nLab local algebra

Idea

A local algebra over an algebraic theory is to an algebra over an algebraic theory as a local ring is to a ring:

a local algebra in a sheaf topos is an algebra object / sheaf of algebras, which is determined by its local restrictions, for a sense of local determined both by the Grothendieck topology of any site of definition of the topos, as well as by a coverage on the category of finitely presented algebras.

Definition

Let $\mathbb{T}$ be an essentially algebraic theory and write $\mathcal{C}_{\mathbb{T}}$ for its syntactic category: the category of finitely presented $\mathbb{T}$-algebras

$$\mathcal{C}_{\mathbb{T}} \simeq \mathbb{T}Alg^{fp} \,.$$

Let $J$ be a coverage on $\mathcal{C}_{\mathbb{T}}$.

Definition

For $\mathcal{E}$ a topos, a $J$-local $\mathbb{T}$-algebra in $\mathcal{E}$ is a functor

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

that

1. preserves finite limit;

2. sends $J$-coverings in $\mathcal{C}_{\mathbb{T}}$ to epimorphisms in $\mathcal{E}$.

A topos equipped with a local algebra object is a locally algebra-ed topos.

Properties

Proposition

A theory of local algebras is a geometric theory and every geometric theory is the theory of some local algebras.

For the moment see classifying topos for details.

Examples

• A local ring is a local algebra for the theory of rings.

  A topos equipped with a local ring is a locally ringed topos.

Related concepts

• algebraic theory, essentially algebraic theory

• algebra over an algebraic theory

The (∞,1)-category theory-analog of a theory of local algebras is (except for the additional choice of “admissible morphisms”) a

• geometry (for structured (∞,1)-toposes)