geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
A torsor (in the category of sets) is, roughly speaking, a group that has forgotten its identity element; given any (non-empty) torsor with respect to a group , we recover a group isomorphic to by making what is known as a trivialisation of the torsor, which roughly corresponds to choosing an identity element. That we wish to keep track of the choice is precisely the reason for working with torsors.
Something analogous is present in the theory of fibrations, where it can be important to make a choice of lifts (‘cloven fibrations’).
The notion of a torsor can be internalised to any category with products, and more generally to any category in which the notion of an internal group can be made good sense of. We discuss this general notion below, after first discussing the notion in the category of sets.
Let be a group. A -torsor is a set together with an action of on such that the shear map is an isomorphism, where is the canonical projection map .
If here is not required to be inhabited (possibly empty) one also speaks of a pseudo-torsor.
If is non-empty, we shall prove below that it follows from the definition that is isomorphic to the underlying set of . There are many such isomorphisms, and where torsors are used it is often important to choose/fix one. Such a choice is known as a trivialisation of . See below for more details.
As a consequence of Remark , a torsor with respect to some group can be thought of as a heap.
Asking that the shear map be an isomorphism is the same as to say that the action is both free and transitive, hence regular if inhabited: free-ness corresponds to injectivity of the shear map, and transitivity corresponds to surjectivity of the shear map.
An affine space of dimension over a field is a torsor for the additive group : this acts by translation.
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
Let be a group. The action of on itself equips the underlying set of with the structure of a -torsor.
Given two isomorphic objects and in any category , all isomorphisms between and form a torsor (both for and for , which are mutually isomorphic but not canonically). This is an insight used in (M. Kontsevich, Operads and motives in deformation quantization, Lett.Math.Phys.48:35-72 (1999) arXiv:math/9904055 doi) explaining period matrices from the point of view of a coordinate ring of an affine torsor.
It is possible to define torsors using a single-sorted algebraic theory. This is entirely analogous to how affine spaces can be defined either as sets with a free and transitive action of a vector space, or, equivalently, as sets equipped with operations that take arbitrary affine combinations with coefficients in a given ring.
More precisely, a torsor (also known as a heap when stated in a single-sorted form) is a set equipped with a ternary operation
such that
A homomorphism of torsors is a map of sets that preserves this operation.
The equivalence with the two-sorted definition is demonstrated as follows.
Given a -torsor , we send it to the set equipped with the ternary operation , where is the unique element of such that .
Given a torsor , we send it to the pair , where is a subgroup of the group of bijections on the set comprising precisely the bijections of the form for some elements . The group acts on by evaluation: .
Mapping gives back the same torsor that we started from.
Mapping produces a torsor that is naturally isomorphic to via the isomorphism
Let be a group, and let be a -torsor. If is non-empty, it is isomorphic to the underlying set of .
Let be an element of . The following diagram is cartesian.
Since is a -torsor, we have that is an isomorphism. The proposition thus follows immediately from the fact that pullbacks of isomorphisms are isomorphisms (as proven at pullback).
Let be a group, and let be a -torsor. A trivialisation of is an isomorphism between and the underlying set of .
By Proposition , if is non-empty, is always isomorphic to the underlying set of . Thus the notion of a trivialisation of is a question of a choice of isomorphism between and the underlying set of .
The proof of Proposition shows that any choice of element of gives rise to a trivialisation. Some of these may of course coincide.
Let be a trivialisation of a torsor . The map
where is the canonical map, can be interpreted as a notion of division , ‘dividing’ one element of by another to obtain an element of . If we further compose with
we can think of the resulting map as a ‘division structure’ on for which behaves as an identity, namely such that for all . In this way, acquires a group structure isomorphic to that of .
Thus a trivialisation of a torsor equips with it a choice of group structure amongst all of those isomorphic to .
Torsors can be transported, or in other words pushed forward, along group homomorphisms, as we shall now show. (See also at G-space – change of structure group).
Let be a group homomorphism, and let be a -torsor. Observe that defines an action of on . Then , the quotient of with respect to this -action, defines a -torsor with respect to the action of induced by left multiplication, namely that given by , where , for some , denotes the orbit of with respect to the action of on .
We shall demonstrate that the map
given by
induces a map
That is, it respects, in its left factor, the passage to the quotient by the action of .
Suppose indeed that we have and such that . We make the following observations.
Putting these together, we obtain that
as required.
We now observe that defines an inverse to , where is the action of on induced by left multiplication, and is the projection map . In one direction, for any and , we have that
as required. In the other direction, suppose that we have . Since the map is an isomorphism, there is a such that , and we have that . We then have that applied to
is
as required.
Let be a group homomorphism, and let be a -torsor. We refer to the -torsor constructed from using as in Proposition as the torsor obtained from by change of structure group, and denote it .
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).
A left -torsor is an inhabited object equipped with a -action, (subject to the usual laws for actions) such that the map
is an isomorphism.
More generally, suppose is finitely complete, and let be an object. Then the slice is finitely complete, and the pullback functor preserves finite limits. Thus acquires a group structure in .
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.
A left -torsor over is a bundle over 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 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).)
Let Top, so that all objects are topological spaces and groups are topological groups.
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).
Let be a category of sheaves over a site .
The canonical example for a torsor in is the trivial torsor over a sheaf of groups, .
Every group extension canonically equips with the structure of an -torsor over . See Group extensions as torsors for details
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 .
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.
Elementary exposition:
John Baez, Torsors made easy, (web)
(discussion for discrete groups)
Textbook accounts:
James Milne, Prop. III.4.1 in: Étale Cohomology, Princeton Mathematical Series 33 (1980) [jstor:j.ctt1bpmbk1, ISBN:9780691082387]
(discussion for algebraic groups in algebraic geometry)
For more see the references at principal bundle (which are torsors in the generality internal to slice categories).
A general topos theoretic account is in
See also the references at Diaconescu's theorem.
Discussion in homotopy type theory/univalent mathematics:
Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson, Def. 4.8.1 in: Symmetry [pdf]
David Wärn, Eilenberg-MacLane spaces and stabilisation in homotopy type theory [arXiv:2301.03685]
Some further category theoretic articles discussing torsors:
Tomasz Brzeziński, 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 -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.
Much further material is also in Giraud’s book on nonabelian cohomology.
In a model theoretic context of definable sets, principal homogeneous spaces are studied in
See also
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