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\begin{document}

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\section*{torsor}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_sets}{In sets}\dotfill \pageref*{in_sets} \linebreak
\noindent\hyperlink{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \linebreak
\noindent\hyperlink{in_sheaves}{In sheaves}\dotfill \pageref*{in_sheaves} \linebreak
\noindent\hyperlink{GroupExtensions}{Group extensions}\dotfill \pageref*{GroupExtensions} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{torsors_in_}{Torsors in $Set$}\dotfill \pageref*{torsors_in_} \linebreak
\noindent\hyperlink{LocalTrivialization}{Local trivialization}\dotfill \pageref*{LocalTrivialization} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $G$ a [[group]], a $G$-\textbf{torsor} (also called a \textbf{principal homogeneous space}) is an [[inhabited object]]/[[space]] $P$ with an [[action]] $\rho : G \times P \to P$ by $G$ that is

\begin{itemize}%
\item [[free action|free]]: only the identity element acts with fixed points;

\end{itemize}
and

\begin{itemize}%
\item [[transitive action|transitive]]: for every two points in (a fiber of) the space, there is an element of the group taking one to the other.

\end{itemize}
The second axiom says that $\langle \rho, \pi_2 \rangle: G \times P \to P \times P$ is surjective, and the first says it is injective.

In other words, in the classical case where we are working in the category of sets over the point, a torsor is a [[heap]]:

a $G$-set $P$ with action $\rho: G \times P \to P$ such that every choice of point $p \in P$ induces an isomorphism of $G$-sets

\begin{displaymath}
\rho(-,p) : G \stackrel{\simeq}{\to} P
  \,.
\end{displaymath}
This says equivalently that \emph{after picking any point of $P$ as the identity} , $P$ acquires a group structure isomorphic to $G$. But this is a non-canonical isomorphism: every choice of point of $P$ yields a different isomorphism.

As a \textbf{slogan} we can summarize this as: \emph{A torsor is like a group that has forgotten its neutral element.}

Again, this applies to torsors ``over the point'' in $Set$. More generally, one may consider torsors over some base space $B$ (in other words, working in the [[topos]] of sheaves over $B$ instead of $Set$). In this case the term \textbf{$G$-torsor} is often used more or less a synonym for the term $G$-[[principal bundle]], but torsors are generally understood in contexts much wider than the term ``principal bundle'' is usually taken to apply. And a principal bundle is strictly speaking a torsor that is required to be \emph{locally trivial} . Thus, while the terminology `principal bundle' is usually used in the setting of [[topological spaces]] or [[smooth manifold]]s, the term \emph{torsor} is traditionally used in the more general contex of [[Grothendieck topology|Grothendieck topologies]] (faithfully flat and \'e{}tale topology in particular), [[topos|topoi]] and for generalizations in various category-theoretic setups. While in the phrase `$G$-principal bundle' $G$ is usually a (topological) [[group]] or [[groupoid]], when we say `$G$-torsor', $G$ is usually a [[presheaf]] or [[sheaf]] of group(oid)s, or $G$ is a plain [[category]] (not necessarily even a groupoid).

A \textbf{$G$-torsor}, without any base space given, can also simply be an inhabited transitive free $G$-[[action|set]], which is the same as a principal $G$-bundle over the [[point]]. The notion may also be defined in any category with products: a torsor over a [[group object]] $G$ is a [[well-supported object]] $E$ together with a $G$-action $\alpha: G \times E \to E$ such that the arrow

\begin{displaymath}
\langle \pi_1, \alpha \rangle: G \times E \to E \times E
\end{displaymath}
is an [[isomorphism]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $G$ be a [[group]] object in some [[category]] $C$, that in the following is assumed, for simplicity, to be a [[cartesian monoidal category]]. The [[object]]s of $C$ we sometimes call [[space]]s. Examples to keep in mind are $C =$ [[Set]] (in which case $G$ is an ordinary [[group]]) or [[Top]] (in which case it is a [[topological group]]) or [[Diff]] (in which case it is a [[Lie group]]).

\begin{udefn}
A left \textbf{$G$-torsor} is an [[inhabited object]] $P$ equipped with a $G$-[[action]], $\rho: G \times P \to P$ (subject to the usual laws for actions) such that the map

\begin{displaymath}
\langle \rho, \pi_2 \rangle: G \times P \to P \times P
\end{displaymath}
is an [[isomorphism]].

\end{udefn}
More generally, suppose $C$ is [[finitely complete category|finitely complete]], and let $B$ be an object. Then the [[slice category|slice]] $C/B$ is finitely complete, and the pullback functor $- \times B: C \to C/B$ preserves finite limits. Thus $\pi_2: G \times B  \to B$ acquires a group structure in $C/B$.

\begin{udefn}
A left \textbf{$G$-torsor over $B$} is a $G$-torsor in $C/B$.

\end{udefn}
Thus, if $B = 1$ is a point, a torsor over a point is the same as an ordinary torsor in $C$, but sometimes the additional ``over a point'' is convenient for the sake of emphasis.

We restate this definition equivalently in more nuts-and-bolts terms. The ambient category is $C$, as before.

\begin{udefn}
A left $G$-torsor over $B \in C$ is a [[bundle]] $P\stackrel{\pi}{\to} B$ over $B$ together with a left group [[action]]

\begin{displaymath}
\rho : G\times_B P \to P
\end{displaymath}
which in terms of [[generalized element]]s we write

\begin{displaymath}
(g,p)\to g.p
\end{displaymath}
such that the induced morphism of [[product]]s

\begin{displaymath}
\phi := (\rho, p_2) : G\times_B P \to P\times_B P
\end{displaymath}
which on elements acts as

\begin{displaymath}
(g,p)\to (g.p,p)
\end{displaymath}
is an [[isomorphism]].

\end{udefn}
\begin{uremark}
As we explain \hyperlink{LocalTrivialization}{below}, a torsor is in some tautological sense \textbf{locally trivial}, but some care must be taken in interpreting this. One sense is that there is a cover $U$ of $1$ (so that $U \to 1$ is epi, i.e., $U$ is inhabited) such that the torsor, when pulled back to $U$, becomes trivial (i.e., isomorphic to $G$ as $G$-torsor). But this is a very general notion of ``cover''. A more restrictive sense frequently encountered in the literature is that ``cover'' means a coproduct of subterminal objects $U_i \hookrightarrow 1$ such that $U = \sum_i U_i$ is inhabited (e.g., an open cover of a space $B$ seen as the terminal object of the sheaf topos $Sh(B)$), and ``torsor'' would then refer to the local triviality condition for some such $U$. This is the more usual sense when referring to principal bundles as torsors. Or, ``cover'' could refer to a covering sieve in a [[Grothendieck topology]].

(The condition on the action can be translated to give transitivity etc. in the case of $B$ is a point (left as a standard exercise).)

\end{uremark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_sets}{}\subsubsection*{{In sets}}\label{in_sets}

Let $C =$ [[Set]].

\begin{itemize}%
\item An [[affine space]] of dimension $n$ over a [[field]] $k$ is a torsor for the additive group $k^n$: this acts by \emph{translation}.


\item A [[unit of measurement]] is (typically) an element in an $\mathbb{R}^\times$-torsor, for $\mathbb{R}^\times$ the multiplicative group of non-zero [[real number]]s: for $u$ any unit and $r \in \mathbb{R}$ any non-vanishing real number, also $r u$ is a unit. And for $u_1$ and $u_2$ two units, one is expressed in terms of the other by a unique $r \neq 0$ as $u_1 = r u_2$. For instance for units of [[mass]] we have the unit of [[kilogram]] and that of gram and there is a unique number, $r = 1000$ with

\begin{displaymath}
kg = 1000 g
  \,.
\end{displaymath}


\end{itemize}
\hypertarget{in_topological_spaces}{}\subsubsection*{{In topological spaces}}\label{in_topological_spaces}

Let $C =$ [[Top]], so that all objects are [[topological space]]s and groups $G$ are [[topological group]]s.

A topological $G$-[[principal bundle]] $\pi: P \to B$ is an example of a torsor over $B$ in $Top$. This becomes a definition of principal bundle if we demand local triviality with respect to some open cover of $B$ (see the remarks \hyperlink{LocalTrivialization}{below}).

\hypertarget{in_sheaves}{}\subsubsection*{{In sheaves}}\label{in_sheaves}

Let $C = Sh(S)$ be a [[category of sheaves]] over a [[site]] $S$.

The canonical example for a torsor in $C$ is the [[trivial torsor]] over a [[sheaf]] of groups, $G$.

(\ldots{})

\hypertarget{GroupExtensions}{}\subsubsection*{{Group extensions}}\label{GroupExtensions}

Every [[group extension]] $A \to \hat G \to G$ canonically equips $\hat G$ with the structure of an $A$-torsor over $G$. See  for details

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{torsors_in_}{}\subsubsection*{{Torsors in $Set$}}\label{torsors_in_}

Let $P$ be a $G$-torsor over the point in the category $C =$ [[Set]]. Then as objects of $C$, $P$ is [[isomorphism|isomorphic]] to $G$:

since $P$ is [[inhabited set|inhabited]] (here meaning non-empty), we may pick an point $p : * \to P$ of $P$. Write $\{p\} \to P$ for this morphism, for emphasis. One sees that the diagram

\begin{displaymath}
\itexarray{
  G \simeq G \times \{p\} &\stackrel{(Id, p)}{\to}& G \times P
  \\
  \downarrow^{\mathrlap{\rho(-,p)}} && \downarrow^{\mathrlap{\langle \rho, \pi_2}}
  \\
  P \times \{p\} &\stackrel{(Id,p)}{\to}& P \times P
  }
\end{displaymath}
is a [[pullback]] diagram. But since $\rho$ is by assumption an [[isomorphism]], and since pullbacks of isomorphisms are isomorphisms, also $\rho(-,p) : G \to P$ is an isomorphism.

In other language, we say $P$ is \textbf{trivial} if it is isomorphic to $G$ as $G$-torsor, and a choice of isomorphism such as $\rho(-, p): G \to P$ is a \textbf{trivialization}. Notice that the composite

\begin{displaymath}
P \times P \stackrel{\rho^{-1}}{\to} G \times P \stackrel{\pi_1}{\to} G
\end{displaymath}
can be interpreted as ``division'' $d: P \times P \to G$, dividing one element of $P$ by another to get an element of $G$. If we further compose division with a choice of trivialization,

\begin{displaymath}
P \times P \stackrel{d}{\to} G \stackrel{\rho(-, p)}{\to} P,
\end{displaymath}
then we get a division structure $D$ on $P$ for which $p$ behaves as an identity (i.e., $D(x, x) = p$ for all $x \in P$), so that $P$ acquires a group structure isomorphic to that of $G$.

\hypertarget{LocalTrivialization}{}\subsubsection*{{Local trivialization}}\label{LocalTrivialization}

In other categories $C$ besides $Set$, we cannot just ``pick a point'' of $P$ even if $P \to 1$ is an [[epimorphism]], so this argument cannot be carried out, and indeed trivializations may not exist. However, it is possible to construct a local trivialization of a torsor, following a general philosophy from [[topos theory]] that a statement is ``locally true'' in a category $C$ if it becomes true when reinterpreted in a slice after pulling back $C \to C/U$, where $U$ is inhabited. (This in some sense is the basis of [[Kripke-Joyal semantics]].)

In the present case, we may take $U = P$. Although we cannot ``pick a point'' of $P$ (= global section of $P \to 1$), we can pick a point of $P$ if we reinterpret it by pulling back to $C/P$. In other words, $\pi_2: P \times P \to 1 \times P \cong P$ does have a global section regarded as an arrow in $C/P$. In fact, there is a ``generic point'': the diagonal $\Delta: P \to P \times P$. Then, we may mimic the argument above, and consider the pullback diagram

\begin{displaymath}
\itexarray{
G \times P & \to & G \times P \times P \\
\downarrow & & \downarrow \mathrlap{\langle \rho, \pi_2 \rangle \times id} \\
P \times P & \underset{id \times \Delta}{\to} & P \times P \times P
}
\end{displaymath}
living in $C/P$. As argued above, the vertical arrow on the left is an isomorphism; in fact, it is the isomorphism $\langle \rho, \pi_2 \rangle: G \times P \to P \times P$ we started with!

Thus, a $G$-torsor in a category with products can be tautologically interpreted in terms of $G$-actions on objects $P$ which become trivialized upon pulling back to the slice $C/P$.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\begin{itemize}%
\item Instead of a torsor over a group, one can consider a torsor over a [[category]]. See [[torsor with structure category]].


\item In [[noncommutative algebraic geometry]], faithfully flat [[Hopf-Galois extension]]s are considered a generalization of (affine) torsors in algebraic geometry.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[homogeneous space]]


\item [[principal bundle]] / [[associated bundle]]


\item [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]


\item [[principal 3-bundle]] / [[bundle 2-gerbe]]


\item [[principal ∞-bundle]] / [[associated ∞-bundle]]


\item [[descent along a torsor]], [[Schneider's descent theorem]]


\item [[Hopf-Galois extension]], [[quantum homogeneous space]],


\item [[noncommutative principal bundle]], [[quantum heap]]


\item [[physical unit]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For elementary examples of torsors over the point in [[Set]] , see:

\begin{itemize}%
\item [[John Baez]], \emph{Torsors made easy}, (\href{http://math.ucr.edu/home/baez/torsors.html}{web})

\end{itemize}
A general [[topos]]-theoretic account is in section B3.2 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]} .

\end{itemize}
See also the references at \emph{[[Diaconescu's theorem]]}.

Some categorically-oriented articles discussing torsors are

\begin{itemize}%
\item [[Tomasz Brzeziński]], \emph{On synthetic interpretation of quantum principal bundles}, AJSE D - Mathematics 35(1D): 13-27, 2010 \href{http://arxiv.org/abs/0912.0213}{arXiv:0912.0213}


\item D. H. Van Osdol, \emph{Principal homogeneous objects as representable functors}, Cahiers Topologie G\'e{}om. Diff\'e{}rentielle 18 (1977), no. 3, 271--289, \href{http://www.numdam.org/item?id=CTGDC_1977__18_3_271_0}{numdam}


\item K. T. S. Mohapeloa, \emph{A $2$-colimit characterization of internal categories of torsors}, J. Pure Appl. Algebra 71 (1991), no. 1, 75--91, \href{http://dx.doi.org/10.1016/0022-4049%2891%2990041-Y}{doi}


\item Thomas Booker, Ross Street, \emph{Torsors, herds and flocks}, \href{http://arxiv.org/abs/0912.4551}{arXiv:0912.4551}


\item J. Duskin, \emph{Simplicial methods and the interpretation of `triple' cohomology}, Memoirs AMS \textbf{3}, issue 2, n\textdegree{} 163, 1975. MR393196


\item A. Vistoli, \emph{Grothendieck topologies, fibered categories and descent theory}, in: [[FGA explained]], 1--104, Math. Surveys Monogr., 123, AMS 2005, \href{http://front.math.ucdavis.edu/0412.5512}{math.AG/0412512}


\item [[Ieke Moerdijk]], \emph{Introduction to the language of stacks and gerbes}, \href{http://arxiv.org/abs/math/0212266}{math.AT/0212266}.



\end{itemize}
A standard elementary discussion of torsors in algebraic geometry is in J. Milne's book \emph{Etale cohomology}. Much material is also in Giraud's book on nonabelian cohomology.

MathOverflow: \href{http://mathoverflow.net/questions/25863/torsors-for-monoids/25886}{torsors-for-monoids}

[[!redirects torsors]]

[[!redirects principal homogeneous space]] [[!redirects principal homogeneous spaces]]



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