# nLab locally algebra-ed topos

## Theorems

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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 $\left(X,{𝒪}_{X}\right)$ 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 $\left({𝒞}_{𝕋},J\right)$ be a standard site and every standard site will do.

Then the sheaf topos $\mathrm{Sh}\left({𝒞}_{𝕋},J\right)$ 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

$\left({𝒪}_{X}⊣{A}_{*}\right):ℰ\stackrel{\stackrel{{𝒪}_{X}}{←}}{\underset{{A}_{*}}{\to }}\mathrm{Sh}\left({𝒞}_{𝕋},J\right)\phantom{\rule{thinmathspace}{0ex}}.$(\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 $\left(ℰ,{𝒪}_{X}\right)$ is called a locally $𝕋$-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 June 7, 2013 15:09:53 by Urs Schreiber (129.173.234.174)