start writing on torsors

3 files changed, 61 insertions, 0 deletions

diff --git a/lists/zz_all/004 b/lists/zz_all/004
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--- /dev/null
+++ b/lists/zz_all/004
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+../../posts/torsors.md
\ No newline at end of file
diff --git a/posts.sh b/posts.sh
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+++ b/posts.sh
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+#!/usr/bin/env bash
+
+shopt -s extglob nullglob
+
+
+new_post() {
+    filename="$1"
+    shift 1
+    cat >posts/$filename <<EOF
+% $@
+EOF
+    last_link=$(find ./lists/zz_all -regex '.*/[0-9]*$' -printf '%f\n' | sort | tail -n1)
+    new_link=$(printf '%03d\n' $(($last_link + 1)))
+    ln -s ../../posts/"$filename" ./lists/zz_all/"$new_link"
+}
+
+
+. ./getopts_long.sh
+
+while getopts_long ":n:" opt \
+                   "" "$@"
+do
+    case $opt in
+        n)
+            shift "$(($OPTLIND - 1))"
+            new_post "$OPTLARG" "$@"
+            exit 0;;
+        :)
+            printf >&2 '%s: %s\n' "${0##*/}" "$OPTLERR"
+            exit 1;;
+    esac
+done
diff --git a/posts/torsors.md b/posts/torsors.md
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+% 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.
+
+<defn> 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.  </defn>
+
+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:
+
+<defn> 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 $U \times G$.  </defn>