topos theory

# Contents

## 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

$\mathrm{Topos}\left(\mathrm{Sh}\left(X\right),\left[BG,\mathrm{Set}\right]\right)\simeq G\mathrm{Tors}\left(X\right)$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

$\mathrm{PSh}\left(C\right):=\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$PSh(C) := [C^{op}, Set]

for its presheaf topos.

For $ℰ$ any topos, write

$\mathrm{FlatFunc}\left(C,ℰ\right)↪\left[C,ℰ\right]$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

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

between the category of geometric morphisms $f:ℰ\to \mathrm{PSh}\left(C\right)$ and the category of internally flat functors $C\to ℰ$.

This equivalence takes $f$ to the composite

$C\stackrel{j}{\to }\mathrm{PSh}\left(C\right)\stackrel{{f}^{*}}{\to }ℰ\phantom{\rule{thinmathspace}{0ex}},$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 $C\to ℰ$ 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

$\mathrm{Topos}\left(ℰ,\mathrm{PSh}\left(C\right)\right)\simeq {𝕋}_{C}\mathrm{Mod}\left(ℰ\right)$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 $\mathrm{PSh}\left(C\right)$ is the classifying topos for ${𝕋}_{C}$.

###### Remark

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

In this case an internally flat functor $C=BG\to ℰ$ 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)