Torsor Under A Groupoid
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a principal homogeneous space, or torsor, for a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
''G'' is a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
''X'' for ''G'' in which the
stabilizer subgroup In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under func ...
of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-empty set ''X'' on which ''G''
acts The Acts of the Apostles (, ''Práxeis Apostólōn''; ) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message to the Roman Empire. Acts and the Gospel of Luke make up a two-par ...
freely Freely is a British free-to-air IPTV service launched in 2024 by Everyone TV, a joint venture between the country's public broadcasters BBC, ITV, Channel 4 and 5. The service offers the ability to watch live television and on demand media from t ...
and transitively (meaning that, for any ''x'', ''y'' in ''X'', there exists a unique ''g'' in ''G'' such that , where · denotes the (right) action of ''G'' on ''X''). An analogous definition holds in other
categories Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *Category (Vais ...
, where, for example, *''G'' is a
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
, ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
and the action is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
, *''G'' is a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
, ''X'' is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
and the action is smooth, *''G'' is an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
, ''X'' is an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
and the action is
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
.


Definition

If ''G'' is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the definition more explicitly, ''X'' is a ''G''-torsor or ''G''-principal homogeneous space if ''X'' is nonempty and is equipped with a map (in the appropriate category) such that :''x''·1 = ''x'' :''x''·(''gh'') = (''x''·''g'')·''h'' for all and all , and such that the map given by :(x,g) \mapsto (x,x\cdot g) is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
(of sets, or topological spaces or ..., as appropriate, i.e. in the category in question). Note that this means that ''X'' and ''G'' are isomorphic (in the category in question; not as groups: see the following). However—and this is the essential point—there is no preferred 'identity' point in ''X''. That is, ''X'' looks exactly like ''G'' except that which point is the identity has been forgotten. (This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.) Since ''X'' is not a group, we cannot multiply elements; we can, however, take their "quotient". That is, there is a map that sends to the unique element such that . The composition of the latter operation with the right group action, however, yields a
ternary operation In mathematics, a ternary operation is an ''n''- ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operato ...
, which serves as an affine generalization of group multiplication and which is sufficient to both characterize a principal homogeneous space algebraically and intrinsically characterize the group it is associated with. If we denote x/y \cdot z \,:=\, x \cdot (y\backslash z) the result of this ternary operation, then the following identities :x/y \cdot y = x = y/y \cdot x :v/w \cdot (x/y \cdot z) = (v/w \cdot x)/y \cdot z will suffice to define a principal homogeneous space, while the additional property :x/y \cdot z = z/y \cdot x identifies those spaces that are associated with abelian groups. The group may be defined as formal quotients x \backslash y subject to the equivalence relation :x \backslash y = u \backslash v \quad \text \quad v = u/x \cdot y, with the group product, identity and inverse defined, respectively, by :(x \backslash y) \cdot (u \backslash v) = x \backslash (y/u \cdot v) = (u/y \cdot x)\backslash v, :e = x \backslash x, :(x \backslash y)^ = y \backslash x, and the group action by :x\cdot (y \backslash z) = x/y \cdot z.


Examples

Every group ''G'' can itself be thought of as a left or right ''G''-torsor under the natural action of left or right multiplication. Another example is the
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
concept: the idea of the affine space ''A'' underlying a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'' can be said succinctly by saying that ''A'' is a principal homogeneous space for ''V'' acting as the additive group of translations. The
flags A flag is a piece of fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and flags have ...
of any
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
form a torsor for its symmetry group. Given a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'' we can take ''G'' to be the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
GL(''V''), and ''X'' to be the set of all (ordered) bases of ''V''. Then ''G'' acts on ''X'' in the way that it acts on vectors of ''V''; and it acts transitively since any basis can be transformed via ''G'' to any other. What is more, a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
fixing each vector of a basis will fix all ''v'' in ''V'', and hence be the neutral element of the general linear group GL(''V'') : so that ''X'' is indeed a ''principal'' homogeneous space. One way to follow basis-dependence in a
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
argument is to track variables ''x'' in ''X''. Similarly, the space of
orthonormal bases In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, th ...
(the
Stiefel manifold In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one ...
V_n(\mathbf^n) of ''n''-frames) is a principal homogeneous space for the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
. In
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, if two objects ''X'' and ''Y'' are isomorphic, then the isomorphisms between them, Iso(''X'',''Y''), form a torsor for the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of ''X'', Aut(''X''), and likewise for Aut(''Y''); a choice of isomorphism between the objects gives rise to an isomorphism between these groups and identifies the torsor with these two groups, giving the torsor a group structure (as it has now a
base point In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
).


Applications

The principal homogeneous space concept is a special case of that of
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
: it means a principal bundle with base a single point. In other words the local theory of principal bundles is that of a family of principal homogeneous spaces depending on some parameters in the base. The 'origin' can be supplied by a
section Section, Sectioning, or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of the bundle—such sections are usually assumed to exist ''locally on the base''—the bundle being ''locally trivial'', so that the local structure is that of a
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
. But sections will often not exist globally. For example a
differential manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
''M'' has a principal bundle of frames associated to its
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. A global section will exist (by definition) only when ''M'' is
parallelizable In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist Smooth function, smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a Basis of a vector space, ...
, which implies strong topological restrictions. In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
there is a (superficially different) reason to consider principal homogeneous spaces, for
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
s ''E'' defined over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K'' (and more general
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...
). Once this was understood, various other examples were collected under the heading, for other
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s:
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s for
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
s, and Severi–Brauer varieties for
projective linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
s being two. The reason of the interest for
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s, in the elliptic curve case, is that ''K'' may not be
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...
. There can exist curves ''C'' that have no point defined over ''K'', and which become isomorphic over a larger field to ''E'', which by definition has a point over ''K'' to serve as identity element for its addition law. That is, for this case we should distinguish ''C'' that have
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
1, from elliptic curves ''E'' that have a ''K''-point (or, in other words, provide a Diophantine equation that has a solution in ''K''). The curves ''C'' turn out to be torsors over ''E'', and form a set carrying a rich structure in the case that ''K'' is a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
(the theory of the
Selmer group In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties. Selmer group of an isogeny The Selmer group of an abelian variety ''A'' with respect to an isogeny ''f'' ...
). In fact a typical plane cubic curve ''C'' over Q has no particular reason to have a
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
; the standard Weierstrass model always does, namely the point at infinity, but you need a point over ''K'' to put ''C'' into that form ''over'' ''K''. This theory has been developed with great attention to
local analysis In algebraic geometry and related areas of mathematics, local analysis is the practice of looking at a problem relative to each prime number ''p'' first, and then later trying to integrate the information gained at each prime into a 'global' pictur ...
, leading to the definition of the
Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group \mathrm(A/K) = H^1(G_K, A), where G_K = \mathrm(K ...
. In general the approach of taking the torsor theory, easy over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
, and trying to get back 'down' to a smaller field is an aspect of
descent Descent may refer to: As a noun Genealogy and inheritance * Common descent, concept in evolutionary biology * Kinship, one of the major concepts of cultural anthropology **Pedigree chart or family tree **Ancestry **Lineal descendant **Heritage ** ...
. It leads at once to questions of
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
, since the torsors represent classes in
group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...
''H''1.


Other usage

The concept of a principal homogeneous space can also be globalized as follows. Let ''X'' be a "space" (a
scheme Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'', a BBC Scotland documentary TV series * The Scheme (band), an English pop band * ''The Scheme'', an action role-playing video game for the PC-8801, made by Quest Corporation * ...
/
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
/
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
etc.), and let ''G'' be a group over ''X'', i.e., a
group object In category theory, a branch of mathematics, group objects are certain generalizations of group (mathematics), groups that are built on more complicated structures than Set (mathematics), sets. A typical example of a group object is a topological gr ...
in the
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
of spaces over ''X''. In this case, a (right, say) ''G''-torsor ''E'' on ''X'' is a space ''E'' (of the same type) over ''X'' with a (right) ''G''
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
such that the morphism :E \times_X G \rightarrow E \times_X E given by :(x,g) \mapsto (x,xg) is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
in the appropriate
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
, and such that ''E'' is locally trivial on ''X'', in that acquires a section locally on ''X''. Isomorphism classes of torsors in this sense correspond to classes in the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
group ''H''1(''X'',''G''). When we are in the smooth manifold
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
, then a ''G''-torsor (for ''G'' a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
) is then precisely a principal ''G''-
bundle Bundle or Bundling may refer to: * Bundling (packaging), the process of using straps to bundle up items Biology * Bundle of His, a collection of heart muscle cells specialized for electrical conduction * Bundle of Kent, an extra conduction path ...
as defined above. Example: if ''G'' is a compact Lie group (say), then EG is a ''G''-torsor over the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
BG.


See also

*
Homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
*
Heap (mathematics) In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set ''H'' with a ternary operation denoted ,y,z\in H that satisfies a modified associativity property: \forall a,b,c,d,e \in H \quad a,b,cd,e] = ,[d,c,be= [a,b,[ ...


Notes


Further reading

* * {{cite book , last=Skorobogatov , first=A. , title=Torsors and rational points , series=Cambridge Tracts in Mathematics , volume=144 , location=Cambridge , publisher=Cambridge University Press , year=2001 , isbn=0-521-80237-7 , zbl=0972.14015 , url-access=registration , url=https://archive.org/details/torsorsrationalp0000skor


External links


Torsors made easy
by John Baez Group theory Topological groups Lie groups Algebraic homogeneous spaces Diophantine geometry Vector bundles