Examples of topoi that are not ordinary spaces

Question

In [SGA6] we find:

Mais nous lui conseillons néanmoins, de préférence, de s'assimiler le langage des topos, qui fournit un principe d'unification extrêmement commode. (DeepL translate: However, we nevertheless advise him, preferably, to assimilate the language of toposes, which provides an extremely convenient principle of unification.)

Lets motivate this advice following some examples of topoi that are not “ordinary” spaces:

Comme autres exemples remarquables de topos qui ne sont pas des espaces ordinaires, et pour lesquels il ne semble pas y avoir non plus de substitut satisfaisant en termes des notions “admises”, je signalerai : les topos quotients d’un espace topologique par une relation d’équivalence locale (par exemple des feuilletages de variétés, auquel cas le topos quotient est même une “multiplicité” i.e. est localement une variété) ; les topos “classifiants” pour à peu près n’importe quelle espèce de structure mathématique (tout au moins celles “s’exprimant en termes de limites projectives finies et de limites inductives quelconques”). Quand on prend une structure de “variété” (topologique, différentiable, analytique réelle ou complexe, de Nash, etc. … ou même schématique lisse sur une base donnée) on trouve dans chaque cas un topos particulièrement alléchant, qui mérite le nom de “variété universelle” (de l’espèce envisagée). Ses invariants homotopiques (et notamment sa cohomologie, qui mérite le nom de “cohomologie classifiante” pour l’espèce de variété envisagée) devraient être étudiés et connus depuis longtemps, mais pour le moment ça n’en prend nullement le chemin…. [ReS]

DeepL translation:

As other remarkable examples of topos that are not ordinary spaces, and for which there also seems to be no satisfactory substitute in terms of “accepted” notions, I would point out: the topos quotients of a topological space by a local equivalence relation (e.g. foliations of manifolds, in which case the topos quotient is even a “multiplicity” i. e. is locally a manifold); “classifying” topos for almost any species of mathematical structure (at least those “expressed in terms of finite projective limits and any inductive limits”). When we take a “manifold” structure (topological, differentiable, real or complex analytical, Nash, etc. … or even smooth schematic on a given base) we find in each case a particularly attractive topos, which deserves the name of “universal variety” (of the species considered). Its homotopic invariants (and in particular its cohomology, which deserves the name of “classifying cohomology” for the species of variety considered) should have been studied and known for a long time, but for the moment it does not take any way….

What are some other examples of topoi that are not “ordinary” spaces?

---

[ReS] Récoltes et Semailles, A. Grothendieck

[SGA6] SGA6 Théorie des intersections et théorème de Riemann-Roch, 1966–1967. Séminaire de Géométrie Algébrique du Bois Marie doi:10.1007/BFb0066283

Answer 1

The bicategory of Grothendieck toposes is equivalent to the bicategory of localic groupoids, with appropriately defined 1-morphisms and 2-morphisms.

"Ordinary spaces" are topological spaces, or, perhaps, locales. There is a forgetful functor from the category of locales to the above bicategory of localic groupoids that sends a locale L to the groupoid that has L as its space of objects and morphisms and identity maps as structure maps.

Thus, any localic groupoid that is not equivalent to a locale gives rise to such a topos.

For instance, one can take a localic group G (e.g., a locally compact Hausdorff topological group) and consider the delooping groupoid BG. The topos of sheaves of sets over BG is an example of a topos that does not come from an ordinary space.

Foliation groupoids and quotient groupoids of nonfree actions of groups on spaces provide additional examples.

Answer 2

This is a translation of Dmitri Pavlov's answer into a more intrinsic, more geometric, and more elementary language. In particular, I will show that the étale topos of a positive-dimensional variety is never the topos of a topological space.

If $X$ is a topological space, then the associated topos $E = \mathbf{Sh}(X)$ is generated by subobjects of the final object $\mathbf 1_X$. In fact we may take the sheaves $h_U = j_{U,!}(\mathbf 1_U)$ for $U \subseteq X$ open: if two maps $\mathscr F \rightrightarrows \mathscr G$ are different, there is an open $U \subseteq X$ such that $\mathscr F(U) \rightrightarrows \mathscr G(U)$ don't agree, so we get a map $j_{U,!}(\mathbf 1_U) \to \mathscr F$ such that the compositions $j_{U,!}(\mathbf 1_U) \to \mathscr F \rightrightarrows \mathscr G$ differ.

On a general site $\mathscr C$, the same argument shows that the $j_{U,!}(\mathbf 1_U)$ for $U \in \mathscr C$ generate the topos $E = \mathbf{Sh}(\mathscr C)$. But the difference is that we cannot do this with subobjects of $\mathbf 1$. For example, if $G$ is a discrete group, then the terminal object $\mathbf 1$ of $BG = G\text{-}\mathbf{Set}$ is a point with a trivial $G$-action, which does not have any subobjects except $\varnothing$ and $\mathbf 1$.

More generally¹, let $\mathscr C$ be a subcanonical site with fibre products and a terminal object $X$, such that every object $V$ covers a subobject $W \subseteq X$ (for example, the (small) étale site $X_{\operatorname{\acute et}}$ on a scheme $X$, since $W = \operatorname{im}(V \to X)$ is open and $V \to W$ a covering). Assume $\mathscr C$ has an object $U$ that does not admit a map from a subobject of $X$ (for example, the étale site of any positive-dimensional variety $X$ over a field $k$, or of $\operatorname{Spec} k$ when $k$ is not separably closed). Then I claim that $h_U = j_{U,!}(\mathbf 1_U)$ does not admit any map from a nonempty subobject of $\mathbf 1_X$.

Indeed, suppose $\mathscr F \subseteq \mathbf 1_X$ maps to $h_U$. Let $V \in \mathscr C$, and choose a covering $V \to W$ of a subobject $W \subseteq X$. By assumption, $h_U(W) = \varnothing$, so $\mathscr F(W) = \varnothing$. But since $V \to W$ is a covering and $\mathscr F \subseteq \mathbf 1_X$, for any section in $\mathscr F(V)$ the restrictions $\mathscr F(V) \rightrightarrows \mathscr F(V \times_W V)$ vacously agree, hence they glue to a section of $\mathscr F(W)$, which is impossible. We conclude that $\mathscr F(V) = \varnothing$, so $\mathscr F = \varnothing$ as $V$ was arbitrary. $\square$

So the étale topos on a reasonable scheme $X$ is basically never a topos of sheaves on a topological space.

---

¹For example, the case $\mathscr C = (\operatorname{Spec} \mathbf R)_{\operatorname{\acute et}}$ recovers the example $B(\mathbf Z/2)$ above.