My first blog post

So, I heard you can earn points by doing blog posts. So here we go.

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.