nLab ultracategory

Idea

Ultracategories are categories endowed with extra structure meant to abstract the possibility of forming ultraproducts. They were introduced by Makkai (1987) in order to prove strong conceptual completeness for coherent logic: while the category of models of a coherent theory, on its own, is not sufficient to determine the theory up to any useful notion of equivalence, it becomes sufficient to reconstruct the theory up to Morita equivalence once it is endowed with the structure of an ultracategory.

Theorem

(Makkai) Let $P$ be a small pretopos and let $\operatorname{Mod}(P) = \mathbf{Pretop}(P, Set)$ be its category of models. Then, the evaluation functor

$$ev \colon P \to \mathbf{Ult}\big( \operatorname{Mod}(P) , Set \big)$$

is an equivalence of categories, where $\mathbf{Ult}$ is the 2-category of ultracategories, ultrafunctors, and transformations thereof.

In other words, ultracategories determine a 2-fully-faithful embedding

$$\mathbf{Pretop}^{op} \hookrightarrow \mathbf{Ult} .$$

One of the key motivations to introduce ultracategories lies in the fact that, given two coherent theories $T_1$ and $T_2$, the existence of an equivalence of categories $\operatorname{Mod} (T_1) \simeq \operatorname{Mod} (T_2)$ does not necessarily imply that the two theories are Morita equivalent, i.e. that the syntactic pretoposes $P_1$ and $P_2$ of $T_1$ and $T_2$, obtained by pretopos completion of their coherent syntactic categories, are equivalent. Indeed, it is only when an equivalence $\operatorname{Mod} (T_2) \to \operatorname{Mod} (T_1)$ comes from a pretopos morphism $I : P_1 \to P_2$ – corresponding to a suitable kind of interpretation of $T_1$ in $T_2$ – that we can apply the usual conceptual completeness theorem, deducing that $I$ is an equivalence itself. The strong conceptual completeness theorem allowed by ultracategories, instead, characterizes the functors $\operatorname{Mod} (T_2) \to \operatorname{Mod} (T_1)$ coming from a pretopos morphism $P_1 \to P_2$ in terms of preservation of ultraproducts, in a suitable sense: that is, as the ultrafunctors $\operatorname{Mod} (T_2) \to \operatorname{Mod} (T_1)$. Thus, the existence of an equivalence $\operatorname{Mod} (T_1) \simeq \operatorname{Mod} (T_2)$ in $\mathbf{Ult}$ corresponds to the existence of an equivalence of categories $P_1 \simeq P_2$; in particular, this directly implies the usual conceptual completeness theorem, simply by noting that an ultrafunctor is an equivalence in $\mathbf{Ult}$ if (and only if) it is an equivalence of categories.

To provide intuition on ultracategories, let us start by recalling that a compact Hausdorff space $K$ can be seen as a relational $\beta$-module where each ultrafilter $\mu$ on $K$ converges to a unique limit, namely the unique point $x \in K$ such that every open neighborhood of $x$ in $K$ lies in $\mu$, so as to determine a function $\beta K \to K$. More generally, we can say that a function $f \colon I \to K$ for some set $I$ converges with respect to some ultrafilter $\mu$ on $I$ to the limit of the ultrafilter $f_*(\mu) = \{ S \subseteq K | f^{-1}(K) \in \mu\}$, i.e. the unique point $x \in K$ such that every open neighborhood of $x$ in $K$ contains a $\mu$-large subset of values of $f$. This way, the convergence relation on $K$ can be seen as a family of functions

$$K^I \times \beta I \to K$$

for each set $I$: in the same spirit, an ultrastructure (called pre-ultracategory by Makkai ‘87) on a category $\mathcal{M}$ is given at its core by a family of functors

$$\int_I - d - : \mathcal{M}^I \times \beta I \to \mathcal{M}$$

for each set $I$. Intuitively, for an $I$-indexed family $(m_i)_{i\in I}$ of objects in $\mathcal{M}$ and an ultrafilter $\mu$ on $I$, the object $\int_I m_i d \mu$ is the “abstract ultraproduct” of the family $(m_i)_{i\in I}$ with respect to $\mu$. Different, inequivalent, axiomatizations for the concept of ultracategory exist in the literature: most notably, Lurie‘s definition of ultracategories (together with the introduction of left ultrafunctors) allowed to extend Makkai’s result to a reconstruction theorem for coherent toposes, and it formalizes the idea that ultracategories categorify compact Hausdorff spaces (see Lurie, Thm 3.1.5).

Theorem

(Lurie) Let $P$ be a small pretopos and let $\mathcal{E} = Sh(P)$ be its coherent topos of sheaves. Then, the evaluation functor $ev : P \to \mathbf{Ult}^L (\operatorname{Mod}(P) , Set )$ induces an equivalence

$$\mathcal{E} \simeq \mathbf{Ult}^L ( \operatorname{Mod}(P) , Set )$$

where $\mathbf{Ult}^L$ is the $2$-category of ultracategories, left ultrafunctors, and transformations thereof. In particular, $ev$ is an equivalence $P \simeq \mathbf{Ult} (\operatorname{Mod}(P) , Set )$, where $\mathbf{Ult}$ is the locally full sub-2-category of $\mathbf{Ult}^L$ determined by ultrafunctors.

Lurie’s axiomatization was proved by Hamad to coincide with that of normal colax algebras for a pseudomonad on the 2-category of locally small categories. Tarantino and Wrigley weakened Lurie’s definition to obtain an axiomatization of ultracategories sharing the same properties, but emerging universally from the ultrafilter monad. Instead, Saadia and Hamad independently introduced virtual ultracategories as a multicategorical generalization of ultracategories which allows to extend Lurie’s result to toposes with enough points; the former, in particular, introduces yet another axiomatization of ultracategories, satisfying the same reconstruction theorem above, as a particular case of virtual ultracategories. If ultracategories categorify compact Hausdorff spaces, virtual ultracategories categorify all topological spaces, allowing for maps into an abstract ultraproduct which may not exist in the underlying category.

Related concepts

• Clementino and Tholen introduced a different concept of ultracategory as an instance of a generalized multicategory.

• conceptual completeness

References

• Mihaly Makkai, Stone duality for first-order logic, Adv. Math. 65 2 (1987) 97–170 [doi:10.1016/0001-8708(87)90020-X, MR89h:03067]

• Marek W. Zawadowski, Descent and duality, Annals of Pure and Applied Logic 71, n.2 (1995), 131–188

• Jacob Lurie, Ultracategories [pdf]

On a 2-monadic treatment of ultracategories:

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

As colax algebras for a pseudomonad:

• Ali Hamad, Ultracategories as colax algebras for a pseudo-monad on CAT [arXiv:2502.20597]

In terms of Kan extensions of relative monads:

• Umberto Tarantino, Joshua Wrigley, Ultracategories via Kan extensions of relative monads [arXiv:2506.09788]

For a generalization to virtual ultracategories, a categorification of relational $\beta$-modules designed to allow reconstruction of any topos with enough points:

• Gabriel Saadia, Extending conceptual completeness via virtual ultracategories [arXiv:2506.23935]

• Ali Hamad, Generalised ultracategories and conceptual completeness of geometric logic [arXiv:2507.07922]

For a different notion of ultracategory:

• 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).