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.

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.

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:

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.

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 $U \times G$ .

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

Epimorphisms in a topos are stable under pullback.

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.

A morphism $f$ is an isomorphism if and only if $f$ is both monic and epic.

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:

In a topos, any local epimorphism is an epimorphism, any local monomorphism is a monomorphism, and any local isomorphism is an isomorphism.

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.

Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4↩