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 
A\times U \ar[into, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
A \ar[r, "f"'] & B
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 
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
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 
A\times U \ar[onto, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
A \ar[r, "f"'] & B
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.

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