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torsor

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Idea

For GG a group, a GG-torsor (also called a principal homogeneous space) is an inhabited object/space PP with an action ρ:G×PP\rho : G \times P \to P by GG that is

  • free: only the identity element acts trivially;

and

  • transitive: for every two points in (a fiber of) the space, there is an element of the group taking one to the other.

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 GG-set PP with action ρ:G×PP\rho: G \times P \to P such that every choice of point pPp \in P induces an isomorphism of GG-sets

ρ(,p):GP. \rho(-,p) : G \stackrel{\simeq}{\to} P \,.

This says equivalently that after picking any point of PP as the identity , PP acquires a group structure isomorphic to GG. But this is a non-canonical isomorphism: every choice of point of PP 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 SetSet. More generally, one may consider torsors over some base space BB (in other words, working in the topos of sheaves over BB instead of SetSet). In this case the term GG-torsor is often used more or less a synonym for the term GG-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 ‘GG-principal bundle’ GG is usually a (topological) group or groupoid, when we say ‘GG-torsor’, GG is usually a presheaf or sheaf of group(oid)s, or GG is a plain category (not necessarily even a groupoid).

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

π 1,α:G×EE×E\langle \pi_1, \alpha \rangle: G \times E \to E \times E

is an isomorphism.

Definition

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

Definition

A left GG-torsor is an inhabited object PP equipped with a GG-action, ρ:G×PP\rho: G \times P \to P (subject to the usual laws for actions) such that the map

ρ,π 2:G×PP×P\langle \rho, \pi_2 \rangle: G \times P \to P \times P

is an isomorphism.

More generally, suppose CC is finitely complete, and let BB be an object. Then the slice C/BC/B is finitely complete, and the pullback functor ×B:CC/B- \times B: C \to C/B preserves finite limits. Thus π 2:G×BB\pi_2: G \times B \to B acquires a group structure in C/BC/B.

Definition

A left GG-torsor over BB is a GG-torsor in C/BC/B.

Thus, if B=1B = 1 is a point, a torsor over a point is the same as an ordinary torsor in CC, 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 CC, as before.

Equivalent definition

A left GG-torsor over BCB \in C is a bundle PπBP\stackrel{\pi}{\to} B over BB together with a left group action

ρ:G× BPP \rho : G\times_B P \to P

which in terms of generalized elements we write

(g,p)g.p (g,p)\to g.p

such that the induced morphism of products

ϕ:=(ρ,p 2):G× BPP× BP \phi := (\rho, p_2) : G\times_B P \to P\times_B P

which on elements acts as

(g,p)(g.p,p) (g,p)\to (g.p,p)

is an isomorphism.

Remark

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 UU of 11 (so that U1U \to 1 is epi, i.e., UU is inhabited) such that the torsor, when pulled back to UU, becomes trivial (i.e., isomorphic to GG as GG-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 i1U_i \hookrightarrow 1 such that U= iU iU = \sum_i U_i is inhabited (e.g., an open cover of a space BB seen as the terminal object of the sheaf topos Sh(B)Sh(B)), and “torsor” would then refer to the local triviality condition for some such UU. 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 BB is a point (left as a standard exercise).)

Examples

In sets

Let C=C = Set.

  • An affine space of dimension nn over a field kk is a torsor for the additive group k nk^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 uu any unit and rr \in \mathbb{R} any non-vanishing real number, also rur u is a unit. And for u 1u_1 and u 2u_2 two units, one is expressed in terms of the other by a unique r0r \neq 0 as u 1=ru 2u_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=1000r = 1000 with

    kg=1000g. kg = 1000 g \,.

In topological spaces

Let C=C = Top, so that all objects are topological spaces and groups GG are topological groups.

A topological GG-principal bundle π:PB\pi: P \to B is an example of a torsor over BB in TopTop. This becomes a definition of principal bundle if we demand local triviality with respect to some open cover of BB (see the remarks below).

In sheaves

Let C=Sh(S)C = Sh(S) be a category of sheaves over a site SS.

The canonical example for a torsor in CC is the trivial torsor over a sheaf of groups, GG.

(…)

Group extensions

Every group extension AG^GA \to \hat G \to G canonically equips G^\hat G with the structure of an AA-torsor over GG. See Group extensions as torsors for details

Properties

Torsors in SetSet

Let PP be a GG-torsor over the point in the category C=C = Set. Then as objects of CC, PP is isomorphic to GG:

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

GG×{p} (Id,p) G×P ρ(,p) ρ,π 2 P×{p} (Id,p) P×P \array{ 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 }

is a pullback diagram. But since ρ\rho is by assumption an isomorphism, and since pullbacks of isomorphisms are isomorphisms, also ρ(,p):GP\rho(-,p) : G \to P is an isomorphism.

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

P×Pρ 1G×Pπ 1GP \times P \stackrel{\rho^{-1}}{\to} G \times P \stackrel{\pi_1}{\to} G

can be interpreted as “division” d:P×PGd: P \times P \to G, dividing one element of PP by another to get an element of GG. If we further compose division with a choice of trivialization,

P×PdGρ(,p)P,P \times P \stackrel{d}{\to} G \stackrel{\rho(-, p)}{\to} P,

then we get a division structure DD on PP for which pp behaves as an identity (i.e., D(x,x)=pD(x, x) = p for all xPx \in P), so that PP acquires a group structure isomorphic to that of GG.

Local trivialization

In other categories CC besides SetSet, we cannot just “pick a point” of PP even if P1P \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 CC if it becomes true when reinterpreted in a slice after pulling back CC/UC \to C/U, where UU is inhabited. (This in some sense is the basis of Kripke-Joyal semantics.)

In the present case, we may take U=PU = P. Although we cannot “pick a point” of PP (= global section of P1P \to 1), we can pick a point of PP if we reinterpret it by pulling back to C/PC/P. In other words, π 2:P×P1×PP\pi_2: P \times P \to 1 \times P \cong P does have a global section regarded as an arrow in C/PC/P. In fact, there is a “generic point”: the diagonal Δ:PP×P\Delta: P \to P \times P. Then, we may mimic the argument above, and consider the pullback diagram

G×P G×P×P ρ,π 2×id P×P id×Δ P×P×P\array{ 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 }

living in C/PC/P. As argued above, the vertical arrow on the left is an isomorphism; in fact, it is the isomorphism ρ,π 2:G×PP×P\langle \rho, \pi_2 \rangle: G \times P \to P \times P we started with!

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

Generalizations

References

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 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 22-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

Revised on November 22, 2013 00:13:49 by Urs Schreiber (77.251.114.72)