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Diaconescu’s theorem asserts that any presheaf topos is the classifying topos for internally flat functors on its site.
Often a special case of this is considered, which asserts that for every topological space $X$ and discrete group $G$ there is an equivalence of categories
between the geometric morphisms from the sheaf topos over $X$ to the category of permutation representations of $G$ and the category of $G$-torsors on $X$.
For $C$ a category, write
for its presheaf topos.
For $\mathcal{E}$ any topos, write
for the full subcategory of the functor category on the internally flat functors.
(Diaconescu’s theorem)
There is an equivalence of categories
between the category of geometric morphisms $f : \mathcal{E} \to PSh(C)$ and the category of internally flat functors $C \to \mathcal{E}$.
This equivalence takes $f$ to the composite
where $j$ is the Yoneda embedding and $f^*$ is the inverse image of $f$.
See for instance (Johnstone, theorem B3.2.7).
If $C$ is a finitely complete category we may think of it as the syntactic category and in fact the syntactic site of an essentially algebraic theory $\mathbb{T}_C$. An internally flat functor $C \to \mathcal{E}$ is then precisely a finite limit preserving functor, hence is precisely a $\mathbb{T}$-model in $\mathcal{E}$.
Therefore the above theorem says in this case that there is an equivalence of categories
between the geometric morphisms and the $\mathbb{T}$-models in $\mathcal{E}$.
This says that $PSh(C)$ is the classifying topos for $\mathbb{T}_C$.
If $G$ is a discrete group and $C = \mathbf{B}G$ is its delooping groupoid, $PSh(C) \simeq [\mathbf{B}G, Set]$ is the category of permutation representations of $G$, also called the classifying topos of $G$.
In this case an internally flat functor $C = \mathbf{B}G \to \mathcal{E}$ may be identified with a $G$-torsor object in $\mathcal{E}$.
For this reason one sees in the literature sometimes the term “torsor” for internally flat functors out of any category $C$. It is however not so clear in which sense this terminology is helpful in cases where $C$ is not a delooping groupoid or at least some groupoid.
A standard reference is section B3.2 in
The first proof of this result can be found in:
Another proof is in
Last revised on July 29, 2015 at 08:03:47. See the history of this page for a list of all contributions to it.