Étendues in Topos Theory

• Étendues are Grothendieck topoi that become localic when sliced over an inhabited object, capturing local coordinates and symmetries.
• They admit multiple representations via étale groupoids, Ehresmann sites, and left-cancellative categories, facilitating descent and site comparison techniques.
• Recent work characterizes étendues model-theoretically, linking uniform co-ordinatization and rigidity to their localic nature and invariant extraction.

Searching arXiv for papers on étendues to ground the article in the cited literature. An étendue is a Grothendieck topos that is locally localic: there exists an object $A$ whose support $A\to 1$ is an epimorphism and such that the slice topos $E/A$ is localic. In the standard SGA4 formulation, an étendue is thus a topos that becomes localic after passage to a sufficiently large étale cover. In recent work, étendues appear in at least three technically distinct but conceptually related roles: as a structural class of Grothendieck topoi admitting a model-theoretic characterization (Wrigley, 6 Jul 2025), as invariants attached to combinatorial spaces whose internal logic detects dimension (Menni, 2024), and as toposes represented via ordered groupoids, left-cancellative categories, and Ehresmann sites (DeWolf et al., 2019). A different usage in observational astronomy refers to the optical quantity $G=A\Omega$, also called “étendue,” measuring throughput; that notion is unrelated to the topos-theoretic concept except by name (Pál et al., 2018).

1. Definition and categorical position

Let $E$ be a Grothendieck topos with terminal object $1$. A geometric morphism $f:F\to E$ is étale or locally homeomorphic if its inverse-image functor $f^*$ preserves monomorphisms and is open; equivalently, $F\simeq E/A$ for some object $A\in E$ via the slice construction (Wrigley, 6 Jul 2025). A geometric morphism is localic if its direct-image functor is full and faithful; equivalently, $A\to 1$0 for some locale $A\to 1$1 (Wrigley, 6 Jul 2025).

Against that background, a topos $A\to 1$2 is an étendue if there exists $A\to 1$3 such that:

1. the support $A\to 1$4 is an epimorphism, and
2. the slice topos $A\to 1$5 is localic (Wrigley, 6 Jul 2025).

When these conditions hold, one says that $A\to 1$6 is an étendue over $A\to 1$7. Equivalently there is an étale geometric morphism

$A\to 1$8

whose inverse-image sends $A\to 1$9 to $E/A$0 (Wrigley, 6 Jul 2025).

This definition places étendues between general Grothendieck topoi and localic topoi. Localic topoi are immediate examples: they are trivially étendues over $E/A$1 (Wrigley, 6 Jul 2025). More generally, the notion isolates topoi that are “not far” from localic ones in the sense that locality emerges after slicing over an inhabited object. This suggests that étendues form a natural environment for analyzing toposes whose geometric content is controlled by local coordinates, witnesses, or local symmetries.

Two basic examples recur in the literature. The topos $E/A$2 of $E/A$3-sets for a discrete group $E/A$4 is an étendue because $E/A$5 (Wrigley, 6 Jul 2025). More generally, if $E/A$6 is a localic étale groupoid, then $E/A$7 is an étendue, with $E/A$8 localic (Wrigley, 6 Jul 2025).

2. Representation by groupoids, sites, and descent

A foundational structural fact is that every étendue arises, up to equivalence, as the topos of sheaves on an étale localic groupoid by Joyal–Tierney descent (Wrigley, 6 Jul 2025). This gives a representation theorem: étendues can be treated as geometric realizations of suitable groupoid objects in locales.

The representation theory has been developed further through ordered groupoids and Ehresmann sites. DeWolf and Pronk describe ordered groupoids as a particular type of double category and use this to lift Lawson’s correspondence between ordered groupoids and left-cancellative categories to a biequivalence (DeWolf et al., 2019). In their formulation, an ordered groupoid carries a partial order on arrows compatible with inversion, composition, and unique restriction. The resulting double-categorical viewpoint supports a site-theoretic treatment of étendue representations.

An Ehresmann topology on an ordered groupoid $E/A$9 assigns covering vertical sieves satisfying maximality, closure under restriction, and local character; an ordered groupoid equipped with such a topology is an Ehresmann site (DeWolf et al., 2019). The paper identifies which double functors between Ehresmann sites induce geometric morphisms between the associated sheaf categories: this occurs exactly for double functors that are covering preserving and covering flat (DeWolf et al., 2019). It also establishes a Comparison Lemma for Ehresmann sites, paralleling the usual Grothendieck-site comparison theorem (DeWolf et al., 2019).

These results are not merely formal. They show that étendues admit multiple interoperable presentations:

Presentation	Structural datum	Associated topos
Localic groupoid	localic étale groupoid $G=A\Omega$0	$G=A\Omega$1
Slice-localic form	inhabited $G=A\Omega$2 with $G=A\Omega$3 localic	$G=A\Omega$4
Ehresmann-site form	ordered groupoid with Ehresmann topology	$G=A\Omega$5

This range of presentations clarifies why étendues sit at an intersection of descent theory, localic geometry, and noncommutative or ordered categorical structures. A plausible implication is that the notion is robust under changes of site and hence well-suited to classification results phrased either syntactically or geometrically.

3. Model-theoretic characterization

A central recent advance is the model-theoretic characterization of the geometric theories classified by an étendue. For a geometric theory $G=A\Omega$6 with classifying topos $G=A\Omega$7, Wrigley proves that three conditions are equivalent: $G=A\Omega$8 is uniformly co-ordinatised, $G=A\Omega$9 is uniformly rigid, and $E$0 is an étendue (Wrigley, 6 Jul 2025).

The setting begins with a set $E$1 of geometric formulae $E$2. For a model $E$3, one writes

$E$4

(Wrigley, 6 Jul 2025).

The theory $E$5 is uniformly co-ordinatised over $E$6 if there exist families $E$7 of geometric formulae $E$8 such that, for each $E$9,

1. $1$0,
2. $1$1,
3. $1$2

(Wrigley, 6 Jul 2025).

Informally, each witness $1$3 co-ordinates the entire model $1$4 in a uniform way via one of the functional formulas in $1$5 (Wrigley, 6 Jul 2025).

The theory $1$6 is uniformly rigid over $1$7 if, for any model $1$8 in any topos and any global element $1$9, the only automorphism of $f:F\to E$0 fixing $f:F\to E$1 pointwise is the identity (Wrigley, 6 Jul 2025).

The main theorem states that for a geometric theory $f:F\to E$2 and its classifying topos $f:F\to E$3, the following are equivalent:

• $f:F\to E$4 is uniformly co-ordinatised;
• $f:F\to E$5 is uniformly rigid;
• $f:F\to E$6 is an étendue (Wrigley, 6 Jul 2025).

This theorem supplies both a syntactic and a semantic criterion for the “locally localic” property. Rather than identifying étendues only by existence of a localic slice, it characterizes them by the extent to which a model is determined by a witness to a fixed family of formulas. The phrase “each model is determined, syntactically and semantically, by any witness of a fixed collection of formulae” is the paper’s summary of this phenomenon (Wrigley, 6 Jul 2025).

Several intermediate results sharpen the picture. If $f:F\to E$7 is localic, then $f:F\to E$8 is uniformly co-ordinatised over the formulas indexing $f:F\to E$9 (Wrigley, 6 Jul 2025). If $f^*$0 is uniformly co-ordinatised over $f^*$1, then it is uniformly rigid over $f^*$2 (Wrigley, 6 Jul 2025). If $f^*$3 is uniformly rigid over $f^*$4, then the slice $f^*$5 has at most one isomorphism between any two points, hence is localic, and therefore $f^*$6 is an étendue over $f^*$7 (Wrigley, 6 Jul 2025).

A key localicity criterion used here is that a topos $f^*$8 is localic if and only if for any base topos $f^*$9, every pair of geometric morphisms $F\simeq E/A$0 admits at most one natural transformation between them (Wrigley, 6 Jul 2025). This can be read as a “no extra points” condition: localic topoi do not support nontrivial 2-categorical ambiguity between generalized points.

4. Examples and non-examples in classification theory

The model-theoretic characterization becomes concrete through examples. Localic topoi, equivalently classifying topoi of propositional theories, are trivially étendues over $F\simeq E/A$1, uniformly co-ordinatised over $F\simeq E/A$2, and uniformly rigid (Wrigley, 6 Jul 2025). This is the degenerate case in which the theory already has no genuine individual-variable structure.

A more informative example is the theory of finite-dimensional vector spaces over a finite field $F\simeq E/A$3. Let $F\simeq E/A$4, where $F\simeq E/A$5 says that $F\simeq E/A$6 is a basis of length $F\simeq E/A$7. The coordinating formulas say that $F\simeq E/A$8 is a given linear combination of the basis $F\simeq E/A$9. The paper states that this makes the theory uniformly co-ordinatised and rigid, and that its classifying topos is the disjoint union of the classifying topoi for $A\in E$0-dimensional spaces, equivalently the presheaf topos on the groupoid $A\in E$1, which is an étendue over $A\in E$2 (Wrigley, 6 Jul 2025).

Another standard example is the theory of $A\in E$3-torsors. Torsors under a group $A\in E$4 form the classifying topos $A\in E$5, an étendue over the formula $A\in E$6. The coordinating formulas are $A\in E$7, which exhibit uniform co-ordinatisation and rigidity in any topos (Wrigley, 6 Jul 2025).

The literature also records a significant non-example. Atomic topoi, which classify atomic theories, are in general not étendues unless they are localic. The reason given is that atomic topoi admit richer automorphism behavior on points, associated with Ryll–Nardzewski phenomena, so they fail uniform rigidity unless trivial (Wrigley, 6 Jul 2025). This distinguishes étendue-classified theories from other well-studied logical classes: atomicity and localic-by-slice behavior are independent in general.

One corollary is a smallness statement: any étendue has only a small set of points up to isomorphism (Wrigley, 6 Jul 2025). Syntactically, this is linked to the observation that a uniformly co-ordinatised theory with infinite coordinate families can only have models of bounded finite size (Wrigley, 6 Jul 2025). This suggests that étendues impose a strong cardinality or orbit-control constraint on their points.

5. Étendues of combinatorial spaces

A distinct development constructs an étendue from a simplicial set or, more generally, from an object of a presheaf topos. For a small category $A\in E$8 and presheaf topos $A\in E$9, an object $A\to 1$00 has a category of elements $A\to 1$01. Inside $A\to 1$02, one considers the full subcategory $A\to 1$03 on the minimal objects, namely representable maps $A\to 1$04 such that whenever $A\to 1$05 and $A\to 1$06 satisfy $A\to 1$07, the map $A\to 1$08 must be monic in $A\to 1$09 (Menni, 2024).

Menni defines the étendue of $A\to 1$10 by

$A\to 1$11

and shows that the composite geometric morphism from this presheaf topos to $A\to 1$12 is an étendue (Menni, 2024). Equivalently, $A\to 1$13 is the level $A\to 1$14 subtopos of $A\to 1$15 maximal among those whose composite to $A\to 1$16 is an étendue (Menni, 2024). The construction can also be recovered internally: in the logic of $A\to 1$17, a subterminal object classifies minimality, and $A\to 1$18 is the closed subtopos cut out by that subterminal (Menni, 2024).

For simplicial sets, the internal logic of $A\to 1$19 encodes dimension. If $A\to 1$20 is the topos of simplicial sets and $A\to 1$21 is non-singular, then for non-singular simplicial sets $A\to 1$22 and $A\to 1$23,

$A\to 1$24

(Menni, 2024).

The paper also introduces bounded-depth formulas $A\to 1$25 in the internal Heyting algebra of subterminals: $A\to 1$26 For each $A\to 1$27,

$A\to 1$28

under the hypotheses stated in the paper, with the converse requiring $A\to 1$29 to be strongly regular and the site to satisfy a mild well-foundedness hypothesis (Menni, 2024).

The mechanism is explicit. The subterminals $A\to 1$30 correspond to sieves on $A\to 1$31 consisting of those minimal simplices whose “height” is at most $A\to 1$32, while the $A\to 1$33-skeleton is characterized via an idempotent comonad $A\to 1$34 whose counit detects simplices factoring through an object of height $A\to 1$35 (Menni, 2024). Thus internal logical depth in the étendue recovers external simplicial dimension.

Standard examples illustrate the construction. For the standard simplex $A\to 1$36, $A\to 1$37 is equivalent to the finite chain $A\to 1$38, so

$A\to 1$39

and $A\to 1$40 holds exactly when $A\to 1$41 (Menni, 2024). For the boundary sphere $A\to 1$42, the minimal simplices form a finite poset of height $A\to 1$43, so $A\to 1$44 detects dimension $A\to 1$45 through the validity of $A\to 1$46 and failure of $A\to 1$47 (Menni, 2024).

The construction extends verbatim to presheaf topoi $A\to 1$48 when $A\to 1$49 has strong epi/mono factorizations and no infinite properly descending chain of non-invertible strong epimorphisms (Menni, 2024). The stated scope includes sites such as finite sets, globular cells, cells in a Ball-complex, and various Gaeta topoi (Menni, 2024).

6. Related meanings and terminological caution

The word étendue also has a standard meaning in optics and astronomy: the throughput or $A\to 1$50 product of an imaging system. In that setting,

$A\to 1$51

for entrance-pupil area $A\to 1$52 and simultaneously imaged solid angle $A\to 1$53, with cumulative optical étendue

$A\to 1$54

for multi-camera systems (Pál et al., 2018). The TESS/Kepler comparison computes $A\to 1$55 and $A\to 1$56, then introduces a “net étendue” $A\to 1$57 reflecting the fraction of the optical field actually downlinked (Pál et al., 2018).

This optical notion is entirely distinct from the topos-theoretic notion. The shared term reflects historical French mathematical and physical usage rather than a substantive connection. In mathematical writing, especially across category theory and mathematical physics, disambiguation is therefore necessary.

Within topos theory itself, another possible source of confusion is the relation between localic topoi, étendues, and atomic topoi. Localic topoi are always étendues, but atomic topoi are generally not; the presence of abundant point automorphisms obstructs uniform rigidity (Wrigley, 6 Jul 2025). Likewise, a topos represented by a groupoid need not be merely a groupoid topos in the naive sense: the étendue condition specifically requires a localic étale groupoid or an equivalent slice-localic presentation (Wrigley, 6 Jul 2025).

Current work also indicates that not every geometric theory embeds into an étendue in the ordinary sense, though every theory can be “reducibly embedded” in a co-ordinatisable theory (Wrigley, 6 Jul 2025). This suggests that étendues are not universal ambient objects for geometric logic, but they may still provide a controlled framework for studying broader classes of theories and possible non-commutative Stone-type dualities.

7. Significance and directions

Étendues occupy a notable position because they admit parallel descriptions in categorical, logical, and geometric terms. Categorically, they are topoi locally modeled by locales and represented by étale localic groupoids (Wrigley, 6 Jul 2025). Logically, they classify precisely those geometric theories whose models are determined by witnesses to a fixed family of formulas, expressed as uniform co-ordinatisation or uniform rigidity (Wrigley, 6 Jul 2025). Geometrically, they can encode invariants of combinatorial spaces, including simplicial dimension, through the internal logic of a canonically associated topos $A\to 1$58 (Menni, 2024). Site-theoretically, they can be approached via ordered groupoids, double categories, and Ehresmann topologies, with induced geometric morphisms controlled by covering-preserving and covering-flat functors (DeWolf et al., 2019).

These convergences make étendues a focal point for several active themes in contemporary topos theory. One theme is the syntactic detection of geometric structure: the equivalence between localic slices and rigidity properties suggests new criteria for recognizing when a classifying topos is locally localic (Wrigley, 6 Jul 2025). Another is invariant extraction from combinatorial or higher-dimensional data: the étendue $A\to 1$59 shows that internal logical formulas can recover external dimension data in presheaf settings (Menni, 2024). A third is representation theory beyond ordinary sites, where ordered and double-categorical structures provide refined models of sheaf semantics (DeWolf et al., 2019).

Open questions recorded in the recent literature include whether classical topoi of open subpolyhedra can be realized as étendues $A\to 1$60 of simplicial approximations, how such étendues relate to other fiberwise “discrete bundle” topoi used in higher category theory, and whether one can develop a fully site-independent account of $A\to 1$61 in a “gros” topos of spaces (Menni, 2024). A plausible implication is that étendues may continue to serve as a bridge concept linking localic geometry, logical definability, and combinatorial structure in settings where each of these viewpoints alone is incomplete.