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]

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). A groupoid G is sometimes denoted as  , where   is the set of all morphisms, and the two arrows   represent the source and the target.

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 groups and orbitsEdit

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.

The orbit of a groupoid G at a point   is given by the set   containing every point that can be joined to x by an morphism in G. If two points   and   are in the same orbits, their vertex groups   and   are isomorphic: if   is any morphism from   to  , then the isomorphism is given by the mapping  .

Orbits form a partition of the set X, and a groupoid is called transitive if it has only one orbit (equivalently, if it is connected as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section below for counterexamples).

Subgroupoids and morphismsEdit

A subgroupoid of   is a subcategory   that is itself a groupoid. It is called wide or full if it is wide or full as a subcategory i.e., respectively, if   or   for every  .

A groupoid morphism is simply a functor between two (category-theoretic) groupoids.

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

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  .

The orbits of the fundamental groupoid   are the path-connected components of  . Accordingly, the fundamental groupoid of a path-connected space is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as categories (see the section below for the general theory).

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

Equivalence relationEdit

If   is a setoid, i.e. a set with an equivalence relation  , 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   (denote by  ) if and only if  ;
  • The composition of   and   is  .

The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:

  • If every element of   is in relation with every other element of  , we obtain the pair groupoid of  , which has the entire   as set of arrows, and which is transitive.
  • If every element of   is only in relation with itself, one obtains the unit groupoid, which has   as set of arrows,  , and which is completely intransitive (every singleton   is an orbit).

ExamplesEdit

For example, if   is a smooth surjective submersion of smooth manifolds, then   is an equivalence relation[6] since   has a topology isomorphic to the quotient topology of   under the surjective map of topological spaces. If we write,   then we get a groupoid

 

which is sometimes called the banal groupoid of a surjective submersion of smooth manifolds.

Čech groupoidEdit

A Čech groupoid[6]pg 5 is a special kind of groupoid associated to an equivalence relation given by an open cover   of some manifold  . It's objects are given by the disjoint union

 

and its arrows are the intersections

 

The source and target maps are then given by the induced maps

 

and the inclusion map

 

giving the structure of a groupoid. In fact, this can be further extended by setting

 

as the  -iterated fiber product where the   represents  -tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since

 

is a cartesian diagram where the maps to   are the target maps. This construction can be seen as a model for some ∞-groupoids. Also, another artifact of this construction is k-cocycles

 

for some constant sheaf of abelian groups can be represented as a function

 

giving an explicit representation of cohomology classes.

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   and with source and target maps   and  . It is often denoted   (or   for a right action). 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). Similarly, the orbits of the action groupoid are the orbit of the group action, and the groupoid is transitive if and only if the group action is transitive.

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  .

Finite setEdit

Consider the group action of   on the finite set   which takes each number to its negative, so   and  . The quotient groupoid   is the set of equivalence classes from this group action  , and   has a group action of   on it.

Quotient varietyEdit

Any finite group   which maps to   give a group action on the affine space   (since this is the group of automorphisms). Then, a quotient groupoid can be of the forms  , which has one point with stabilizer   at the origin. Examples like these form the basis for the theory of orbifolds. Another commonly studied family of orbifolds are weighted projective spaces   and subspaces of them, such as Calabi-Yau orbifolds.

Fiber product of groupoidsEdit

Given a diagram of groupoids with groupoid morphisms

 

where   and  , we can form the groupoid   whose objects are triples  , where  ,  , and   in  . Morphisms can be defined as a pair of morphisms   where   and   such that for triples  , there is a commutative diagram in   of  ,   and the  .[7]

Homological algebraEdit

A two term complex

 

of objects in a concrete Abelian category can be used to form a groupoid. It has as objects the set   and as arrows the set  ; the source morphism is just the projection onto   while the target morphism is the addition of projection onto   composed with   and projection onto  . That is, given  , we have

 

Of course, if the abelian category is the category of coherent sheaves on a scheme, then this construction can be used to form a presheaf of groupoids.

PuzzlesEdit

While puzzles such as the Rubik's Cube can be modeled using group theory (see Rubik's Cube group), certain puzzles are better modeled as groupoids.[8]

The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).[9][10][11] 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
Unital magma Required Unneeded Required Unneeded Unneeded
Semigroup Required Required Unneeded Unneeded Unneeded
Loop Required Unneeded Required Required Unneeded
Inverse semigroup Required Required Unneeded Required Unneeded
Monoid Required Required Required Unneeded Unneeded
Commutative monoid Required Required Required Unneeded Required
Group Required Required Required Required Unneeded
Abelian group Required Required Required Required Required
The closure axiom, used by many sources and defined differently, is equivalent.

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

Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above)  . By transitivity, there will only be one orbit under the action.

Note that the isomorphism just mentioned is not unique, and there is no natural choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object  , a group isomorphism   from   to  , and for each   other than  , a morphism in   from   to  .

If a groupoid is not transitive, then it is isomorphic to a disjoint union of groupoids of the above type, also called its connected components (possibly with different groups   and sets   for each connected component).

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 does not need to specify the sets  , but only the groups   For example,

  • 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 entire groupoid. Otherwise, one must choose a way to view each   in terms of a single group, and this choice can be arbitrary. In the example from topology, one 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.

Category of groupoidsEdit

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, and is denoted by Grpd.

The category Grpd is, like the category of small categories, Cartesian closed: for any groupoids   we can construct 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.

Another important property of Grpd is that it is both complete and cocomplete.

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 a Kan complex.

The nerve has a left adjoint

 

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

Groupoids in GrpdEdit

There is an additional structure which can be derived from groupoids internal to the category of groupoids, double-groupoids.[13][14] Because Grpd is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids   with functors

 

and an embedding given by an identity functor

 

One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares

  and  

with   the same morphism, they can be vertically conjoined giving a diagram

 

which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.

Groupoids with geometric structuresEdit

When studying geometrical objects, the arising groupoids often carry a topology, turning them into topological groupoids, or even some differentiable structure, turning them into Lie groupoids. These last objects can be also studied in terms of their associated Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.

Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in Poisson geometry one has the notion of a symplectic groupoid, which is a Lie groupoid endowed with a compatible symplectic form. Similarly, one can have groupoids with a compatible Riemannian metric, or complex structure, etc.

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 semi-group", Encyclopedia of Mathematics, EMS Press, 2001 [1994], 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. ^ a b Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection". arXiv:0803.1529 [math.QA].
  7. ^ "Localization and Gromov-Witten Invariants" (PDF). p. 9. Archived (PDF) from the original on February 12, 2020.
  8. ^ An Introduction to Groups, Groupoids and Their Representations: An Introduction; Alberto Ibort, Miguel A. Rodriguez; CRC Press, 2019.
  9. ^ Jim Belk (2008) Puzzles, Groups, and Groupoids, The Everything Seminar
  10. ^ The 15-puzzle groupoid (1) Archived 2015-12-25 at the Wayback Machine, Never Ending Books
  11. ^ The 15-puzzle groupoid (2) Archived 2015-12-25 at the Wayback Machine, Never Ending Books
  12. ^ 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..
  13. ^ Cegarra, Antonio M.; Heredia, Benjamín A.; Remedios, Josué (2010-03-19). "Double groupoids and homotopy 2-types". arXiv:1003.3820 [math.AT].
  14. ^ Ehresmann, Charles (1964). "Catégories et structures : extraits". Séminaire Ehresmann. Topologie et géométrie différentielle. 6: 1–31.

ReferencesEdit