Ultracategories are categories with extra structure, called an ultrastructure (see Lurie, Sec 1.3). For an ultracategory, 𝒜\mathcal{A}, its ultrastructure assigns to a set of objects of 𝒜\mathcal{A} indexed by a set, SS, equipped with an ultrafilter, μ\mu, the categorical ultraproduct, SA sdμ\int_S A_s d \mu, an object of 𝒜\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

Last revised on July 3, 2020 at 04:58:49. See the history of this page for a list of all contributions to it.