classifying topos of a localic groupoid


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The classifying topos of a localic groupoid 𝒢\mathcal{G} is a an incarnation of a localic groupoid (possibly a topological groups) 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/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:xxf \colon x \to x 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

Proposition (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}).

Proposition (Moerdijk 88, theorem 5)

This construction extends to an equivalence of 2-categories between that of Grothendieck toposes Topos and a localization of that of localic groupoids.

Proposition (Butz-Moerdijk 98) If \mathcal{E} has enough points, then there exists in fact 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.

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

Revised on June 3, 2017 06:23:31 by Urs Schreiber (