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 -set with action such that every choice of point induces an isomorphism of -sets
This says equivalently that after picking any point of as the identity , acquires a group structure isomorphic to . But this is a non-canonical isomorphism: every choice of point of 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 . More generally, one may consider torsors over some base space (in other words, working in the topos of sheaves over instead of ). In this case the term -torsor is often used more or less a synonym for the term -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 ‘-principal bundle’ is usually a (topological) group or groupoid, when we say ‘-torsor’, is usually a presheaf or sheaf of group(oid)s, or is a plain category (not necessarily even a groupoid).
A -torsor, without any base space given, can also simply be an inhabited transitive free -set, which is the same as a principal -bundle over the point. The notion may also be defined in any category with products: a torsor over a group object is a well-supported object together with a -action such that the arrow
is an isomorphism.
Let be a group object in some category , that in the following is assumed, for simplicity, to be a cartesian monoidal category. The objects of we sometimes call spaces. Examples to keep in mind are Set (in which case is an ordinary group) or Top (in which case it is a topological group) or Diff (in which case it is a Lie group).
is an isomorphism.
A left -torsor over is a -torsor in .
Thus, if is a point, a torsor over a point is the same as an ordinary torsor in , 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 , as before.
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 of (so that is epi, i.e., is inhabited) such that the torsor, when pulled back to , becomes trivial (i.e., isomorphic to as -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 such that is inhabited (e.g., an open cover of a space seen as the terminal object of the sheaf topos ), and “torsor” would then refer to the local triviality condition for some such . 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 is a point (left as a standard exercise).)
A unit of measurement is (typically) an element in an -torsor, for the multiplicative group of non-zero real numbers: for any unit and any non-vanishing real number, also is a unit. And for and two units, one is expressed in terms of the other by a unique as . For instance for units of mass we have the unit of kilogram and that of gram and there is a unique number, with
A topological -principal bundle is an example of a torsor over in . This becomes a definition of principal bundle if we demand local triviality with respect to some open cover of (see the remarks below).
since is inhabited (here meaning non-empty), we may pick an point of . Write for this morphism, for emphasis. One sees that the diagram
In other language, we say is trivial if it is isomorphic to as -torsor, and a choice of isomorphism such as is a trivialization. Notice that the composite
can be interpreted as “division” , dividing one element of by another to get an element of . If we further compose division with a choice of trivialization,
then we get a division structure on for which behaves as an identity (i.e., for all ), so that acquires a group structure isomorphic to that of .
In other categories besides , we cannot just “pick a point” of even if 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 if it becomes true when reinterpreted in a slice after pulling back , where is inhabited. (This in some sense is the basis of Kripke-Joyal semantics.)
In the present case, we may take . Although we cannot “pick a point” of (= global section of ), we can pick a point of if we reinterpret it by pulling back to . In other words, does have a global section regarded as an arrow in . In fact, there is a “generic point”: the diagonal . Then, we may mimic the argument above, and consider the pullback diagram
living in . As argued above, the vertical arrow on the left is an isomorphism; in fact, it is the isomorphism we started with!
Thus, a -torsor in a category with products can be tautologically interpreted in terms of -actions on objects which become trivialized upon pulling back to the slice .
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
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 -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 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.