In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. They are often used to capture information about geometrical objects such as manifolds. Mathematics is the study of quantity, structure, space and change. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... Categories: Wikipedia cleanup | Stub ... 1926 was a common year starting on Friday (link will take you to calendar). ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, a differentiable manifold is a topological space that looks locally like the Euclidean space Rn, and the Euclidean space indeed provides the simplest example of a manifold. ...

The term "groupoid" is also used for a magma: a set with any sort of binary operation on it. We do not use the term for that concept in this encyclopedia. In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...

Definitions

From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, a groupoid G is: Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

• A set G₀ of objects;
• For each pair of objects x and y in G₀, a set G(x,y) of morphisms (or arrows) from x to y -- we write f: x -> y to indicate that f is an element of G(x,y);

equipped with: The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ...

• An element id_x of G(x,x);
• For each triple of objects x, y, and z, a binary function comp_x,y,z from G(x,y) and G(y,z) to G(x,z) -- we write gf for comp_x,y,z(f,g);
• A function inv_x,y from G(x,y) to G(y,x) -- we write f^-1 for inv_x,y(f);

such that: In mathematics, a binary function, or function of two variables, is like a function, except that it has two inputs instead of one. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

• If f: x -> y, then fid_x = f and id_yf = f;
• If f: x -> y, g: y -> z, and h: z -> w, then (hg)f = h(gf);
• If f: x -> y, then ff^-1 = id_y and f^-1f = id_x.

One can also define a groupoid as a certain algebraic structure. To be specific, let G be a set and let comp be a partially defined binary operation on G. That is, given elements f and g of G, comp(f,g) may be an element of G, or it may be undefined. We write gf for comp(f,g). There is also a total (everywhere defined) function inv on G. We write f^-1 for the inverse inv(f) of f. Then G is a groupoid if: In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics and computer science, a partial function from the domain X to the codomain Y is a binary relation over X and Y which associates with every element in the set X at most one element in the set Y. If a partial function associates with every element in... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P...

• Whenever fg and gh are both defined, then (fg)h and f(gh) are also defined, and they are equal;
• f^-1f and ff^-1 are always defined;
• Whenever fg is defined, then fgg^-1 = f and f^-1fg = g -- we already know that these expressions are unambiguously defined by the previous conditions.

The relation between this definitions is as follows: Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y). Then inv and comp become partially defined operations on G, and inv will in fact be defined everywhere. Explicit reference to G₀ (and hence to id) can be dropped. In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

On the other hand, given a groupoid in the algebraic sense, let G₀ be the set of all elements of the form ff^-1 for some element f of G. In other words, the objects are identified with the identity morphisms, and id_x is just x. Let G(x,y) be the set all elements f such that yfx is defined. Then inv and comp break up into several functions on the various G(x,y).

While we have referred to sets in the definitions above, one may instead want to use classes, in the same way as for other categories. In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

Examples

From linear algebra: Given a field K, the general linear groupoid GL_*(K) consists of all invertible matrices with entries from K, with composition given by matrix multiplication. If G = GL_*(K), then G₀ can be identified with the set of natural numbers, since there is one identity matrix for each natural number. G(m,n) is empty unless m = n, in which case it is the set of n by n matrices. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... For the square matrix section, see square matrix. ... This article gives an overview of the various ways to multiply matrices. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In mathematics, the empty set is the set with no elements. ...

From topology: Start with a topological space X and let G₀ be the set X. The morphisms from the point p to the point q are equivalence classes of continuous paths from p to q, with two paths being considered equivalent if they are homotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of X, denoted Π₁(X). Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

If X is a set and ~ is an equivalence relation on X, then we can form a groupoid representing this equivalence relation as follows: The objects are the elements of X, and for any two elements x and y in X, there is a single morphism from x to y if and only if x ~ y. The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...

If the group G acts on the set X, then we can form a groupoid representing this group action as follows: The objects are the elements of X, and for any two elements x and y in X, there is a morphism from x to y for every element g of G such that g.x = y. Composition of morphisms is given by the group operation in G. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, groups are often used to describe symmetries of objects. ...

Relation to groups

If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theory can be generalized to groupoids, with the notion of group homomorphism being replaced by that of functor. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Group theory is that branch of mathematics concerned with the study of groups. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In category theory, a functor is a special type of mapping between categories. ...

If x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism f from x to y, then the groups G(x) and G(y) are isomorphic, with an isomorphism given by mapping g to f'g'f^-1. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to a groupoid of the following form: Pick a group G and a set (or class) X. Let the objects of the groupoid be the elements of X. For elements x and y of X, let the set of morphisms from x to y be G. Composition of morphisms is the group operation of G. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups G per connected component). Thus, any groupoid may be given (up to isomorphism) by a set of ordered pairs (X,G). Connected is a category theory term referring to the existence of one or more morphisms between two or more objects. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... Look up Up to in Wiktionary, the free dictionary Modern Slang In modern slang, up to means you are either willing to engage in an act (Sally is up to going to the park), capable of an act (Im sorry, Im just not up to it) or are... An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object x₀, a group isomorphism h from G(x₀) to G, and for each x other than x₀ a morphism in G from x₀ to x. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset of unrelated groups. In other words, for equivalence instead of isomorphism, you don't have to specify the sets X, only the groups G. In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a cardinal number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ...

Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear groups GL_n(F). On the other hand, the fundamental groupoid of X is equivalent to the collection of the fundamental groups of each path-connected component of X, but for an isomorphism you must also specify the set of points in each component. The set X with the equivalence relation ~ is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but for an isomorphism you must also specify what each equivalence class is. Finally, the set X equipped with an action of the group G is equivalent (as a groupoid) to one copy of G for each orbit of the action, but for an isomorphism you must also specify what set each orbit is. In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... In mathematics, groups are often used to describe symmetries of objects. ...

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it's not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. If you don't, then you must choose a way to view each G(x) in terms of a single group, and this can be rather arbitrary. In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point p to each point q in the same path-connected component. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Covariance in special relativity

An example of this phenomenon that is well known in physics is covariance in special relativity. Working with a single group corresponds to picking a specific frame of reference, and you can do all of physics in this fashion. But it's more natural to describe physics in a way that makes no mention of any particular frame of reference, and this corresponds to using the entire groupoid. (I need to go into more detail about this. It really is a precise correspondence -- the particular group involved is the Poincaré group -- but I'm not sure how best to explain it yet.) The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... In probability theory and statistics, the covariance between two real_valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity(SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ...

Lie groupoids and Lie algebroids

When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoids. These can be studied in terms of Lie algebroids, in analogy to the relation between Lie groups and Lie algebras. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics, a Lie algebroid can be thought of as a restricted Lie module that has both a Lie bracket and a Lie algebra morphism, known as an anchor map, given as ( with denoting the restricted tangent space), associated to it. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

Explain this

Related topics

Heinrich Brandt Categories: Wikipedia cleanup | Stub ...

External links

• Alan Weinstein, Groupoids: unifying internal and external symmetry, available as Groupoids.ps or weinstein.pdf
• Part VI of Geometric Models for Noncommutative Algebras, by A. Cannas da Silva and A. Weinstein PDF file.