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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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 XX and discrete group GG 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 from the sheaf topos over XX to the category of permutation representations of GG and the category of GG-torsors on XX.


For CC a category, write

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

for its presheaf topos.

For \mathcal{E} any topos, write

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

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


(Diaconescu’s theorem)

There is an equivalence of categories

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

between the category of geometric morphisms f:PSh(C)f : \mathcal{E} \to PSh(C) and the category of internally flat functors CC \to \mathcal{E}.

This equivalence takes ff to the composite

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

where jj is the Yoneda embedding and f *f^* is the inverse image of ff.

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


If CC 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\mathbb{T}_C. An internally flat functor CC \to \mathcal{E} is then precisely a finite limit preserving functor, hence is precisely a 𝕋\mathbb{T}-model in \mathcal{E}.

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

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

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

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


If GG is a discrete group and C=BGC = \mathbf{B}G is its delooping groupoid, PSh(C)[BG,Set]PSh(C) \simeq [\mathbf{B}G, Set] is the category of permutation representations of GG, also called the classifying topos of GG.

In this case an internally flat functor C=BG C = \mathbf{B}G \to \mathcal{E} may be identified with a GG-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 CC. It is however not so clear in which sense this terminology is helpful in cases where CC 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

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

Last revised on June 6, 2019 at 17:18:53. See the history of this page for a list of all contributions to it.