nLab
models in presheaf toposes

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Model theory

Contents

Statement

Proposition

Let 𝕋\mathbb{T} be a geometric theory over a signature Σ\Sigma and let 𝒞\mathcal{C} be a small category. Then there is an equivalence of categories

𝕋Mod([𝒞,Set])[𝒞,𝕋Mod(Set)] \mathbb{T}Mod([\mathcal{C}, Set]) \simeq [\mathcal{C}, \mathbb{T}Mod(Set)]

between the category of models in the presheaf topos over 𝒞 op\mathcal{C}^{op} and the category of presheaves with values in 𝕋\mathbb{T}-models.

For instance (Johnstone, cor. D1.2.14).

Note that this continues to work for theories which involve infinitary limits as well. (The key observation is just that limits in [𝒞,Set][\mathcal{C}, Set] are taken pointwise.)

Examples

Example

A group object in a presheaf topos is equivalently a presheaf of groups. This is a fact used a lot for instance in homological algebra (for abelian groups). See at group object for more.

References

Around cor. D1.2.14 in

Last revised on September 23, 2016 at 18:07:03. See the history of this page for a list of all contributions to it.