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@@ -24,7 +24,7 @@ ideas and results. As such, it is woefully unpolished, unreliable and generally
24to be used at your own risk. 24to be used at your own risk.
25 25
26Still, have fun! And please send email with suggestions, corrections, 26Still, have fun! And please send email with suggestions, corrections,
27criticism or general rants to <vkleen+math@17220103.de> if you like. 27criticism or general rants to <kleen@usc.edu> if you like.
28 28
29I have the following categories of posts: 29I have the following categories of posts:
30 30
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1% Torsors and Their Classification 1% Torsors and Their Classification I
2
3# Torsors in Grothendieck Toposes #
2 4
3Let $\ca C$ be some Grothendieck topos, that is a category of sheaves on some 5Let $\ca C$ be some Grothendieck topos, that is a category of sheaves on some
4Grothendieck topology. This text will look at some properties of *torsors* in 6Grothendieck topology. This text will look at some properties of *torsors* in
@@ -20,9 +22,79 @@ corresponding topos on $S$.
20Let's now assume we have a group object $G$ in a Grothendieck topos $\ca 22Let's now assume we have a group object $G$ in a Grothendieck topos $\ca
21C$. Then we can define torsors over $G$ as follows: 23C$. Then we can define torsors over $G$ as follows:
22 24
23<defn> A *trivial torsor* over $G$ is an object $X$ with a left $G$--action 25<defn> A *trivial $G$--torsor* is an object $X$ with a left $G$--action which is
24which is isomorphic to $G$ itself with the action given by left 26isomorphic to $G$ itself with the action given by left multiplication. </defn>
25multiplication. A *torsor* over $G$ is an object $X\in\ca C$ with a left 27
26$G$--action which is locally isomorphic to a trivial torsor; that is, there is 28<defn> A *$G$--torsor* is an object $X\in\ca C$ with a left $G$--action which is
27an epimorphism $U\to *$ such that $U\times X$ is a trivial torsor in $\ca C/U$ 29locally isomorphic to a trivial torsor; that is, there is an epimorphism $U\to
28over $U \times G$. </defn> 30*$ such that $U\times X$ is a trivial torsor in $\ca C/U$ over $U \times G$.
31</defn>
32
33I want to show that for any $G$--torsor $X$ according to this definition the
34left action of $G$ on $X$ is free and transitive, that is the map
35$$ f\colon G\times X \to X\times X $$
36given (on generalized elements) by $f(g, x) = (gx, x)$ is an isomorphism. This
37is going to be some relatively elementary category theory but I think it's worth
38writing it up. First a few facts about isomorphisms in toposes, they can be
39found for example in Sheaves in Geometry and Logic.[^1]
40
41<lem>
42Epimorphisms in a topos are stable under pullback.
43</lem>
44<lem>
45In a topos every morphism $f\colon X\to Y$ has a functorial factorization $f =
46m\circ e$ with $m$ a monomorphism and $e$ and epimorphism.
47</lem>
48<lem>
49A morphism $f$ is an isomorphism if and only if $f$ is both monic and epic.
50</lem>
51
52Now, let $f\colon A\to B$ be a *local monomorphism*, i.e. there is an epimorphism
53$U\to *$ such that the pullback $f\times U$ of $f$ to $U$ is a
54monomorphism. Then, since epimorphisms are stable under pullback, it follows that
55in the commutative square
56$$
57\begin{tikzcd}
58A\times U \ar[into, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
59A \ar[r, "f"'] & B
60\end{tikzcd}
61$$
62both vertical maps are epimorphisms. Now let $\varphi,\psi\colon T\to A$ be a pair of morphisms
63such that $f\varphi = f\psi$. Then, denoting by $\varphi_U$ and $\psi_U$ the
64pullbacks to $U$, we have $f_U\varphi_U = f_U\psi_U$. but $f_U$ is a
65monomorphism by assumption, so $\varphi_U = \psi_U$. So we have a commutative
66diagram
67$$
68\begin{tikzcd}
69T\times U \ar[r, "\varphi_U = \psi_U"] \ar[onto, d] & A\times U \ar[onto, d] \\
70T \ar[r, "\varphi", shift left] \ar[r, "\psi"', shift right] & A
71\end{tikzcd}
72$$
73in which the vertical maps are epimorphisms. It follows that $\varphi =
74\psi$.
75
76Now, if $f$ is a local epimorphism, then again we have the diagram
77$$
78\begin{tikzcd}
79A\times U \ar[onto, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
80A \ar[r, "f"'] & B
81\end{tikzcd}
82$$
83and it is immediate that $f$ is an epimorphism. In summary:
84
85<prop>
86In a topos, any local epimorphism is an epimorphism, any local monomorphism is a
87monomorphism, and any local isomorphism is an isomorphism.
88</prop>
89
90Now, let's check that $G$--torsors $X$ as defined above are free and
91transitive. Take an epimorphism $U\onto *$ such that $X\times U$ is trivial in
92$\ca C/U$. The action of $G$ on itself by left multiplication is plainly free
93and transitive, so in $\ca C/U$ we have the isomorphism
94$$(G\times U)\times_U (X\times U) \iso (X\times U)
95\times_U (X\times U)$$
96and $(G\times U)\times_U(X\times U) = (G\times X)\times U$ and $(X\times
97U)\times_U (X\times U)$ because pullback preserves products. So, $G\times X\to
98X\times X$ is a local isomorphism, hence an isomorphism.
99
100[^1]: Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4