Torsors in Grothendieck Toposes
Let be some Grothendieck topos, that is a category of sheaves on some Grothendieck topology. This text will look at some properties of torsors in 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 is an object such that the associated Yoneda functor actually takes values in the category of groups.
A trivial kind of example would be just a group considered as an object in . A more involved example would be a group scheme over some base . Such a thing is essentially defined as a group object in the category . If we have any subcanonical topology on , then defines a sheaf on the associated site and we obtain in this way a group object in the corresponding topos on .
Let’s now assume we have a group object in a Grothendieck topos . Then we can define torsors over as follows:
A trivial –torsor is an object with a left –action which is isomorphic to itself with the action given by left multiplication.
A –torsor is an object with a left –action which is locally isomorphic to a trivial torsor; that is, there is an epimorphism such that is a trivial torsor in over .
I want to show that for any –torsor according to this definition the left action of on is free and transitive, that is the map given (on generalized elements) by 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 has a functorial factorization with a monomorphism and and epimorphism.
A morphism is an isomorphism if and only if is both monic and epic.
Now, let be a local monomorphism, i.e. there is an epimorphism such that the pullback of to is a monomorphism. Then, since epimorphisms are stable under pullback, it follows that in the commutative square both vertical maps are epimorphisms. Now let be a pair of morphisms such that . Then, denoting by and the pullbacks to , we have . but is a monomorphism by assumption, so . So we have a commutative diagram in which the vertical maps are epimorphisms. It follows that .
Now, if is a local epimorphism, then again we have the diagram and it is immediate that 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 –torsors as defined above are free and transitive. Take an epimorphism such that is trivial in . The action of on itself by left multiplication is plainly free and transitive, so in we have the isomorphism and and because pullback preserves products. So, is a local isomorphism, hence an isomorphism.
Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4↩