group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $G$ a group, a $G$-torsor (also called a principal homogeneous space) is an inhabited object/space $P$ with an action $\rho : G \times P \to P$ by $G$ that is
and
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
This says equivalently that 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 slogan we can summarize this as: 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 $G$-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 locally trivial . Thus, while the terminology ‘principal bundle’ is usually used in the setting of topological spaces or smooth manifolds, the term torsor is traditionally used in the more general contex of Grothendieck topologies (faithfully flat and étale topology in particular), 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 $G$-torsor, without any base space given, can also simply be an inhabited transitive free $G$-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
is an isomorphism.
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 objects of $C$ we sometimes call spaces. 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).
A left $G$-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
is an isomorphism.
More generally, suppose $C$ is finitely complete, and let $B$ be an object. Then the 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 $C/B$.
A left $G$-torsor over $B$ is a $G$-torsor in $C/B$.
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.
A left $G$-torsor over $B \in C$ is a bundle $P\stackrel{\pi}{\to} B$ over $B$ together with a left group action
which in terms of generalized elements we write
such that the induced morphism of products
which on elements acts as
is an isomorphism.
As we explain below, a torsor is in some tautological sense 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).)
Let $C =$ Set.
An affine space of dimension $n$ over a field $k$ is a torsor for the additive group $k^n$: this acts by translation.
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 numbers: 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
Let $C =$ Top, so that all objects are topological spaces and groups $G$ are topological groups.
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 below).
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, $G$.
(…)
Every group extension $A \to \hat G \to G$ canonically equips $\hat G$ with the structure of an $A$-torsor over $G$. See Group extensions as torsors for details
Let $P$ be a $G$-torsor over the point in the category $C =$ Set. Then as objects of $C$, $P$ is isomorphic to $G$:
since $P$ is 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
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 trivial if it is isomorphic to $G$ as $G$-torsor, and a choice of isomorphism such as $\rho(-, p): G \to P$ is a trivialization. Notice that the composite
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,
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 $G$.
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
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 $C/P$.
Instead of a torsor over a group, one can consider a torsor over a category. See torsor with structure category.
In noncommutative algebraic geometry, faithfully flat Hopf-Galois extensions are considered a generalization of (affine) torsors in algebraic geometry.
For elementary examples of torsors over the point in Set , see:
A general topos-theoretic account is in section B3.2 of
See also the references at Diaconescu's theorem.
Some categorically-oriented articles discussing torsors are
Tomasz Brzezi?ski?, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arXiv:0912.0213
D. H. Van Osdol, Principal homogeneous objects as representable functors, Cahiers Topologie Géom. Différentielle 18 (1977), no. 3, 271–289, numdam
K. T. S. Mohapeloa, A $2$-colimit characterization of internal categories of torsors, J. Pure Appl. Algebra 71 (1991), no. 1, 75–91, doi
Thomas Booker, Ross Street, Torsors, herds and flocks, arXiv:0912.4551
J. Duskin, Simplicial methods and the interpretation of ‘triple’ cohomology, Memoirs AMS 3, issue 2, n° 163, 1975. MR393196
A. Vistoli, Grothendieck topologies, fibered categories and descent theory, in: FGA explained, 1–104, Math. Surveys Monogr., 123, AMS 2005, math.AG/0412512
Ieke Moerdijk, Introduction to the language of stacks and gerbes, math.AT/0212266.
A standard elementary discussion of torsors in algebraic geometry is in J. Milne’s book Etale cohomology. Much material is also in Giraud’s book on nonabelian cohomology.
MathOverflow: torsors-for-monoids