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+% Torsors and Their Classification
+
+Let $\ca C$ be some Grothendieck topos, that is a category of sheaves on some
+Grothendieck topology. This text will look at some properties of *torsors* in
+$\ca C$ and how they might be classified using cohomology and homotopy
+theory. For this we will first need to define group objects. Torsors will then
+be associated to those.
+
+<defn> A *group object* in a category $\ca C$ is an object $G\in\ca C$ such that
+the associated Yoneda functor $\hom(\_,G)\colon \op{\ca C}\to\Set$ actually
+takes values in the category $\Grp$ of groups.  </defn>
+
+A trivial kind of example would be just a group considered as an object in
+$\Set$. A more involved example would be a *group scheme* $G$ over some base
+$S$. Such a thing is essentially defined as a group object in the category
+$\sch[S]$. If we have any subcanonical topology on $\sch[S]$, then $G$ defines a
+sheaf on the associated site and we obtain in this way a group object in the
+corresponding topos on $S$.
+
+Let's now assume we have a group object $G$ in a Grothendieck topos $\ca
+C$. Then we can define torsors over $G$ as follows:
+
+<defn> A *trivial torsor* over $G$ is an object $X$ with a left $G$--action
+which is isomorphic to $G$ itself with the action given by left
+multiplication. A *torsor* over $G$ is an object $X\in\ca C$ with a left
+$G$--action which is locally isomorphic to a trivial torsor; that is, there is
+an epimorphism $U\to *$ such that $U\times X$ is a trivial torsor in $\ca C/U$
+over $U \times G$.  </defn>