indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
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
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.)
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.
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.