nLab ultracategory

Idea

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.

References

• 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

• Mike Shulman, blog comment

For a different notion of ultracategory see

• Maria Manuel Clementino, Walter Tholen, Metric, topology and multicategory–a common approach, J. Pure Appl. Algebra 179 (2003) 13-47, (doi:10.1016/S0022-4049(02)00246-3).

On a 2-monadic treatment of ultracategories:

• Francisco Marmolejo, Ultraproducts and continuous families of models, PhD thesis (1995) [hdl:10222/55092; pdf]