Generalised Ultracategories

• Generalised ultracategories are relational extensions of ultracategories that use formal ultraproduct data and convergence morphisms to bridge model theory with topology.
• They enable a reconstruction of toposes with enough points via a conceptual completeness theorem, illustrating the deep interplay between topology and geometric logic.
• The framework unifies approaches such as left ultrafunctors, virtual ultracategories, and profunctorial ultraconvergence, extending classical ultraproduct semantics.

Generalised ultracategories are relational extensions to ultracategories as defined by Lurie. In the theory introduced by Hamad, topological spaces are an essential example, and points of toposes furnish another central example. The subject is organized around formal ultraproduct data, convergence-style morphisms into such formal ultraproducts, and left ultrafunctors that preserve this structure. Its main reconstruction result is a conceptual completeness theorem for toposes with enough points: any such topos can be recovered from its generalised ultracategory of points (Hamad, 10 Jul 2025). In this sense, generalised ultracategories lie at the intersection of model-theoretic ultraproducts, categorical topology, and geometric logic, extending the ultrastructures studied by Makkai and Lurie (Liberti, 2022).

1. Historical and categorical setting

Makkai’s original ultracategory was aimed at first-order structures, while Lurie gave a streamlined version. In parallel, categorical analyses of ultrafilters clarified how ultraproducts, ultrapowers, tensor products of ultrafilters, and the Rudin--Keisler ordering can be encoded inside a single categorical framework. Garner proved that

$$FC(\mathrm{Set},\mathrm{Set}) \simeq [UF,\mathrm{Set}],$$

where $UF$ is Blass’s category of ultrafilters, and showed that this perspective draws connections with Makkai’s and Lurie’s work on conceptual completeness for first-order logic via ultracategories (Garner, 2018).

A second strand came from topos theory. Coherent topoi were shown to be right Kan injective with respect to flat embeddings, and from this universal property the ultrastructure on the category of points $\mathsf{pt}(\mathcal E)$ emerges naturally. For a coherent theory $\mathbb T$, this recovers the familiar ultraproduct functor

$$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$

now interpreted through geometric morphisms and Kan extension data rather than only through model-theoretic syntax (Liberti, 2022).

Generalised ultracategories extend this picture in a specifically relational direction. They were introduced to cover both ordinary ultracategories and topological spaces in a uniform formalism, and to make points of Grothendieck toposes into objects of the same theory. The extension is not merely terminological: it replaces a representable ultraproduct operation by hom-sets into formal ultraproducts, thereby allowing topological convergence phenomena to enter the theory directly (Hamad, 10 Jul 2025).

2. Formal definition

A generalised ultracategory $\mathcal C$ consists of the following data. First, there is a class $\mathrm{Obj}(\mathcal C)$. Second, for every object $A$, every set $I$, every family $\{M_i\}_{i\in I}\subseteq \mathrm{Obj}(\mathcal C)$, and every ultrafilter $UF$0 on $UF$1, one has a set

$UF$2

where $UF$3 is regarded as a formal ultraproduct. Third, for every function $UF$4 and ultrafilter $UF$5 on $UF$6, there is an injective-base map

$UF$7

Fourth, there is a unit element

$UF$8

Finally, there is a composition operation

$UF$9

These data satisfy the analogues of Lurie’s ultracategory axioms: injectivity of $\mathsf{pt}(\mathcal E)$0, functoriality $\mathsf{pt}(\mathcal E)$1, left-unit, right-unit, two compatibilities between $\mathsf{pt}(\mathcal E)$2 and $\mathsf{pt}(\mathcal E)$3, and one associativity axiom (Hamad, 10 Jul 2025).

The key structural point is that ordinary ultracategories become a special case in which the formal ultraproducts are representable. Generalised ultracategories keep the formal ultraproduct notation, but treat morphisms into it as primitive. This is what makes genuinely relational examples possible.

A fundamental degeneration is the posetal one. When all the $\mathsf{pt}(\mathcal E)$4-sets have at most one element and each $\mathsf{pt}(\mathcal E)$5 is an isomorphism, one obtains an ultrapreorder; these are exactly topological spaces, via the specialization preorder. Thus the theory does not merely resemble topology: in its thin case it recovers $\mathsf{pt}(\mathcal E)$6 itself (Hamad, 10 Jul 2025).

3. Fundamental examples

The first basic example is a topological space $\mathsf{pt}(\mathcal E)$7. Writing $\mathsf{pt}(\mathcal E)$8 when $\mathsf{pt}(\mathcal E)$9 converges to $\mathbb T$0, one defines

$\mathbb T$1

The required $\mathbb T$2, $\mathbb T$3, and $\mathbb T$4 then follow from standard ultrafilter-convergence axioms. Hamad cites Wyler’s theorem that topologies on a set $\mathbb T$5 are in bijection with convergence-relations

$\mathbb T$6

satisfying $\mathbb T$7 and closure under iterated limits; this is the mechanism by which the assignment $\mathbb T$8 becomes fully faithful onto the 2-category of ultrapreorders (Hamad, 10 Jul 2025).

The second basic example is the category of points of a Grothendieck topos. If $\mathbb T$9 is a Grothendieck topos with site of definition $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$0, then

$$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$1

Because $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$2 embeds fully in $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$3, one can define

$$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$4

with the ultraproduct taken pointwise in $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$5. This again satisfies the axioms, so $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$6 becomes a generalised ultracategory (Hamad, 10 Jul 2025).

These examples show the precise role of generalisation. In an ordinary ultracategory one has an actual ultraproduct object together with comparison morphisms and Fubini isomorphisms. In a generalised ultracategory the formal ultraproduct remains syntactic, but its incoming morphisms encode convergence or realization data. This allows the same formalism to treat points of a topos and points of a topological space on the same footing.

4. Left ultrafunctors and representational constructions

A left ultrafunctor $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$7 between generalised ultracategories consists of a map on objects together with, for each $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$8, each family $$\int_X(-)\,dU \;:\; \mathsf{Mod}(\mathbb T)^X \longrightarrow \mathsf{Mod}(\mathbb T),$$9, and each ultrafilter $\mathcal C$0, a function

$\mathcal C$1

These functions must respect the unit $\mathcal C$2, the change-of-base maps $\mathcal C$3, and the composition $\mathcal C$4. On the underlying ordinary categories, such a morphism preserves composition and identities, so left ultrafunctors are genuine categorical morphisms (Hamad, 10 Jul 2025).

This notion recovers classical representation theorems. For any topological space $\mathcal C$5, the category of sheaves $\mathcal C$6 is equivalent to $\mathcal C$7, the category of left ultrafunctors from $\mathcal C$8 to $\mathcal C$9. In the same framework, objects of $\mathrm{Obj}(\mathcal C)$0 correspond to left ultrafunctors $\mathrm{Obj}(\mathcal C)$1, where $\mathrm{Obj}(\mathcal C)$2 is the generalised ultracategory of points of a topos $\mathrm{Obj}(\mathcal C)$3 (Hamad, 10 Jul 2025).

A closely related metric analogue appears in the theory of ultracategories of complete metric structures. For any continuous theory $\mathrm{Obj}(\mathcal C)$4, there is an equivalence

$\mathrm{Obj}(\mathcal C)$5

for compact Hausdorff $\mathrm{Obj}(\mathcal C)$6. The notion of bundle of $\mathrm{Obj}(\mathcal C)$7-models introduced there recovers classical examples such as Banach bundles, continuous fields of $\mathrm{Obj}(\mathcal C)$8-algebras, Hilbert bundles, and $\mathrm{Obj}(\mathcal C)$9-bundles with trace (Hamad, 2024). This does not redefine generalised ultracategories, but it shows how left ultrafunctor technology organizes geometric objects well beyond the discrete first-order setting.

5. Conceptual completeness for geometric logic

Let $A$0 and $A$1 be Grothendieck toposes with enough points, and let $A$2 and $A$3 denote their generalised ultracategories of points. Hamad’s conceptual completeness theorem states that there is a canonical equivalence

$A$4

In particular,

$A$5

This extends Lurie’s coherent-topos result from ordinary ultracategories to a geometric setting that includes topological spaces and toposes with enough points (Hamad, 10 Jul 2025).

The proof is decomposed into three equivalences. First,

$A$6

where $A$7 is the lax slice 2-category of topological spaces over $A$8. Second,

$A$9

as 2-fibred categories over $I$0; this uses the identification of sheaves with left ultrafunctors from spaces to $I$1. Third,

$I$2

The final step uses the representation of $I$3 as a 2-colimit of its logical groupoids in the sense of Butz--Moerdijk (Hamad, 10 Jul 2025).

The significance of the theorem is twofold. On the logical side, it is a conceptual completeness statement: the ambient topos can be reconstructed from the structure carried by its points. On the topological side, it shows that the relational ultraproduct-convergence data encoded in the generalised ultracategory of points is sufficient to recover geometric morphisms. Hamad formulates this by saying that no further information beyond the ultraproduct convergence relations is needed to recover geometric morphisms (Hamad, 10 Jul 2025).

6. Relations to adjacent formalisms

Generalised ultracategories belong to a broader family of attempts to recast ultraproduct phenomena in categorical terms. One nearby notion is that of a virtual ultracategory. A virtual ultracategory consists of objects, hom-sets of the form

$I$4

identities, and a composition indexed by sums of ultrafilters. Virtual ultracategories generalize ultracategories in the same way that multicategories generalize monoidal categories, and the points of an arbitrary Grothendieck topos form a virtual ultracategory. When the ambient topos is coherent, the points form a genuine ultracategory, recovering the Makkai--Lurie theory (Saadia, 30 Jun 2025). This places generalised ultracategories next to, rather than identical with, another relational enlargement of the classical notion.

Ordinary ultracategories themselves have acquired several equivalent categorical formulations. They are normal colax algebras for a pseudo-monad $I$5, with left and right ultrafunctors arising as lax and colax algebra morphisms (Hamad, 27 Feb 2025). They also arise as colax algebras for the ultracompletion pseudomonad obtained by left oplax Kan extension of the relative ultrafilter 2-monad, yielding the identification

$I$6

(Tarantino et al., 11 Jun 2025). These results concern ultracategories rather than generalised ultracategories, but they clarify the algebraic background against which the relational extension is defined.

A further extension appears in profunctorial form. The ultracompletion pseudomonad on $I$7 extends to $I$8, and the normalized lax algebras of this profunctorial extension are ultraconvergence spaces, described as a categorification of topological spaces (Aristote et al., 30 Jan 2026). This is consonant with the role of topological spaces inside generalised ultracategories: in both cases the central issue is how ultrafilter-based convergence can be internalized categorically without requiring strict representability.

Finally, the foundational role of ultrafilter categories remains visible. Garner’s equivalence

$I$9

shows that representable presheaves on $\{M_i\}_{i\in I}\subseteq \mathrm{Obj}(\mathcal C)$0 correspond to ultrapower functors, and that composition corresponds to Day convolution, with the induced monoidal structure on $\{M_i\}_{i\in I}\subseteq \mathrm{Obj}(\mathcal C)$1 given by tensor product of ultrafilters (Garner, 2018). This suggests that the passage from ultrafilters to generalized ultrastructures is not an isolated construction but part of a systematic categorical elaboration of ultraproduct semantics.

In sum, generalised ultracategories provide a relational enhancement of Lurie’s ultracategories that encompasses topological spaces and categories of points of toposes, supports a sheaf-theoretic and topos-theoretic theory of left ultrafunctors, and yields a conceptual completeness theorem for geometric logic in the presence of enough points (Hamad, 10 Jul 2025).