indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
Ultracategories are categories with extra structure, called an ultrastructure (see Lurie, Sec 1.3). For an ultracategory, $\mathcal{A}$, its ultrastructure assigns to a set of objects of $\mathcal{A}$ indexed by a set, $S$, equipped with an ultrafilter, $\mu$, the categorical ultraproduct, $\int_S A_s d \mu$, an object of $\mathcal{A}$.
Ultracategories were introduced in Makkai 87 in order to prove conceptual completeness, but note that Lurie’s definition slightly differs from Makkai’s (Lurie, Warning 1.0.4).
(For a conjecture that ultracategories are a kind of generalized multicategory, see Shulman.)
In (Clementino-Tholen 03), a different concept of ultracategory is introduced as an instance of a generalized multicategory.
Mihaly Makkai, Stone duality for first-order logic, Adv. Math. 65 (1987) no. 2, 97–170, doi, MR89h:03067
Marek W. Zawadowski, Descent and duality, Annals of Pure and Applied Logic 71, n.2 (1995), 131–188
Jacob Lurie, Ultracategories (pdf)
For a conjecture that ultracategories are a kind of generalized multicategory see
For a different notion of ultracategory see
For a 2-monadic treatment of ultracategories, see