Generalised Ultracategory of Points

• Generalised ultracategories of points are enhanced point objects for topoi that retain ultraproduct data beyond the classical coherent regime.
• They package additional ultraconvergence data through various formalizations such as ultraconvergence spaces, virtual ultracategories, and condensed points.
• Reconstruction theorems using these enriched structures recover the topos and associated homotopy types, unifying geometric, logical, and topological perspectives.

Generalised ultracategory of points denotes an enhanced point object attached to a topos, or more generally to a coherent $\infty$-topos, which extends the classical ultracategory of points beyond the coherent regime. The common problem is that, for a general Grothendieck topos $\mathcal E$ with enough points, the ordinary category $\mathrm{pt}(\mathcal E)$ does not by itself retain the ultraproduct data needed for reconstruction: pointwise ultraproducts of points need not again be points. Recent work therefore replaces honest ultraproduct objects by formal arrows into ultraproduct functors, by $\mathbf{Set}$-valued ultraconvergence data, or by condensed families of $K$-valued points, and proves reconstruction or comparison theorems from these richer structures (Gool et al., 13 Aug 2025, Saadia, 30 Jun 2025, Hamad, 10 Jul 2025, Haine, 24 Feb 2026).

1. Classical origin and the obstruction beyond coherence

A point of a Grothendieck topos $\mathcal E$ is a geometric morphism

$\mathbf{Set}\to \mathcal E,$

systematically identified with its inverse image functor

$x:\mathcal E\to \mathbf{Set}.$

Such a functor preserves finite limits and small colimits. If $X$ is a class of points, evaluation defines

$$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$

The class $\mathcal E$0 is separating when this evaluation functor is conservative, and $\mathcal E$1 has enough points when it admits a separating class of points (Gool et al., 13 Aug 2025).

In the coherent case, the category of points or models carries an ultracategory structure because ultraproducts of models are again models by Łoś’s theorem. For an ultrafilter $\mathcal E$2 on $\mathcal E$3, an $\mathcal E$4-family $\mathcal E$5 of points has an ultraproduct

$\mathcal E$6

which is again a point, so one has actual morphisms

$\mathcal E$7

This is the setting of Makkai duality and Lurie’s extension to coherent topoi (Gool et al., 13 Aug 2025).

The obstruction in geometric logic is that this closure under ultraproducts fails in general: for arbitrary geometric theories, or arbitrary topoi with enough points, the pointwise ultraproduct functor

$\mathcal E$8

need not itself be a point. The ordinary category of points is therefore too rigid, and classical ultracategory structure no longer suffices (Gool et al., 13 Aug 2025).

A geometric precursor of this viewpoint appeared in the result that coherent topoi are right Kan injective with respect to flat embeddings of topoi. In particular, for the embeddings

$\mathcal E$9

right Kan extension recovers functorial ultraproduct operators on the category of points, yielding an ultrastructure and, for $\mathrm{pt}(\mathcal E)$0-complete topoi, an ultracategory of points (Liberti, 2022).

2. Principal formalizations

Recent papers isolate several closely related generalizations. They agree that one should remember more than the underlying ordinary category of points, but they package the extra data in different ways.

Formalism	Basic point object	Core additional datum
Ultraconvergence space (Gool et al., 13 Aug 2025)	a class $\mathrm{pt}(\mathcal E)$1 of points	natural transformations $\mathrm{pt}(\mathcal E)$2, with reindexing and composition
Virtual ultracategory (Saadia, 30 Jun 2025)	objects are points	ultraarrows $\mathrm{pt}(\mathcal E)$3, with homsets $\mathrm{pt}(\mathcal E)$4
Generalised ultracategory (Hamad, 10 Jul 2025)	objects are points	generalised Hom-sets $\mathrm{pt}(\mathcal E)$5, together with $\mathrm{pt}(\mathcal E)$6, $\mathrm{pt}(\mathcal E)$7, and $\mathrm{pt}(\mathcal E)$8
Condensed category of points (Haine, 24 Feb 2026)	$\mathrm{pt}(\mathcal E)$9	all $\mathbf{Set}$0-valued points, compared via condensed classifying anima

In the ultraconvergence-space formulation, an ultra-arrow from $\mathbf{Set}$1 to a $\mathbf{Set}$2-family $\mathbf{Set}$3 is a natural transformation

$\mathbf{Set}$4

The structure includes identities, reindexing along the category $\mathbf{Set}$5 of ultrafilters, and composition. The resulting 2-category is $\mathbf{Set}$6, and every ultraconvergence space has a specialization category $\mathbf{Set}$7 recovering an ordinary category from the ultraconvergence data (Gool et al., 13 Aug 2025).

In the virtual-ultracategory formulation, one does not ask that the codomain ultrafamily $\mathbf{Set}$8 be represented by an actual object $\mathbf{Set}$9. Instead one has generalized arrows

$K$0

The paper explicitly says that virtual ultracategories are to ultracategories what multicategories are to monoidal categories, and it presents them as a categorification of relational $K$1-modules (Saadia, 30 Jun 2025).

In the generalised-ultracategory formulation, the basic datum is a set of objects together with generalized Hom-sets

$K$2

composition maps $K$3, change-of-base maps $K$4, and units $K$5. Topological spaces are a decisive example: the corresponding generalized Hom-set is empty or singleton according to ultrafilter convergence, and the underlying ordinary category is the specialization preorder (Hamad, 10 Jul 2025).

The condensed $K$6-categorical formulation replaces a single category of points by the condensed object

$K$7

on extremally disconnected profinite sets $K$8. Its global sections recover the ordinary category of points, and for coherent $K$9-topoi one may restrict to the full subcategory of coherent points

$\mathcal E$0

The comparison is then made at the level of classifying anima rather than literal equivalence of categories (Haine, 24 Feb 2026).

3. Reconstruction theorems

The central point of the theory is not merely the existence of extra structure on points, but the recovery of the ambient topos from it.

For a topos $\mathcal E$1 and a separating set $\mathcal E$2 of points equipped with the canonical ultraconvergence structure, the evaluation functor is an equivalence

$\mathcal E$3

equivalently

$\mathcal E$4

and, using the equivalence between $\mathcal E$5-valued continuous maps and étale spaces,

$\mathcal E$6

At the level of all points, if $\mathcal E$7 has enough points then

$\mathcal E$8

is an equivalence of categories. The same work also gives a 2-categorical embedding of $\mathcal E$9 into $\mathbf{Set}\to \mathcal E,$0 (Gool et al., 13 Aug 2025).

The virtual-ultracategory formulation proves a parallel theorem. For a topos $\mathbf{Set}\to \mathcal E,$1 and a class $\mathbf{Set}\to \mathcal E,$2, one has

$\mathbf{Set}\to \mathcal E,$3

where $\mathbf{Set}\to \mathcal E,$4 is the restriction of $\mathbf{Set}\to \mathcal E,$5 to the class $\mathbf{Set}\to \mathcal E,$6 of points. When $\mathbf{Set}\to \mathcal E,$7 has enough points and $\mathbf{Set}\to \mathcal E,$8, this specializes to

$\mathbf{Set}\to \mathcal E,$9

In this language, the topos is reconstructed as the category of ultrasheaves on its virtual ultracategory of points (Saadia, 30 Jun 2025).

The generalised-ultracategory formulation proves a conceptual-completeness statement in terms of left ultrafunctors: $x:\mathcal E\to \mathbf{Set}.$0 Specializing $x:\mathcal E\to \mathbf{Set}.$1 to $x:\mathcal E\to \mathbf{Set}.$2 gives

$x:\mathcal E\to \mathbf{Set}.$3

Here the proof proceeds by comparing left ultrafunctors from topological spaces into the generalised ultracategory of points with geometric morphisms from sheaf topoi, and by paralleling this with the representation of a topos with enough points as a colimit of a topological groupoid (Hamad, 10 Jul 2025).

In the ultraconvergence-space approach, the étale description is fully explicit. For $x:\mathcal E\to \mathbf{Set}.$4, the associated étale space has points $x:\mathcal E\to \mathbf{Set}.$5 with $x:\mathcal E\to \mathbf{Set}.$6 and $x:\mathcal E\to \mathbf{Set}.$7, projection $x:\mathcal E\to \mathbf{Set}.$8, and ultra-arrows determined by the condition

$x:\mathcal E\to \mathbf{Set}.$9

Thus the representing objects of the topos are precisely $X$0-valued continuous maps on the generalized point space (Gool et al., 13 Aug 2025).

4. Condensed and $X$1-categorical variants

For an $X$2-topos $X$3, the ordinary category of points is

$X$4

where $X$5 denotes the $X$6-category of left exact left adjoints. The condensed enhancement replaces this by the functor

$X$7

defined on extremally disconnected profinite sets $X$8, and preserving finite products; hence it is a condensed category. For coherent $X$9-topoi, one also has the smaller condensed category of coherent points

$$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$0

The associated invariant is the classifying anima $$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$1, defined as the left adjoint to the inclusion of anima into categories, and applied pointwise to condensed categories (Haine, 24 Feb 2026).

The central comparison theorem states that if $$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$2 is a spectral topos, then for each extremally disconnected profinite set $$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$3, the inclusion

$$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$4

admits a left adjoint. Consequently, the inclusion

$$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$5

induces an equivalence on condensed classifying anima (Haine, 24 Feb 2026).

The proof reduces to $$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$6 by Gleason’s theorem and uses the sheaf-theoretic form of Łoś’s theorem. For a set $$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$7 with Čech–Stone compactification $$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$8, the pushforward

$$\llbracket - \rrbracket:\mathcal E\to \mathbf{Set}^X,\qquad \phi\mapsto (x\mapsto x(\phi)).$$9

preserves finite limits, finite coproducts, and effective epimorphisms. This is the $\mathcal E$00-categorical descendant of Lurie’s ultracategory technology and the mechanism behind the condensed point construction (Haine, 24 Feb 2026).

This comparison is deliberately weaker than literal equivalence of categories of points. The paper emphasizes that the significant invariant is the classifying anima, because if an inclusion $\mathcal E$01 admits a left adjoint, then

$\mathcal E$02

is an equivalence. The condensed classifying anima therefore retains the homotopy-theoretic information relevant to fundamental groups even when the two point constructions remain categorically different (Haine, 24 Feb 2026).

5. Topological, logical, and geometric interpretations

A persistent theme is that the generalized point structure is a categorification of ultrafilter convergence. Barr’s classical theorem encodes a topology by a relation between points and ultrafilters, equivalently a relational module for the ultrafilter monad $\mathcal E$03. In the ultraconvergence-space approach this becomes a $\mathcal E$04-valued relation on points and ultrafamilies, while in the virtual-ultracategory approach it becomes a distributional $\mathcal E$05-module. In the generalised-ultracategory approach, a topological space $\mathcal E$06 gives generalized Hom-sets

$\mathcal E$07

iff the pushforward of $\mathcal E$08 by $\mathcal E$09 converges to $\mathcal E$10, and $\mathcal E$11 otherwise (Gool et al., 13 Aug 2025, Saadia, 30 Jun 2025, Hamad, 10 Jul 2025).

The logical interpretation is equally direct. If $\mathcal E$12 is a geometric theory with classifying topos $\mathcal E$13, then points of $\mathcal E$14 are precisely $\mathcal E$15-models of $\mathcal E$16. In the ultraconvergence formulation, the generalized point structure on models is

$\mathcal E$17

which is defined even when the ultraproduct is not again a model of $\mathcal E$18. The resulting theorem is described as a strong conceptual completeness theorem in Makkai’s sense for geometric theories with enough $\mathcal E$19-models, and the proof also yields that open subclasses of models stable under ultraconvergence are definable by geometric sentences (Gool et al., 13 Aug 2025).

The condensed $\mathcal E$20-categorical picture produces group-valued invariants. For a qcqs scheme $\mathcal E$21, the étale $\mathcal E$22-topos $\mathcal E$23 is spectral, and a mild completion of the fundamental group of the condensed anima of coherent points recovers Bhatt–Scholze’s proétale fundamental group. For a complete first-order theory $\mathcal E$24, forthcoming work with Damaj and Zhang proves an isomorphism of condensed groups

$\mathcal E$25

where $\mathcal E$26 for the classifying topos $\mathcal E$27 of $\mathcal E$28. The abstract consequence is that, up to a mild completion, the proétale fundamental group of a scheme and the Lascar group of a complete first-order theory are both special cases of the same classifying-anima construction (Haine, 24 Feb 2026).

6. Comparisons, misconceptions, and present usage

A frequent misconception is that the ordinary category of points should suffice. The recent literature rejects this in several ways. In the ultraconvergence-space formulation, the specialization category $\mathcal E$29 recovers the ordinary point category, but the ultraconvergence data are strictly richer. In the generalised-ultracategory formulation, the underlying category is recovered by the singleton index

$\mathcal E$30

but reconstruction uses the full generalized Hom-data rather than only this underlying category (Gool et al., 13 Aug 2025, Hamad, 10 Jul 2025).

A second misconception is that the generalized theory abandons the classical coherent case. In fact, each formalism is designed to collapse back to ordinary ultracategorical behavior when ultraproducts of points remain representable. In the ultraconvergence-space approach, if $\mathcal E$31 is again a point, then the generalized notion recovers ordinary ultracategorical arrows. In the virtual-ultracategory approach, ordinary ultracategories appear as the representable virtual ultracategories, exactly as multicategories become monoidal when multihoms are represented by tensor products (Gool et al., 13 Aug 2025, Saadia, 30 Jun 2025).

A third point concerns the strength of the completeness statement. The ultraconvergence-space work explicitly distinguishes strong conceptual completeness from plain conceptual completeness: an equivalence on categories of points alone does not force an equivalence of topoi. What is fully faithful is the embedding into ultraconvergence spaces, that is, into points equipped with the full ultraconvergence structure. In the condensed $\mathcal E$32-categorical setting, the analogous phenomenon is that the all-points and coherent-points constructions may differ as categories even when they determine the same classifying anima (Gool et al., 13 Aug 2025, Haine, 24 Feb 2026).

The present terminology is not completely uniform. One finds ultraconvergence spaces, virtual ultracategories, generalised ultracategories, ultrastructures, ultracategories, and condensed categories of points. This suggests that the expression “generalised ultracategory of points” currently designates a research program rather than a single standardized formal definition. What is stable across these formulations is the principle that, for topoi with enough points, one must retain formal maps into pointwise ultraproducts—or their topological, profunctorial, or condensed analogues—in order to reconstruct the topos or its associated homotopy type (Saadia, 30 Jun 2025, Hamad, 10 Jul 2025).

The proofs also differ substantially. The ultraconvergence-space proof extends and simplifies Makkai’s original proof and does not assume groupoid representations of topoi, whereas the independent proofs via virtual ultracategories and generalised ultracategories rely on groupoid representations. The earlier right-Kan-injectivity approach provides a different geometric source for ultrastructure in the coherent setting, showing that the modern formulations sit within a broader geometric account of ultraproduct operations on points (Gool et al., 13 Aug 2025, Liberti, 2022).