

Context

Higher geometry

Higher algebra

Contents

Idea

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

For that reason in (Lurie) such pairs (X,𝒪 X)(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 TT-modelled topos, where TT the given algebraic theory.

Definition

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

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

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

(𝒪 XA *):A *𝒪 XSh(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 \mathcal{O}_X : \mathcal{C}_{\mathbb{T}} \to \mathcal{E}

such that

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

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

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

Examples

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

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

References

Revised on October 11, 2014 16:15:30 by Urs Schreiber (185.26.182.27)