nLab
ultracategory

Contents

Idea

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.

References

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