nLab
classifying topos of a localic groupoid

under construction – am being interrupted…

Idea

The classifying topos of a localic groupoid $𝒢$ is a an incarnation of a localic groupoid in the world of toposes. At least in good cases, geometric morphisms into it classify $𝒢$-principal bundles.

Recall that a localic groupoid is a groupoid $𝒢=({𝒢}_1\overrightarrow{\to }{𝒢}_0)$ internal to locales/Grothendieck-(0,1)-toposes.

Let $N_{•}𝒢:{\Delta }^{\mathrm{op}}\to \mathrm{Locales}$ be the simplicial object in locales given by the nerve of $𝒢$. By applying the sheaf topos functor $\mathrm{Sh}:\mathrm{Locale}\to \mathrm{Topos}$ to this, we obtain a simplicial topos $\mathrm{Sh}(N𝒢):[n]\mapsto \mathrm{Sh}(N_n𝒢)$. Let ${\mathrm{tr}}_2\mathrm{Sh}(N𝒢)$ be its 2-truncation, then the 2-colimit

$$\mathrm{Sh}(𝒢):=\underset{{\to }_{[n]}}{\lim }{\mathrm{tr}}_2\mathrm{Sh}(N_{•}𝒢)$$Sh(\mathcal{G}) := \lim_{\to_{[n]}} tr_2 Sh(N_\bullet \mathcal{G})

in the 2-category of toposes? is called the classifying topos of $𝒢$.

This has an explicit description along the lines discussed at sheaves on a simplicial topological space.

Proposition (Joyal-Tierney)

For every Grothendieck topos $E$ there is a localic groupoid $𝒢$ such that $E\simeq \mathrm{Sh}(𝒢)$.

References

The original result appears in

• Andre Joyal, M. Tierney, An extension of the Galois theory of Grothendieck Mem. Amer. Math. Soc. no 309 (1984)

An extension of the equivalence to morphisms is discussed in

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

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 II, Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168.