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 $G$, we recover a group isomorphic to $G$ 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 $G$ be a group. A $G$-torsor is a set $T$ together with an action $a: G \times T \rightarrow T$ of $G$ on $T$ such that the shear map $a \times p_2: G \times T \rightarrow T \times T$ is an isomorphism, where $p_2$ is the canonical projection map $G \times T \rightarrow T$.
If here $T$ is not required to be inhabited (possibly empty) one also speaks of a pseudo-torsor.
If $T$ is non-empty, we shall prove below that it follows from the definition that $T$ is isomorphic to the underlying set of $G$. 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 $T$. 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 $a \times p_2$ 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 the shear map, and transitivity corresponds to surjectivity of the shear map.
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 $G$ be a group. The action of $G$ on itself equips the underlying set of $G$ with the structure of a $G$-torsor.
Given two isomorphic objects $X$ and $Y$ in any category $C$, all isomorphisms between $X$ and $Y$ form a torsor (both for $Aut(X)$ and for $Aut(Y)$, 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.
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 $T$ 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 $G$-torsor $T$, we send it to the set $T$ equipped with the ternary operation $t(a,b,c)=g(a,b)c$, where $g(a,b)$ is the unique element of $G$ such that $g(a,b)b=a$.
Given a torsor $(T,t)$, we send it to the pair $(LTrans(T),T)$, where $LTrans(T)$ is a subgroup of the group of bijections on the set $T$ comprising precisely the bijections of the form $c\mapsto t(a,b,c)$ for some elements $a,b\in T$. The group $G$ acts on $T$ by evaluation: $gt=g(t)$.
Mapping $(T,t)\mapsto (LTrans(T),T)\mapsto (T,t)$ gives back the same torsor $(T,t)$ that we started from.
Mapping $(G,T)\mapsto (T,t)\mapsto (LTrans(T),t)$ produces a torsor $(LTrans(T),t)$ that is naturally isomorphic to $(G,T)$ via the isomorphism
Let $G$ be a group, and let $\underline{T} = \left( T, a: G \times T \rightarrow T \right)$ be a $G$-torsor. If $T$ is non-empty, it is isomorphic to the underlying set of $G$.
Let $t$ be an element of $T$. The following diagram is cartesian.
Since $\underline{T}$ is a $G$-torsor, we have that $a \times p_2$ is an isomorphism. The proposition thus follows immediately from the fact that pullbacks of isomorphisms are isomorphisms (as proven at pullback).
Let $G$ be a group, and let $\underline{T} = \left( T, a: G \times T \rightarrow T \right)$ be a $G$-torsor. A trivialisation of $\underline{T}$ is an isomorphism between $T$ and the underlying set of $G$.
By Proposition , if $T$ is non-empty, $T$ is always isomorphic to the underlying set of $G$. Thus the notion of a trivialisation of $\underline{T}$ is a question of a choice of isomorphism between $T$ and the underlying set of $G$.
The proof of Proposition shows that any choice of element of $T$ gives rise to a trivialisation. Some of these may of course coincide.
Let $\rho : G \rightarrow T$ be a trivialisation of a torsor $\underline{T}$. The map
where $p_1$ is the canonical map, can be interpreted as a notion of division $d : T \times T \to G$, ‘dividing’ one element of $T$ by another to obtain an element of $G$. If we further compose $d$ with $\rho$
we can think of the resulting map $D$ as a ‘division structure’ on $T$ for which $\rho$ behaves as an identity, namely such that $D(g, g) = \rho$ for all $g \in G$. In this way, $T$ acquires a group structure isomorphic to that of $G$.
Thus a trivialisation of a torsor equips with it a choice of group structure amongst all of those isomorphic to $G$.
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 $h : G_1 \rightarrow G_2$ be a group homomorphism, and let $\underline{T} = \left( T, a \right)$ be a $G_1$-torsor. Observe that $\left(g_1, g_2, t \right) \mapsto \left( g_2 \cdot h(g_1)^{-1}, a(g_1, t) \right)$ defines an action $G_1 \times G_2 \times T \rightarrow G_2 \times T$ of $G_1$ on $G_2 \times T$. Then $\left(G_2 \times T \right) / G_1$, the quotient of $G_2 \times X$ with respect to this $G_1$-action, defines a $G_2$-torsor with respect to the action of $G_2$ induced by left multiplication, namely that given by $\left( g, \left[ (g', t') \right] \right) \mapsto \left[ \left(g g', t' \right) \right]$, where $\left[ p \right]$, for some $p \in G_2 \times T$, denotes the orbit of $p$ with respect to the action of $G_1$ on $G_2 \times T$.
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 $G_1$.
Suppose indeed that we have $(g, t) \in G_2 \times T$ and $\left( g_1, g'', t'' \right) \in G_1 \times G_2 \times T$ such that $(g, t) = \left( g'' \cdot h(g_1)^{-1}, a(g_1, t'') \right)$. We make the following observations.
Putting these together, we obtain that
as required.
We now observe that $\overline{i}$ defines an inverse to $a' \times p_{2}$, where $a'$ is the action of $G_2$ on $\left( G_2 \times T \right) / G_1$ induced by left multiplication, and $p_2$ is the projection map $G_2 \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow \left( \left( G_2 \times T \right) / G_1 \right)$. In one direction, for any $g, g' \in G_2$ and $t \in T$, we have that
as required. In the other direction, suppose that we have $\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) \in \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right)$. Since the map $a \times p_2 : G_1 \times T \rightarrow T \times T$ is an isomorphism, there is a $g_1 \in G_1$ such that $t' = a(g_1, t)$, and we have that $\left[ \left(g, t \right) \right] = \left[ \left( g h(g_1)^{-1}, t' \right) \right]$. We then have that $a' \times p_2$ applied to
is
as required.
Let $h : G_1 \rightarrow G_2$ be a group homomorphism, and let $\underline{T} = \left( T, a \right)$ be a $G_1$-torsor. We refer to the $G_2$-torsor constructed from $\underline{T}$ using $h$ as in Proposition as the torsor obtained from $\underline{T}$ by change of structure group, and denote it $h_{*}\left(\underline{T}\right)$.
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 $C =$ 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
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.
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 $2$-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.
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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