--- title: Étendues in Topos Theory url: https://www.emergentmind.com/topics/etendues type: topic --- # Étendues in Topos Theory Searching arXiv for recent 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 [2507.04526], as invariants attached to combinatorial spaces whose internal logic detects dimension [2411.19863], and as toposes represented via ordered groupoids, left-cancellative categories, and Ehresmann sites [1910.02540]. 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 [1809.07403]. ## 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 [2507.04526]. A geometric morphism is *localic* if its direct-image functor is full and faithful; equivalently, \(F\simeq \mathrm{Sh}(L)\) for some locale \(L\) [2507.04526]. Against that background, a topos \(E\) is an **étendue** if there exists \(A\in E\) such that: 1. the support \(A\to 1\) is an epimorphism, and 2. the slice topos \(E/A\) is localic [2507.04526]. When these conditions hold, one says that \(E\) is an étendue over \(A\). Equivalently there is an étale geometric morphism \[ \pi_A:E/A\to E \] whose inverse-image sends \(B\) to \(B\times A\to A\) [2507.04526]. This definition places étendues between general Grothendieck topoi and localic topoi. Localic topoi are immediate examples: they are trivially étendues over \(1\) [2507.04526]. 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 \(\widehat{G}\) of \(G\)-sets for a discrete group \(G\) is an étendue because \(\widehat{G}/G\simeq \mathrm{Set}\) [2507.04526]. More generally, if \(X=(X_1\rightrightarrows X_0)\) is a localic étale groupoid, then \(\mathrm{Sh}(X)\) is an étendue, with \(\mathrm{Sh}(X)/X_1\simeq \mathrm{Sh}(X_0)\) localic [2507.04526]. ## 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 [2507.04526]. 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 [1910.02540]. 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 \(G\) assigns covering vertical sieves satisfying maximality, closure under restriction, and local character; an ordered groupoid equipped with such a topology is an **Ehresmann site** [1910.02540]. 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 [1910.02540]. It also establishes a Comparison Lemma for Ehresmann sites, paralleling the usual Grothendieck-site comparison theorem [1910.02540]. These results are not merely formal. They show that étendues admit multiple interoperable presentations: | Presentation | Structural datum | Associated topos | |---|---|---| | Localic groupoid | localic étale groupoid \(X\) | \(\mathrm{Sh}(X)\) | | Slice-localic form | inhabited \(A\in E\) with \(E/A\) localic | \(E\) | | Ehresmann-site form | ordered groupoid with Ehresmann topology | \(\mathrm{Sh}(G,T)\) | 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 \(T\) with classifying topos \(E_T\), Wrigley proves that three conditions are equivalent: \(T\) is uniformly co-ordinatised, \(T\) is uniformly rigid, and \(E_T\) is an étendue [2507.04526]. The setting begins with a set \(\Psi\) of geometric formulae \(\psi(\bar{x})\). For a model \(M\models T\), one writes \[ \psi(M)=\{\bar{m}\in M^{|\bar{x}|}\mid M\models \psi(\bar{m})\}, \qquad \Psi(M)=\coprod_{\psi\in\Psi}\psi(M) \] [2507.04526]. The theory \(T\) is **uniformly co-ordinatised over** \(\Psi\) if there exist families \(\Theta_\psi\) of geometric formulae \(\theta(\bar{x},y)\) such that, for each \(\psi\in\Psi\), 1. \(T\vdash \theta(\bar{x},y)\Rightarrow \psi(\bar{x})\), 2. \(T\vdash \theta(\bar{x},y)\wedge \theta(\bar{x},y')\Rightarrow (y=y')\), 3. \(T\vdash \psi(\bar{x})\wedge (y=y)\Rightarrow \bigvee_{\theta\in\Theta_\psi}\theta(\bar{x},y)\) [2507.04526]. Informally, each witness \(\bar{m}\in \psi(M)\) co-ordinates the entire model \(M\) in a uniform way via one of the functional formulas in \(\Theta_\psi\) [2507.04526]. The theory \(T\) is **uniformly rigid over** \(\Psi\) if, for any model \(M\models T\) in any topos and any global element \(\bar{m}:1\to \Psi(M)\), the only automorphism of \(M\) fixing \(\bar{m}\) pointwise is the identity [2507.04526]. The main theorem states that for a geometric theory \(T\) and its classifying topos \(E_T\), the following are equivalent: - \(T\) is uniformly co-ordinatised; - \(T\) is uniformly rigid; - \(E_T\) is an étendue [2507.04526]. 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 [2507.04526]. Several intermediate results sharpen the picture. If \(E_T/\Psi\) is localic, then \(T\) is uniformly co-ordinatised over the formulas indexing \(\Psi\) [2507.04526]. If \(T\) is uniformly co-ordinatised over \(\Psi\), then it is uniformly rigid over \(\Psi\) [2507.04526]. If \(T\) is uniformly rigid over \(\Psi\), then the slice \(E_T/\Psi\) has at most one isomorphism between any two points, hence is localic, and therefore \(E_T\) is an étendue over \(\Psi\) [2507.04526]. A key localicity criterion used here is that a topos \(E\) is localic if and only if for any base topos \(F\), every pair of geometric morphisms \(F\rightrightarrows E\) admits at most one natural transformation between them [2507.04526]. 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 \(1\), uniformly co-ordinatised over \(\Psi=\{\top\}\), and uniformly rigid [2507.04526]. 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 \(F\). Let \(\Psi=\{B_n(\bar{x})\mid n\in \mathbb{N}\}\), where \(B_n(\bar{x})\) says that \(\bar{x}\) is a basis of length \(n\). The coordinating formulas say that \(y\) is a given linear combination of the basis \(\bar{x}\). 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 \(n\)-dimensional spaces, equivalently the presheaf topos on the groupoid \(\coprod_n GL_n(F)\), which is an étendue over \(\Psi\) [2507.04526]. Another standard example is the theory of \(G\)-torsors. Torsors under a group \(G\) form the classifying topos \(\widehat{G}\), an étendue over the formula \((x=x)\). The coordinating formulas are \(\{(y=g\cdot x)\mid g\in G\}\), which exhibit uniform co-ordinatisation and rigidity in any topos [2507.04526]. 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 [2507.04526]. 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 [2507.04526]. 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 [2507.04526]. 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 \(C\) and presheaf topos \(\widehat{C}=\mathrm{Set}^{C^{op}}\), an object \(X\in \widehat{C}\) has a category of elements \(C/X\). Inside \(C/X\), one considers the full subcategory \((C/X)^*\) on the **minimal objects**, namely representable maps \(x:C(-,C)\to X\) such that whenever \(y:C(-,D)\to X\) and \(f:D\to C\) satisfy \(y\circ f=x\), the map \(f\) must be monic in \(C\) [2411.19863]. Menni defines the **étendue of \(X\)** by \[ E(X)=\widehat{(C/X)^*} \] and shows that the composite geometric morphism from this presheaf topos to \(\mathrm{Set}\) is an étendue [2411.19863]. Equivalently, \(E(X)\) is the level \(\varepsilon\) subtopos of \(\widehat{C}/X\) maximal among those whose composite to \(\mathrm{Set}\) is an étendue [2411.19863]. The construction can also be recovered internally: in the logic of \(\widehat{C}\), a subterminal object classifies minimality, and \(E(X)\) is the closed subtopos cut out by that subterminal [2411.19863]. For simplicial sets, the internal logic of \(E(X)\) encodes dimension. If \(A=\widehat{\Delta}\) is the topos of simplicial sets and \(X\) is **non-singular**, then for non-singular simplicial sets \(X\) and \(Y\), \[ E(X)\simeq E(Y)\quad\Longleftrightarrow\quad \dim X=\dim Y \] [2411.19863]. The paper also introduces bounded-depth formulas \(IBD_n\) in the internal Heyting algebra of subterminals: \[ \begin{aligned} IBD_{-\infty} &= \bot, \\ IBD_0 &= \forall x:\Omega\;[\,x\lor(x\Rightarrow\bot)=x\,], \\ IBD_{n+1} &= \forall x:\Omega\;[\,x\lor(x\Rightarrow IBD_n)=x\,]. \end{aligned} \] For each \(n\in \mathbb{N}\), \[ \dim X\le n \quad\Longleftrightarrow\quad E(X)\models IBD_n \] under the hypotheses stated in the paper, with the converse requiring \(X\) to be strongly regular and the site to satisfy a mild well-foundedness hypothesis [2411.19863]. The mechanism is explicit. The subterminals \(IBD_n\) correspond to sieves on \((\Delta/X)^*\) consisting of those minimal simplices whose “height” is at most \(n\), while the \(n\)-skeleton is characterized via an idempotent comonad \(Sk_n\) whose counit detects simplices factoring through an object of height \(\le n\) [2411.19863]. Thus internal logical depth in the étendue recovers external simplicial dimension. Standard examples illustrate the construction. For the standard simplex \(\Delta[n]\), \((\Delta/\Delta[n])^*\) is equivalent to the finite chain \(\{0<1<\cdots