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In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed , , say. Composition is then a total function: , so that .

Special cases include:

Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.[2]

Contents

DefinitionsEdit

A groupoid is an algebraic structure   consisting of a non-empty set   and a binary partial function ' ' defined on  .

AlgebraicEdit

A groupoid is a set   with a unary operation   and a partial function  . Here * is not a binary operation because it is not necessarily defined for all pairs of elements of  . The precise conditions under which   is defined are not articulated here and vary by situation.

  and −1 have the following axiomatic properties: For all  ,  , and   in  ,

  1. Associativity: If   and   are defined, then   and   are defined and are equal. Conversely, if one of   and   is defined, then so are both   and   as well as   =  .
  2. Inverse:   and   are always defined.
  3. Identity: If   is defined, then  , and  . (The previous two axioms already show that these expressions are defined and unambiguous.)

Two easy and convenient properties follow from these axioms:

  •  ,
  • If   is defined, then  .[3]

Category theoreticEdit

A groupoid is a small category in which every morphism is an isomorphism, i.e. invertible.[1] More precisely, a groupoid G is:

  • A set G0 of objects;
  • For each pair of objects x and y in G0, there exists a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y. We write f : xy to indicate that f is an element of G(x,y).
  • For every object x, a designated element   of G(x,x);
  • For each triple of objects x, y, and z, a function  ;
  • For each pair of objects x, y a function  ;

satisfying, for any f : xy, g : yz, and h : zw:

  •   and  ;
  •  ;
  •   and  .

If f is an element of G(x,y) then x is called the source of f, written s(f), and y is called the target of f, written t(f).

More generally, one can consider a groupoid object in an arbitrary category admitting finite fiber products.

Comparing the definitionsEdit

The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then   and   become partial operations on G, and   will in fact be defined everywhere. We define ∗ to be   and −1 to be  , which gives a groupoid in the algebraic sense. Explicit reference to G0 (and hence to  ) can be dropped.

Conversely, given a groupoid G in the algebraic sense, define an equivalence relation   on its elements by   iff aa−1 = bb−1. Let G0 be the set of equivalence classes of  , i.e.  . Denote aa−1 by   if   with  .

Now define   as the set of all elements f such that   exists. Given   and   their composite is defined as  . To see that this is well defined, observe that since   and   exist, so does  . The identity morphism on x is then  , and the category-theoretic inverse of f is f−1.

Sets in the definitions above may be replaced with classes, as is generally the case in category theory.

Vertex groupsEdit

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Category of groupoidsEdit

A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.

It is useful that this category is, like the category of small categories, Cartesian closed. That is, we can construct for any groupoids   a groupoid   whose objects are the morphisms   and whose arrows are the natural equivalences of morphisms. Thus if   are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids   there is a natural bijection

 

This result is of interest even if all the groupoids   are just groups.

Fibrations and coveringsEdit

Particular kinds of morphisms of groupoids are of interest. A morphism   of groupoids is called a fibration if for each object   of   and each morphism   of   starting at   there is a morphism   of   starting at   such that  . A fibration is called a covering morphism or covering of groupoids if further such an   is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.[4]

It is also true that the category of covering morphisms of a given groupoid   is equivalent to the category of actions of the groupoid   on sets.

ExamplesEdit

Linear algebraEdit

Given a field K, the corresponding general linear groupoid GL*(K) consists of all invertible matrices, of any size, whose entries range over K. Matrix multiplication interprets composition. If G = GL*(K), then the set of natural numbers is a proper subset of G0, since for each natural number n, there is a corresponding identity matrix of dimension n. G(m,n) is empty unless m=n, in which case it is the set of all nxn invertible matrices.

TopologyEdit

Given a topological space  , let   be the set  . The morphisms from the point   to the point   are equivalence classes of continuous paths from   to  , with two paths being 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  , denoted   (or sometimes,  ).[5] The usual fundamental group   is then the vertex group for the point  .

An important extension of this idea is to consider the fundamental groupoid   where   is a chosen set of "base points". Here, one considers only paths whose endpoints belong to  .   is a sub-groupoid of  . The set   may be chosen according to the geometry of the situation at hand.

Equivalence relationEdit

If   is a set with an equivalence relation denoted by infix  , then a groupoid "representing" this equivalence relation can be formed as follows:

  • The objects of the groupoid are the elements of  ;
  • For any two elements   and   in  , there is a single morphism from   to   if and only if  .

Group actionEdit

If the group   acts on the set  , then we can form the action groupoid (or transformation groupoid) representing this group action as follows:

  • The objects are the elements of  ;
  • For any two elements   and   in  , the morphisms from   to   correspond to the elements   of   such that  ;
  • Composition of morphisms interprets the binary operation of  .

More explicitly, the action groupoid is a small category with   and   with source and target maps   and  . It is often denoted   (or  ). Multiplication (or composition) in the groupoid is then   which is defined provided  .

For   in  , the vertex group consists of those   with  , which is just the isotropy subgroup at   for the given action (which is why vertex groups are also called isotropy groups).

Another way to describe  -sets is the functor category  , where   is the groupoid (category) with one element and isomorphic to the group  . Indeed, every functor   of this category defines a set   and for every   in   (i.e. for every morphism in  ) induces a bijection   :  . The categorical structure of the functor   assures us that   defines a  -action on the set  . The (unique) representable functor   :    is the Cayley representation of  . In fact, this functor is isomorphic to   and so sends   to the set   which is by definition the "set"   and the morphism   of   (i.e. the element   of  ) to the permutation   of the set  . We deduce from the Yoneda embedding that the group   is isomorphic to the group  , a subgroup of the group of permutations of  .

Fifteen puzzleEdit

The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).[6][7][8] This groupoid acts on configurations.

Mathieu groupoidEdit

The Mathieu groupoid is a groupoid introduced by John Horton Conway acting on 13 points such that the elements fixing a point form a copy of the Mathieu group M12.

Relation to groupsEdit

Group-like structures
Totalityα Associativity Identity Invertibility Commutativity
Semigroupoid Unneeded Required Unneeded Unneeded Unneeded
Small Category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Magma Required Unneeded Unneeded Unneeded Unneeded
Quasigroup Required Unneeded Unneeded Required Unneeded
Loop Required Unneeded Required Required Unneeded
Semigroup Required Required Unneeded Unneeded Unneeded
Inverse Semigroup Required Required Unneeded Required Unneeded
Monoid Required Required Required Unneeded Unneeded
Group Required Required Required Required Unneeded
Abelian group Required Required Required Required Required
Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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.[9] Many concepts of group theory generalize to groupoids, with the notion of functor replacing that of group homomorphism.

If   is an object of the groupoid  , then the set of all morphisms from   to   forms a group   (called the vertex group, defined above). If there is a morphism   from   to  , then the groups   and   are isomorphic, with an isomorphism given by the mapping  .

Every connected groupoid - that is, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above)  . By connectedness, there will only be one orbit under the action. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups   and sets   for each connected component).

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  , a group isomorphism   from   to  , and for each   other than  , a morphism in   from   to  .

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, one need not specify the sets  , only the groups  .

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  . On the other hand:

  • The fundamental groupoid of   is equivalent to the collection of the fundamental groups of each path-connected component of  , but an isomorphism requires specifying the set of points in each component;
  • The set   with the equivalence relation   is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an isomorphism requires specifying what each equivalence class is:
  • The set   equipped with an action of the group   is equivalent (as a groupoid) to one copy of   for each orbit of the action, but an isomorphism requires specifying what set each orbit is.

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is 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. Otherwise, one must choose a way to view each   in terms of a single group, and this choice can be arbitrary. In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point   to each point   in the same path-connected component.

As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup   of a group   yields an action of   on the set of cosets of   in   and hence a covering morphism   from, say,   to  , where   is a groupoid with vertex groups isomorphic to  . In this way, presentations of the group   can be "lifted" to presentations of the groupoid  , and this is a useful way of obtaining information about presentations of the subgroup  . For further information, see the books by Higgins and by Brown in the References.

Properties of the category GrpdEdit

  • Grpd is both complete and cocomplete
  • Grpd is a cartesian closed category

Relation to CatEdit

The inclusion   has both a left and a right adjoint:

 
 

Here,   denotes the localization of a category that inverts every morphism, and   denotes the subcategory of all isomorphisms.

Relation to sSetEdit

The nerve functor   embeds Grpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always Kan complex.

The nerve has a left adjoint

 

Here,   denotes the fundamental groupoid of the simplicial set X.

Lie groupoids and Lie algebroidsEdit

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.

See alsoEdit

NotesEdit

  1. ^ a b Dicks & Ventura (1996). The Group Fixed by a Family of Injective Endomorphisms of a Free Group. p. 6.
  2. ^ Brandt semigroup in Springer Encyclopaedia of Mathematics - ISBN 1-4020-0609-8
  3. ^ Proof of first property: from 2. and 3. we obtain a−1 = a−1 * a * a−1 and (a−1)−1 = (a−1)−1 * a−1 * (a−1)−1. Substituting the first into the second and applying 3. two more times yields (a−1)−1 = (a−1)−1 * a−1 * a * a−1 * (a−1)−1 = (a−1)−1 * a−1 * a = a. ✓
    Proof of second property: since a * b is defined, so is (a * b)−1 * a * b. Therefore (a * b)−1 * a * b * b−1 = (a * b)−1 * a is also defined. Moreover since a * b is defined, so is a * b * b−1 = a. Therefore a * b * b−1 * a−1 is also defined. From 3. we obtain (a * b)−1 = (a * b)−1 * a * a−1 = (a * b)−1 * a * b * b−1 * a−1 = b−1 * a−1. ✓
  4. ^ J.P. May, A Concise Course in Algebraic Topology, 1999, The University of Chicago Press ISBN 0-226-51183-9 (see chapter 2)
  5. ^ "fundamental groupoid in nLab". ncatlab.org. Retrieved 2017-09-17.
  6. ^ Jim Belk (2008) Puzzles, Groups, and Groupoids, The Everything Seminar
  7. ^ The 15-puzzle groupoid (1), Never Ending Books
  8. ^ The 15-puzzle groupoid (2), Never Ending Books
  9. ^ Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of homotopy theory, see "delooping in nLab". ncatlab.org. Retrieved 2017-10-31..

ReferencesEdit