nLab classifying topos of a localic groupoid

Idea

The classifying topos of a localic groupoid $\mathcal{G}$ 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 $\mathcal{G}$-groupoid principal bundles, whence the name.

Definition

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

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

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

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

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

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

Properties

Exhaustion of the category of all toposes

Proposition

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

Proposition

(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.

Proposition

(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 $\mathcal{G}$ such that $\mathcal{E} \simeq Sh(\mathcal{G})$.

Related entries

• Grothendieck's Galois theory

References

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

• André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309 (1984) [ISBN:978-1-4704-0722-3]

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

• Carsten Butz, Ieke Moerdijk, Representing topoi by topological groupoids, Journal of pure and applied algebra 130 (1998) 223-235 (pdf)