From 94956f26a68f129a4d4076ad83498d38629ad609 Mon Sep 17 00:00:00 2001 From: Viktor Kleen Date: Fri, 6 Mar 2015 20:48:09 +0000 Subject: show torsors are torsors --- index.md.do | 2 +- posts/torsors.md | 86 +++++++++++++++++++++++++++++++++++++++++++++++++++----- 2 files changed, 80 insertions(+), 8 deletions(-) diff --git a/index.md.do b/index.md.do index e36b596..b75845a 100644 --- a/index.md.do +++ b/index.md.do @@ -24,7 +24,7 @@ ideas and results. As such, it is woefully unpolished, unreliable and generally to be used at your own risk. Still, have fun! And please send email with suggestions, corrections, -criticism or general rants to if you like. +criticism or general rants to if you like. I have the following categories of posts: diff --git a/posts/torsors.md b/posts/torsors.md index 22bf7af..b87c2df 100644 --- a/posts/torsors.md +++ b/posts/torsors.md @@ -1,4 +1,6 @@ -% Torsors and Their Classification +% 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 @@ -20,9 +22,79 @@ 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 $U \times G$. + 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 $(X\times +U)\times_U (X\times U)$ 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 -- cgit v1.2.3