From f81be5dfb7720994f6deca3bc738c63f48f26309 Mon Sep 17 00:00:00 2001 From: Viktor Kleen Date: Tue, 3 Mar 2015 05:55:33 +0000 Subject: start writing on torsors --- posts/torsors.md | 28 ++++++++++++++++++++++++++++ 1 file changed, 28 insertions(+) create mode 100644 posts/torsors.md (limited to 'posts/torsors.md') diff --git a/posts/torsors.md b/posts/torsors.md new file mode 100644 index 0000000..22bf7af --- /dev/null +++ b/posts/torsors.md @@ -0,0 +1,28 @@ +% Torsors and Their Classification + +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 torsor* over $G$ is an object $X$ with a left $G$--action +which is isomorphic to $G$ itself with the action given by left +multiplication. A *torsor* over $G$ 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 $U \times G$. -- cgit v1.2.3