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-% Torsors and Their Classification I
-
-# Torsors in Grothendieck Toposes #
-
-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 $G$--torsor* is an object $X$ with a left $G$--action which is
-isomorphic to $G$ itself with the action given by left multiplication.  </defn>
-
-<defn> A *$G$--torsor* 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>
-
-I want to show that for any $G$--torsor $X$ according to this definition the
-left action of $G$ on $X$ is free and transitive, that is the map
-$$ f\colon G\times X \to X\times X $$
-given (on generalized elements) by $f(g, x) = (gx, x)$ is an isomorphism. This
-is going to be some relatively elementary category theory but I think it's worth
-writing it up. First a few facts about isomorphisms in toposes, they can be
-found for example in Sheaves in Geometry and Logic.[^1]
-
-<lem>
-Epimorphisms in a topos are stable under pullback.
-</lem>
-<lem>
-In a topos every morphism $f\colon X\to Y$ has a functorial factorization $f =
-m\circ e$ with $m$ a monomorphism and $e$ and epimorphism.
-</lem>
-<lem>
-A morphism $f$ is an isomorphism if and only if $f$ is both monic and epic.
-</lem>
-
-Now, let $f\colon A\to B$ be a *local monomorphism*, i.e. there is an epimorphism
-$U\to *$ such that the pullback $f\times U$ of $f$ to $U$ is a
-monomorphism. Then, since epimorphisms are stable under pullback, it follows that
-in the commutative square
-$$
-\begin{tikzcd}
-A\times U \ar[into, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
-A \ar[r, "f"'] & B
-\end{tikzcd}
-$$
-both vertical maps are epimorphisms. Now let $\varphi,\psi\colon T\to A$ be a pair of morphisms
-such that $f\varphi = f\psi$. Then, denoting by $\varphi_U$ and $\psi_U$ the
-pullbacks to $U$, we have $f_U\varphi_U = f_U\psi_U$. but $f_U$ is a
-monomorphism by assumption, so $\varphi_U = \psi_U$. So we have a commutative
-diagram
-$$
-\begin{tikzcd}
-T\times U \ar[r, "\varphi_U = \psi_U"] \ar[onto, d] & A\times U \ar[onto, d] \\
-T \ar[r, "\varphi", shift left] \ar[r, "\psi"', shift right] & A
-\end{tikzcd}
-$$
-in which the vertical maps are epimorphisms. It follows that $\varphi =
-\psi$.
-
-Now, if $f$ is a local epimorphism, then again we have the diagram
-$$
-\begin{tikzcd}
-A\times U \ar[onto, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
-A \ar[r, "f"'] & B
-\end{tikzcd}
-$$
-and it is immediate that $f$ is an epimorphism. In summary:
-
-<prop>
-In a topos, any local epimorphism is an epimorphism, any local monomorphism is a
-monomorphism, and any local isomorphism is an isomorphism.
-</prop>
-
-Now, let's check that $G$--torsors $X$ as defined above are free and
-transitive. Take an epimorphism $U\onto *$ such that $X\times U$ is trivial in
-$\ca C/U$. The action of $G$ on itself by left multiplication is plainly free
-and transitive, so in $\ca C/U$ we have the isomorphism
-$$(G\times U)\times_U (X\times U) \iso (X\times U)
-\times_U (X\times U)$$
-and $(G\times U)\times_U(X\times U) = (G\times X)\times U$ and $(X\times
-U)\times_U (X\times U) = (X\times X)\times U$ because pullback preserves products. So, $G\times X\to
-X\times X$ is a local isomorphism, hence an isomorphism.
-
-[^1]: Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4