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

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

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

Definition

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

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

that

  1. preserves finite limit;

  2. sends JJ-coverings in 𝒞 𝕋\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)