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authorViktor Kleen <viktor@kleen.org>2015-03-03 05:55:33 +0000
committerViktor Kleen <viktor@kleen.org>2015-03-03 05:55:33 +0000
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start writing on torsors
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1% Torsors and Their Classification
2
3Let $\ca C$ be some Grothendieck topos, that is a category of sheaves on some
4Grothendieck topology. This text will look at some properties of *torsors* in
5$\ca C$ and how they might be classified using cohomology and homotopy
6theory. For this we will first need to define group objects. Torsors will then
7be associated to those.
8
9<defn> A *group object* in a category $\ca C$ is an object $G\in\ca C$ such that
10the associated Yoneda functor $\hom(\_,G)\colon \op{\ca C}\to\Set$ actually
11takes values in the category $\Grp$ of groups. </defn>
12
13A trivial kind of example would be just a group considered as an object in
14$\Set$. A more involved example would be a *group scheme* $G$ over some base
15$S$. Such a thing is essentially defined as a group object in the category
16$\sch[S]$. If we have any subcanonical topology on $\sch[S]$, then $G$ defines a
17sheaf on the associated site and we obtain in this way a group object in the
18corresponding topos on $S$.
19
20Let's now assume we have a group object $G$ in a Grothendieck topos $\ca
21C$. Then we can define torsors over $G$ as follows:
22
23<defn> A *trivial torsor* over $G$ is an object $X$ with a left $G$--action
24which is isomorphic to $G$ itself with the action given by left
25multiplication. A *torsor* over $G$ is an object $X\in\ca C$ with a left
26$G$--action which is locally isomorphic to a trivial torsor; that is, there is
27an epimorphism $U\to *$ such that $U\times X$ is a trivial torsor in $\ca C/U$
28over $U \times G$. </defn>