nLab models in presheaf toposes

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.

Related concepts

• categorical semantics

References

Around cor. D1.2.14 in

• Peter Johnstone, Sketches of an Elephant