nLab classifying topos of a localic groupoid



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The classifying topos of a localic groupoid 𝒢\mathcal{G} is an incarnation of a localic groupoid (possibly a topological group) in the category of toposes. At least in good cases, geometric morphisms into it classify 𝒢\mathcal{G}-groupoid principal bundles, whence the name.


A localic groupoid is a groupoid object 𝒢=(𝒢 1𝒢 0)\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0) internal to locales (which we can think of as Grothendieck-(0,1)-toposes). If both 𝒢 0\mathcal{G}_0 and 𝒢 1\mathcal{G}_1 happen to be spatial locales, hence topological spaces, then this is a topological groupoid.

Let N 𝒢:Δ opLocalesN_\bullet \mathcal{G} : \Delta^{op} \to Locales be the simplicial object in locales given by the nerve of 𝒢\mathcal{G}. By applying the sheaf topos functor Sh:LocaleToposSh : Locale \to Topos to this, we obtain a simplicial topos Sh(N𝒢):[n]Sh(N n𝒢)Sh(N \mathcal{G}) : [n] \mapsto Sh(N_n \mathcal{G}). Let tr 2Sh(N𝒢)tr_2 Sh(N \mathcal{G}) be its 2-truncation, then the 2-colimit

Sh(𝒢)lim ntr 2Sh(N 𝒢) Sh(\mathcal{G}) \coloneqq \underset{\longrightarrow}{\lim}_n tr_2 Sh(N_\bullet \mathcal{G})

in the 2-category Topos is called the classifying topos of 𝒢\mathcal{G}.

This has a more explicit description along the lines discussed at sheaves on a simplicial topological space:

For ESh(𝒢 0)E \in Sh(\mathcal{G}_0) a sheaf on the topological space of its objects, say that a 𝒢 1\mathcal{G}_1-action on EE is a continuous groupoid action of 𝒢 \mathcal{G}_\bullet on the etale space Sp(E)𝒢 0Sp(E) \to \mathcal{G}_0 over 𝒢 0\mathcal{G}_0 that corresponds to the sheaf EE, hence for each morphisms f:xyf \colon x \to y in 𝒢\mathcal{G} a continuous function ρ(f):Sp(E) xSp(E) y\rho(f) \colon Sp(E)_x \to Sp(E)_y that satisfies the usual action property. These sheaves with 𝒢 1\mathcal{G}_1-action and with the evident homomorphisms between them form a category, and this is Sh(𝒢)Sh(\mathcal{G}).


Exhaustion of the category of all toposes


(Joyal-Tierney 84)
For every Grothendieck topos \mathcal{E} there is a localic groupoid 𝒢\mathcal{G} such that Sh(𝒢)\mathcal{E} \simeq Sh(\mathcal{G}).


(Moerdijk 88, theorem 5)
The construction in Prop. extends to an equivalence of 2-categories between that of Grothendieck toposes Topos and a localization of that of localic groupoids.


(Butz-Moerdijk 98)
If \mathcal{E} has enough points, then the localic groupoid in Prop. may in fact be taken to be a topological groupoid 𝒢\mathcal{G} such that Sh(𝒢)\mathcal{E} \simeq Sh(\mathcal{G}).


The original result for localic groupoids and arbitrary Grothendieck toposes is due to:

The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in

  • Ieke Moerdijk, The classifying topos of a continuous groupoid I , Trans. Amer. Math. Soc. Volume 310, Number 2, (1988) (pdf)

  • Ieke Moerdijk, The classifying topos of a continuous groupoid II , Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168. (pdf)

The restriction to topological groupoids and Grothendieck toposes with enough points is due to

Last revised on August 19, 2022 at 13:42:02. See the history of this page for a list of all contributions to it.