nLab
torsor

Context

Representation theory

Bundles

Contents

Idea

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 GG, we recover a group isomorphic to GG 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.

In the category of sets

Two-sorted definition

Definition

Let GG be a group. A GG-torsor is a set TT together with an action a:G×TTa: G \times T \rightarrow T of GG on TT such that the shear map a×p 2:G×TT×Ta \times p_2: G \times T \rightarrow T \times T is an isomorphism, where p 2p_2 is the canonical projection map G×TTG \times T \rightarrow T.

If here TT is not required to be inhabited (possibly empty) one also speaks of a pseudo-torsor.

Remark

If TT is non-empty, we shall prove below that it follows from the definition that TT is isomorphic to the underlying set of GG. 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 TT. See below for more details.

Remark

As a consequence of Remark , a torsor with respect to some group can be thought of as a heap.

Remark

Asking that the shear map a×p 2a \times p_2 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 a×p 2a \times p_2, and transitivity corresponds to surjectivity of the shear map.

Example

An affine space of dimension nn over a field kk is a torsor for the additive group k nk^n: this acts by translation.

Example

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.

Example

Let GG be a group. The action of GG on itself equips the underlying set of GG with the structure of a GG-torsor.

Remark

We shall see in Remark that all torsors actually arise as in Example .

Example

Given two isomorphic objects XX and YY in any category CC, all isomorphisms between XX and YY form a torsor (both for Aut(X)Aut(X) and for Aut(Y)Aut(Y), 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.

Single-sorted definition

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 TT equipped with a ternary operation

t:T 3Tt\colon T^3 \to T

such that

t(a,a,b)=t(b,a,a)=b,t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).t(a,a,b)=t(b,a,a)=b,\qquad t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).

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 GG-torsor TT, we send it to the set TT equipped with the ternary operation t(a,b,c)=g(a,b)ct(a,b,c)=g(a,b)c, where g(a,b)g(a,b) is the unique element of GG such that g(a,b)b=ag(a,b)b=a.

Given a torsor (T,t)(T,t), we send it to the pair (LTrans(T),T)(LTrans(T),T), where LTrans(T)LTrans(T) is a subgroup of the group of bijections on the set TT comprising precisely the bijections of the form ct(a,b,c)c\mapsto t(a,b,c) for some elements a,bTa,b\in T. The group GG acts on TT by evaluation: gt=g(t)gt=g(t).

Mapping (T,t)(LTrans(T),T)(T,t)(T,t)\mapsto (LTrans(T),T)\mapsto (T,t) gives back the same torsor (T,t)(T,t) that we started from.

Mapping (G,T)(T,t)(LTrans(T),t)(G,T)\mapsto (T,t)\mapsto (LTrans(T),t) produces a torsor (LTrans(T),t)(LTrans(T),t) that is naturally isomorphic to (G,T)(G,T) via the isomorphism

(G,T)(LTrans(T),T),g(tgt),tt.(G,T)\to(LTrans(T),T),\qquad g\mapsto (t\mapsto gt),\qquad t\mapsto t.

Trivialisation

Proposition

Let GG be a group, and let T̲=(T,a:G×TT)\underline{T} = \left( T, a: G \times T \rightarrow T \right) be a GG-torsor. If TT is non-empty, it is isomorphic to the underlying set of GG.

Proof

Let tt be an element of TT. The following diagram is cartesian.

Since T̲\underline{T} is a GG-torsor, we have that a×p 2a \times p_2 is an isomorphism. The proposition thus follows immediately from the fact that pullbacks of isomorphisms are isomorphisms (as proven at pullback).

Definition

Let GG be a group, and let T̲=(T,a:G×TT)\underline{T} = \left( T, a: G \times T \rightarrow T \right) be a GG-torsor. A trivialisation of T̲\underline{T} is an isomorphism between TT and the underlying set of GG.

Remark

By Proposition , if TT is non-empty, TT is always isomorphic to the underlying set of GG. Thus the notion of a trivialisation of T̲\underline{T} is a question of a choice of isomorphism between TT and the underlying set of GG.

The proof of Proposition shows that any choice of element of TT gives rise to a trivialisation. Some of these may of course coincide.

Remark

Let ρ:GT\rho : G \rightarrow T be a trivialisation of a torsor T̲\underline{T}. The map

p 1(ρ 1×id):T×TG×TGp_{1} \circ \left( \rho^{-1} \times id \right): T \times T \rightarrow G \times T \rightarrow G

where p 1p_1 is the canonical map, can be interpreted as a notion of division d:T×TGd : T \times T \to G, ‘dividing’ one element of TT by another to obtain an element of GG. If we further compose dd with ρ\rho

ρd:T×TGT,\rho \circ d : T \times T \rightarrow G \rightarrow T,

we can think of the resulting map DD as a ‘division structure’ on TT for which ρ\rho behaves as an identity, namely such that D(g,g)=ρD(g, g) = \rho for all gGg \in G. In this way, TT acquires a group structure isomorphic to that of GG.

Thus a trivialisation of a torsor equips with it a choice of group structure amongst all of those isomorphic to GG.

Functoriality (change of structure group)

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).

Proposition

Let h:G 1G 2h : G_1 \rightarrow G_2 be a group homomorphism, and let T̲=(T,a)\underline{T} = \left( T, a \right) be a G 1G_1-torsor. Observe that (g 1,g 2,t)(g 2h(g 1) 1,a(g 1,t))\left(g_1, g_2, t \right) \mapsto \left( g_2 \cdot h(g_1)^{-1}, a(g_1, t) \right) defines an action G 1×G 2×TG 2×TG_1 \times G_2 \times T \rightarrow G_2 \times T of G 1G_1 on G 2×TG_2 \times T. Then (G 2×T)/G 1\left(G_2 \times T \right) / G_1, the quotient of G 2×XG_2 \times X with respect to this G 1G_1-action, defines a G 2G_2-torsor with respect to the action of G 2G_2 induced by left multiplication, namely that given by (g,[(g,t)])[(gg,t)]\left( g, \left[ (g', t') \right] \right) \mapsto \left[ \left(g g', t' \right) \right], where [p]\left[ p \right], for some pG 2×Tp \in G_2 \times T, denotes the orbit of pp with respect to the action of G 1G_1 on G 2×TG_2 \times T.

Proof

We shall demonstrate that the map

i:(G 2×T)×((G 2×T)/G 1)G 2×((G 2×T)/G 1)i : \left( G_2 \times T \right) \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow G_2 \times \left( \left( G_2 \times T \right) / G_1 \right)

given by

((g,t),[(g,t)])(g(g) 1,[(g,t)])\left( (g, t), \left[ (g', t') \right] \right) \mapsto \left( g\left(g'\right)^{-1}, \left[ (g', t') \right] \right)

induces a map

i¯:((G 2×T)/G 1)×((G 2×T)/G 1)G 2×((G 2×T)/G 1).\overline{i} : \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow G_2 \times \left( \left( G_2 \times T \right) / G_1 \right).

That is, it respects, in its left factor, the passage to the quotient by the action of G 1G_1.

Suppose indeed that we have (g,t)G 2×T(g, t) \in G_2 \times T and (g 1,g,t)G 1×G 2×T\left( g_1, g'', t'' \right) \in G_1 \times G_2 \times T such that (g,t)=(gh(g 1) 1,a(g 1,t))(g, t) = \left( g'' \cdot h(g_1)^{-1}, a(g_1, t'') \right). We make the following observations.

  1. gh(g 1) 1(g) 1=g(gh(g 1)) 1g'' h(g_1)^{-1} \left(g'\right) ^{-1} = g'' \left( g' h(g_1) \right)^{-1}
  2. [(g,t)]=[(gh(g 1 1) 1,a(g 1 1,t))]=[(gh(g 1),a(g 1 1,t))]\left[ (g', t') \right] = \left[ \left( g' h\left(g_1^{-1}\right)^{-1}, a(g_1^{-1}, t') \right) \right] = \left[ \left( g' h(g_1), a\left(g_1^{-1}, t'\right) \right) \right]

Putting these together, we obtain that

i((g,t),[(g,t)]) =i((g,t),[(gh(g 1),a(g 1 1,t))]) =(gh(g 1) 1(g) 1,[(gh(g 1),a(g 1 1,t))]) =(g(g) 1,[(g,t)]) =i(g,[(g,t)]), \begin{aligned} i\left( \left( g'', t'' \right), \left[ \left( g', t' \right) \right] \right) &= i\left( \left( g'', t'' \right), \left[ \left( g' h(g_1), a\left( g_1^{-1}, t' \right) \right) \right] \right) \\ &= \left( g'' h(g_1)^{-1} \left(g' \right)^{-1}, \left[ \left( g' h(g_1), a\left( g_1^{-1}, t' \right) \right) \right] \right) \\ &= \left( g \left(g' \right)^{-1}, \left[ \left( g', t' \right) \right] \right) \\ &= i\left( g, \left[ \left( g', t' \right) \right] \right), \end{aligned}

as required.

We now observe that i¯\overline{i} defines an inverse to a×p 2a' \times p_{2}, where aa' is the action of G 2G_2 on (G 2×T)/G 1\left( G_2 \times T \right) / G_1 induced by left multiplication, and p 2p_2 is the projection map G 2×((G 2×T)/G 1)((G 2×T)/G 1)G_2 \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow \left( \left( G_2 \times T \right) / G_1 \right). In one direction, for any g,gG 2g, g' \in G_2 and tTt \in T, we have that

i¯([(gg,t)],[(g,t)])=(ggg 1,[(g,t)])=(g,[(g,t)]),\overline{i}\left( \left[ \left(g' g, t \right) \right], \left[ \left( g, t \right) \right] \right) = \left( g' g g^{-1}, \left[ \left( g, t \right) \right] \right) = \left( g', \left[ \left( g, t \right) \right] \right),

as required. In the other direction, suppose that we have ([(g,t)],[(g,t)])((G 2×T)/G 1)×((G 2×T)/G 1)\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) \in \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right). Since the map a×p 2:G 1×TT×Ta \times p_2 : G_1 \times T \rightarrow T \times T is an isomorphism, there is a g 1G 1g_1 \in G_1 such that t=a(g 1,t)t' = a(g_1, t), and we have that [(g,t)]=[(gh(g 1) 1,t)]\left[ \left(g, t \right) \right] = \left[ \left( g h(g_1)^{-1}, t' \right) \right]. We then have that a×p 2a' \times p_2 applied to

i¯([(g,t)],[(g,t)])=i¯([(gh(g 1) 1,t)],[(g,t)])=(gh(g 1) 1(g) 1,[(g,t)])\overline{i}\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) = \overline{i}\left( \left[ \left(g h(g_1)^{-1}, t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( g h(g_1)^{-1} \left(g'\right)^{-1}, \left[ \left( g', t' \right) \right] \right)

is

([(gh(g 1) 1(g) 1g,t)],[(g,t)])=([(gh(g 1) 1,t)],[(g,t)])=([(g,t)],[(g,t)]),\left( \left[ \left(g h(g_1)^{-1} \left( g' \right)^{-1} g', t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( \left[ \left(g h(g_1)^{-1}, t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right),

as required.

Definition

Let h:G 1G 2h : G_1 \rightarrow G_2 be a group homomorphism, and let T̲=(T,a)\underline{T} = \left( T, a \right) be a G 1G_1-torsor. We refer to the G 2G_2-torsor constructed from T̲\underline{T} using hh as in Proposition as the torsor obtained from T̲\underline{T} by change of structure group, and denote it h *(T̲)h_{*}\left(\underline{T}\right).

In general

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.

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 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

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

Elementary exposition:

Textbook accounts:

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:

  • Def. 4.6.1 in: Symmetry (pdf)

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 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.

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

  • Anand Pillay, Remarks on Galois cohomology and definability, The Journal of Symbolic Logic 62:2 (1997) 487-492 doi

See also

Last revised on April 16, 2021 at 19:01:51. See the history of this page for a list of all contributions to it.