nLab Diaconescu's theorem

This entry is about the theorem in topos theory. For the theorem in logic that often goes by the same name, see at Diaconescu-Goodman–Myhill theorem.

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Idea

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

$$Topos(Sh(X),[\mathbf{B}G, Set]) \simeq G Tors(X)$$

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 $X$.

Statement

For $C$ a category, write

$$PSh(C) := [C^{op}, Set]$$

for its presheaf topos.

For $\mathcal{E}$ any topos, write

$$FlatFunc(C, \mathcal{E}) \hookrightarrow [C, \mathcal{E}]$$

for the full subcategory of the functor category on the internally flat functors.

Theorem

(Diaconescu’s theorem)

There is an equivalence of categories

$$Topos(\mathcal{E}, PSh(C)) \simeq FlatFunc(C, \mathcal{E})$$

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

$$C \stackrel{j}{\to} PSh(C) \stackrel{f^*}{\to} \mathcal{E} \,,$$

where $j$ is the Yoneda embedding and $f^*$ is the inverse image of $f$.

See for instance (Johnstone, theorem B3.2.7).

Remark

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

$$Topos(\mathcal{E}, PSh(C)) \simeq \mathbb{T}_C Mod(\mathcal{E})$$

between the geometric morphisms and the $\mathbb{T}$-models in $\mathcal{E}$.

This says that $PSh(C)$ is the classifying topos for $\mathbb{T}_C$.

Remark

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.

References

A standard reference is section B3.2 in

• Peter Johnstone, Sketches of an Elephant

The first proof of this result can be found in:

• Radu Diaconescu, Change of base for toposes with generators , J. Pure Appl. Algebra 6 (1975), no. 3, 191-218 (doi:10.1016/0022-4049(75)90015-8)

Another proof is in

• Ieke Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Mathematics 1616, Springer 1995. vi+94 pp. ISBN: 3-540-60319-0