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Diaconescu's 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),[BG,Set])≃GTors(X)Topos(Sh(X),[\mathbf{B}G, Set]) \simeq G Tors(X)

between the geometric morphisms form 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]PSh(C) := [C^{op}, Set]

for its presheaf topos.

For ℰ any topos, write

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

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

Theorem

(Diaconescu’s theorem)

There is an equivalence of categories

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

between the category of geometric morphisms f:ℰ→PSh(C) and the category of internally flat functors C→ℰ.

This equivalence takes f to the composite

C→jPSh(C)→f *ℰ,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 𝕋 C. An internally flat functor C→ℰ is then precisely a finite limit preserving functor, hence is precisely a 𝕋-model in ℰ.

Therefore the above theorem says in this case that there is an equivalence of categories

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

between the geometric morphisms and the 𝕋-models in ℰ.

This says that PSh(C) is the classifying topos for 𝕋 C.

Remark

If G is a discrete group and C=BG is its delooping groupoid, PSh(C)≃[BG,Set] is the category of permutation representations of G, also called the classifying topos of G.

In this case an internally flat functor C=BG→ℰ may be identified with a G-torsor object in 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 textbook references is section B3.2 in

The first proof of this result can be found in:

  • R. Diaconescu, Change of base for toposes with generators , J. Pure Appl. Algebra 6 (1975), no. 3, 191-218.

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

Revised on May 15, 2013 18:06:40 by Anonymous Coward (129.132.146.160)
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