# nLab local algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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

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

Created on April 27, 2011 14:40:56 by Urs Schreiber (131.211.239.144)