nLab models in presheaf toposes

Contents

topos theory

model theory

Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

Contents

Statement

Proposition

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

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

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